Towards Jet Cross Sections at NNLO for hadron colliders Aude Gehrmann-De Ridder 15.09.2010 HP2.3 Firenze
Expectations at LHC proton - (anti)proton cross sections � � Large production rates for 9 9 10 10 Standard Model processes 8 8 10 10 � tot 7 7 10 10 Tevatron LHC � � jets 6 6 10 10 � � top quark pairs 5 5 10 10 -1 -2 s � b 4 4 10 10 33 cm � � vector bosons 3 3 10 10 events / sec for L = 10 jet > � s/20) � jet (E T � � Allow precision measurements 2 2 10 10 � �� nb � 1 1 � W 10 10 � � masses � Z 0 0 10 10 jet > 100 GeV) � jet (E T -1 -1 � � couplings 10 10 -2 -2 10 10 � � parton distributions -3 -3 10 10 � t jet > � s/4) � jet (E T -4 -4 10 10 � � Require precise theory: NNLO � Higgs (M H =120 GeV) -5 -5 10 10 200 GeV -6 -6 10 10 500 GeV WJS2009 -7 -7 10 10 0.1 1 10 J. Stirling � s (TeV) Aude Gehrmann-De Ridder HP2.3 Firenze
Where are NNLO corrections needed? � � Processes measured to few per cent accuracy � � e + e - � 3 jets, 2+1 jet production in DIS � � hadron collider processes: � � jet production � � vector boson (+jet) production � � top quark pair production � � Processes with potentially large perturbative corrections � � Higgs or vector boson pair production � � prediction stable only at NNLO Aude Gehrmann-De Ridder HP2.3 Firenze
NNLO calculations � � Require three principal ingredients (here: pp � 2j) � � two-loop matrix elements � � explicit infrared poles from loop integral � � known for all massless 2 � 2 processes � � one-loop matrix elements � � explicit infrared poles from loop integral � � and implicit poles from soft/collinear emission � � usually known from NLO calculations � � tree-level matrix elements � � implicit poles from two partons unresolved � � known from LO calculations � � Challenge: combine contributions into parton-level generator � � need method to extract implicit infrared poles Aude Gehrmann-De Ridder HP2.3 Firenze
NNLO calculations � � Solutions � � sector decomposition: expansion in distributions, numerical integration (T. Binoth, G. Heinrich; C. Anastasiou, K. Melnikov, F. Petriello; M. Czakon) � � applied to Higgs and vector boson production (C. Anastasiou, K. Melnikov, F. Petriello) � � subtraction: add and subtract counter-terms: process- independent approximations in all unresolved limits, analytical integration � � several well-established methods at NLO � � q T subtraction applied to Higgs and vector boson production (S. Catani, M. Grazzini; with L. Cieri, G. Ferrera, D. de Florian) � � antenna subtraction for jet observables in e + e - processes (T. Gehrmann, E.W.N. Glover, AG) Aude Gehrmann-De Ridder HP2.3 Firenze
� s from three-jet rate at NNLO � � NNLO corrections small 0.75 � 3 jet / � had NNLO (T. Gehrmann, E.W.N. Glover, G. Heinrich, AG; S. Weinzierl) NLO LO � � stable perturbative prediction 0.5 6 � ( % ) � � resummation not needed 4 0.25 2 Q = M Z � � theory error below 2% � s (M Z ) = 0.1189 0 -2 -1.5 -1 ALEPH data � � hadronization corrections 0 -4 -3 -2 -1 0 log 10 (y cut ) � � much smaller than for event shapes 0.125 � s (M Z ) 0.1225 � � data with different jet resolution 0.12 Q=M Z 0.1175 0.115 0.1125 correlated 0.11 central result with stat. uncertainty 0.1075 0.105 total uncertainty 0.1025 � � fit at y cut = 0.02 0.1 -6 -5 -4 -3 -2 -1 ln(y cut ) � � consistent results with other resolution 0.005 total error perturbative hadronisation 0.004 experimental statistical � s = 0.1175±0.0020(exp)±0.0015(th) � 0.003 �� s (M Z ) 0.002 (G. Dissertori, T. Gehrmann, E.W.N. Glover, G. Heinrich, H. Stenzel, AG) 0.001 0 -6 -5 -4 -3 -2 -1 ln(y cut ) Aude Gehrmann-De Ridder HP2.3 Firenze
NNLO Subtraction � � Structure of NNLO m-jet cross section at hadron colliders � � � σ R σ S dˆ σ NNLO = dˆ NNLO − dˆ NNLO dΦ m +2 � � � σ V, 1 σ MF, 1 σ V S, 1 + dˆ NNLO + dˆ NNLO − dˆ NNLO dΦ m +1 � � � � � σ V, 2 σ MF, 2 σ V S, 1 σ S + dˆ NNLO + dˆ + dˆ NNLO + dˆ NNLO NNLO dΦ m dΦ m +2 dΦ m +1 � � with: σ V, 1 σ V, 2 σ R � � Partonic contributions: dˆ dˆ dˆ NNLO NNLO NNLO σ V S, 1 � � Subtraction terms: σ S dˆ dˆ NNLO NNLO � � Mass factorization terms: σ MF, 1 σ MF, 2 dˆ dˆ NNLO NNLO � � Challenge: construction and integration of subtraction terms Aude Gehrmann-De Ridder HP2.3 Firenze
Antenna subtraction at NLO � � real radiation contribution to m-jet cross section � d σ R = N dΦ m +1 |M m +1 | 2 J ( m +1) ( p 1 , . . . , p m +1 ) m � � antenna subtraction term: � d σ S NLO = N dΦ m +1 ( p 1 , . . . , p m +1 ; q ) � ijk |M m | 2 J ( m ) X 0 m ( p 1 , . . . , ˜ p I , ˜ p K , . . . , p m +1 ) j i 1 1 j i I k j I k K m+1 m+1 K � � antenna describes soft and collinear radiation off a hard parton pair X 0 ijk Aude Gehrmann-De Ridder HP2.3 Firenze
Colour-ordered antenna functions � � colour-ordered pair of hard partons (radiators) � � quark-antiquark pair � � quark-gluon pair � � gluon-gluon pair � � three-parton antenna � one unresolved parton � � four-parton antenna � two unresolved partons � � at tree-level or at one loop � � radiators in initial or final state: � � three types of antennae: final-final, initial-final, initial-initial � � all antenna functions derived from physical |M| 2 Aude Gehrmann-De Ridder HP2.3 Firenze
Subtraction for hadronic processes at NNLO � � Colour-connected double unresolved case � � final-final i j i k I j l I k L l � � initial-final L j k i l j JKL k i l I JKL � � initial-initial I j i j i I k l I k K l K Aude Gehrmann-De Ridder HP2.3 Firenze
Hadron collider processes at NNLO Double real radiation at NNLO for pp → 2 j � � Contributions from all tree-level 2 � 4 processes � � Test case: (E.W.N. Glover, J. Pires) gg → gggg � � α s � 2 N 2 N born d σ R = dΦ 4 ( p 3 , . . . , p 6 ; p 1 , p 2 ) NNLO 2 π � 2 2 g , i g , j g , k g , l g ) J (4) A 0 6 (ˆ 1 g , ˆ 2 ( p i , . . . , p l ) 4! P ( i,j,k,l ) ∈ (3 , 4 , 5 , 6) � + 2 2 g , j g , k g , l g ) J (4) A 0 6 (ˆ 1 g , i g , ˆ 2 ( p i , . . . , p l ) 4! P ( i,j,k,l ) ∈ (3 , 4 , 5 , 6) � � + 2 2 g , k g , l g ) J (4) A 0 6 (ˆ 1 g , i g , j g , ˆ 2 ( p i , . . . , p l ) 4! P C ( i,j,k,l ) ∈ (3 , 4 , 5 , 6) � � three topologies according to initial state gluon positions � � antenna subtraction terms constructed, implemented and tested in all unresolved limits Aude Gehrmann-De Ridder HP2.3 Firenze
Antenna subtraction for gg � gggg � � Subtraction terms involve only gluon-gluon antennae � F 0 � � in final-final, initial-final, initial-initial, � 4 for colour-connected double unresolved limits � � F 0 3 ⊗ F 0 � � � in all configurations, 3 for oversubtracted single unresolved limits and colour unconnected double unresolved limits F 0 � � for single unresolved limits 3 � � Need to identify hard radiators for phase space mapping Aude Gehrmann-De Ridder HP2.3 Firenze
Antenna subtraction for gg � gggg � � Identification of hard radiators � � Problem: each parton is radiator or unresolved � � due to colour cyclicity of quark-gluon and gluon-gluon antennae � � only in final-final case (initial state fixes radiators) � � decompose into sub-antennae, e.g. F 0 3 (1 , 2 , 3) = f 0 3 (1 , 3 , 2) + f 0 3 (3 , 2 , 1) + f 0 3 (2 , 1 , 3) � � each sub-antenna has � � different phase space mapping, fixed hard and unresolved partons � � was done for four-parton quark-gluon antenna functions previously � � based on N=1 SUSY relations among splitting functions � � achieved now for gluon-gluon antenna (E.W.N. Glover, J. Pires) F 0 4 � � eight sub-antennae contained in F 0 4 Aude Gehrmann-De Ridder HP2.3 Firenze
Antenna subtraction for gg � gggg � � Check of the subtraction terms (E.W.N. Glover, J. Pires) � � choose scaling parameter x for each limit � � generate phase space trajectories into each limit � � require reconstruction of two hard jets � � compute ratio (matrix element)/(subtraction term): | M RR | 2 /S term s ij � s � � Example: double soft limit : 6000 10 20 x=10 -4 i double soft limit for gg � gggg 10 18 x=10 -5 double soft limit for gg � gggg ratio |M RR | 2 /S term matrix element |M RR | 2 (in GeV -2 ) x=10 -6 5000 10 16 #PS points=10000 subtr.term S term (in GeV -2 ) x=(s-s ij )/s 1487 outside the plot 10 14 x=(s-s ij )/s 317 outside the plot 10 12 4000 59 outside the plot l 10 10 2 10 8 3000 1 p t =100 GeV 10 6 p t =200 GeV p t =300 GeV 10 4 2000 k 10 2 1000 1.2 ratio |M RR | 2 /S term j 1.1 1 0.9 0 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 0.99997 0.99998 0.99999 1 1.00001 1.00002 1.00003 x Aude Gehrmann-De Ridder HP2.3 Firenze
Recommend
More recommend
