THREE-JET CALCULATION AT NLO LEVEL FOR HADRON COLLIDERS CDF/D0/Theory Jet Workshop Fermilab, December 16, 2002 Zoltán Nagy ITS, University of Oregon Contents: Introduction, Method Results in hadron hadron collision One-jet inclusive Inclusive three-jet cross sections, Energy fraction distributions Subjet rates Event shapes (Transverse thrust and jet broadening, Transverse thrust of the two leading jets) Summary, Outlook
INTRODUCTION, MOTIVATION 2 INTRODUCTION, MOTIVATION In the point of the QCD the multi-jet cross section can be used to test the prediction of perturbative QCD determine the α s (Q 2 ) strong coupling and the parton distribution function of proton (f a (x, Q 2 )) In the point of the new physics search when QCD particles are presented the QCD contributions could be very important both to the signal and background process (e.g.: pp H + 0, 1jet γγ + 0, 1jet). Because of the large strong coupling the LO order predictions strongly depend on the unphysical renormalization and factorization scales thus the higher order corrections are important to stabilize the theoretical prediction.
NLO CROSS SECTION 3 NLO CROSS SECTION σ(p A, p B ) = a,b 1 0 dη a dη b f a (η b, µ 2 F )f b (η b, µ 2 F ) [ˆσ LO a,b (η a p A, η b p B ) + ˆσ NLO a,b (η a p A, η b p B ) ] where ˆσ LO a,b (p a, p b ) = m dˆσ B a (p a, p b ) = dγ (m) M a,b 2 F (m) J (p a, p b, p 1,.., p m ) and the NLO correction ˆσ NLO a (p a, p b ) = m+1 ˆσ R a,b(p a, p b ) + m ˆσ V a,b(p a, p b ) + m ˆσ C a,b(p a, p b ) This integrals (R,V,C) are separately divergent but their sum is finite in d = 4 dimension.
NLO CROSS SECTION: SUBTRACTION TERM 4 NLO CROSS SECTION: SUBTRACTION TERM We used the dipole method to regularize the divergent integrals [Catani- Seymour] dσ NLO a,b = [dσ R a,b dσ A a,b] + dσ A a,b + dσ V a,b + dσ C a,b, where dσ A a,b is a local approximation of dσr a,b and it has to exactly match the singular behaviour of dσ R a,b it has to be exactly integrable in d = 4 2ɛ dimension over the single parton subspaces leading to soft and collinear divergences σ NLO a,b = m+1 [dσ R a,b ɛ=0 dσ A a,b ɛ=0 ] + m [dσ V a,b + dσ C a,b + dσa,b] A ɛ=0 1 The cancelation of the divergences is guaranteed only for that jet observables which fulfill the condition F (m+1) J F (m) J in the kinematically denegerate phase space regions (soft and collinear).
JET ALGORITHM 5 JET ALGORITHM: k ALGORITHM (ELLIS-SOPER) The algorithm starts with a list of the particles and the empty list of the jets. 1. For each particle (pseudo-particle) i in the list and for each pair (i, j) define d i = p 2 T,i, d ij = min(p 2 T,i, p 2 T,j) (y i y j ) 2 + (φ i φ j ) 2 D 2, 2. Find the minimum of all the d i and d ij and label it d min. 3. If d min = d ij then merge the two particles by the recombination scheme p (ij) = p i + p j. 4. If d min = d i, remove particle (pseudo-particle) i from the list of particle and add it to list of jets. 5. If any particles remain, go to step 1.
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JETS IN HADRON-HADRON COLLISIONS 7 KINEMATICS OF THE HADRON-HADRON COLLISIONS p A f a (1 η ) a p A ηap A ^ σ a....... p 1 PDF and α s : LHAPDF in CTEQ6 mode, α s (M Z ) = 0.118 p B η b p B f b (1 η ) b p B p n Matrix elements: Six parton tree level: 0 gggggg, 0 q qgggg, 0 q qq Qgg and 0 q qq Qr r Five parton tree and 1-loop level: 0 ggggg, 0 q qggg and 0 q qq Qg Gunion, Kunszt, Berends, Giele & Kuijf Bern, Dixon, Kosower Kunszt, Signer, Trócsányi
JETS IN HADRON HADRON COLLISIONS 8 ONE-JET INCLUSIVE CROSS SECTION AT NLO (Data - Theory)/Theory 0.6 0.5 0.4 0.3 0.2 0.1 0.0-0.1-0.2-0.3-0.4-0.5 One-jet inclusive k MRST98 CTEQ4M CTEQ4HJ algorithm -0.6 50 100 150 200 250 300 350 400 450 500 p T [GeV] Difference between the data and the NLOJET++ prediction.
JETS IN HADRON HADRON COLLISIONS 9 ONE-JET INCLUSIVE CROSS SECTION AT NLO Difference between the data and the JETRAD prediction.
JETS IN HADRON HADRON COLLISIONS 10 ONE-JET INCLUSIVE CROSS SECTION AT NLO K(E T ) 1.6 1.4 1.2 1.0 0.8 0.6 0.4 JETRAD NLOJET++ NLOJET++ vs. JETRAD s( R ) = 0.1 R = F = 100 GeV s = (1800 GeV) 2 < 0.5 50 100 150 200 250 300 350 400 450 500 E T [GeV] Comparison of the JETRAD and NLOJET++ programs using the MRSD parton densities and k algorithm.
JETS IN HADRON HADRON COLLISIONS 11 ONE-JET INCLUSIVE CROSS SECTION AT NLO Comparison of the EKS and NLOJET++ programs using the CTEQ4M parton densities and k algorithm.
JETS IN HADRON HADRON COLLISIONS 12 THREE-JET IN HADRON HADRON COLLISIONS AT NLO k algorithm (1) d /de T [nb/gev] 10 1 10 0 10-1 10-2 10-3 10-4 10-5 LO NLO E T > 20GeV, < 2 E T > 80GeV Q HS = E T /3 0.5 < x R,F < 2 K(E T (1) ) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 100 200 300 (1) E T 10-6 0 100 200 300 400 500 E T (1) [GeV] Differential 3-jet cross section as a function of tranverse energy of the leading jet
JETS IN HADRON HADRON COLLISIONS 13 THREE-JET IN HADRON HADRON COLLISIONS AT NLO midcone algorithm 10 1 10 0 1.6 1.4 [nb/gev] 10-1 10-2 LO NLO K(E T (1) ) 1.2 1.0 0.8 d /de T (1) 10-3 10-4 R = 0.7, f = 0.5 E T > 20GeV, < 2 E T > 80GeV 0.6 0 100 200 300 400 500 (1) E T 10-5 10-6 Q HS = E T /3 0.5 < x R,F < 2 0 100 200 300 400 500 E T (1) [GeV] Differential 3-jet cross section as a function of tranverse energy of the leading jet
JETS IN HADRON HADRON COLLISIONS 14 THREE-JET IN HADRON HADRON COLLISIONS AT NLO Midcone and inclusive k algorithms 50 40 LO E T > 20GeV & < 2 E T > 175GeV 3jet [nb] 30 20 Q HS = E T /3 R = F = NLO 10 0 Midcone algorithm Inclusive k algorithm 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 /Q HS Scale dependence of the 3-jet cross section
JETS IN HADRON HADRON COLLISIONS 15 DALITZ VARIABLES The Daltiz variables are defined in the rest frame of the three leading (leading in transverse energy) jets where the jets are labeled by E 1 > E 2 > E 3. X i = 2E i m 3J, i = 1, 2, 3, The jets are defined by using the inclusive k and midcone algorithm. Kinematical region: E T > 20GeV, η < 2, ET > 175GeV.
JETS IN HADRON HADRON COLLISIONS 16 DALITZ VARIABLES Midcone algorithm Inclusive k algorithm 20 17.5 15 12.5 10 7.5 5 2.5 0 20 18 16 14 12 10 8 6 4 2 0 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 X 2 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 X 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.65 0.7 0.75 Next-to-leading order perturbative prediction for normalized double differention distribution (1/σdσ/dX 1 dx 2 ) of the energy fraction variables X 1 and X 2 using the inclusive k and midcone algorithm. X 2 0.8 0.85 0.9 0.95 1 X 1
JETS IN HADRON HADRON COLLISIONS 17 DALITZ VARIABLES 1/ d /dx i (k alg.) 8 7 6 5 4 3 2 1 0 Midcone and iclusive k E T > 20GeV & < 2 E T > 175GeV R = F = E T /3 X 2 algorithms X 1 1/ d /dx i (cone alg.) 6 5 4 3 2 1 LO NLO X 2 0 0.5 0.6 0.7 0.8 0.9 1.0 X i The energy fraction distribution of the leading (X 1 ) and second leading (X 2 ) jets. The upper figure is result with the inclusive k algorithm and the lower figures shows the midcone result. X 1
JET ALGORITHM 18 JET ALGORITHM: k ALGORITHM (CATANI et. al.) The algorithm starts with a list of the particles and the empty list of the jets. 1. For each particle (pseudo-particle) i in the list and for each pair (i, j) define d i and d ij. 2. Find the minimum of all the d i and d ij and label it d min. 3. If d min = d ij then merge the two particles by the recombination scheme. 4. If d min = d i, remove particle (pseudo-particle) i from the list of particle and add it to list of beam jets. 5. Repeate this algorithm until all objects have d i, d ij larger than some stopping parameter d cut. To define the sub-jet structure we redifine the resolution variable y cut = Q 0 /d cut < 1 and rerun the algorithm only for those partons which are assigned to any hard final state jet.
JET ALGORITHM 19 JET ALGORITHM: k ALGORITHM (CATANI et. al.) One can define event shape parameter E tn. The variable E 2 tn variable when the event has n hard final state jets. being the value of the smallest resolution The n-jet exclusive cross section is given by σ n jet (E cut ) = E cut de tn dσ de tn dσ de tn+1 E cut de tn+1 We can define event shape variable similary in the second step y n.
JETS IN HADRON HADRON COLLISIONS 20 E t3 EVENT SHAPE VARIABLE exclusive k algorithm E t3 d /de t3 10 4 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 10-6 LO NLO < 2 Q HS = E T /3 0.5 < x R,F < 2 K(E t3 ) 1.8 1.6 1.4 1.2 1.0 0.8 0 100 200 300 E t3 0 50 100 150 200 250 300 E t3 Perturbative QCD prediction for E t3 distribution.
JETS IN HADRON HADRON COLLISIONS 21 SUB-JET RATES & SUB-JET MULTIPLICITY In this analysis we use two step jet algorithm: 1. Resolve two hard macro-jet in the final state. This step is characterized E t2 jet shape variable. 2. Resolve the subjet structure of these two macro-jets. This step is charecterized by the y cut variable. In the fix order calculation the sub-jet fraction is: R n (y cut ) = σ(0) n (y cut ) + σ n (1) (y cut ) + σ 0 (1 + K 1 + ), n 2, and the sub-jet multiplicity N(y cut ) = nr n (y cut ), n=2
JETS IN HADRON HADRON COLLISIONS 22 SUB-JET RATES & SUB-JET MULTIPLICITY k algorithm 1.0 0.8 3-jet 2-jet R n (y cut ) 0.6 0.4 4-jet 50 GeV < E t2 < 75 GeV 75 GeV < E t2 < 100 GeV 100 GeV < E t2 0.2 0.0-3.0-2.5-2.0-1.5-1.0-0.5 0.0 log 10 (y cut ) Perturbative QCD prediction for the 2-, 3-, 4-subjet rates.
JETS IN HADRON HADRON COLLISIONS 23 SUB-JET RATES & SUB-JET MULTIPLICITY k algorithm 5.0 4.5 4.0 50 GeV < E t2 < 75 GeV 75 GeV < E t2 < 100 GeV 100 GeV < E t2 N(y cut ) 3.5 3.0 2.5 2.0-3.0-2.5-2.0-1.5-1.0-0.5 0.0 log 10 (y cut ) Perturbative QCD prediction for the sub-jet multiplicity.
JETS IN HADRON HADRON COLLISIONS 24 EVENT SHAPES ON THE TRANSVERSE PLANE One can define event shapes on the transverse plane. Important example is the transeverse thrust which is defined by T = max n where C N denote any well defined selection criteria i C N p,i n i C N p,i C N = {i : η i < 1, i = 1,..., N},, We can define similary the transverse jet broadening: B = i C N p,i n 2 i C N p,i. we can calculate the differential distribution of the event shape or average values Σ(O 3 ) = O 3 σ dσ do 3, O 3 = 1 0 do 3 Σ(O 3 )
JETS IN HADRON HADRON COLLISIONS 25 EVENT SHAPES ON THE TRANSVERSE PLANE 1.2 Event shapes on the transverse plane 1.6 O 3 d /do 3 1.0 0.8 0.6 0.4 0.2 0.0 1-T B K(O 3 ) 1.4 1.2 1.0 0.8 1-T B 0.6 0.0 0.1 0.2 0.3 1-T, B LO NLO < 1 H T > 100GeV 0.0 0.1 0.2 0.3 0.4 0.5 1-T, B NLOJET++ predictions for transverse thrust and jet broadening distribution.
JETS IN HADRON HADRON COLLISIONS 26 EVENT SHAPES ON THE TRANSVERSE PLANE Event shapes on the transverse plane <1-T >, <B > 0.2 0.15 0.1 LO NLO < 1 <B > K(H T ) 1.2 1.0 0.8 0.6 <1-T > <B > 100 200 300 400 500 H T [GeV] 0.05 0.0 <1-T > 100 200 300 400 500 H T [GeV] NLOJET++ predictions for average value of transverse event shapes.
JETS IN HADRON HADRON COLLISIONS 27 TRANSVERSE THRUST DISTRIBUTION DØ measured the transverse thrust distribution of the two leading jets. The jets were defined by k algorithm. T T = max n For tree partons in the final state : 3/2 < TT2 < 1 For four partons in the final state : 2/2 < TT2 < 1 a p T,a n a p T,a Kinematical regions: Pseudo-rapidity cuts : η jet 1,2 < 1 Transverse energy cuts : H T = E (1) T This distribution was measured over four H T regions: 160 < H T < 260 GeV, 260 < H T < 360 GeV, 360 < H T < 430 GeV, 430 < H T < 700 GeV. + E(2) T + E(3) T
JETS IN HADRON HADRON COLLISIONS 28 TRANSVERSE THRUST DISTRIBUTION 10 1 Transverse thrust (1-T T2 )d /d(1-t T2 ) [nb] 10 0 10-1 10-2 10-3 10-4 10-5 10-6 s = (1800 GeV) 2 1,2 < 1 LO NLO 160 GeV < H T3 < 260 GeV 260 GeV < H T3 < 360 GeV 360 GeV < H T3 < 430 GeV 430 GeV < H T3 < 700 GeV 0.0 0.05 0.1 0.15 0.2 0.25 0.3 1-T T2 NLOJET++ prediction for transverse thrust distribution of the two leading jets at NLO.
NLOJET++ 29 PROGRAM: NLOJET++ A program for calculating next-to-leading order jet cross sections. 1. It knows several process: (a) 3,4-jet in e + e at NLO and 5-jet at LO (b) 2,3-jet production in DIS at NLO and 4-jet at LO (c) 1,2,3-jet in hadron-hadron collision at NLO and 4-jet at LO (d) 2γ + 1-jet in hadron-hadron collision at NLO (e) 1,2,3-jet in photoproduction at NLO and 4-jet at LO 2. It implements the dipole method generally process independet way. 3. It s written in C++ but there are Fortran interfaces to the user defined parts. 4. This program could be a good starting point of a NNLO general program. 5. http://www.cpt.dur.ac.uk/ nagyz/nlo++/
NLOJET++ 30 CPU TIME All the presented distribution was calculated on a PC farm in the Theoretical Department in the Fermilab. I used 24 processors (550MHz Pentium III) and every jobs run in the 1day queue.