e + e − → 3 jets at NNLO Thomas Gehrmann in collaboration with: A. Gehrmann-De Ridder, E.W.N. Glover, G. Heinrich Universit¨ at Z¨ urich S T A U T R I S I C R E E N V I S N I S U MDCCC XXXIII RADCOR 2007 e + e − → 3 jets at NNLO – p.1
e + e − → 3 jets and event shapes Classical QCD observable testing ground for QCD: perturbation theory, power corrections and logarithmic resummation precision measurement of strong coupling constant α s current error on α s from jet observables dominated by theoretical uncertainty: S. Bethke, 2006 α s ( M Z ) = 0 . 121 ± 0 . 001( experiment ) ± 0 . 005( theory ) theoretical uncertainty largely from missing higher orders current status: NLO plus NLL resummation Theoretical description easier than at hadron colliders, since coloured partons only in final state: no initial state emission, no parton distributions new calculational methods first developed for e + e − , then extended to hadronic processes e + e − → 3 jets at NNLO – p.2
e + e − → 3 jets and event shapes Event shape variables assign a number x to a set of final state momenta: { p } i → x E cm =206 GeV 10 7 ALEPH E cm =200 GeV 10 6 E cm =189 GeV e.g. Thrust in e + e − 10 5 1/ � d � /dT E cm =183 GeV 10 4 P n i =1 | � p i · � n | E cm =172 GeV T = max � 10 3 n P n i =1 | � p i | E cm =161 GeV 10 2 E cm =133 GeV limiting values: 10 E cm =91.2 GeV 1 back-to-back (two-jet) limit: T = 1 -1 10 spherical limit: T = 1 / 2 2 ) + NLLA O( � s -2 10 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 T e + e − → 3 jets at NNLO – p.3
e + e − → 3 jets and event shapes Standard Set of LEP Thrust (E. Farhi) n n ! ! X X T = max � | � p i · � n | / | � p i | n DELPHI EEC i =1 i =1 AEEC x µ = 1 Heavy jet mass (L. Clavelli, D. Wyler) JCEF 1-Thr « 2 1 „ X O M 2 i /s = | � p k | C E 2 B Max vis B Sum k ∈ H i ρ H C -parameter: eigenvalues of the tensor (G. Parisi) ρ S ρ D k p β k p α P 1 D 2E0 Θ αβ = k D 2P0 P P k | � p k | k | � p k | D 2P D 2Jade D 2Durham D 2Geneva Jet broadenings (S. Catani, G. Turnock, B. Webber) D 2Cambridge „ X « „ « X B i = | � p k × � n T | / 2 | � p k | w. average : α S (M Z2 ) = 0.1232 ± 0.0116 χ 2 /n df = 71 / 17 ρ eff = 0.635 k ∈ H i k f err = 3.38 B W = max( B 1 , B 2 ) B T = B 1 + B 2 0.06 0.08 0.1 0.12 0.14 0.16 0.18 α S (M Z2 ) 3 j → 2 j transition parameter in Durham algorithm y D 23 S.Catani, Y.L.Dokshitzer, M.Olsson, G.Turnock, B.Webber e + e − → 3 jets at NNLO – p.4
e + e − → 3 jets and event shapes Current status: NLO and NLL NLO calculations of event shapes and 3 j R.K. Ellis, D.A. Ross, A.E. Terrano; Z. Kunszt J. Vermaseren, K.F . Gaemers, S.J. Oldham; L. Clavelli, D. Wyler K. Fabricius, I. Schmitt, G. Kramer, G. Schierholz NLO parton level event generators for 3 j EVENT: Z. Kunszt, P . Nason EERAD: W. Giele, E.W.N. Glover EVENT2: S. Catani, M. Seymour NLO parton level event generators for 4 j MenloParc: L.D. Dixon, A. Signer EERAD2: J. Campbell, M. Cullen, E.W.N. Glover Debrecen: Z. Nagy, Z. Trocsanyi Mercurito: D. Kosower, S. Weinzierl NLL resummation S. Catani, L. Trentadue, G. Turnock, B. Webber Power corrections G. Korchemsky, G. Sterman; Y. Dokshitzer, B.R. Webber e + e − → 3 jets at NNLO – p.5
Ingredients to NNLO e + e − → 3-jet Two-loop matrix elements |M| 2 explicit infrared poles from loop integrals 2-loop , 3 partons L. Garland, N. Glover, A. Koukoutsakis, E. Remiddi, TG (RADCOR 00/02); S. Moch, P . Uwer, S. Weinzierl One-loop matrix elements |M| 2 explicit infrared poles from loop integral and 1-loop , 4 partons implicit infrared poles due to single unresolved radiation Z. Bern, L. Dixon, D. Kosower, S. Weinzierl; J. Campbell, D.J. Miller, E.W.N. Glover Tree level matrix elements |M| 2 implicit infrared poles due to double unresolved radiation tree , 5 partons K. Hagiwara, D. Zeppenfeld; F .A. Berends, W.T. Giele, H. Kuijf; N. Falck, D. Graudenz, G. Kramer Infrared Poles cancel in the sum e + e − → 3 jets at NNLO – p.6
NNLO Infrared Subtraction Structure of NNLO m -jet cross section: Z “ ” d σ R NNLO − d σ S d σ NNLO = NNLO dΦ m +2 Z “ ” d σ V, 1 NNLO − d σ V S, 1 + NNLO dΦ m +1 Z Z Z d σ V, 2 d σ V S, 1 d σ S + NNLO + NNLO + NNLO , dΦ m dΦ m +2 dΦ m +1 d σ S NNLO : real radiation subtraction term for d σ R NNLO d σ V S, 1 NNLO : one-loop virtual subtraction term for d σ V, 1 NNLO d σ V, 2 NNLO : two-loop virtual corrections Each line above is finite numerically and free of infrared ǫ -poles − → numerical programme e + e − → 3 jets at NNLO – p.7
Numerical Implementation Structure of e + e − → 3 jets program: Monte Carlo Histograms Cross section Definition of Observables Phase Space d σ V, 2 NNLO � { p i } 3 { p i } 3 , w w, { C, S, T } d σ V S, 1 3 parton 3 parton + NNLO dΦ X 3 σ 3 j dΦ q ¯ qg ✲ ✲ ✲ → 3 jet channel � d σ S + NNLO dΦ X 4 d σ/ d T ✲ { p i } 4 { p i } 4 , w w, { C, S, T } ⊕ 4 parton 4 parton d σ V, 1 NNLO − d σ V S, 1 dΦ q ¯ qgg ✲ ✲ → 3 jet channel NNLO d σ/ d S ✲ { p i } 5 { p i } 5 , w w, { C, S, T } 5 parton 5 parton d σ/ d C dΦ q ¯ d σ R NNLO − d σ S qggg ✲ ✲ ✲ → 3 jet channel NNLO e + e − → 3 jets at NNLO – p.8
Numerical Implementation Antenna subtraction NLO: M. Cullen, J. Campbell, E.W.N. Glover; D. Kosower; A. Daleo, D. Maitre, TG NNLO: A. Gehrmann-De Ridder, E.W.N. Glover, TG (RADCOR 05) construct subtraction terms from physical 1 → 3 and 1 → 4 matrix elements each antenna function interpolates between all limits associated to one or two unresolved partons integrated subtraction terms cancel infrared pole structure of two-loop matrix element S. Catani; G. Sterman, M.E. Yeomans-Tejeda Checks cancellation of infrared poles in 3-parton and 4-parton channel convergence of subtraction terms towards matrix elements along phase space trajectories distributions in raw phase space variables independence on phase space cut y 0 e + e − → 3 jets at NNLO – p.9
Colour structure at NNLO Decomposition into leading and subleading colour terms " 1 ( N 2 − 1) N 2 A NNLO + B NNLO + d σ NNLO = N 2 C NNLO + NN F D NNLO „ 4 # + N F « E NNLO + N 2 F F NNLO + N F,γ N − N G NNLO N last term: closed quark loop coupling to vector boson ” 2 “P q e q N F,γ = q e 2 P q was found to be O (10 − 4 ) in NLO 4 j final states L.D. Dixon, A. Signer will be negelected here e + e − → 3 jets at NNLO – p.10
Event shapes at NNLO NNLO expression for Thrust 1 d σ “ α s “ α s ” 2 ” (1 − T ) = A ( T ) + ( B ( T ) − 2 A ( T )) σ had d T 2 π 2 π “ α s ” 3 + ( C ( T ) − 2 B ( T ) − 1 . 64 A ( T )) 2 π with LO contribution A ( T ) , NLO contribution B ( T ) 30 300 (1-T) d A (1-T) d B d T d T 20 200 10 100 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 1-T 1-T e + e − → 3 jets at NNLO – p.11
Event shapes at NNLO Individual colour structures 20000 7500 100 N 2 N 0 5000 0 10000 2500 -100 1/N 2 0 0 -2500 -200 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 500 6000 N 2 F N F /N 250 4000 (1-T)d C d T -10000 0 2000 N F N -250 0 -20000 -500 -2000 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 1-T 1-T 1-T dominated by leading colour N 2 and N F N sizable contributions from N 0 , N F /N and N 2 F negligible contribution from 1 /N 2 e + e − → 3 jets at NNLO – p.12
Results NNLO thrust distribution 0.5 8000 (1-T) 1/ σ had d σ /d T (1-T) d C NNLO 0.4 NLO 6000 d T LO 0.3 4000 Q = M Z 0.2 α s (M Z ) = 0.1189 2000 0.1 0 0 0.1 0.2 0.3 0.4 0 0 0.1 0.2 0.3 0.4 1-T 1-T NNLO corrections sizable theory error reduced by 30–40 % large 1 − T : need hadronization corrections small 1 − T : two-jet region, need matching onto NLL resummation Work in progress: G. Luisoni, TG mean value � 1 − T � : A = 2 . 101 B = 44 . 98 C = 1095 ± 130 � 1 − T � ( α s = 0 . 1189) = 0 . 0398 + 0 . 0146 + 0 . 0068 e + e − → 3 jets at NNLO – p.13
Results NNLO heavy jet mass and C -parameter heavy jet mass M 2 H /s C -parameter 0.5 0.5 M H 1/ σ had d σ /d M H C 1/ σ had d σ /d C NNLO NNLO 0.4 0.4 NLO NLO LO LO 0.3 0.3 Q = M Z 0.2 0.2 α s (M Z ) = 0.1189 Q = M Z 0.1 0.1 α s (M Z ) = 0.1189 0 0 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 M H C heavy jet mass (closely related to thurst) has very small NNLO corrections NNLO corrections for C large again require matching onto NLL resummation and hadronization corrections Sudakov shoulder in C = 0 . 75 also requires resummation S. Catani, B. Webber e + e − → 3 jets at NNLO – p.14
Results NNLO jet broadenings wide jet boadening B W total jet boadening B T 0.6 0.7 B W 1/ σ had d σ /d B W B T 1/ σ had d σ /d B T NNLO NNLO 0.6 NLO NLO LO 0.5 LO 0.4 0.4 Q = M Z Q = M Z α s (M Z ) = 0.1189 α s (M Z ) = 0.1189 0.3 0.2 0.2 0.1 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 B W B T relative magnitude of NNLO corrections smaller than for thurst NNLO corrections for B W smaller than for B T again require matching onto NLL resummation and hadronization corrections e + e − → 3 jets at NNLO – p.15
Recommend
More recommend
