Writing your own Monte Carlo Integrator

Transcription

1 Writing your own Monte Carlo Integrator Thomas Morgan 18/12/2014 Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 1/ 23

2 Disclaimer What I won t be talking about... Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 2/ 23

3 Disclaimer What I won t be talking about... Sherpa Herwig++ Pythia Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 2/ 23

4 Disclaimer What I will be talking about... Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 3/ 23

5 Disclaimer What I will be talking about... Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 3/ 23

6 Ingredients Things we need VEGAS + Phase Space Generator Parton Distribution Functions(PDFs) Matrix elements Jet algorithm + observables Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 4/ 23

7 Ingredients Things we need VEGAS + Phase Space Generator Parton Distribution Functions(PDFs) Matrix elements Jet algorithm + observables Subtraction terms(beyond Leading Order(LO)) Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 4/ 23

8 VEGAS Older than the known universe. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 5/ 23

9 VEGAS Use the idea of importance sampling. Generatesasetofrandomnumbers,theserandomnumbersarefedintothe phase space generator to generate a unique phase space point. Start with a uniform unit n-dimensional grid, where n is the required number of variables to define your phase space point. Oncetheinitialsampleiscomplete,VEGASadaptsthegridtofocusonthe dominant features. Rinse and repeat multiple times until you have a good convergence. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 6/ 23

10 from Protons to Partons TocomputecrosssectionsweneedtoexploitthefactorisationofQCDintolow energy physics and high energy physics. dσ = i,j Parton Distribution Functions dξ1 dξ {}}{ 2 f i (ξ 1,µ 2 F ξ 1 ξ )f j(ξ 2,µ 2 F ) dˆσ ij(α s (µ R ),µ R,µ F ) 2 }{{} Partonic Cross Section (1) ξ 1 andξ 2 arethemomentumfractionsofparton1and2respectively. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 7/ 23

11 Parton Distribution Functions Parton Distribution Functions(PDFs) give us the probability of finding a parton with a certain momentum fraction within a proton. Theydescribelowenergyphysicsoftheproton.Thisisimpossibletomodelin any meaningful way, we fit PDFs using known results from previous collider experiments. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 8/ 23

12 Partonic Cross sections We re interested in computing cross sections for massless QCD in a perturbative regime, dˆσ ij = dˆσ LO ij + ( αs (µ R ) 2π ) dˆσ NLO ij + ( αs (µ R ) 2π ) 2 dˆσ NNLO ij +O(α 3 s). (2) Whatdoweneedtocalculateacrosssectionforagivenorder? Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/2014 9/ 23

13 Matrix Elements Describes the high energy physics of the event(the interesting bit). Numerous tools on the market for tree(helicity amplitudes, colour decompositions, recursion relations,...) and one loop(integrand reduction, generalised unitarity,...) scattering amplitudes. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

14 NLO Corrections Oneextrapowerofα s.thisimpliesweareleftwithtwopossibilities Real corrections Jetfunctionmaps3partons 2jets. Virtual corrections Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

15 IR singularities in Real Corrections Considertherealradiationcorrectionstotheγ 2jetprocess,thematrix element behaves like M(1 q,i g,2 q ) 2 1 E g (1 cos(θ qg ))(1 cos(θ qg )) (3) Singularities E g 0, softsingularity θ qg 0, collinearsingularity θ qg 0, collinearsingularity Singularities are bad! Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

16 IR singularities in Virtual Corrections An explicit pole structure appears from dimensional regularisation. Catani pole structure {}}{ M2 1 (1 q,2 q ) =2I qq (ǫ,µ;s 12 )M2 0 (1 q,2 q )+O(ǫ 0 ) (4) Explicitpolestructuresarereallybad... Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

17 Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

18 Subtraction ( ) ( ) dˆσ q q NLO = dˆσ R,NLO q q dˆσ q q S,NLO + dˆσ V,NLO q q dˆσ q q T,NLO, (5) dσ 3 dσ 2 whereeachsetofbracketsisfreeofirpoles.also dˆσ q q T,NLO = dˆσ q q S,NLO. (6) dσ 1 Unlike UV poles, your cross section and all IR-safe observables are not dependent on your subtraction scheme. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

19 NLO Subtraction Subtraction schemes Catani-Seymour(CS) dipole subtraction Frixione-Krunszt-Signer(FKS) subtraction Phase space slicing Sector Decomposition Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

20 Antenna Subtraction Follows a very similar idea to CS dipole subtraction. Exploit the IR universal factorisation of QCD M 0 3(1 q,i g,2 q ) Antenna function i g {}}{ unresolved A 0 3(1 q,i g,2 q ) M2( (1i) 0 q, (i2) q ). (7) }{{} reduced matrix element The antenna function only depends on the momentum configuration and flavours of the unresolved parton and the hard radiators. By construction it contains all the unresolved limits between the two hard radiators and the unresolved parton. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

21 Antenna Subtraction M 0 3 (1 q,3 g,2 q ) M 0 2 ( (13) q, (32) q ) A0 3(1 q,3 g,2 q ). (8) Once we have an antenna we can recycle this for arbitrarily complicated processes with the same unresolved limits. Problems The resulting antenna must be suitably simple such that we can integrate it analytically. Weneedtodefineamapfromthen+1 nphasespacesuchthatwe correctly intepolate between the limits in your antenna. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

22 Processes quark-antiquarkantenna:γ qg q quark-gluonantenna: χ ggg, χ gq q gluon-gluonantenna:h ggg,h gq q Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

23 NNLO Corrections Nowwehavetwoextrapowersofα s and3possiblecorrections. Real-Real corrections Real-Virtual corrections Virtual-Virtual corrections Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

24 NNLO cross section dˆσ q q NNLO = + + ( dˆσ RR,NNLO q q dσ 4 ( dˆσ RV,NNLO q q dσ 3 ( dˆσ VV,NNLO q q dσ 2 dˆσ q q S,NNLO ) ) dˆσ q q T,NNLO dˆσ q q U,NNLO ) (9) Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

25 New Integredients for NNLO New Integredients n +2partonphasespacegenerator-trivial(ish) a numerically stable one loop matrix element- usually ok new subtraction terms- very hard twoloopmatrixelement-veryhard Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23

26 Conclusions ThedifferencebetweenanNLOandNNLOMCintegratoristrivialoncewe have the subtraction scheme and two loop matrix elements. Antenna subtraction provides a numerically efficient and relatively simple approach to dealing with IR singularities at NNLO. Hopefullywe llbeabletoproviderealphysicsresultsinthenottoodistant future. Thomas Morgan (IPPP, Durham) Writing your own Monte Carlo Integrator 18/12/ / 23