Jet production in the colorful NNLO framework

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1 Jet production in the colorful NNLO framework Adam Kardos! University of Debrecen, MTA-DE Particle Physics Research Group and NCSR Demokritos", Athens!

2 QCD matters High precision experiments demand high precision predictions. Relatively large coupling final state is QCD dominated. Key processes have irreducible QCD background and/or QCD corrections. 3.0 Durham clustering at y cut =0.0 d j d 1 [MeV] µ [0., ]m H LO NLO NNLO Large scale uncertaint(y)(ies). - H bb NNLO Del Duca et al. arxiv:

3 Road to high precision predictions How to increase precision? Go one step further in the perturbative expansion: = LO + NLO + NNLO +... The aim: numerical calculation kinematical singularities due to soft/collinear parton emissions are treated with local Z NLO = m+1 h d R mj m+1 Singly unresolved i d R,A m+1 J m + Z m apple d V m + Z 1 d R,A m+1 J m

4 The Colorful NNLO framework

5 The Colorful NNLO framework The Colorful NNLO framework was worked out for colorless initial states by Del Duca, Somogyi, Trocsanyi et al., for detailed information see Gabor s NNLO = = + + Z Z Z m+ m+1 m ( n d RR m+j m+ d RR,A m+ J m ( d RV m+1 + d VV m + Z Z 1 d RR,A 1 m+ J m+1 " h i d RR,A m+ d RR,A 1 m+ h io d RR,A 1 m+ J m+1 d RR,A 1 m+ J m + d RV,A 1 m+1 + Z 1 " Z d RV,A 1 m+1 + d RR,A 1 m+ 1 Z 1 d RR,A 1 m+ "=0 A1 # J m ) A1 #) "=0 "=0 J m

6 @NNLO: m-parton line: Z m ( d VV m + Z The Colorful NNLO framework h i Z " Z #) A1 d RR,A m+ d RR,A 1 m+ + d RV,A 1 m+1 + d RR,A 1 m+ J m 1 1 "=0 Free of kinematical singularities (jet-function!). Contains the integrals of the subtraction terms ( see the talk of Gabor on thursday afternoon, too). State of the art: Two-loop amplitude ( & integration of subtraction terms).

7 @NNLO: m+1-parton line: Z m+1 ( d RV m+1 + The Colorful NNLO framework Z 1 d RR,A 1 m+ J m+1 " d RV,A 1 m+1 + Z d RR,A 1 m+ 1 A1 # J m ) "=0 Only singly unresolved regions present. But RV has different factorization properties (one-loop amplitude is involved). State of the art: fast and precise evaluation of the involved one-loop amplitude (even in the singly unresolved region!).

8 The Colorful NNLO m+-parton line: Z m+ n d RR m+j m+ d RR,A m+ J m h io d RR,A 1 m+ J m+1 d RR,A 1 m+ J m "=0 Both singly and doubly unresolved regions present. Also spurious ones! e.g.: kinematical singularities of A in singly and kinematical singularities of A 1 in doubly unresolved regions.

9 The Colorful NNLO framework m+-parton line: Bottleneck(s): RR SME still tree-level, but contains two partons more compared to the Born one. Several subtraction terms. For e + e! u ūggg O(100). Generation of UB PS points (momentum mappings) and the corresponding UB SMEs. Application of physical cuts. Filling histograms.

10 The Colorful NNLO framework Bottom line: Several subprocesses contribute. Various subtraction terms have to be calculated. Subtraction terms present a twofold problem: heavy bookkeeping due to the immense number of terms, a clever grouping is needed to ease up (a bit) the computational burden. More profitable to automatize subtractions!

11 A numerical implementation of the Colorful NNLO framework

12 A numerical implementation For demonstration we take e+ e- to 3-jet production. The code is organized directory-wise, for a new process a new folder has to be created:

13 A numerical implementation The generation of all subprocesses is automatic:

14 A numerical implementation Investigating for possible numerical relations between SMES:

15 A numerical implementation Automatic detection of all singular regions:

16 A numerical implementation An NNLO calculation is extremely complex. Due to this complexity it is good practice to make as much checks as possible. In our code the following ones are built in: Check upon individual subtraction terms, e.g.: C irs lim p i p r p s M RR =1 Checking bookkeeping and overall consistency by checking complete lines, e.g.: lim p i p r,p s!0 A 1 + A A 1 M RR =1

17 A numerical implementation Testing the subtraction terms in all limits (even in quad precision):

18 A numerical implementation Testing the whole m+ parton line: Doubly unresolved Singly unresolved:

19 A numerical implementation e + e! q qggg A 1 +A A 1,p M RR p 6 p 7 y 67 =10 6 y 67 =10 8 y 67 = e + e! q qggg A 1 +A A 1 M RR p p 6,p 4 p 7 y 6 = y 47 =10 4 y 6 = y 47 =10 y 6 = y 47 =10 6 #ofevents 10 #ofevents ratio ratio Behavior of the m+ parton line in a triple and double collinear limit.

20 A numerical implementation e + e! q qggg A 1 +A A 1,p M RR p 6 y 6 =10 6 y 6 =10 8 y 6 = e + e! q qggg A 1 +A A 1 M RR p 6! 0 y 6 =10 4 y 6 =10 y 6 =10 6 #ofevents 10 #ofevents ratio ratio Behavior of the m+ parton line in a collinear and a soft limit.

21 Towards phenomenology

22 e + e - 3 jets Considering only e + + e!!3 jets NNLO* : no VV and I operators, hence non-physical, though still informative. p s = 90 GeV µ R = m Z m Z = GeV,m W = GeV s (m Z )=0.118, 1/ EM = 13.3 Calculating key event shape variables. For a precise definition, see: Gehrmann-De Ridder et al., arxiv: and also Kunszt et al. QCD at LEP, ETH-PT The whole calculation took 4+4 hours on 300 cores.

23 e + e - 3 jets Total and wide jet broadening: BT d db [pb] T NNLO e + e!! 3jet p s =90GeV µ R = m Z B T BW d db [pb] W NNLO e + e!! 3jet p s =90GeV µ R = m Z B W Preliminary

24 Y3 d dy3 [pb] e + e - 3 jets The jet transition variable and large hemisphere invariant mass: NNLO e + e!! 3jet p s =90GeV µ R = m Z log Y 3 d d [pb] NNLO e + e!! 3jet p s =90GeV µ R = m Z Preliminary

25 (1 T ) d dt [pb] e + e - 3 jets The jet transition variable and large hemisphere invariant mass: NNLO e + e!! 3jet p s =90GeV µ R = m Z T C d dc [pb] e + e!! 3jet p s =90GeV µ R = m Z NNLO C Preliminary

26 e + e - 3 jets and even a completely new one, never appeared in the literature before: Energy-energy correlation: sin d EEC d cos [pb] e + e!! 3jet p s =90GeV µ R = m Z NNLO cos Calculation cost: cores. Preliminary

27 Conclusions

28 Conclusions A general numerical NNLO framework is introduced to manage subtractions for the case of e + e - annihilation. It contains all subtraction terms, the user only has to provide his/her matrix elements (the I operators will also be included). The framework was applied to 3-jet production to obtain some key distributions. The inclusion of the I operators and the VV part for 3-jet production is in progress.

29 Thank you for your attention!