Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3

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1 Electroweak Physics at the LHC Lecture 3 Electroweak Di-boson Production Stefan Dittmaier Albert-Ludwigs-Universita t Freiburg Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

2 Contents Electroweak di-boson production brief overview / production / / production Gauge-invariance issues in E multi-boson production Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

3 Electroweak di-boson production brief overview Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

4 E di-boson production V V V V V,V =,, ± Physics issues: triple-gauge-boson couplings, especially at high momentum transfer E corrections significant anomalous TGC: formfactor approach to switch off unitarity violation element of arbitrariness, avoid when possible important background processes to Higgs production, H / 4f invariant masses below VV thresholds, proper description of off-shell V V 4f production required! to searches at high invariant masses E corrections Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

5 State-of-the-art predictions / (with leptonic decays) NNLO QCD Grazzini, Kallweit, Rathlev 14, 15 NLO E Denner, S.D., Hecht, Pasold 14, 15,, NNLO QCD (on-shell and off-shell) Cascioli et al. 14; Grazzini, Kallweit, Rathlev 15 (on-shell) Gehrmann et al. 14 gg V V 4 leptons LO: Binoth et al. 05, 06; NLO: Caola et al. 15, 16 NLO E stable / bosons Bierweiler, Kasprzik, Kühn 12/ 13 Baglio, Le, eber 13 pp 4 leptons in DPA Billoni, S.D., Jäger, Speckner 13 approximative inclusion in HERIG++ pp / 4 leptons fully off-shell Gieseke, Kasprzik, Kühn 14 Biedermann et al. 16 Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

6 / production Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

7 Example of production q q q ν l l q q q ν l l q q l l ν l q q ν l l Issues / physics goals: clean photon jet separation zk q q quark-to-photon fragmentation function Glover, Morgan 94 or Frixione isolation Frixione 98 k q stronger bounds on anomalous coupling: + µ (q) ν ( q) ρ(p) = e {q µ g νρ ( + ( q ρ q ρ) λ M 2 κ + λ q 2 M 2 ) q ν g µρ ( ( p µ p ν 1 2 gµν p 2 ) } κ + λ q 2 ( M M2 Λ 2 ) ) 2 ATLAS limits 12: κ = 0.41, λ = for Λ = 2TeV Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

8 Photon jet separation via photon fragmentation function D q Glover, Morgan 94 hy? u l + ν l QCD radiation cannot be suppressed by cuts d treat at least soft/collinear jets inclusively d g separation of collinear quarks and photons u l + leads to IR-unstable corrections ln(m 2 q/q 2 ) recombine collinear quarks and photons g d d ν l d quark and gluon jets cannot be distinguished event by event common recombination required for quarks/gluons with photons (g hard + soft ) }{{} E corr. to X+jet and (g soft + hard ) }{{} QCD corr. to X+ both appear as 1 jet u d l + ν l d Problem: signatures of X+jet and X+ overlap! d g Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

9 Photon jet separation via photon fragmentation function D q Glover, Morgan 94 Solution: idea: declare photon/jet systems as photon or jet according to energy share determine photon energy fraction z = event selection: z > z 0 : photon E E jet +E of photon/jet system z < z 0 : jet (typical value z 0 = 0.7) but: cut on z destroys inclusiveness needed for KLN theorem collinear singularity αlnm q remains (but are universal!) absorb universal collinear singularity in fragmentation function D q (z ) subtract convolution of LO cross section with [ q (z,µ fact ) = αq2 q mass.reg. 2π P q (z ) ln m2 q µ 2 fact D MS where P q (z ) = 1+(1 z ) 2 + 2lnz + 1 ] cancels coll. singularities + Dq ALEPH (z,µ fact ) non-perturbative part fitted to ALEPH data z = quark-to-photon splitting function Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

10 Alternative: photon jet separation via Frixione isolation Frixione 98 Idea: suppress jets inside collinear cone around photons: ( ) 1 cosrjet p T,jet < εp T, 1 cosr 0 photon and jet collinear (R jet 0) (R 0 = fixed cone size) event discarded photon soft or collinear to beams (p T, 0) event discarded jet soft or collinear beams (p T,jet 0) event kept IR safety Comments: Frixione isolation simple to implement theoretically, but problematic experimentally cleaner isolation of non-perturbative effects by fragmentation function approximate relation between the two methods: z p T, p T, +p T,jet > ε 1 cosr jet 1 cosr 0 1+ε for R jet R 0 methods yield quite similar results for z ε Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

11 data/theory data/theory dσ/dp T [fb/gev] dσ/dp T [fb/gev] production QCD theory versus experiment Grazzini, Kallweit, Rathlev pp ll l s =7 TeV NLO NNLO ATLAS 10 2 pp ll l s =7 TeV NLO NNLO ATLAS N jet N jet = p T [GeV] p T [GeV] good agreement of experimental results with NNLO QCD (no E corrections included) QCD uncertainties: (for small/moderate p T, ) scale: 4 5%, PDF: 1 2% (increasing with p T, ) LHC run 2: higher energy reach & higher statistics E corrections important Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

12 production E corrections Denner, S.D., Hecht, Pasold 14 NLO E corrections calculated with full off-shell/decay effects (complex-mass scheme) more + more complicated diagrams than in QCD u u / l + l + ν l d etc. particular focus on: high energies (e.g. large p T ): large E corrections sensitivity to anomalous couplings missing corrections could fake anomalous couplings photon-induced contributions Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

13 Rapidity distributions in production Denner, S.D., Hecht, Pasold pp l + ν l ( jet ) 800 dσ/dy [fb] σ LO NLO QCD σ NLO QCD σveto δe,qq [%] s = 14TeV δ NCS E,qq δ CS E,qq δqcd [%] δ QCD δ veto QCD y δe,q [%] δ E,q δ veto E,q y huge QCD corrections ( 100%), only mildly reduced by jet veto p T,jet < 100GeV E corrections and q channels (few %) small and flat resemble corrections to integrated cross section (CS=collinear-safe, NCS=non-collinear-safe) Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

14 p T distributions in production E corrections Denner, S.D., Hecht, Pasold 14 dσ/dpt,[fb/gev] s = 14TeV pp l + ν l (/jet) NLO QCD σ σ NLO NCS NLO QCD veto σ NLO NCS,veto σ dσ/dpt,l +[fb/gev] s = 14TeV pp l + ν l (/jet) NLO QCD σ σ NLO NCS NLO QCD σveto σ NLO NCS,veto δe,qq [%] δe,q [%] δ E,q /10 δ veto E,q p T, [GeV] δ NCS E,qq δ CS E,qq E corrections 30% in TeV range δe,qq [%] δe,q [%] p T,l + [GeV] (CS=collinear-safe, NCS=non-collinear-safe) δ NCS E,qq δ CS E,qq δ E,q /10 δ veto E,q -induced corrections non-negligible in TeV range (even with jet veto) reduction of PDF uncertainties mandatory! Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

15 production anomalous couplings Denner, S.D., Hecht, Pasold 14 dσ/dpt,[fb/gev] σ σ s = 14TeV NLO QCD AC NLO QCD AC NLO QCD σac NLO QCD σ,λ=1tev,λ=2tev,λ pp l + ν l (jet) dσ/dpt,l +[fb/gev] s = 14TeV pp l + ν l (jet) σ σ NLO QCD AC NLO QCD AC NLO QCD σac NLO QCD σ,λ=1tev,λ=2tev,λ δac [%] δ AC,1TeV δ AC,2TeV / δac [%] δ AC,1TeV δ AC,2TeV / p T, [GeV] p T,l +[GeV] results shown without and with jet veto on p T,jet > 100GeV ATLAS values of 2012 used: κ = 0.41, λ = much tighter limits expected at LHC run 2 Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

16 / / production Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

17 Complementarity in / / production production: / g g production: production: g g Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

18 Complementarity in / / production production: / g g production: production: g g Sensitivity to different PDF combinations: q q in / u d/dū in + / in Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

19 Complementarity in / / production production: / g g production: production: / g g Sensitivity to different anomalous TGCs: overlay of / in only in / in Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

20 Complementarity in / / production production: / g g production: production: g g Background to Higgs production in channel H / 4f off-shell calculation particularly important for /! H / / Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

21 QCD corrections to,,,, production NLO QCD calculated (including leptonic / decays) Baur, Han, Ohnemus Dixon, Kunszt, Signer 99 Campbell, R.K.Ellis 99 DeFlorian, Signer 00 Campbell, R.K.Ellis et al. 99 Haywood et al. 00 Large positive corrections due to jet radiation, i.e. VV +jet production reduction of corrections and scale dependence by jet veto: p T,jet < cut? include QCD resummation for veto NNLO QCD corrections important Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

22 production NNLO QCD theory versus experiment Gehrmann et al. 14 σ[pb] pp + +X ATLAS CMS gg H added to all predictions NNLO+gg NLO+ggN NLON+gg LONN+gg Subtlety: Separation of single-t and t t NNLO QCD b-jet veto, etc σ/σ NLO 7 8 s[tev] good agreement of experimental results with NNLO QCD NNLO QCD correction 8(13)TeV, scale uncertainty < 3% gg contribution 8(13)TeV LHC run 2: higher energy & higher statistics E corrections important Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

23 production NNLO QCD theory versus experiment Cascioli et al. 14 good agreement of experimental results with NNLO QCD NNLO QCD correction 8(13)TeV, scale uncertainty < 3% gg contribution 8(13) TeV LHC run 2: higher energy & higher statistics E corrections important Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

24 production NLO QCD theory versus experiment Baglio, Le, eber et al σ(pp + + ) [pb] NLO QCD+E, µ 0 = M +M tot ATLAS CMS s [TeV] good agreement of experimental results with NLO QCD NLO QCD scale uncertainty 3%, PDF+αs 4% LHC run 2: higher energy & higher statistics NNLO QCD and NLO E corrections important Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

25 E corrections to massive di-boson production (stable/on-shell bosons) Bierweiler, Kasprzik, Kühn 13 t E corrections small for integrated XS growing in distributions for larger scales t Note: M VV not accessible for final states on-shell approximation not applicable for M VV < M V1 +M V2 Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

26 Survey of corrections to production (stable/on-shell bosons) dσ/dy(pb) 0.1 LHC at 14 TeV, M > 1000 GeV σ q q, LO σ q q,+ LO σ gg LO 10 σ LO δ(%) δe δ δ gg δ QCD /10 Bierweiler, Kasprzik, Kühn 12 Note: large contribution by channel for high invariant masses! y y 1 2 dσ/dpt [pb/gev] pp + s = 14 TeV, µ0 = 2M [GeV] p + T q q (NLO QCD) q q (NLO E) q q (LO) gg (LO) (LO) (NLO) 10 6 (LO) (NLO) d σ/dσ Born [%] pp + s = 14 TeV, µ0 = 2M q q (E NLO) q q (E virtual) induced radiated p + T Baglio, Le, eber [GeV] Large impact of q collisions? Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

27 E corrections with leptonic / decays Double-pole approximation (DPA) vs. Full off-shell q q 4f calculation q V V q on-shell production V V on-shell decays f 1 f 2 f 3 f 4 q q / f 3 f 1 f 2 f 4 f 3 d d u f 2 f 3 f 1 f 2 / f 3 f 4 expansion about resonance poles factorizable & non-factorizable corrs. not many diagrams (2 2 production) + numerically fast validity only for ŝ > 2M V +O(Γ V ) error estimate for ŝ < 0.5 1TeV: α π Γ V M V log(...) 0.5 2% off-shell calculation with complex-mass scheme many off-shell diagrams ( 10 3 /channel) CPU intensive + NLO accuracy everywhere global error estimate: δ NNLOE δ 2 NLOE Approaches compared for e + e 4f Denner, S.D., Roth, ieders 05 New: pp 4f Biedermann et al. 16 Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

28 Details of the full 4f NLO calculation Biedermann et al. 16 Virtual corrections one version diagrammatically as for e + e 4f Denner et al. 05 another version based on recursive method with RECOLA Actis et al. 13 some checks done with FEYNARTS/FORMCALC in the framework of POLE Accomando et al. 05 / resonances treated in the complex-mass scheme loop integrals evaluated with COLLIER Real corrections and Monte Carlo integration IR singularities treated with dipole subtraction Catani, Seymour 96; S.D. 99; S.D. et al. 08 collinear-unsafe ( bare ) and dressed leptons supported multi-channel Monte Carlo integration -induced contributions collisions included in LO (small contributions) q contributions taken into account Two independent calculations of all ingredients Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

29 Details of the full 4f NLO calculation Biedermann et al. 16 Collier Hepforge Virtual corrections Collier is hosted by Hepforge, IPPP Durham one version diagrammatically as for e + e 4f Denner et al. 05 another version based on recursive method A Complex with One-Loop RECOLA LIbrary Actis et al. 13 some checks done with FEYNARTS/FORMCALC Regularizations in the framework of POLE Authors / resonances Ansgar Denner treated Universität in ürzburg, the complex-mass Germany! scheme loop integrals evaluated with COLLIER Real corrections and Monte Carlo integration Accomando et al. 05!! scalar and tensor integrals for high particle multiplicities IR singularities treated with dipole subtraction Catani, Seymour 96; S.D. 99; S.D. et al. 08 collinear-unsafe!! dimensional ( bare ) regularization and for dressed soft infrared divergences leptons supported multi-channel Monte Carlo integration -induced contributions Stefan Dittmaier!!!Universität Freiburg, Germany! Lars Hofer Universitat de Barcelona, Spain! Features of the library collisions included in LO (small contributions)!! cache system to speed up calculations q contributions taken into account with Extended COLLIER is a fortran library for the numerical evaluation of one-loop scalar and tensor integrals appearing in perturbative relativistic quantum field theory with the following features:!! dimensional regularization for ultraviolet divergences!!!!! (mass regularization for abelian soft divergences is supported as well)!! dimensional regularization or mass regularization for collinear mass singularities!! complex internal masses (for unstable particles) fully supported!!!!! (external momenta and virtualities are expected to be real)!! numerically dangerous regions (small Gram or other kinematical determinants)!!!!! cured by dedicated expansions!! two independent implementations of all basic building blocks allow for internal cross-checks If you use Collier for a publication, please cite all the references listed here! Two independent calculations of all ingredients Released on April 25! Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

30 Details of the full 4f NLO calculation Biedermann et al. 16 Virtual corrections one version diagrammatically as for e + e 4f Denner et al. 05 another version based on recursive method with RECOLA Actis et al. 13 some checks done with FEYNARTS/FORMCALC in the framework of POLE Accomando et al. 05 / resonances treated in the complex-mass scheme loop integrals evaluated with COLLIER Real corrections and Monte Carlo integration IR singularities treated with dipole subtraction Catani, Seymour 96; S.D. 99; S.D. et al. 08 collinear-unsafe ( bare ) and dressed leptons supported multi-channel Monte Carlo integration -induced contributions collisions included in LO (small contributions) q contributions taken into account Two independent calculations of all ingredients Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

31 DPA versus full off-shell E correction in pp ν µ µ + e ν e +X Biedermann et al. 16 Rapidity and invariant-mass distributions 75 pp ν µ µ + e ν e +X spp = 13TeV dσ dy(e ) [fb] dσ [ ] fb dm e µ + GeV ATLAS cuts µ treated coll. unsafe 1 pp ν µ µ + e ν e +X spp = 13TeV ATLAS cuts µ treated coll. unsafe LO qq LO DPA 10 3 LO qq LO DPA δ[%] δ qq δ qq DPA δ[%] δ qq δ qq DPA y(e ) M e µ + [GeV] Level of agreement as expected (dominance of doubly-resonant diagrams) Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

32 DPA versus full off-shell E correction in pp ν µ µ + e ν e +X Biedermann et al. 16 Transverse-momentum distribution of a single lepton 10 [ ] dσ fb dp T (e ) GeV ν µ µ pp ν µ µ + e ν e +X spp = 13TeV ATLAS cuts µ treated coll. unsafe ν e p T (µ + ν µ ν e ) 10 2 q e 10 3 q / 10 4 LO qq LO DPA p T (e ) δ[%] δ qq δ qq DPA e Impact of singly-resonant diagrams wheree takes recoil from(µ + ν µ ν e ) p T (e )[GeV] Agreement degrades for p T > 300GeV, since off-shell diagrams get enhanced Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

33 DPA versus full off-shell E correction in pp ν µ µ + e ν e +X Biedermann et al. 16 Transverse-momentum distribution of the charged lepton pair [ ] dσ fb dp T (e µ + ) GeV pp ν µ µ + e ν e +X spp = 13TeV ATLAS cuts µ treated coll. unsafe q µ e ν e µ + p T (e µ + ) p T ( ν e ) 10 4 q / LO qq LO DPA p T (ν µ ) δ[%] δ qq δ qq DPA Double resonance extremely suppressed! Dominance of singly-resonant diagrams where (e µ + ) recoil against (ν µ ν e ) ν µ p T (e µ + )[GeV] DPA fails for p T > 200GeV, since off-shell production dominates! Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

34 Gauge-invariance issues in E multi-boson production Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

35 Gauge invariance implies... Slavnov Taylor or ard identites = algebraic relations of or between Greens functions guarantee cancellation of unitarity-violating terms, crucial for proof of unitarity of S-matrix Nielsen identities (compensation of gauge-fixing artefacts) gauge-parameter independence of S-matrix although Greens function (e.g. self-energies) are gauge dependent Both statements hold order by order in standard perturbation theory! Implications: Resonances require Dyson summation of resonant propagators perturbative orders mixed gauge invariance jeopardized! Gauge-invariance-violating terms Γ are formally of higher order, but can be dramatically enhanced if unitarity cancellations disturbed Anomalous couplings potentially enhanced if effective operator not gauge invariant Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

36 Important ard identities for processes with E gauge bosons: Elmg. U(1) gauge invariance implies k F 1 k µ µ = 0 for any on-shell fields F l F n Identity becomes crucial for collinear light fermions: for fermion momenta p 1 cp 2 : p 1 k = p 1 p 2 = ū 2 (p 2 ) µ u 1 (p 1 ) k µ A typical situation: quasi-real space-like photons e e p 2 k 1 k 2 kµ T µ for k 2 O(m 2 e) E 2 Identity k µ T µ = 0 needed to cancel 1/k 2, otherwise gauge-invariance-breaking terms enhanced bye 2 /m 2 e ( for LEP2) Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

37 Electroweak SU(2) gauge invariance implies k µ µ F 1 F n = im χ F 1 F n F l = on-shell fields χ,φ ± = would-be Goldstone fields F 1 F 1 k µ ± µ F n = ±M φ ± F n A typical situation: high-energetic quasi-real longitudinal vector bosons fermion current attached to V(k) again k µ k V 1 k 2 M 2 V k µ T V µ for k 0 M V Identity k µ T V µ = c V M V T S needed to cancel factor k 0, otherwise gauge-invariance/unitarity-breaking terms enhanced by k 0 /M V For on-shell V : ε µ V L (k) = kµ M V +O(M V /k 0 ) Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

38 d u Illustration of unitarity cancellations for V production (V = /) Leading behaviour of amplitudes with ε µ + L d V V + L ie2 g V 2 2s M [ v d µ ω u u ] (k) = kµ M V +... for k 0 M : { g V 1 [ [ ]} ε µ V kµ ε V ]+κ k s V ε µ V +kµ ε V k s u + L ie2 g V dd 2s M [ v d/ε V ω u u ], g dd = s c Q d 1 2s c, g dd = Q d d + L u V ie2 g V uu 2s M [ v d/ε V ω u u ], g uu = s c Q u + 1 2s c, g uu = Q u Cancellation (unitarity!) of sum demands: g V dd g V uu g V 2 (g V 1 +κ V )! = 0, g V 1! = κ V SM provides unique solution: g 1 = κ = g 1 = κ = 1 Note: no constraint on coupling λ V, since effective operator gauge invariant! Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

39 idth schemes for LO calculations and gauge invariance Naive propagator substitutions in full tree-level amplitudes: 1 k 2 m 1 in all propagators 2 k 2 m 2 +imγ(k 2 ) constant width Γ(k 2 ) = const. U(1) respected, SU(2) mildly violated running width Γ(k 2 ) const. U(1) and SU(2) violated Fudge factor approaches: results can be totally wrong! Multiply full amplitudes without widths with p 2 m 2 factors for each potentially resonant propagator p 2 m 2 +imγ gauge invariant, but spurious factors of O(Γ/m) Complex-mass scheme: (see lecture 1) Consistent use of complex masses everywhere (including couplings) For / bosons: M 2 V µ 2 V = M 2 V im V Γ V, V =, complex weak mixing angle: gauge invariance fully respected c 2 = 1 s 2 = µ2 µ 2 Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

40 An example: e e + e ν e u d result of Kurihara, Perret-Gallix, Shimizu 95 s = 180GeV solid: gauge-invariant (fudge factor) scheme dashed: constant width only in resonant propagator crude U(1) gauge-invariance violation Dominant diagrams: nearly real photon! e e + + e u d νe e e + + e e u d νe e + e e u d νe e + e u d νe Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

41 Example continued: e e e e e e + + e u d ν e e e + + e u d νe e + u d νe e + u d νe Partial amplitude from above photon diagrams : M = Q e eū e (k e ) µ u e (p e ) 1 T k 2 µ Elmg. ard identity: 0! = k µ T µ (p 2 + p 2 )Q P w (p 2 +)P w (p 2 )+Q e P w (p 2 +) (Q d Q u )P w (p 2 ) ith Q = Q e = Q d Q u and P w (p 2 ) = [p 2 M 2 +im Γ (p 2 )] 1 one obtains: Γ (p 2 +)! = Γ (p 2 ) Elmg. gauge invariance demands common width on s- and t-channel propagators in naive fixed width scheme Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

42 Examples from e + e physics: RACOON (Denner et al ) and LUSIFER (S.D.,Roth 02) σ[fb] for e + e u dµ ν µ s 189GeV 500GeV 2TeV 10TeV constant width 703.5(3) 237.4(1) 13.99(2) 0.624(3) running width 703.4(3) 238.9(1) 34.39(3) 498.8(1) complex mass 703.1(3) 237.3(1) 13.98(2) 0.624(3) σ[fb] for e + e u dµ ν µ + (separation cuts for visible : E,θ f > cut) s = 189GeV 500GeV 2TeV 10TeV constant width 224.0(4) 83.4(3) 6.98(5) 0.457(6) running width 224.6(4) 84.2(3) 19.2(1) 368(6) complex mass 223.9(4) 83.3(3) 6.98(5) 0.460(6) σ[fb] for e + e ν e ν e µ ν µ u d (phase-space cuts applied) s 500GeV 800GeV 2TeV 10TeV constant width 1.633(1) 4.105(4) 11.74(2) 26.38(6) running width 1.640(1) 4.132(4) 12.88(1) 12965(12) complex mass 1.633(1) 4.104(3) 11.73(1) 26.39(6) Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep

43 Gauge-invariant width NLO Problem much more complicated than at LO! (would fill own lectures) Complex-Mass Scheme (CMS) Denner, S.D., Roth, ieders 05 complex, but straightforward renormalization NLO everywhere in phase space loop integrals with complex masses Pole Approximation (PA) (= leading term of pole expansion) corrections decomposed into two types factorizable: corrections to on-shell production / decay non-factorizable: soft photon/gluon exchange between production / decays NLO in neighbourhood of resonances PA involves less diagrams than CMS higher multiplicities possible Effective Field Theories Beneke et al. 03, 04; Hoang,Reisser 04 involves pole expansions NLO in neighbourhood of resonances formal elegance e.g. combination with resummations For details & examples see literature... Stefan Dittmaier, Electroweak Physics at the LHC Lecture 3 Corfu Summer School, Sep