Basics of Event Generators II

Transcription

1 Basics of Event Generators II Leif Lönnblad Department of Astronomy and Theoretical Physics Lund University MCnet School on Event Generators São Paulo Event Generators II 1 Leif Lönnblad Lund University

2 Outline of Lectures Lecture I: Basics of Monte Carlo methods, the event generator strategy, matrix elements, LO/NLO,... Lecture II: Parton showers, initial/final state, matching/merging,... Lecture III: Matching/merging (cntd.), underlying events, multiple interactions, minimum bias, pile-up, hadronization, decays,... Lecture IV: Protons vs. heavy ions, summary,... Buckley et al. (MCnet collaboration), Phys. Rep. 504 (2011) 145. Event Generators II 2 Leif Lönnblad Lund University

3 Outline Final-State Showers Angular Ordering Evolution Variables The Sudakov Veto Algorithm Backwards Evolution k -Factorization The Basic Idea Tree-level matching NLO Matching Multi-leg NLO Matching Event Generators II 3 Leif Lönnblad Lund University

4 Angular Ordering Evolution Variables The Sudakov Veto Algorithm The purpose of parton showers is to generate real exclusive events on parton level down to a very low (almost non-perutbative) jet resolution scale µ. Starting from an initial hard scattering eg. e + e q q or q q Z 0, we basically need σ +0 = σ 0 (1+C 01 α s + C 02 αs 2 + C 03 αs ) σ +1 = σ 0 (C 11 α s + C 12 αs 2 + C 13 αs ) σ +2 = σ 0 (C 22 αs 2 + C 23 αs 3 + C 24 αs ). Tree-level generators only gives us inclusive events. NLO generators only gives us one extra parton. Event Generators II 4 Leif Lönnblad Lund University

5 Angular Ordering Evolution Variables The Sudakov Veto Algorithm The purpose of parton showers is to generate real exclusive events on parton level down to a very low (almost non-perutbative) jet resolution scale µ. Starting from an initial hard scattering eg. e + e q q or q q Z 0, we basically need σ +0 = σ 0 (1+C 01 α s + C 02 αs 2 + C 03 αs ) σ +1 = σ 0 (C 11 α s + C 12 αs 2 + C 13 αs ) σ +2 = σ 0 (C 22 αs 2 + C 23 αs 3 + C 24 αs ). Tree-level generators only gives us inclusive events. NLO generators only gives us one extra parton. Event Generators II 4 Leif Lönnblad Lund University

6 Angular Ordering Evolution Variables The Sudakov Veto Algorithm The purpose of parton showers is to generate real exclusive events on parton level down to a very low (almost non-perutbative) jet resolution scale µ. Starting from an initial hard scattering eg. e + e q q or q q Z 0, we basically need σ +0 = σ 0 (1+C 01 α s + C 02 αs 2 + C 03 αs ) σ +1 = σ 0 (C 11 α s + C 12 αs 2 + C 13 αs ) σ +2 = σ 0 (C 22 αs 2 + C 23 αs 3 + C 24 αs ). Tree-level generators only gives us inclusive events. NLO generators only gives us one extra parton. Event Generators II 4 Leif Lönnblad Lund University

7 Final-State Showers Angular Ordering Evolution Variables The Sudakov Veto Algorithm The tree-level matrix element for an n-parton state can be approximated by a product of splitting functions corresponding to a sequence of one-parton emissions from the zeroth order state. e e + q _ q Event Generators II 5 Leif Lönnblad Lund University

8 Final-State Showers Angular Ordering Evolution Variables The Sudakov Veto Algorithm The tree-level matrix element for an n-parton state can be approximated by a product of splitting functions corresponding to a sequence of one-parton emissions from the zeroth order state. We can then order the emissions acording to some resolution scale, ρ, so that ρ 1 ρ 2 ρ 3... e e q 5 _ q Event Generators II 5 Leif Lönnblad Lund University

9 Angular Ordering Evolution Variables The Sudakov Veto Algorithm We have the standard DGLAP splitting kernels P q qg (ρ, z)dρdz P g gg (ρ, z)dρdz P g q q (ρ, z)dρdz = α s dρ dz 2π ρ C F 1+z 2 1 z = α s dρ dz 2π ρ N (1 z(1 z)) 2 C z(1 z) = α s dρ dz 2π ρ T R(z 2 +(1 z) 2 ) where ρ is the squared invariant mass or transverse momentum, and z is the energy (or light-cone) fraction taken by one of the daugthers. (We ignore the φ-dependence here). Event Generators II 6 Leif Lönnblad Lund University

10 Angular Ordering Evolution Variables The Sudakov Veto Algorithm We now to make the events exclusive. This is done by saying that the first emission at some ρ 1 is given by the splitting kernel multiplied by the probability that there has been no emission above that scale. In a given interval dρ we have the no-emission probability 1 dρ dz P a bc (z,ρ) bc Integrating from ρ 1 up to some maximum scale, ρ 0 we get ( (ρ 0,ρ 1 ) = exp ρ0 ) dρ dz P a bc (z,ρ) bc ρ 1 Event Generators II 7 Leif Lönnblad Lund University

11 Angular Ordering Evolution Variables The Sudakov Veto Algorithm In the same way we get the probability to have the nth emission at some scale ρ n P(ρ n ) = dz P a bc (ρ n, z) abc ( exp ρn 1 ) dρ dz P a bc (z,ρ ) abc ρ n Event Generators II 8 Leif Lönnblad Lund University

12 Angular Ordering Evolution Variables The Sudakov Veto Algorithm Integrating we get schematically σ +0 σ +1 σ +2 = σ 0 S0 = σ 0 (1+C01 PS α s + C02 PS α2 s +...) = σ 0 C11 PS α s S1 = σ 0 (C11 PS α s + C12 PS α2 s + C13 PS α3 s +...) = σ 0 C22 PS α2 s S2 = σ 0 (C22 PS α2 s + C23 PS α3 s + C24 PS α4 s +...). We still need a cutoff, ρ cut, and the coefficients C PS nn diverges as log 2n ρ max /ρ cut but the Sudakovs corresponds to the an approximate resummation of all virtual terms and makes things finite, and we can use ρ cut 1 GeV. Event Generators II 9 Leif Lönnblad Lund University

13 Angular Ordering Evolution Variables The Sudakov Veto Algorithm Integrating we get schematically σ +0 σ +1 σ +2 = σ 0 S0 = σ 0 (1+C01 PS α s + C02 PS α2 s +...) = σ 0 C11 PS α s S1 = σ 0 (C11 PS α s + C12 PS α2 s + C13 PS α3 s +...) = σ 0 C22 PS α2 s S2 = σ 0 (C22 PS α2 s + C23 PS α3 s + C24 PS α4 s +...). We still need a cutoff, ρ cut, and the coefficients C PS nn diverges as log 2n ρ max /ρ cut but the Sudakovs corresponds to the an approximate resummation of all virtual terms and makes things finite, and we can use ρ cut 1 GeV. Event Generators II 9 Leif Lönnblad Lund University

14 Angular Ordering Evolution Variables The Sudakov Veto Algorithm The divergencies comes from the soft and collinear poles in the splitting kernels, eg. ρ0 ρ0 α s dρ dρ dz P q qg (ρ, z) ln(ρ 0 /ρ) α s ln 2 (ρ 0 /ρ c ) ρ ρ c Parton showers systematically resums all orders of α n s ln 2n (ρ 0 /ρ c ) which is the main part of the higher order corrections. (Also important terms α n s ln 2n 1 (ρ 0 /ρ c ) are resummed.) ρ c Event Generators II 10 Leif Lönnblad Lund University

15 Angular Ordering Evolution Variables The Sudakov Veto Algorithm The divergencies comes from the soft and collinear poles in the splitting kernels, eg. ρ0 ρ0 α s dρ dρ dz P q qg (ρ, z) ln(ρ 0 /ρ) α s ln 2 (ρ 0 /ρ c ) ρ ρ c Parton showers systematically resums all orders of α n s ln 2n (ρ 0 /ρ c ) which is the main part of the higher order corrections. (Also important terms α n s ln 2n 1 (ρ 0 /ρ c ) are resummed.) ρ c However if there is no strong ordering, ρ 1 ρ 2 ρ 3..., the PS approximation breaks down Parton showers cannot model several hard jets very well. Especially the correlations between hard jets are poorly described. Event Generators II 10 Leif Lönnblad Lund University

16 Angular Ordering Angular Ordering Evolution Variables The Sudakov Veto Algorithm The splitting probabilities means that coherence effects are not taken into account 2 + Event Generators II 11 Leif Lönnblad Lund University

17 Angular Ordering Angular Ordering Evolution Variables The Sudakov Veto Algorithm The splitting probabilities means that coherence effects are not taken into account Most coherence effects can be taken into account by angular ordering. Event Generators II 11 Leif Lönnblad Lund University

18 Angular Ordering Angular Ordering Evolution Variables The Sudakov Veto Algorithm The splitting probabilities means that coherence effects are not taken into account Most coherence effects can be taken into account by angular ordering. Some angular correlations can also be taken into account by adjusting the azimuthal angles after a shower is generated. Event Generators II 11 Leif Lönnblad Lund University

19 Angular Ordering Evolution Variables The Sudakov Veto Algorithm Coherence effects can be included directly, by considering gluon radiation from colour dipoles between colour-connected partons = Event Generators II 12 Leif Lönnblad Lund University

20 Angular Ordering Evolution Variables The Sudakov Veto Algorithm Coherence effects can be included directly, by considering gluon radiation from colour dipoles between colour-connected partons = Rather than iterating 1 2 parton splitting we iterate 2 3 splittings. Each emission from a dipole will create two new dipoles, each of which may continue radiating. This was first implemented in the ARIADNE generator. Recently similar schemes have been implemented in PYTHIA, HERWIG++, SHERPA and VINCIA. Event Generators II 12 Leif Lönnblad Lund University

21 Evolution Variables Angular Ordering Evolution Variables The Sudakov Veto Algorithm How do we choose the evolution variable, ρ? The most natural choice is to choose a variable which isolates both the soft and collinear poles in the splitting kernel. This is the case for ρ = p 2 as used in eg. ARIADNE. In old versions of PYTHIA and SHERPA the evolution variable is the virtuality Q 2 which in principle is fine except that α s (p 2 ) may diverge for any given Q 2. Also angular ordering needs to be imposed in separately. In HERWIG the ordering is in angle, which ensures angular ordering, but does not isolate the soft pole, and an additional cutoff is needed. Event Generators II 13 Leif Lönnblad Lund University

22 Angular Ordering Evolution Variables The Sudakov Veto Algorithm Transverse momentum ρ = p 2 ln p Virtuality ρ = Q 2 p2 z(1 z) ln p Angle ρ E 2 θ 2 p2 z 2 (1 z) 2 ln p y y y Event Generators II 14 Leif Lönnblad Lund University

23 Angular Ordering Evolution Variables The Sudakov Veto Algorithm The Fifth Commandment of Event Generation Thou shalt always be independent of Lorentz frame Event Generators II 15 Leif Lönnblad Lund University

24 Angular Ordering Evolution Variables The Sudakov Veto Algorithm Final-state parton showers did really well at LEP corr. fac. 1/N dn/d(1-t) Thrust charged & neutral particles JT 7.3 PS JT 7.4 PS AR 4.06 H 5.8C JT 7.4 ME corr. fac. 1/N dn/dp t out p out t Thr. charged & neutral particles JT 7.3 PS JT 7.4 PS AR 4.06 H 5.8C JT 7.4 ME DELPHI 10-2 DELPHI (MC-Data)/Data (MC-Data)/Data (1-T) p out 3.5 t [GeV] Event Generators II 16 Leif Lönnblad Lund University

25 ˆ Evolution Variables The Sudakov Veto Algorithm How do we generate a parton shower emission? ( P(t) = P(t) exp tmax t ) dt P(t ) P(t) is a probability distribiution, so we can do the standard transformation method 1 tmax ( tmax ) 1 r = dt p R (t) = dtp(t) = 1 exp dt P(t ) r So if P has a simple primitive function F we get but P is never simple... t t = F 1 (F(t max ) ln r) t Event Generators II 17 Leif Lönnblad Lund University

26 ˆ Evolution Variables The Sudakov Veto Algorithm How do we generate a parton shower emission? ( P(t) = P(t) exp tmax t ) dt P(t ) P(t) is a probability distribiution, so we can do the standard transformation method 1 tmax ( tmax ) 1 r = dt p R (t) = dtp(t) = 1 exp dt P(t ) r So if P has a simple primitive function F we get but P is never simple... t t = F 1 (F(t max ) ln r) t Event Generators II 17 Leif Lönnblad Lund University

27 ˆ Evolution Variables The Sudakov Veto Algorithm How do we generate a parton shower emission? ( P(t) = P(t) exp tmax t ) dt P(t ) P(t) is a probability distribiution, so we can do the standard transformation method 1 tmax ( tmax ) 1 r = dt p R (t) = dtp(t) = 1 exp dt P(t ) r So if P has a simple primitive function F we get but P is never simple... t t = F 1 (F(t max ) ln r) t Event Generators II 17 Leif Lönnblad Lund University

28 ˆ Evolution Variables The Sudakov Veto Algorithm Assume g is a simple function with a simple primitive function G such that g(t) P(t), t. Then we can use the following algorithm start with t 0 = t max ; select t i = G 1 (G(t i 1 ) ln R), compare a (new) R with the ratio P(t i )/g(t i ); if P(t i )/g(t i ) R, then return to point 2 for a new try, i i + 1; otherwise t i is retained as final answer. If t i < t cut, there is no emission and the shower is done. Event Generators II 18 Leif Lönnblad Lund University

29 ˆ Evolution Variables The Sudakov Veto Algorithm Consider the various ways in which one can select a specific scale t. The probability that the first try works, t = t 1, i.e. that no intermediate t values need be rejected, is given by p 0 (t) = e tmax t g(t ) dt g(t) P(t) g(t) = tmax P(t)e t g(t ) dt The probability that we have thrown away one intermediate value t 1 tmax [ p 1 (t) = dt 1 e tmax t g(t ) dt 1 g(t 1 ) 1 P(t ] 1) t g(t 1 ) e t 1 t g(t ) dt g(t) P(t) g(t) Event Generators II 19 Leif Lönnblad Lund University

30 ˆ Evolution Variables The Sudakov Veto Algorithm Similarly we get tmax p 2 (t) = p 0 (t) tmax p 1 (t) = p 0 (t) dt 1 [g(t 1 ) P(t 1 )] t t t1 dt 1 [g(t 1 ) P(t 1 )] ( ) 2 = p 0 (t) 1 tmax [g(t ) P(t )] dt 2 t t dt 2 [g(t 2 ) P(t 2 )] p tot (t) = ( ) n 1 tmax p n (t) = p 0 (t) [g(t ) P(t )] dt n! t n=0 n=0 = P(t)e tmax t g(t ) dt e tmax t [g(t ) P(t )]dt = P(t)e tmax t P(t ) dt Event Generators II 20 Leif Lönnblad Lund University

31 ˆ Evolution Variables The Sudakov Veto Algorithm Also if several things may happen, P 1 (t), P 2 (t), P 3 (t),... the probability of i happening first is P i (t) j e tmax t P j (t ) dt Simply generate a scale for each i according to P i (t) e tmax t P i (t ) dt and pick the process with the largest scale. Event Generators II 21 Leif Lönnblad Lund University

32 Backwards Evolution k -Factorization For incoming hadrons, we need to consider the evolution of the parton densities. Using collinear factorization and DGLAP evolution we have (with t = log k 2 /Λ2 ) df b (x, t) dt = a dx x f a (x, t) α s 2π P a b ( x x ) We can interpret this as during a small increase dt there is a probability for parton a with momentum fraction x to become resolved into parton b at x = zx and another parton c at x x = (1 z)x. Event Generators II 22 Leif Lönnblad Lund University

33 Backwards Evolution k -Factorization c s W + u d g p p/ p Event Generators II 23 Leif Lönnblad Lund University

34 Backwards Evolution k -Factorization In a backward evolution scenario we start out with the hard sub-process at some scale t max σ 0 ˆσ ab X f a (x a, t max )f b (x b, t max ) and we get the relative probability for the parton a to be unresolved into parton c during a decrease in scale dt dp a = df a(x a, t) f a (x a, t) = dt c dx x f c (x, t) f a (x a, t) α ( s 2π P xa ) c a x Summing up the cumulative effect of many small changes dt, the probability for no radiation exponentiates and we get a Sudakov S+a (x a, t max, t) = exp { tmax t dt c dx x f c (x, t ) f a (x a, t ) } α s (t ) ( 2π P xa ) c a x Event Generators II 24 Leif Lönnblad Lund University

35 Backwards Evolution k -Factorization This now gives us the probability for the first backwards initial-state splitting dp ca = α s 2π P ac(z) f c(x a /z, t) dz dt f a (x a, t) z S +a (x a, t max, t) In a hadronic collision we first generate the hard scattering, then evolve the incoming partons backward to lower scales, and then alow for a final-state shower from all partons from the hard scattering and the initial-state shower. This is like undoing the evolution of the PDFs Event Generators II 25 Leif Lönnblad Lund University

36 The small-x problem Backwards Evolution k -Factorization DGLAP evolution is not applicable if the hard scale is much smaller than the total energy and the virtuality of the incoming partons are not much smaller than the hard scale. (small x) Collinear factorization = k -factorization dx a dx bˆσ ab X f a (x a, Q 2 )f b (x b, Q 2 ) = dx a dx b dk a dk bˆσ ab X F a(x a, k a, Q 2 )F b (x b, k b, Q 2 ) F an unintegrated parton density. ˆσ is the off-shell matrix element Event Generators II 26 Leif Lönnblad Lund University

37 Backwards Evolution k -Factorization lepton proton P k0 qγ kn k2 k1 qn+1 q3 q2 q1 In DIS, the cross section is dominated by events with small Q 2 = q 2 γ and small x. The available phase space for emitting partons is not limited by Q 2, but rather by the total hadronic energy, W 2 Q 2 /x. The 1/z pole in the gluon splitting function makes it possible to emit many initial-state gluons even for small Q 2. We need to take into account unordered evolution. Event Generators II 27 Leif Lönnblad Lund University

38 Backwards Evolution k -Factorization lepton proton P k0 k1 k2 qγ kn q1 q2 qn+1 q3 In DIS, the cross section is dominated by events with small Q 2 = q 2 γ and small x. The available phase space for emitting partons is not limited by Q 2, but rather by the total hadronic energy, W 2 Q 2 /x. The 1/z pole in the gluon splitting function makes it possible to emit many initial-state gluons even for small Q 2. We need to take into account unordered evolution. Forward jets at HERA cannot be reproduced by DGLAP based initial-state parton showers. Event Generators II 27 Leif Lönnblad Lund University

39 Backwards Evolution k -Factorization lepton proton P k0 k1 k2 qγ kn q1 q2 qn+1 q3 In DIS, the cross section is dominated by events with small Q 2 = q 2 γ and small x. The available phase space for emitting partons is not limited by Q 2, but rather by the total hadronic energy, W 2 Q 2 /x. The 1/z pole in the gluon splitting function makes it possible to emit many initial-state gluons even for small Q 2. We need to take into account unordered evolution. Forward jets at HERA cannot be reproduced by DGLAP based initial-state parton showers. We need BFKL or CCFM. Event Generators II 27 Leif Lönnblad Lund University

40 Backwards Evolution k -Factorization dσ Det /de T,Jet [nb/gev] ZEUS data ARIADNE LEPTO HERWIG 5.9 ZEUS 1995 dσ Det /dx Jet [nb] b) 10 1 a) E T,Jet [GeV] 10-2 x Jet 10-1 dσ Det /dη [nb] c) dσ Det /d(e 2 /Q 2 ) [nb] T,Jet d) η Jet E 2 1 /Q 2 10 T,Jet Event Generators II 28 Leif Lönnblad Lund University

41 However, at the LHC... Backwards Evolution k -Factorization Event Generators II 29 Leif Lönnblad Lund University

42 The Basic Idea Tree-level matching NLO Matching lepton The power of k -factorization is the resummation of un-ordered emissions, which cannot be done by DGLAP-based showers. proton P k0 qγ kn k2 k1 qn+1 q3 q2 q1 BFKL- or CCFM-based showers do not yet describe data. But we now have matrix element generators which can produce many un-ordered emissions. But to get the complete picture we need to combine many multiplicities, and we still need parton showers. Event Generators II 30 Leif Lönnblad Lund University

43 The Basic Idea Tree-level matching NLO Matching Tree-level matrix element generators and are good for a handful hard, well separated partons, but bad for many soft and collinear partons. Parton shower generators are not good for a handful hard, well separated partons, but good for many soft and collinear partons. Let s try to get the best of both. Event Generators II 31 Leif Lönnblad Lund University

44 The Basic Idea Tree-level matching NLO Matching Fixed-order expansion of a parton shower (using P i = F i F i 1 P i ) [ dσ0 ex ρ0 ( = F 0 M α dφ S dρdzp 1 + α2 ρ0 ) ] 2 S dρdzp 1 0 ρ MS 2 ρ MS dσ ex 1 = F 0 M 0 2 α dφ S P 1 dρ 1 dz 1 0 [ ρ0 1 α S ρ 1 dρdzp 1 α S ρ1 ρ MS dρdzp 2 dσ 2 dφ 0 = F 0 M 0 2 α 2 S P 1dρ 1 dz 1 P 2 dρ 2 dz 2 Θ(ρ 1 ρ 2 ) Unitary to all orders in α S total cross section is F 0 M jet cross section will not even be correct to LO. Event Generators II 32 Leif Lönnblad Lund University ]

45 ME reweighting The Basic Idea Tree-level matching NLO Matching We really want to improve our parton shower. The easiest thing is P i P ME i M i 2 dφ i M i 1 2 dφ i 1 dρdz This has been around quite a while in PYTHIA for the first splitting in some processes. Preserves the unitarity of the shower Event Generators II 33 Leif Lönnblad Lund University

46 The Basic Idea Tree-level matching NLO Matching [ dσ0 ex ρ0 ( = F 0 M α dφ S dρdzp1 ME + α2 ρ0 ) ] 2 S dρdzp1 ME 0 ρ MS 2 ρ MS dσ1 ex = F 0 M 0 2 α dφ S P1 ME dρ 1 dz 1 0 [ ρ0 ρ1 ] 1 α S dρdzp1 ME α S dρdzp 2 ρ 1 ρ MS dσ 2 = F 0 M 0 2 αs 2 dφ PME 1 dρ 1 dz 1 P 2 dρ 2 dz 2 Θ(ρ 1 ρ 2 ) 0 Still unitary to all orders of α S. We can decrease ρ MS to the non-perturbative boundary ρ cut. Going to higher multiplicities turns out to be difficult. Event Generators II 34 Leif Lönnblad Lund University

47 The Basic Idea Tree-level matching NLO Matching Matching: The Basic Idea A fixed-order ME-generator gives the first few orders in α s exactly. The parton shower gives approximate (N)LL terms to all orders in α s through the Sudakov form factors. Take a parton shower and correct the first few terms in the resummation with (N)LO ME. Take events generated with (N)LO ME with subtracted Parton Shower terms. Add parton shower. Take events samples generated with (N)LO ME, reweight and combine with Parton showers: Event Generators II 35 Leif Lönnblad Lund University

48 Tree-level Merging The Basic Idea Tree-level matching NLO Matching Has been around the whole millennium: CKKW(-L), MLM,... Combines samples of tree-level (LO) ME-generated events for different jet multiplicities. Reweight with proper Sudakov form factors (or approximations thereof). Needs a merging scales to separate ME and shower region and avoid double counting. Only observables involving jets above that scale will be correct to LO. Typically the merging scale dependence is beyond the precision of the shower: O(L 3 αs) 2 1 +O(L 2 α N s). 2 c 2 Event Generators II 36 Leif Lönnblad Lund University

49 Tree-level Merging The Basic Idea Tree-level matching NLO Matching Has been around the whole millennium: CKKW(-L), MLM,... Combines samples of tree-level (LO) ME-generated events for different jet multiplicities. Reweight with proper Sudakov form factors (or approximations thereof). Needs a merging scales to separate ME and shower region and avoid double counting. Only observables involving jets above that scale will be correct to LO. Typically the merging scale dependence is beyond the precision of the shower: O(L 3 αs) 2 1 +O(L 2 α N s). 2 c 2 Event Generators II 36 Leif Lönnblad Lund University

50 CKKW(-L) Final-State Showers The Basic Idea Tree-level matching NLO Matching Generate inclusive few-jet samples according to exact tree-level M n 2 using some merging scale ρ MS. These are then made exclusive by reweighting no-emission probabilities (in CKKW-L generated by the shower itself) Add normal shower emissions below ρ MS. Add all samples together. Dependence on the merging scale cancels to the precision of the shower. If the merging scale is not defined in terms of the shower ordering variable, we need vetoed and truncated showers. Breaks the unitarity of the shower. Event Generators II 37 Leif Lönnblad Lund University

51 The Basic Idea Tree-level matching NLO Matching The Second Commandment of Event Generation Thou shalt always cover the whole of phase space exactly once. Event Generators II 38 Leif Lönnblad Lund University

52 The Basic Idea Tree-level matching NLO Matching The Second Commandment of Event Generation Thou shalt always cover the whole of phase space exactly once. Event Generators II 38 Leif Lönnblad Lund University

53 The Basic Idea Tree-level matching NLO Matching Multi-jet tree-level matching [ dσ0 ex ρ0 ( = F 0 M α dφ S dρdzp 1 + α2 ρ0 ) ] 2 S dρdzp 1 0 ρ MS 2 ρ MS dσ ex 1 dφ 0 = F 0 M 0 2 α S P1 ME dρ 1 dz 1 [ ρ0 1 α S dρdzp 1 α S ρ 1 ρ1 ρ MS dρdzp 2 dσ 2 dφ 0 = F 0 M 0 2 α 2 S PME 1 dρ 1 dz 1 P ME 2 dρ 2 dz 2 Θ(ρ 1 ρ 2 ) NOT unitary. Gives artificial dependence of ρ MS. e.g. extra contribution to α S P ME 1 is α 2 S L3. ] Event Generators II 39 Leif Lönnblad Lund University

54 The Basic Idea Tree-level matching NLO Matching Mature procedure. Available in HERWIG++, SHERPA, PYTHIA8. The MLM-procedure (ALPGEN + HERWIG/PYTHIA) is similar, but even less control over the perturbative expansion. There are recent procedures to restore unitarity: Vincia exponentiates the full n-parton matrix elements. UMEPS uses a add/subtract procedure combined with a re-clustering algorithm. Event Generators II 40 Leif Lönnblad Lund University

55 The Basic Idea Tree-level matching NLO Matching UMEPS Restoring unitarity [ dσ0 ex ρ0 ( = F 0 M α dφ S dρdzp 1 + α2 ρ0 ) ] 2 S dρdzp 1 0 ρ MS 2 ρ MS dσ ex 1 dφ 0 = F 0 M 0 2 α S P ME 1 dρ 1 dz 1 [ 1 α S ρ0 ρ 1 dρdzp 1 α S ρ1 ρ MS dρdzp 2 ] dσ 2 dφ 0 = F 0 M 0 2 α 2 S PME 1 dρ 1 dz 1 P ME 2 dρ 2 dz 2 Θ(ρ 1 ρ 2 ) Event Generators II 41 Leif Lönnblad Lund University

56 The Basic Idea Tree-level matching NLO Matching UMEPS Restoring unitarity [ dσ0 fx ρ0 ( = F 0 M α dφ S dρdzp 1 + α2 ρ0 ) ] 2 S dρdzp 1 0 ρ MS 2 ρ MS dσ fx 1 dφ 0 = F 0 M 0 2 α S P ME 1 dρ 1 dz 1 [ 1 α S ρ0 ρ 1 dρdzp 1 α S ρ1 ρ MS dρdzp 2 ] dσ 2 dφ 0 = F 0 M 0 2 α 2 S PME 1 dρ 1 dz 1 P ME 2 dρ 2 dz 2 Θ(ρ 1 ρ 2 ) Event Generators II 41 Leif Lönnblad Lund University

57 The Basic Idea Tree-level matching NLO Matching UMEPS Restoring unitarity [ dσ0 fx ρ0 = F 0 M α dφ S 0 dσ fx 1 dφ 0 dρdzp 1 + α2 S ρ MS 2 dσ fx 1 dρ 1 dz 1 dφ 0 dρ 1 dz 1 [ ρ0 = F 0 M 0 2 α S P1 ME dρ 1 dz 1 1 α S dσ fx 2 ( ρ0 ρ MS dρdzp 1 ρ 1 dρdzp 1 α S dρ 2 dz 2 dφ 0 dρ 1 dz 1 dρ 2 dz 2 dσ 2 = F 0 M 0 2 αs 2 dφ PME 1 dρ 1 dz 1 P2 ME dρ 2 dz 2 Θ(ρ 1 ρ 2 ) 0 ) 2 ] ρ1 ρ MS dρdzp 2 ] Event Generators II 41 Leif Lönnblad Lund University

58 The Basic Idea Tree-level matching NLO Matching In CCKW we need to recreate the sequence of emissions. In CKKW-L this is done by selecting a full parton shower history of an n-parton state. In UMEPS performing the integration is simply to replace the n-jet the state with the one with one jet less in the history. Event Generators II 42 Leif Lönnblad Lund University

59 The Basic Idea Tree-level matching NLO Matching But why worry about unitarity, the cross sections are never better than LO anyway, so scale uncertainties are huge. 1.1 σ merged / σ inclusive CKKW-L UMEPS t MS [GeV] Event Generators II 43 Leif Lönnblad Lund University

60 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching NLO The anatomy of NLO calculations. O = dφ n (B n + V n )O n (φ n )+ dφ n+1 B n+1 O n+1 (φ n+1 ). Not practical, since V n and B n+1 are separately divergent, although their sum is finite. The standard subtraction method: ( O = dφ n B n + V n + ) dψ n,ps (a) (a) n,p O n (φ n ) p ( + dφ n+1 B n+1 O n+1 (φ n+1 ) ) S n,po (a) n ( φ n+1 p ψ (a) ), n,p Event Generators II 44 Leif Lönnblad Lund University

61 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching (Frixione et al.) The subtraction terms must contain all divergencies of the real-emission matrix element. A parton shower splitting kernel does exactly that. Generating two samples, one according to B n + V n + S PS and one according to B n+1 Sn PS, and just add the parton shower from which S n is calculated. n, Event Generators II 45 Leif Lönnblad Lund University

62 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching POWHEG (Nason et al.) Calculate B n = B n + V n + B n+1 and generate n-parton states according to that. Generate a first emission according to B n+1 /B n, and then add any 1 parton shower for subsequent emissions. 1 As long as it is transverse-momentum ordered in the same way as in POWHEG or properly truncated Event Generators II 46 Leif Lönnblad Lund University

63 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching POWHEG and are very similar. They are both correct to NLO, but differ at higher orders POWHEG exponentiates also non singular pieces of the n+1 parton cross section POWHEG multiplies the n+1 parton cross section with B n /B n (the phase-space dependent K -factor). POWHEG may also resum k > µ R, and will then generate additional logarithms, log(s/µ R ) log(1/x). Event Generators II 47 Leif Lönnblad Lund University

64 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching The Sixth Commandment of Event Generation Thou shalt always remember that a NLO generator does not always produce NLO results Event Generators II 48 Leif Lönnblad Lund University

65 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Really NLO? Do NLO-generators always give NLO-predictions? For simple Born-level processes such as h γγ production, all inclusive higgs observables will be correct to NLO. y h y γ p γ But note that for p γ > m h /2 the prediction is only leading order! Event Generators II 49 Leif Lönnblad Lund University

66 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Also p h is LO. To get NLO we need to start with H+jet at Born-level and calculate full α 2 S. But for small p h the NLO cross section diverges due to L 2n α n s, L = log(p h /µ R ). If L 2 α s 1, the α 2 s corrections are parametrically as large as the NLO corrections. Can be alleviated by clever choices for µ R, but in general you need to resum. Event Generators II 50 Leif Lönnblad Lund University

67 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching The Seventh Commandment of Event Generation Thou shalt always resum when NLO corrections are large Event Generators II 51 Leif Lönnblad Lund University

68 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Di-jet decorrelation NLO LO 2 dσ/dφ jj φ jj Measure the azimuthal angle between the two hardest jets. Clearly the 2-jet matrix element will only give back-to back jets, so the three-jet matrix element will give the leading order. And an NLO 3-jet generator will give us NLO. But for φ jj < 120, the two hardest jets needs at least two softer jets to balance. So the NLO becomes LO here. φ Event Generators II 52 Leif Lönnblad Lund University

69 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Multi-leg Matching We need to be able to combine several NLO calculations and add (parton shower) resummation in order to get reliable predictions. No double (under) counting. No parton shower emissions which are already included in (tree-level) ME states. No terms in the PS no-emission resummation which are already in the NLO Dependence of any merging scale must not destroy NLO accuracy. The NLO 0-jet cross section must not change too much when adding NLO 1-jet. Dependence on logarithms of the merging scale should be less than L 3 α 2 s in order for predictions to be stable for small scales. Event Generators II 53 Leif Lönnblad Lund University

70 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching SHERPA First working solution for hadronic collisions. CKKW-like combining of events, fixing up double counting of NLO real and virtual terms. Any jet multiplicity possible. Dependence on merging scale canceled at NLO and parton-shower precision. Residual dependence: L 3 αs/n 2 C 2 can t take merging scale too low. Event Generators II 54 Leif Lönnblad Lund University

71 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching MINLO No merging scale! Take e.g. POWHEG Higgs+1-jet calculation down to very low p. Use clever (nodal) renormalization scales Multiply with (properly subtracted) Sudakov form factor Add non-leading terms to Sudakov form factor to get correct NLO 0-jet cross section. Possible to go to NNLO! Not clear how to go to higher jet multiplicities. Event Generators II 55 Leif Lönnblad Lund University

72 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching UNLOPS Start from UMEPS (unitary version of CKKW-L). Add (and subtract) n-jet NLO samples, fixing up double counting of NLO real and virtual terms. dσ sub 1 dφ 0 [ ρ0 ] = α S P1 ME dρ 1 dz 1 Π 0 (ρ 0,ρ 1 ) 1+α S dρdzp 1 ρ 1 Note that PS uses α S (ρ) and f(x,ρ) rather than α S (µ R ) and f(x,µ F ) Event Generators II 56 Leif Lönnblad Lund University

73 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Any jet multiplicity possible. Although there is a merging scale, the dependence of an n-jet cross section due to addition of higher multiplicities drops out completely. Merging scale can be taken arbitrarily small. Lots of negative weights. Possible to go to NNLO? Available in PYTHIA8 (and HERWIG++ in Simon Plätzer s incarnation) Event Generators II 57 Leif Lönnblad Lund University

74 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching GENEVA Analytic (SCET) resummation of NLO cross section to NLL (or even NNLL!) in the merging scale variable. Only e + e so far (W-production in pp on its way). VINCIA Exponentiate NLO Matrix Elements in no-emission probability no merging scale. Only e + e so far FxFx MLM-like merging of different MC@NLO calculations. Difficult to understand merging scale dependence Event Generators II 58 Leif Lönnblad Lund University

75 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Les Houches comparison Event Generators II 59 Leif Lönnblad Lund University

76 ˆ Tree-level matching NLO Matching Multi-leg NLO Matching Outline of Lectures Lecture I: Basics of Monte Carlo methods, the event generator strategy, matrix elements, LO/NLO,... Lecture II: Parton showers, initial/final state, matching/merging,... Lecture III: Matching/merging (cntd.), underlying events, multiple interactions, minimum bias, pile-up, hadronization, decays,... Lecture IV: Protons vs. heavy ions, summary,... Buckley et al. (MCnet collaboration), Phys. Rep. 504 (2011) 145. Event Generators II 60 Leif Lönnblad Lund University