NLO+PS matching and multijet merging

Transcription

1 NLO-mergng Conclusons NLO+PS matchng and multjet mergng Insttute for Partcle Physcs Phenomenology Cambrdge, 8/0/202 arxv:.220, arxv:2882 arxv: , arxv: arxv: NLO+PS matchng and multjet mergng

2 NLO-mergng Conclusons Introducton Monte-Carlo event generators heavly used tools for fully exclusve SM/BSM predctons on hadron level drectly comparable to expermental data standard LO+PS calculatons have lmted accuracy n regons probed by current collders multple hard objects n fnal state standard NLO calculatons have lmted accuracy n strongly herarchcal confguratons Tevatron and LHC allow for both at the same tme a number of solutons to combne the observable ndependent resummaton of PS wth fxed-order accuracy n hard multparton fnal states exst mperatve to know about mplct choces/assumptons and how to assess uncertantes for phenomenologcal studes NLO+PS matchng and multjet mergng 2

3 NLO-mergng Conclusons Motvaton PS LO LO LO LO pp 2 wth parton showers + exponentaton of large IR logarthms poor hard/wde angle emsson pattern vs. LO pp n matrx elements + domnant terms for hard/wde angle rad. breakdown of α s-expanson n log. regon MEPS schemes: CKKW-type, MLM-type LO+(N)LL accuracy n every jet multplcty scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 3

4 NLO-mergng Conclusons Motvaton ME LO LO LO LO LO pp 2 wth parton showers + exponentaton of large IR logarthms poor hard/wde angle emsson pattern vs. LO pp n matrx elements + domnant terms for hard/wde angle rad. breakdown of α s-expanson n log. regon MEPS schemes: CKKW-type, MLM-type LO+(N)LL accuracy n every jet multplcty scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 3

5 NLO-mergng Conclusons Motvaton ME PS LO LO LO LO LO LO LO LO pp 2 wth parton showers + exponentaton of large IR logarthms poor hard/wde angle emsson pattern vs. LO pp n matrx elements + domnant terms for hard/wde angle rad. breakdown of α s-expanson n log. regon MEPS schemes: CKKW-type, MLM-type LO+(N)LL accuracy n every jet multplcty scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 3

6 NLO-mergng Conclusons Motvaton ME PS LO LO LO LO LO LO LO α k+n s (µ eff ) = α k s(µ) α s (t ) α s (t n ) LO pp 2 wth parton showers + exponentaton of large IR logarthms poor hard/wde angle emsson pattern vs. LO pp n matrx elements + domnant terms for hard/wde angle rad. breakdown of α s-expanson n log. regon MEPS schemes: CKKW-type, MLM-type LO+(N)LL accuracy n every jet multplcty scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 3

7 NLO-mergng Conclusons Motvaton PS NLO NLO NLO promote LOPS to NLOPS (POWHEG, can assess uncertantes (part I) combne NLOPS for successve multplctes nto ncl. sample preserve NLO+(N)LL accuracy n every jet multplcty restore resummaton wrt. to nclusve sample (part II) scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 4

8 NLO-mergng Conclusons Motvaton ME PS NLO NLO NLO NLO NLO NLO NLO promote LOPS to NLOPS (POWHEG, can assess uncertantes (part I) combne NLOPS for successve multplctes nto ncl. sample preserve NLO+(N)LL accuracy n every jet multplcty restore resummaton wrt. to nclusve sample (part II) scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 4

9 NLO-mergng Conclusons Motvaton ME PS NLO NLO NLO NLO NLO NLO NLO promote LOPS to NLOPS (POWHEG, can assess uncertantes (part I) α k+n s (µ eff ) = α k s(µ) α s (t ) α s (t n ) combne NLOPS for successve multplctes nto ncl. sample (MEPS@NLO), preserve NLO+(N)LL accuracy n every jet multplcty restore resummaton wrt. to nclusve sample (part II) scale settng scheme ntegral to preserve PS-resummaton propertes NLO+PS matchng and multjet mergng 4

10 MC@NLO NLO-mergng Conclusons The SHERPA event generator framework JHEP02(2009)007 Two mult-purpose Matrx Element (ME) generators AMEGIC++ JHEP02(2002)044 COMIX JHEP2(2008)039 CS subtracton EPJC53(2008)50 A Parton Shower (PS) generator CSSHOWER++ JHEP03(2008)038 A multple nteracton smulaton à la Pytha AMISIC++ hep-ph/06002 A cluster fragmentaton module AHADIC++ EPJC36(2004)38 A hadron and τ decay package HADRONS++ A hgher order QED generator usng YFS-resummaton PHOTONS++ JHEP2(2008)08 Sherpa s tradtonal strength s the perturbatve part of the event MEPS (CKKW), MC@NLO, MENLOPS, MEPS@NLO full analytc control mandatory for consstency/accuracy NLO+PS matchng and multjet mergng 5

11 MC@NLO NLO-mergng Conclusons The SHERPA event generator framework JHEP02(2009)007 Two mult-purpose Matrx Element (ME) generators AMEGIC++ JHEP02(2002)044 COMIX JHEP2(2008)039 CS subtracton EPJC53(2008)50 A Parton Shower (PS) generator CSSHOWER++ JHEP03(2008)038 A multple nteracton smulaton à la Pytha AMISIC++ hep-ph/06002 A cluster fragmentaton module AHADIC++ EPJC36(2004)38 A hadron and τ decay package HADRONS++ A hgher order QED generator usng YFS-resummaton PHOTONS++ JHEP2(2008)08 Sherpa s tradtonal strength s the perturbatve part of the event MEPS (CKKW), MC@NLO, MENLOPS, MEPS@NLO full analytc control mandatory for consstency/accuracy NLO+PS matchng and multjet mergng 5

12 NLO-mergng Conclusons Short-comngs of fxed-order QCD poor descrpton n phase space regons wth strongly herarchcal scales poor perturbatve jet-modelng (at most two consttuents) no hadronsaton, MPI effects very pronounced n nclusve & djet producton jet-p turn negatve n forward regon unless y-dependent scale s used (e.g. H (y) T ) σ pb Inclusve Jet Multplcty (R=0.4) Sherpa+BlackHat ATLAS data Eur.Phys.J. C7 (20) 763 Sherpa NLO µf = µr = 2 HT Njet Bern et.al. arxv: NLO+PS matchng and multjet mergng 6

13 NLO-mergng Conclusons Short-comngs of fxed-order QCD poor descrpton n phase space regons wth strongly herarchcal scales poor perturbatve jet-modelng (at most two consttuents) no hadronsaton, MPI effects very pronounced n nclusve & djet producton jet-p turn negatve n forward regon unless y-dependent scale s used (e.g. H (y) T ) dσ/dp pb/gev Transverse momentum of the leadng jet (R=0.4) ATLAS data Eur.Phys.J. C7 (20) 763 Sherpa NLO µf = µr = 2 HT Sherpa+BlackHat p (leadng jet) GeV Bern et.al. arxv: NLO+PS matchng and multjet mergng 6

14 NLO-mergng Conclusons Short-comngs of fxed-order QCD poor descrpton n phase space regons wth strongly herarchcal scales poor perturbatve jet-modelng (at most two consttuents) no hadronsaton, MPI effects very pronounced n nclusve & djet producton jet-p turn negatve n forward regon unless y-dependent scale s used (e.g. H (y) T ) dσ/dp pb/gev 0 5 Transverse momentum of the 2nd leadng jet (R=0.4) Sherpa+BlackHat ATLAS data Eur.Phys.J. C7 (20) 763 Sherpa NLO µf = µr = 2 HT p (2nd leadng jet) GeV Bern et.al. arxv: NLO+PS matchng and multjet mergng 6

15 NLO-mergng Conclusons Short-comngs of fxed-order QCD poor descrpton n phase space regons wth strongly herarchcal scales poor perturbatve jet-modelng (at most two consttuents) no hadronsaton, MPI effects very pronounced n nclusve & djet producton jet-p turn negatve n forward regon unless y-dependent scale s used (e.g. H (y) T ) dσ/dp pb/gev Transverse momentum of the 3rd leadng jet (R=0.4) Sherpa+BlackHat ATLAS data Eur.Phys.J. C7 (20) 763 Sherpa NLO µf = µr = 2 HT p (3rd leadng jet) GeV Bern et.al. arxv: NLO+PS matchng and multjet mergng 6

16 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Descrbe wealth of expermental data wth a sngle sample (LHC@7TeV) MC@NLO d-jet producton: Höche, MS arxv: µ R/F = 4 H T, µ Q = 2 p CT0 PDF (α s (m Z ) = 0.8) hadron level calculaton fully hadronsed ncludng MPI vrtual MEs from BLACKHAT Gele, Glover, Kosower Nucl.Phys.B403(993) Bern et.al. arxv: p j > 20 GeV, pj2 > 0 GeV Uncertanty estmates: µ Q varaton MPI varaton µ F, µ R varaton µ R/F 2, 2 µdef R/F µ Q 2, 2 µ def Q MPI actvty n tr. regon ± 0% NLO+PS matchng and multjet mergng 7

17 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Descrbe wealth of expermental data wth a sngle sample (LHC@7TeV) MC@NLO d-jet producton: Höche, MS arxv: µ R/F = 4 H T, µ Q = 2 p CT0 PDF (α s (m Z ) = 0.8) hadron level calculaton fully hadronsed ncludng MPI vrtual MEs from BLACKHAT Gele, Glover, Kosower Nucl.Phys.B403(993) Bern et.al. arxv: p j > 20 GeV, pj2 > 0 GeV Uncertanty estmates: µ R/F 2, 2 µdef R/F µ Q 2, 2 µ def Q MPI actvty n tr. regon ± 0% σ pb Inclusve jet multplcty (ant-kt R=0.4) µr, µf varaton µ Q varaton MPI varaton Sherpa+BlackHat ATLAS data Eur.Phys.J. C7 (20) 763 Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p Njet NLO+PS matchng and multjet mergng 7

18 NLO-mergng Conclusons Case study: Inclusve jet & djet producton dσ/dp pb/gev Jet transverse momenta (ant-kt R=0.4) ATLAS data Eur.Phys.J. C7 (20) 763 Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p µr, µf varaton µ Q varaton MPI varaton st jet.4.2 Höche, MS arxv: st jet nd jet th jet 0 3 Sherpa+BlackHat 3rd jet 0 2 2nd jet rd jet 4th jet p GeV p GeV NLO+PS matchng and multjet mergng 8

19 NLO-mergng Conclusons Case study: Inclusve jet & djet producton R jets over 2 jets rato (ant-kt R=) µf, µr varaton µ Q varaton MPI varaton Sherpa+BlackHat CMS data Phys. Lett. B 702 (20) 336 Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p HT TeV 3-jet-over-2-jet rato Höche, MS arxv: determned from ncl. sample 2-jet rate at NLO+NLL 3-jet rate at LO+LL common scale choces vared smultaneously at large H T large MPI uncertantes better MPI physcs needed (soft QCD) smlar descrpton of related ATLAS observables NLO+PS matchng and multjet mergng 9

20 NLO-mergng Conclusons Case study: Inclusve jet & djet producton dσ/dp pb/gev Inclusve jet transverse momenta n dfferent rapdty ranges Sherpa+BlackHat ATLAS data Phys. Rev. D86 (202) Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p Sherpa MC@NLO µr = µf = 4 H(y) T, µ Q = 2 p µr, µf varaton µ Q varaton MPI varaton p GeV Höche, MS arxv: y < < y < < y < < y < < y < < y < < y < p GeV NLO+PS matchng and multjet mergng 0

21 NLO-mergng Conclusons Case study: Inclusve jet & djet producton dσ/dm2 pb/tev Djet nvarant mass spectra n dfferent rapdty ranges ATLAS data Phys. Rev. D86 (202) Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p Sherpa MC@NLO µr = µf = 4 H(y) T, µ Q = 2 p µr, µf varaton µ Q varaton MPI varaton Sherpa+BlackHat 0 m2 TeV < y < < y < < y < < y < < y < < y < < y <.5 Höche, MS arxv: < y <.0 y < 0 m 2 GeV NLO+PS matchng and multjet mergng

22 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Try dfferent scale µ R/F = 4 H(y) T wth H (y) T = jets p, e 0.3 y boost y wth y boost = /n jets jets y reduces to µ R/F = 2 p e 0.3y wth y = 2 y y 2 for 2 2 and captures real emsson dynamcs Ells, Kunszt, Soper PRD40(989)288 better descrpton of data at large rapdtes, as expected descrpton of most other observables worsened need proper descrpton of forward physcs (e.g. (B)FKL) < y < < y < < y < < y < < y < < y < < y <.5 Höche, MS arxv: < y <.0 y < 0 m 2 GeV NLO+PS matchng and multjet mergng

23 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Try dfferent scale µ R/F = 4 H(y) T wth H (y) T = jets p, e 0.3 y boost y wth y boost = /n jets jets y reduces to µ R/F = 2 p e 0.3y wth y = 2 y y 2 for 2 2 and captures real emsson dynamcs Ells, Kunszt, Soper PRD40(989)288 better descrpton of data at large rapdtes, as expected descrpton of most other observables worsened need proper descrpton of forward physcs (e.g. (B)FKL) Höche, MS arxv: st jet 2nd jet 3rd jet 4th jet p GeV NLO+PS matchng and multjet mergng

24 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Gap Fracton Inclusve jet transverse momenta n dfferent rapdty ranges 0 Leadng djet selecton ATLAS data JHEP 09 (20) 053 Sherpa MC@NLO.5 Höche, MS arxv: GeV < p < 270 GeV 8 µr = µf = 4 HT, µ Q = 2 p µf, µr varaton µ Q varaton MPI varaton.5 20 GeV < p < 240 GeV Sherpa+BlackHat 240 GeV < p < 270 GeV GeV < p < 240 GeV GeV < p < 20 GeV GeV < p < 80 GeV GeV < p < 50 GeV GeV < p < 20 GeV + 70 GeV < p < 90 GeV y.5 80 GeV < p < 20 GeV.5 50 GeV < p < 80 GeV GeV < p < 50 GeV 90 GeV < p < 20 GeV.5 70 GeV < p < 90 GeV y NLO+PS matchng and multjet mergng 2

25 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Gap Fracton Inclusve jet transverse momenta n dfferent rapdty ranges 0 Forward-backward selecton ATLAS data JHEP 09 (20) 053 Sherpa MC@NLO GeV < p < 270 GeV Höche, MS arxv: µr = µf = 4 HT, µ Q = 2 p µf, µr varaton µ Q varaton MPI varaton.5 20 GeV < p < 240 GeV Sherpa+BlackHat 240 GeV < p < 270 GeV GeV < p < 240 GeV GeV < p < 20 GeV GeV < p < 80 GeV GeV < p < 50 GeV GeV < p < 20 GeV + 70 GeV < p < 90 GeV y.5 80 GeV < p < 20 GeV.5 50 GeV < p < 80 GeV GeV < p < 50 GeV 90 GeV < p < 20 GeV.5 70 GeV < p < 90 GeV y NLO+PS matchng and multjet mergng 3

26 NLO-mergng Conclusons Case study: Inclusve jet & djet producton small- y regon small uncertanty on addtonal jet producton large- y regon all uncertantes szable GeV < p < 270 GeV Höche, MS arxv: GeV < p < 240 GeV small- p regon domnated by perturbatve uncertantes large- p regon non-perturbatve uncertantes as large as perturbatve uncertantes Reducton of theoretcal uncertanty necesstates better understandng of soft QCD and nonfactorsable contrbutons GeV < p < 20 GeV 50 GeV < p < 80 GeV 20 GeV < p < 50 GeV 90 GeV < p < 20 GeV 70 GeV < p < 90 GeV y NLO+PS matchng and multjet mergng 3

27 NLO-mergng Conclusons Case study: Inclusve jet & djet producton de/dη GeV Forward energy flow n djet events, p jets > 20 GeV CMS data JHEP (20) 48 Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p Sherpa+BlackHat µf, µr varaton µ Q varaton MPI varaton η Forward energy flow Höche, MS arxv: energy flow n rapdty nterval per event wth a central back-to-back d-jet par normalsaton reduces µ R/F and µ Q dependence domnated by MPI modelng uncertanty NLO+PS matchng and multjet mergng 4

28 NLO-mergng Conclusons W + n jet producton dσ/dp pb/gev Jet transverse momenta Sherpa+BlackHat W+,2,3 jets p (thrd jet) ATLAS data NLO µ F/R = µ/2...2µ MC@NLO PL µ Q = µ/ µ p (second jet) p (frst jet) p GeV Höche, Krauss, MS, Segert arxv:2882 pp W +, 2, 3 jets 3 separate samples/calculatons NLO accuracy for nclusve observables of respectve jet multplcty resummaton of softest/lo jet,.e. 4th jet n pp W + 3 jets no resummaton of sample-defnng jet multplcty,.e. frst 3 jets n pp W + 3 jets scales: µ R/F = 2 Ĥ T, µ Q = p (j n ) Data: ATLAS Phys.Rev.D85(202) NLO+PS matchng and multjet mergng 5

29 NLO-mergng Conclusons Results: e + e hadrons /σ dσ/dln(y23) /σ dσ/dln(y45) Durham jet resoluton 3 2 (E CMS = 9.2 GeV) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo ln(y23) Durham jet resoluton 5 4 (E CMS = 9.2 GeV) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo ln(y45) /σ dσ/dln(y34) /σ dσ/dln(y56) Durham jet resoluton 4 3 (E CMS = 9.2 GeV) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo ln(y34) Durham jet resoluton 6 5 (E CMS = 9.2 GeV) ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo ln(y56) ee hadrons NLO; LO) Jet resolutons (Durham measure) MEPS@NLO vs MENLOPS at y domnated by hadr. effects needs retunng much mproved ren. scale dependence ALEPH data EPJC35(2004) NLO+PS matchng and multjet mergng 6

30 NLO-mergng Conclusons Results: e + e hadrons ALEPH data EPJC35(2004) Thrust (E CMS = 9.2 GeV) Sphercty (E CMS = 9.2 GeV) /σ dσ/dt ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo /σ dσ/ds ALEPH data MePs@Nlo MePs@Nlo µ/2...2µ MEnloPS MEnloPS µ/2...2µ Mc@Nlo T S NLO+PS matchng and multjet mergng 7

31 NLO-mergng Conclusons Results: pp W+jets Inclusve Jet Multplcty 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) pp W +jets NLO; LO) µ R/F 2, 2 µ def scale uncertanty much reduced NLO dependence for pp W +0,,2 jets LO dependence for pp W +3,4 jets Q cut = 30 GeV Njet good data descrpton ATLAS data Phys.Rev.D85(202) NLO+PS matchng and multjet mergng 8

32 NLO-mergng Conclusons Results: pp W+jets dσ/dp pb/gev 0 3 Frst 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 pp W +jets NLO; LO) µ R/F 2, 2 µ def scale uncertanty much reduced NLO dependence for pp W +0,,2 jets LO dependence for pp W +3,4 jets Q cut = 30 GeV good data descrpton ATLAS data Phys.Rev.D85(202) NLO+PS matchng and multjet mergng 8

33 NLO-mergng Conclusons Results: pp W+jets ATLAS data Phys.Rev.D85(202) R Dstance of Leadng Jets Azmuthal Dstance of Leadng 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 R(Frst Jet, Second Jet) φ(frst Jet, Second Jet) NLO+PS matchng and multjet mergng 9

34 NLO-mergng Conclusons Results: pp h+jets dσ/dp (H) fb/gev Rato p dstrbuton of the Hggs boson PRELIMINARY HqT (NLO+NNLL) Sherpa MC@NLO Sherpa MEPS@NLO p (H) GeV pp h+jets NLO; LO) µ R/F 2, 2 µ CKKW scale uncertanty much reduced NLO dependence for pp h+0, jets LO dependence for pp h+2 jets Q cut = 30 GeV HqT: µ R/F 2, 2 2 m h µ exp R ph NLO+PS matchng and multjet mergng 20

35 NLO-mergng Conclusons Results: pp h+jets dσ/dp (j ) fb/gev Rato p dstrbuton of hardest jet: p (j ) PRELIMINARY Sherpa MC@NLO Sherpa MEPS@NLO p (j ) GeV pp h+jets NLO; LO) µ R/F 2, 2 µ CKKW scale uncertanty much reduced NLO dependence for pp h+0, jets LO dependence for pp h+2 jets Q cut = 30 GeV HqT: µ R/F 2, 2 2 m h µ exp R ph NLO+PS matchng and multjet mergng 20

36 NLO-mergng Conclusons Conclusons SHERPA s MC@NLO formulaton allows full evaluaton of perturbatve uncertantes (µ F, µ R, µ Q ) MC@NLO can be easly combned wth MEPS MENLOPS MC@NLO s a necessary nput for NLO mergng MEPS@NLO MEPS@NLO gves full benefts of NLO calculatons (scale dependences, normalsatons) whle also retanng full (N)LL accuracy of parton shower wll be ncluded n SHERPA-2.0.α (upcomng) Current release: SHERPA better descrpton of perturbatve QCD s only part of the story to acheve hgher precsson for (hard) collder observables NLO+PS matchng and multjet mergng 2

37 NLO-mergng Conclusons Thank you for your attenton! NLO+PS matchng and multjet mergng 22

38 NLO-mergng Conclusons Case study: Inclusve jet & djet producton Djet azmuthal decorrelaton n varous p lead bns Höche, MS arxv: /σ dσ/d φ pb CMS data Phys. Rev. Lett. 06 (20) Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p µr, µf varaton µ Q varaton MPI varaton GeV < p lead 200 GeV < p lead < 300 GeV Sherpa+BlackHat GeV < p lead < 200 GeV 0 GeV < p lead < 40 GeV 80 GeV < p lead < 0 GeV φ φ NLO+PS matchng and multjet mergng 23

39 NLO-mergng Conclusons Case study: Inclusve jet & djet producton /N dn/dln(τ,c) pb Central transverse thrust n dfferent leadng jet p ranges.4.2 CMS data Phys. Rev. Lett. 06 (20) Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p µr, µf varaton µ Q varaton MPI varaton Höche, MS arxv: GeV < p jet GeV < p jet GeV < p jet < 200 GeV GeV < p jet < 200 GeV GeV < p jet < 25 GeV +0. Sherpa+BlackHat ln(τ,c) GeV < p jet < 25 GeV ln(τ,c) NLO+PS matchng and multjet mergng 24

40 NLO-mergng Conclusons Case study: Inclusve jet & djet producton /N dn/dln(t m,c) pb Central transverse thrust mnor n dfferent leadng jet p ranges.4.2 CMS data Phys. Rev. Lett. 06 (20) Sherpa MC@NLO µr = µf = 4 HT, µ Q = 2 p µr, µf varaton µ Q varaton MPI varaton Höche, MS arxv: GeV < p jet GeV < p jet GeV < p jet < 200 GeV GeV < p jet < 200 GeV GeV < p jet < 25 GeV +0. Sherpa+BlackHat ln(t m,c) GeV < p jet < 25 GeV ln(t m,c) NLO+PS matchng and multjet mergng 25

41 NLO-mergng Conclusons General NLO calculatons NLO calculaton wth subtracton methods O NLO = dφ B B(Φ B ) + V(Φ B ) + I(Φ B ) O(Φ B ) + dφ R + dφ R R(Φ R ) O(Φ R ) D (S) (Φ R ) O(Φ B ) ntroduce second set of subtracton functons D (A) D (A) and D (S) need to have same momentum maps and IR lmt - full spn-correlatons n collnear lmt - full colour-correlatons n soft lmt NLO+PS matchng and multjet mergng 26

42 NLO-mergng Conclusons General NLO calculatons NLO calculaton wth subtracton methods O NLO = dφ B B(Φ B ) + V(Φ B ) + I(Φ B ) O(Φ B ) + + dφ R dφ R R(Φ R ) D (A) (Φ R ) O(Φ B ) D (A) (Φ R ) O(Φ R ) + O (A) corr ntroduce second set of subtracton functons D (A) D (A) and D (S) D (S) (Φ R ) O(Φ B ) need to have same momentum maps and IR lmt - full spn-correlatons n collnear lmt - full colour-correlatons n soft lmt O (A) corr = dφ R D(A) O(ΦR ) O(Φ B ) NLO+PS matchng and multjet mergng 26

43 NLO-mergng Conclusons General NLO calculatons NLO calculaton wth subtracton methods O NLO = dφ B B(Φ B ) + V(Φ B ) + I(Φ B ) O(Φ B ) + dφ R D (A) (Φ R ) D (S) (Φ R ) + dφ R R(Φ R ) D (A) (Φ R ) O(Φ R ) + O (A) corr O(Φ B ) ntroduce second set of subtracton functons D (A) D (A) and D (S) need to have same momentum maps and IR lmt - full spn-correlatons n collnear lmt - full colour-correlatons n soft lmt NLO+PS matchng and multjet mergng 26

44 NLO-mergng Conclusons General NLO calculatons NLO calculaton wth subtracton methods O NLO = dφ B B(Φ B ) + V(Φ B ) + I(Φ B ) O(Φ B ) + dφ B dφ D (A) + dφ R R(Φ R ) (Φ B, Φ ) D (S) (Φ B, Φ ) O(Φ B ) D (A) (Φ R ) O(Φ R ) + O (A) corr ntroduce second set of subtracton functons D (A) D (A) and D (S) need to have same momentum maps and IR lmt - full spn-correlatons n collnear lmt - full colour-correlatons n soft lmt NLO+PS matchng and multjet mergng 26

45 NLO-mergng Conclusons General NLO calculatons NLO calculaton wth subtracton methods O NLO = dφ B B(Φ B ) + V(Φ B ) + I(Φ B ) O(Φ B ) + dφ B + dφ dφ R R(Φ R ) D (A) (Φ B, Φ ) D (S) (Φ B, Φ ) O(Φ B ) D (A) (Φ R ) O(Φ R ) + O (A) corr ntroduce second set of subtracton functons D (A) D (A) and D (S) need to have same momentum maps and IR lmt - full spn-correlatons n collnear lmt - full colour-correlatons n soft lmt NLO+PS matchng and multjet mergng 26

46 NLO-mergng Conclusons General NLO calculatons NLO calculaton wth subtracton methods O NLO = dφ B(A) B (Φ B ) O(Φ B ) + dφ R R(Φ R ) D (A) (Φ R ) O(Φ R ) + O (A) corr ntroduce second set of subtracton functons D (A) D (A) and D (S) need to have same momentum maps and IR lmt - full spn-correlatons n collnear lmt - full colour-correlatons n soft lmt NLO+PS matchng and multjet mergng 26

47 NLO-mergng Conclusons Parton showers and resummaton parton shower/resummaton kernel K (Φ ), Φ = {t, z, φ} K ncorporates dvergent propagator and DGLAP splttng kernels O PS = dφ B B(Φ B ) O(Φ B ) = dφ B B(Φ B ) O(Φ B ) + Sudakov form factor (K) (t, t ) = exp t dφ t K(Φ ) resummaton features O (K) corr generated by one parton shower step contans NLO+PS matchng and multjet mergng 27

48 NLO-mergng Conclusons Parton showers and resummaton parton shower/resummaton kernel K (Φ ), Φ = {t, z, φ} K ncorporates dvergent propagator and DGLAP splttng kernels O PS = = dφ B B(Φ B ) (K) (t 0, µ 2 F ) O(Φ B ) + dφ B B(Φ B ) O(Φ B ) + µ 2 F t 0 dφ K(Φ ) (K) (t, µ 2 F ) O(Φ R ) Sudakov form factor (K) (t, t ) = exp t dφ t K(Φ ) resummaton features O (K) corr generated by one parton shower step contans NLO+PS matchng and multjet mergng 27

49 NLO-mergng Conclusons Parton showers and resummaton parton shower/resummaton kernel K (Φ ), Φ = {t, z, φ} K ncorporates dvergent propagator and DGLAP splttng kernels O PS = = dφ B B(Φ B ) (K) (t 0, µ 2 F ) O(Φ B ) + dφ B B(Φ B ) O(Φ B ) + µ 2 F µ 2 F t 0 dφ R B K(Φ ) t 0 dφ K(Φ ) (K) (t, µ 2 F ) O(Φ R ) O(Φ R ) O(Φ B ) + O(αs) 2 Sudakov form factor (K) (t, t ) = exp t dφ t K(Φ ) resummaton features O (K) corr generated by one parton shower step contans NLO+PS matchng and multjet mergng 27

50 NLO-mergng Conclusons Parton showers and resummaton parton shower/resummaton kernel K (Φ ), Φ = {t, z, φ} K ncorporates dvergent propagator and DGLAP splttng kernels O PS = = dφ B B(Φ B ) (K) (t 0, µ 2 F ) O(Φ B ) + dφ B B(Φ B ) O(Φ B ) + O (K) corr + O(α 2 s) µ 2 F t 0 dφ K(Φ ) (K) (t, µ 2 F ) O(Φ R ) Sudakov form factor (K) (t, t ) = exp t dφ t K(Φ ) resummaton features O (K) corr generated by one parton shower step contans NLO+PS matchng and multjet mergng 27

51 NLO-mergng Conclusons General NLO+PS matchng O NLO+PS = dφ B(A) B (Φ B ) (A) (t 0, µ 2 Q) O(Φ B ) + dφ R R(Φ R ) µ 2 Q D (A) (Φ B, Φ ) + dφ t 0 B(Φ B ) D (A) (Φ R ) O(Φ R ) + O (A) corr (A) (t, µ 2 Q) O(Φ R ) use D (A) as resummaton kernels resummaton phase space lmted by µ 2 Q = t max startng scale of parton shower evoluton should be of the order of the scale of the hard nteracton POWHEG and MC@NLO now dffer n choce of D (A) and µ 2 Q NLO+PS matchng and multjet mergng 28

52 NLO-mergng Conclusons General NLO+PS matchng O NLO+PS = dφ B(A) B (Φ B ) (A) (t 0, µ 2 Q) O(Φ B ) + dφ R R(Φ R ) µ 2 Q D (A) (Φ B, Φ ) + dφ t 0 B(Φ B ) D (A) (Φ R ) O(Φ R ) (A) (t, µ 2 Q) O(Φ R ) use D (A) as resummaton kernels resummaton phase space lmted by µ 2 Q = t max startng scale of parton shower evoluton should be of the order of the scale of the hard nteracton POWHEG and MC@NLO now dffer n choce of D (A) (A) (t, t ) = exp t dφ t D (A) /B = exp α s log 2 t t +... and µ 2 Q NLO+PS matchng and multjet mergng 28

53 NLO-mergng Conclusons General NLO+PS matchng O NLO+PS = dφ B(A) B (Φ B ) (A) (t 0, µ 2 Q) O(Φ B ) + dφ R R(Φ R ) µ 2 Q D (A) (Φ B, Φ ) + dφ t 0 B(Φ B ) D (A) (Φ R ) O(Φ R ) (A) (t, µ 2 Q) O(Φ R ) use D (A) as resummaton kernels resummaton phase space lmted by µ 2 Q = t max startng scale of parton shower evoluton should be of the order of the scale of the hard nteracton POWHEG and MC@NLO now dffer n choce of D (A) (A) (t, t ) = exp t dφ t D (A) /B = exp α s log 2 t t +... and µ 2 Q NLO+PS matchng and multjet mergng 28

54 NLO-mergng Conclusons POWHEG Specal choces: Nason JHEP(2004)040, Frxone et.al. JHEP(2007)070 exponentaton kernel D (A) = ρ R wth ρ = D (S) / D(S) each ρ R contans only one dvergence structure as defned by D (S) Consequences: no H-events, resummaton scale µ 2 Q at knematc lmt 2 s had n CS-subtracton nstabltes n ρ due to dfferent cuts on R and D (S) exponentaton of R through matrx element corrected parton shower NLO accuracy depends crucally on presence of exact same terms n subtracton and parton shower Modfcatons: ntroduce suppresson functon f(p ) = h 2 /(p 2 + h2 ) Alol et.al. JHEP04(2009)002 D (A) = ρ R f(p ) contnuous dampenng of resummaton kernel at large p NLO+PS matchng and multjet mergng 29

55 NLO-mergng Conclusons POWHEG Specal choces: Nason JHEP(2004)040, Frxone et.al. JHEP(2007)070 exponentaton kernel D (A) = ρ R wth ρ = D (S) / D(S) each ρ R contans only one dvergence structure as defned by D (S) Consequences: no H-events, resummaton scale µ 2 Q at knematc lmt 2 s had n CS-subtracton nstabltes n ρ due to dfferent cuts on R and D (S) exponentaton of R through matrx element corrected parton shower NLO accuracy depends crucally on presence of exact same terms n subtracton and parton shower Modfcatons: ntroduce suppresson functon f(p ) = h 2 /(p 2 + h2 ) Alol et.al. JHEP04(2009)002 D (A) = ρ R f(p ) contnuous dampenng of resummaton kernel at large p NLO+PS matchng and multjet mergng 29

56 NLO-mergng Conclusons POWHEG Specal choces: Nason JHEP(2004)040, Frxone et.al. JHEP(2007)070 exponentaton kernel D (A) = ρ R wth ρ = D (S) / D(S) each ρ R contans only one dvergence structure as defned by D (S) Consequences: no H-events, resummaton scale µ 2 Q at knematc lmt 2 s had n CS-subtracton nstabltes n ρ due to dfferent cuts on R and D (S) exponentaton of R through matrx element corrected parton shower NLO accuracy depends crucally on presence of exact same terms n subtracton and parton shower Modfcatons: ntroduce suppresson functon f(p ) = h 2 /(p 2 + h2 ) Alol et.al. JHEP04(2009)002 D (A) = ρ R f(p ) contnuous dampenng of resummaton kernel at large p NLO+PS matchng and multjet mergng 29

57 NLO-mergng Conclusons tradtonal scheme Specal choces: exponentaton kernel Consequences: Frxone, Webber JHEP06(2002)029 D (A) = B K wth K parton shower kernels resummaton scale µ 2 Q = t max parton shower startng scale n general, D (A) only leadng colour approxmaton and spn-avaraged NLO accuracy depends crucally on correctness of IR-lmt Modfcatons: ntroduce soft modfcaton functon f(p ) such that Frxone, Nason, Webber JHEP08(2003)007 B K f(p ) f(p ) process dependent n general p 0 D (S) NLO+PS matchng and multjet mergng 30

58 NLO-mergng Conclusons tradtonal scheme Specal choces: exponentaton kernel Consequences: Frxone, Webber JHEP06(2002)029 D (A) = B K wth K parton shower kernels resummaton scale µ 2 Q = t max parton shower startng scale n general, D (A) only leadng colour approxmaton and spn-avaraged NLO accuracy depends crucally on correctness of IR-lmt Modfcatons: ntroduce soft modfcaton functon f(p ) such that Frxone, Nason, Webber JHEP08(2003)007 B K f(p ) f(p ) process dependent n general p 0 D (S) NLO+PS matchng and multjet mergng 30

59 NLO-mergng Conclusons tradtonal scheme Specal choces: exponentaton kernel Consequences: Frxone, Webber JHEP06(2002)029 D (A) = B K wth K parton shower kernels resummaton scale µ 2 Q = t max parton shower startng scale n general, D (A) only leadng colour approxmaton and spn-avaraged NLO accuracy depends crucally on correctness of IR-lmt Modfcatons: ntroduce soft modfcaton functon f(p ) such that Frxone, Nason, Webber JHEP08(2003)007 B K f(p ) f(p ) process dependent n general p 0 D (S) NLO+PS matchng and multjet mergng 30

60 NLO-mergng Conclusons D (A) = D (S) scheme Specal choces: exponentaton kernel Consequences: D (A) smplfcaton of B (A) -ntegral = D (S) Θ(µ 2 Q t) SH, FK, MS, FS arxv:.220 resummaton scale µ 2 Q = t max set by phase space lmtaton of subtracton terms subtracton constraned n parton shower t needed for physcal resummaton trvally NLO correct ndependent of the process wthout arbtrary parameter choces NLO+PS matchng and multjet mergng 3

61 NLO-mergng Conclusons D (A) = D (S) scheme Specal choces: exponentaton kernel Consequences: D (A) smplfcaton of B (A) -ntegral = D (S) Θ(µ 2 Q t) SH, FK, MS, FS arxv:.220 resummaton scale µ 2 Q = t max set by phase space lmtaton of subtracton terms subtracton constraned n parton shower t needed for physcal resummaton trvally NLO correct ndependent of the process wthout arbtrary parameter choces NLO+PS matchng and multjet mergng 3

62 NLO-mergng Conclusons D (A) = D (S) scheme Implemented n SHERPA full-colour frst parton shower emsson Trcky pont: D (A) < 0 e.g. for subleadng colour dpoles Use modfed Sudakov veto algorthm SH, FK, MS, FS arxv:.220 Assume f(t) as functon to be generated, and overestmate g(t) Standard probablty for one acceptance wth n rejectons { f(t) t } n tn+ ( g(t) exp d t g( t) dt f(t ) { t+ } ) g(t ) exp d t g( t) g(t) g(t ) t = t t Can splt weght nto MC and analytc part usng auxlary functon h(t) { f(t) h(t) exp g(t) t w(t, t,..., t n ) = g(t) h(t) t } n tn+ d t h( t) n = = t g(t ) h(t ) f(t ) h(t ) g(t ) f(t ) ( dt f(t ) { ) h(t ) exp g(t ) t+ t } d t h( t) NLO+PS matchng and multjet mergng 32

63 NLO-mergng Conclusons D (A) = D (S) scheme Implemented n SHERPA full-colour frst parton shower emsson Trcky pont: D (A) < 0 e.g. for subleadng colour dpoles Use modfed Sudakov veto algorthm SH, FK, MS, FS arxv:.220 Assume f(t) as functon to be generated, and overestmate g(t) Standard probablty for one acceptance wth n rejectons { f(t) t } n tn+ ( g(t) exp d t g( t) dt f(t ) { t+ } ) g(t ) exp d t g( t) g(t) g(t ) t = t t Can splt weght nto MC and analytc part usng auxlary functon h(t) { f(t) h(t) exp g(t) t w(t, t,..., t n ) = g(t) h(t) t } n tn+ d t h( t) n = = t g(t ) h(t ) f(t ) h(t ) g(t ) f(t ) ( dt f(t ) { ) h(t ) exp g(t ) t+ t } d t h( t) NLO+PS matchng and multjet mergng 32

64 NLO-mergng Conclusons D (A) = D (S) scheme Implemented n SHERPA full-colour frst parton shower emsson Trcky pont: D (A) < 0 e.g. for subleadng colour dpoles Use modfed Sudakov veto algorthm SH, FK, MS, FS arxv:.220 Can splt weght nto MC and analytc part usng auxlary functon h(t) { f(t) h(t) exp g(t) t w(t, t,..., t n ) = g(t) h(t) Identfy f(t), g(t), h(t): t } n tn+ d t h( t) n = = t g(t ) h(t ) f(t ) h(t ) g(t ) f(t ) f(t) determned by MC@NLO D (A) h(t) determned by parton shower D (PS) g(t) can be chosen freely const. f constrants: sgn(f) = sgn(g), f g ( dt f(t ) { ) h(t ) exp g(t ) t+ t } d t h( t) NLO+PS matchng and multjet mergng 32

65 NLO-mergng Conclusons NLO mergng LO mergng: LO accuracy for n n max -jet processes preserve LL accuracy of the parton shower NLO mergng: NLO accuracy for n n max -jet processes preserve LL accuracy of the parton shower Catan, Krauss, Kuhn, Webber JHEP(200)063 Lönnblad JHEP05(2002)046 Höche, Krauss, Schumann, Segert JHEP05(2009)053 Hamlton, Rchardson, Tully JHEP(2009)038 Lönnblad, Prestel JHEP03(202)09 Lavesson, Lönnblad JHEP2(2008)070 Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: NLO+PS matchng and multjet mergng 33

66 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

67 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

68 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

69 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

70 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

71 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

72 NLO-mergng Conclusons NLO mergng O MEPS@NLO = + + dφ n B(A) n (A) n (t 0, µ 2 Q) O n µ 2 Q + t 0 dφ n+ R n D (A) dφ n+ B(A) n+ (A) Dn dφ n B n Höche, Krauss, MS, Segert arxv: Gehrmann, Höche, Krauss, MS, Segert arxv: (A) n (t n+, µ 2 Q) Θ(Q cut Q) O n+ Θ(Q cut Q) (PS) n (t n+, µ 2 Q) O n+ + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) tn+ (A) (A) n+ (t Dn+ 0, t n+ ) O n+ + dφ (A) n+ t 0 B (t n+2, t n+ ) O n+2 n+ + dφ n+2 R n+ D (A) n+ (PS) n+ (t n+2, t n+ ) (PS) n (t n+, µ 2 Q) Θ(Q Q cut ) O n+2 NLO+PS matchng and multjet mergng 34

73 NLO-mergng Conclusons NLO mergng Generaton of MC counterterm + B n+ B n+ µ 2 Q t n+ dφ K n same form as exponent of Sudakov form factor (PS) n (t n+, µ 2 Q ) truncated parton shower on n-parton confguraton underlyng n + -parton event no emsson retan n + -parton event as s 2 frst emsson at t wth Q > Q cut, multply event weght wth B n+/ B (A) n+, restart evoluton at t, do not apply emsson knematcs 3 treat every subsequent emsson as n standard truncated vetoed shower generates + B n+ B n+ µ 2 Q t n+ dφ K n (PS) n (t n+, µ 2 Q) dentfy O(α s ) counterterm wth the emtted emsson NLO+PS matchng and multjet mergng 35

74 NLO-mergng Conclusons NLO mergng Renormalsaton scales: determned by clusterng usng PS probabltes and takng the respectve nodal values t α s (µ 2 R )k = k α s (t ) = change of scales µ R µ R n MEs necesstates one-loop counter term ( ) α s ( µ 2 R) k α s( µ 2 R ) k β 0 ln t 2π µ 2 R Factorsaton scale: µ F determned from core n-jet process change of scales µ F µ F n MEs necesstates one-loop counter term ( B n (Φ n ) α s( µ 2 R ) log µ2 n ) F dz 2π µ 2 F x a z P ac(z) f c (x a /z, µ 2 F ) +... c=q,g = NLO+PS matchng and multjet mergng 36