Matching & Merging of Parton Showers and Matrix Elements

Transcription

1 Matching & Merging of Parton Showers and Matrix Elements Frank Krauss Institute for Particle Physics Phenomenology Durham University CTEQ school, Pittsburgh, 2./

2 Outline Introduction: Why precision in event generators? 2 Parton-level calculations 3 Parton showers 4 NLO improvements: Matching 5 LO improvements: Multijet merging 6 NLO improvements: Multijet merging 7 Concluding remarks

3 introduction: why event generators? and why is precision an issue?

4 Physics at the LHC & the need for event generators proton-proton collisions at the LHC: processes with the highest energies ever at accelerator experiments characteristically, signals and their backgrounds from hard interactions, with many particles in the final state (due to strong interaction and huge phase space) complex final states in many channels, hard to gain detailed quantitative understanding from first principles/analytical work need simulation = event generators

5 The dual role of event generators dichotomy in the application/use of event generators by experiments experimental tool : unfolding of the detector, determination of acceptance and corrections,... grasping corrections due to hadronisation, multiple interactions etc. typically many parameters, allowing for more freedom can be improved by improved parametrisations and tuning theory tool : (clearly, an understanding of physics helps in devising successful parametrisations!) (this is the kind of tool SHERPA strives to be) accurate description of signal and background, extrapolation from control to signal region, extraction of physics through comparison with data,... typically few parameters, allowing for less freedom can be improved by improved accuracy of underlying calculations (this yields improved/reduced errors)

6 The inner working of event generators... simulation: divide et impera hard process: fixed order perturbation theory traditionally: Born-approximation bremsstrahlung: resummed perturbation theory hadronisation: phenomenological models hadron decays: effective theories, data underlying event : phenomenological models

7 and possible improvements possible strategies: improving the phenomenological models: tuning (fitting parameters to data) replacing by better models, based on more physics (my hot candidate: minimum bias and underlying event simulation) improving the perturbative description: inclusion of higher order exact matrix elements and correct connection to resummation in the parton shower: NLO-Matching & Multijet-Merging systematic improvement of the parton shower: next-to leading (or higher) logs & colours

8 Reminder: parton-level in perturbation theory

9 Cross sections at the LHC: Born approximation dσ ab N = 0 dx a dx b f a (x a, µ F )f b (x a, µ F ) cuts dφ N 2ŝ M p ap b N(Φ N ; µ F, µ R ) 2 parton densities f a (x, µ F ) (PDFs) phase space Φ N for N-particle final states incoming current /(2ŝ) squared matrix element M pap b N (summed/averaged over polarisations) renormalisation and factorisation scales µ R and µ F complexity demands numerical methods for large N

10 Complexity: factorial growth in e + e q q +ng n # diags Number of diagrams Number of gluons

11 Higher orders: some general thoughts obtained from adding diagrams with additional: loops (virtual corrections) or legs (real corrections) effect: reducing the dependence on µ R & µ F (NLO first order allowing for meaningful estimate of uncertainties) additional difficulties when going NLO: ultraviolet divergences in virtual correction infrared divergences in real and virtual correction enforce UV regularisation & renormalisation IR regularisation & cancellation (Kinoshita Lee Nauenberg Theorem)

12 Structure of an NLO calculation sketch of cross section calculation dσ (NLO) N = dφ N B N }{{} Born approximation = dφ N [B N +V N +B N S + dφ N V N }{{} renormalised virtual correction IR-divergent + dφ N+ R N }{{} real correction IR-divergent [ ] ]+dφ N+ R N B N ds subtraction terms S (integrated) and ds: exactly cancel IR divergence in R process-independent structures result: terms in both brackets separately infrared finite

13 Availability of exact calculations for hadron colliders m loops 2 done for some processes first solutions n legs

14 Parton showers

15 Probabilistic treatment of emissions Sudakov form factor (no-decay probability) (K) ij,k (t,t 0) = exp [ t t 0 dt t α S 2π dz dφ 2π K ij,k (t, z, φ) }{{} splitting kernel for (ij) ij (spectator k) ] evolution parameter t defined by kinematics will replace dt t dzdφ 2π dφ scale choice for strong coupling: α S (k 2) regularisation through cut-off t 0 generalised angle (HERWIG++) or transverse momentum (PYTHIA, SHERPA) resums classes of higher logarithms

16 Emissions off a Born matrix element compound splitting kernels K n and Sudakov form factors (K) n for emission off n-particle final state: K n (Φ ) = α S 2π all{ij,k} [ t ] K ij,k (Φ ij,k ), (K) n (t,t 0 ) = exp dφ K n (Φ ) t 0 consider first emission only off Born configuration dσ B = dφ N B N (Φ N ) { (K) N (µ2 N,t 0 )+ µ 2 N [ ] } K N (Φ ) (K) N (µ2 N,t(Φ )) dφ t 0 }{{} integrates to unity unitarity of parton shower further emissions by recursion with Q 2 = t of previous emission

17 Aside: connection to resummation formalism consider standard Collins-Soper-Sterman formalism (CSS): dσ AB X dydq 2 d 2 b = dφ X B ij (Φ X ) (2π) 2 exp(i b Q ) W ij (b; Φ X ) }{{}}{{} guarantee 4-mom conservation higher orders with W ij (b; Φ X ) = collinear bits loops {}}{{}}{ C i (b; Φ X, α S )C j (b; Φ X, α S ) H ij (α S ) Q 2 X dk 2 ( ))) exp k 2 A(α S (k ))log 2 Q2 X k 2 +B(α S (k 2 /b 2 }{{} Sudakov form factor, A, B expanded in powers of α S

18 Connection to resummation formalism: log accuracy analyse structure of emissions above logarithmic accuracy in log µ N k (a la CSS) possibly up to next-to leading log, if evolution parameter transverse momentum, if argument in α S is k of splitting, if K ij,k terms A,2 and B upon integration (okay, if soft gluon correction is included, and if K ij,k AP splitting kernels) in CSS k typically is the transverse momentum of produced system, in parton shower of course related to the cumulative effect of explicit multiple emissions resummation scale µ N µ F given by (Born) kinematics simple for cases like q q V, gg H,... tricky for more complicated cases m L α n NLL resummed in PS

19 A simple improvement: matrix element corrections (M. Seymour, Comp. Phys. Comm. 90 (995) 95 & E. Norrbin & T. Sjostrand, Nucl. Phys. B603 (2) 297) parton shower ignores interferences typically present in matrix elements pictorially ME : PS : x 2 ME over PS x form many processes R N < B N K N typical processes: q q V, e e + q q, t bw practical implementation: shower with usual algorithm, but reject first/hardest emissions with probability P = R N /(B N K N )

20 analyse first emission, given by dσ B = dφ N B N (Φ N ) { (R/B) N (µ 2 N,t 0 )+ µ 2 N [ RN (Φ N Φ ) ] } (R/B) N (µ 2 N,t(Φ )) dφ B N (Φ N ) t 0 }{{} once more: integrates to unity unitarity of parton shower radiation given by R N (correct at O(α S )) (but modified by logs of higher order in α S from (R/B) ) N emission phase space constrained by µ N also known as soft ME correction hard ME correction fills missing phase space used for power shower : µ N E pp and apply ME correction m L α n NLL resummed in PS

21 NLO improvements: Matching

22 NLO matching: Basic idea parton shower resums logarithms fair description of collinear/soft emissions jet evolution (where the logs are large) matrix elements exact at given order fair description of hard/large-angle emissions jet production (where the logs are small) adjust ( match ) terms: cross section at NLO accuracy & correct hardest emission in PS to exactly reproduce ME at order α S (R-part of the NLO calculation) (this is relatively trivial) maintain (N)LL-accuracy of parton shower (this is not so simple to see) m L α n NLL resummed in PS

23 The POWHEG-trick: modifying the Sudakov form factor (P. Nason, JHEP 04 (24) 040 & S. Frixione, P. Nason & C. Oleari, JHEP 07 (27) 070) reminder: K ij,k reproduces process-independent behaviour of R N /B N in soft/collinear regions of phase space dφ R N (Φ N+ ) B N (Φ N ) IR dφ α S 2π K ij,k(φ ) define modified Sudakov form factor (as in ME correction) (R/B) N (µ 2 N,t 0 ) = exp µ 2 N t 0 dφ R N (Φ N+ ) B N (Φ N ), assumes factorisation of phase space: Φ N+ = Φ N Φ typically will adjust scale of α S to parton shower scale

24 Local K-factors (P. Nason, JHEP 04 (24) 040 & S. Frixione, P. Nason & C. Oleari, JHEP 07 (27) 070) start from Born configuration Φ N with NLO weight: ( local K-factor ) dσ (NLO) N = dφ N B(Φ N ) = dφ N {B N (Φ N )+V N (Φ N )+B N (Φ N ) S }{{} + Ṽ N (Φ N ) dφ [R N (Φ N Φ ) B N (Φ N ) ds(φ )] } by construction: exactly reproduce cross section at NLO accuracy note: second term vanishes if R N B N ds (relevant for MC@NLO)

25 NLO accuracy in radiation pattern (P. Nason, JHEP 04 (24) 040 & S. Frixione, P. Nason & C. Oleari, JHEP 07 (27) 070) generate emissions with (R/B) N (µ 2 N, t 0): dσ (NLO) N = dφ N B(Φ N ) (R/B) N (µ 2 N, t 0 )+ µ 2 N R N (Φ N Φ ) (R/B) N (µ 2 B N (Φ N ) N, k (Φ 2 )) dφ t 0 }{{} integrating to yield - unitarity of parton shower radiation pattern like in ME correction pitfall, again: choice of upper scale µ 2 N apart from logs: which configurations enhanced by local K-factor (this is vanilla POWHEG!) ( K-factor for inclusive production of X adequate for X+ jet at large p?)

26 POWHEG features S. Alioli, P. Nason, C. Oleari, & E. Re, JHEP 0904 (29) 2 large enhancement at high p T,h can be traced back to large NLO correction fortunately, NNLO correction is also large agreement

27 Improved POWHEG split real-emission ME as ( R = R h 2 p 2 +h2 }{{} R (S) + p2 p 2 +h2 }{{} R (F) ) can tune h to mimick NNLO - or maybe resummation result differential event rate up to first emission dσ = dφ B B (R(S) ) [ +dφ R R (F) (Φ R ) S. Alioli, P. Nason, C. Oleari, & E. Re, JHEP 0904 (29) 2 s ] (R(S)/B) R (S) (s, t 0 ) + dφ (S) B (R /B) (s, k ) 2 t 0

28 Resummation in divide R N in soft ( S ) and hard ( H ) part: R N = R (S) N +R(H) N = B N ds +H N identify subtraction terms and shower kernels ds dσ N = dφ N B N (Φ N ) }{{} B+Ṽ +dφ N+ H N [ (K) N (µ2 N, t 0 )+ K ij,k {ij,k} (modify K in st emission to account for colour) µ 2 N effect: only resummed parts modified with local K-factor t 0 dφ K ij,k (Φ ) (K) N (µ2 N, k 2 ) ]

29 Aside: phase space/k-factor effects ( S. Alioli, P. Nason, C. Oleari, & E. Re, JHEP 0904 (29) 2 & S. Hoeche, F. Krauss, M. Schoenherr, & F. Siegert, JHEP 209 (202) 049) problem: impact of subtraction terms on local K-factor (filling of phase space by parton shower) studied in case of gg H above proper filling of available phase space by parton shower paramount

30 Aside : impact of full colour (S. Hoeche, J. Huang, G. Luisoni, M. Schoenherr, & J. Winter, arxiv: [hep-ph]) evaluate effect of full colour treatment, MC@NLO without H-part vs. parton shower with B B

31 for light jets: jet-p (S. Hoeche & M. Schoenherr, Phys. Rev. D86 (202) )

32 for light jets: dijet mass (S. Hoeche & M. Schoenherr, Phys. Rev. D86 (202) )

33 for light jets: azimuthal decorrelations (S. Hoeche & M. Schoenherr, Phys. Rev. D86 (202) )

34 for light jets: R 32 & forward energy flow (S. Hoeche & M. Schoenherr, Phys. Rev. D86 (202) )

35 for light jets: jet vetoes (S. Hoeche & M. Schoenherr, Phys. Rev. D86 (202) )

36 Summary of st lecture

37 Summary of st lecture reviewed higher order calculations and parton shower, with emphasis on accuracy these things are not blackboxes and can be understood systematic improvement of event generators by including higher orders has been at the core of QCD theory and developments in the past decade: ME corrections NLO matching ( MC@NLO, POWHEG )

38 Multijet leading order

39 Multijet merging: basic idea (S. Catani, F. Krauss, R. Kuhn, B. Webber, JHEP 0 (2) 063, L. Lonnblad, JHEP 0205 (22) 046, & F. Krauss, JHEP 0208 (22) 05) parton shower resums logarithms fair description of collinear/soft emissions jet evolution (where the logs are large) matrix elements exact at given order fair description of hard/large-angle emissions jet production (where the logs are small) combine ( merge ) both: result: towers of MEs with increasing number of jets evolved with PS multijet cross sections at Born accuracy maintain (N)LL accuracy of parton shower m L exact ME LO 4jet α n NLL resummed in PS exact ME LO 5jet, but also NLO 4jet separate regions with jet measure Q J ( truncated showering if not identical with evolution parameter)

40 Why it works: jet rates with the parton shower consider jet production in e + e hadrons Durham jet definition: relative transverse momentum k > Q J fixed order: one factor α S and up to log 2 Ec.m. Q J per jet use Sudakov form factor for resummation & replace approximate fixed order by exact expression: R 2 (Q J ) = [ q (E 2 c.m., Q 2 J) ] 2 R 3 (Q J ) = 2 q (E 2 c.m., Q 2 J) E 2 c.m. Q 2 J dk 2 k 2 [ α S (k 2) dzk q (k 2 2π,z) q (E 2 c.m., k 2 ) q (k 2, Q 2 J) g (k 2, Q 2 J) ]

41 First emission(s), again dσ = dφ N B N [ (K) N (µ2 N, t 0 )+ (S. Hoeche, F. Krauss, S. Schumann, F. Siegert, JHEP 0905 (29) 053) µ 2 N t 0 dφ K N (K) N (µ2 N, t N+ )Θ(Q J Q N+ ) +dφ N+ B N+ (K) N (µ2 N+, t N+ )Θ(Q N+ Q J ) ] note: N +-contribution includes also N +2, N +3,... (no Sudakov suppression below t n+, see further slides for iterated expression) potential occurence of different shower start scales: µ N,N+,... unitarity violation in square bracket: B N K N B N+ (cured with UMEPS formalism, L. Lonnblad & S. Prestel, JHEP 302 (203) 094 & S. Platzer, arxiv: [hep-ph] & arxiv: [hep-ph])

42 Iterating the emissions (S. Hoeche, F. Krauss, S. Schumann, F. Siegert, JHEP 0905 (29) 053) dσ = n max n=n (n N) extra jets no emissions off internal lines {}}{{}}{ { [ n ] [ n ] dφ n B n Θ(Q j+ Q J ) (K) j (t j, t j+ ) j=n j=n [ t n (K) n (t n,t 0 ) + } {{ } no emission [ nmax +dφ nmax B nmax [ j=n (K) n max (t nmax,t 0 )+ (K) dφ K n n (t n,t n+ )Θ(Q J Q n+ ) t 0 }{{} next emission no jet & below last ME emission Θ(Q j+ Q J ) t nmax t 0 ][ nmax j=n (K) j (t j, t j+ ) dφ K nmax (K) n max (t nmax,t nmax+) ] ] ]

43 ATLAS: m γγ, p,γγ, and φ γγ in showers (arxiv:2.93 [hep-ex]) [pb/gev] dσ/dm γγ ATLAS s = 7 TeV - Data 20, Ldt = 4.9 fb PYTHIA MCc.2 (MRST27) SHERPA MCc.2 (CTEQ6L) [pb/gev] T,γγ dσ/dp ATLAS s = 7 TeV - Data 20, Ldt = 4.9 fb PYTHIA MCc.3 (MRST27) SHERPA MCc.3 (CTEQ6L) [pb/rad] γγ dσ/d φ ATLAS s = 7 TeV - Data 20, Ldt = 4.9 fb PYTHIA MCc.2 (MRST27) SHERPA MCc.2 (CTEQ6L) data/sherpa data/sherpa data/sherpa data/pythia data/pythia data/pythia m γγ [GeV] p T,γγ [GeV] φ γγ [rad]

44 Aside: Comparison with higher order calculations (arxiv:2.93 [hep-ex]) [pb/gev] dσ/dm γγ ATLAS s = 7 TeV - Data 20, Ldt = 4.9 fb DIPHOX+GAMMA2MC (CT0) 2γNNLO (MSTW28) [pb/gev] T,γγ dσ/dp ATLAS s = 7 TeV - Data 20, Ldt = 4.9 fb DIPHOX+GAMMA2MC (CT0) 2γNNLO (MSTW28) [pb/rad] γγ dσ/d φ ATLAS s = 7 TeV - Data 20, Ldt = 4.9 fb DIPHOX+GAMMA2MC (CT0) 2γNNLO (MSTW28) data/diphox data/diphox data/diphox data/2γnnlo data/2γnnlo data/2γnnlo m γγ [GeV] p T,γγ [GeV] φ γγ [rad]

45 A step towards multijet-merging at NLO: MENLOPS (K. Hamilton & P. Nason, JHEP 6 (200) 039 & S. Hoeche, F. Krauss, M. Schoenherr, & F. Siegert, JHEP 08 (20) 23) combine matching for lowest multiplicity with multijet merging interpolating local K-factor for reweighting hard emissions k N (Φ N+ ) = B ( N H ) N + H { N B N /B N for soft emission B N B N+ B N+ for hard emission dσ = dφ N B N [ (K) N (µ2 N,t 0 )+ µ 2 N +dφ N+ H N (K) N (µ2 N,t N+ )Θ(Q J Q N+ ) t 0 dφ K N (K) N (µ2 N,t N+ )Θ(Q J Q N+ ) +dφ N+ k N B N+ (K) N (µ2 N,t N+ )Θ(Q N+ Q J ) ]

46 Multijet next-to leading order

47 Multijet-merging at NLO: basic idea like at LO: towers of MEs with increasing jet multi (but this time at NLO) combine them into one sample, remove overlap/double-counting maintain NLO and (N)LL accuracy of ME and PS this effectively translates into a merging of MC@NLO simulations and can be further supplemented with LO simulations for even higher final state multiplicities

48 First emission(s), once more dσ = dφ N B N [ (K) N (µ2 N,t 0 )+ µ 2 N +dφ N+ H N (K) N (µ2 N,t N+ )Θ(Q J Q N+ ) +dφ N+ B N+ ( [ + B N+ B N+ (K) N+ (t N+,t 0 )+ t 0 dφ K N (K) N (µ2 N,t N+ )Θ(Q J Q N+ ) µ 2 N t N+ dφ K N t N+ t 0 ) Θ(Q N+ Q J ) dφ K N+ (K) N+ (t N+,t N+2 ) +dφ N+2 H N+ (K) N (µ2 N,t N+ ) (K) N+ (t N+,t N+2 )Θ(Q N+ Q J )+... ] ]

49 example results for e e + hadrons Durham jet resolution 3 2 (E CMS = 9.2 GeV) Durham jet resolution 4 3 (E CMS = 9.2 GeV) /σ dσ/dln(y23) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo /σ dσ/dln(y34) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo MC/data MC/data ln(y23) ln(y34) Durham jet resolution 5 4 (E CMS = 9.2 GeV) Durham jet resolution 6 5 (E CMS = 9.2 GeV) /σ dσ/dln(y45) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo /σ dσ/dln(y56) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo MC/data ln(y45) MC/data ln(y56)

50 example results for e e + hadrons /σ dσ/dt Thrust (E CMS = 9.2 GeV) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo /σdσ/d ( T) n 0 Moments of T at 9 GeV OPAL data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo.4.4 MC/data MC/data T n C-Parameter (E CMS = 9.2 GeV) Moments of C at 9 GeV /σ dσ/dc ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo /σdσ/d C n 0 OPAL data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo MC/data MC/data C n

51 Example: for W +jets (up to two NLO, from BlackHat, see arxiv: [hep-ex]) Inclusive Jet Multiplicity σ(w+ Njet jets) [pb] p jet > 30GeV ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo p jet > 20GeV ( 0) σ( Njet jets)/σ( Njet jets) σ( Njet jets)/σ( Njet jets) p jet > 20GeV p jet > 30GeV Njet Njet

52 dσ/dp [pb/gev] MC/data MC/data MC/data 0 3 First Jet p W+ jet ( ) W+ 2 jets ( 0.) W+ 3 jets ( 0.0) ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo p [GeV] dσ/dp [pb/gev] MC/data MC/data dσ/dp [pb/gev] MC/data Second Jet p W+ 2 jets ( ) W+ 3 jets ( 0.) ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo p [GeV] Third Jet p W+ 3 jets ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo p [GeV]

53 dσ/dh T [pb/gev] W+ jet ( ) W+ 2 jets ( 0.) W+ 3 jets ( 0.0) ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo H T [GeV] R Distance of Leading Jets MC/data MC/data MC/data H T [GeV] Azimuthal Distance of Leading Jets dσ/d R [pb] ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo dσ/d φ [pb] ATLAS data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo MC/data MC/data R(First Jet, Second Jet) φ(first Jet, Second Jet)

54 Example: for W + W +jets (in prep., up to one NLO, virtuals from OPENLOOPS, all interferences, no Higgs)

55 Summary

56 Summary systematic improvement of event generators by including higher orders has been at the core of QCD theory and developments in the past decade: multijet merging ( CKKW, MLM ) NLO matching ( MC@NLO, POWHEG ) MENLOPS combination of matching and merging multijet merging at NLO (MEPS@NLO, FxFx ) (first 3 methods well understood and used in experiments) multijet merging at NLO under scrutiny complete automation of NLO calculations done time to optimise the impact of this gargantuan task