Higgs Effective Field Theory. Giampiero Passarino. Dipartimento di Fisica Teorica, Università di Torino, Italy INFN, Sezione di Torino, Italy

Transcription

1 iggs Effective Field Theory Giampiero Passarino Dipartimento di Fisica Teorica, Università di Torino, Italy INFN, Sezione di Torino, Italy LL 014, Weimar, 7 April May 014

2 dσ off = µ r dσ peak Combined limit PROLOGUE r = Γ Γ SM assume µ = 1 measure r peak, exp resolution / SM width 3 GeV /4 MeV Combined observed expected values r = / SM < 4. 95% CL p-value = 0.0 r = / SM = BINGO! equivalent to: < % CL = MeV 15 R. Covarelli

3 -1 [ pb] dσ dm V V M V V MW WW σ/σ SM ZZ dying line-shape g g g V rising decay s = µ i µ γ CPS required 8 TeV Ý ÑÔ ÖÓ The big 8TeV Peak at Z mass due to singly resonant diagrams. Interference is an important effect. Destructive at large mass, as expected. With the standard model width, $, challenging to see enhancement/deficit due to iggs channel. Events/bin CMS preliminary Data CMS cuts CMS PAS IG x 30 s = 8 TeV, L = 19.7 fb Keith Ellis,CERN, 9 December, 013 gg+vv ZZ Γ = 5 Γ, µ = 1 SM gg+vv ZZ SM qq ZZ Z+X m 4l GeV dynamic QCD scales MZ Mt Mt 1000 M V V [ GeV]

4 OFF SELL I We dene an off-shell production cross-section for all channels as follows: σ prop ij all = 1 π σ s Γ tot ij s s s When the cross-section ij refers to an off-shell iggs boson the choice of the QCD scales should be made according to the virtuality and not to a fixed value. Therefore, for the PDFs and σ ij +X one should select µ µ F = µ R = z s/4 z z s being the invariant mass of the detectable final state.

5 We dene an off-shell production cross-section for all channels as follows: σ prop ij all = 1 s π σij s s Γ tot s When the cross-section ij refers to an off-shell iggs boson the choice of the QCD scales should be made according to the virtuality and not to a fixed value. Therefore, for the PDFs and σij +X one should select µ µ F = µ R = z s/4 z z s being the invariant mass of the detectable final state.

6 OFF SELL II Let us consider the case of a light iggs boson; here, the common belief was that the product of on-shell production cross-section say in gluon-gluon fusion and branching ratios reproduces the correct result to great accuracy. The expectation is based on the well-known result Γ M = 1 s M + Γ M = π M Γ δ ON s M + PV OFF 1 s M where PV denotes the principal value understood as a distribution. Furthermore s is the iggs virtuality and M and Γ should be understood as M = µ and Γ = γ and not as the corresponding on-shell values. In more simple terms, the first term puts you on-shell and the second one gives you the off-shell tail is the iggs propagator, there is no space for anything else in QFT e.g. Breit-Wigner distributions.

7 Let us consider the case of a light iggs boson; here, the common belief was reproduces the correct result to great accuracy. The expectation is based on the well-known result Γ M where PV denotes the principal value understood as a distribution. Furthermore s is the iggs virtuality and M and Γ should be understood as M = µ and Γ = γ and not as the corresponding on-shell values. In more simple terms, that the product of on-shell production cross-section say in gluon-gluon fusion and branching ratios We dene an off-shell production cross-section for all channels as follows: 1 = = s M π δ s M 1 + Γ M + PV M Γ s M the first term puts you on-shell and the second one gives you the off-shell tail σ prop ij all = 1 s π σij s s Γ tot s When the cross-section ij refers to an off-shell iggs boson the choice of the QCD scales should be made according to the virtuality and not to a fixed value. Therefore, for the PDFs and σij +X one should select µ µ F = µ R = z s/4 z z s being the invariant mass of the detectable final state. is the iggs propagator, there is no space for anything else in QFT e.g. Breit-Wigner distributions.

8 OFF SELL III A short istory of beyond ZWA don t try fixing something that is already broken in the first place ➀ There is an enhanced iggs tail Kauer - Passarino arxiv: : away from the narrow peak the propagator and the off-shell width behave like 1 Γ VV MVV, G F M MVV VV M VV µ ➁ Introduce the notion of -degenerate solutions for the iggs couplings to SM particles Dixon - Li arxiv: , Caola - MelnikovarXiv: ➂ Observe that the enhanced tail is obviously γ -independent and that this could be exploited to constrain the iggs width model-independently ➃ Use a matrix element method MEM to construct a kinematic discriminant to sharpen the constraint Campbell, Ellis and Williams arxiv:

9 A short istory of beyond ZWA don t try fixing something that is already broken in the first ➂ ➃ MelnikovarXiv: Observe that the enhanced tail is obviously γ -independent and that this could be exploited to constrain the iggs width model-independently Use a matrix element method MEM to construct a kinematic discriminant to sharpen the constraint Campbell, Ellis and Williams arxiv: reproduces the correct result to great accuracy. The expectation is based on the well-known result Γ M where PV denotes the principal value understood as a distribution. Furthermore s is the iggs virtuality and M and Γ should be understood as M = µ and Γ = γ and not as the corresponding on-shell values. In more simple terms, place ➀ There is an enhanced iggs tail Kauer - Passarino arxiv: : away from the narrow peak the propagator and the off-shell width behave like 1 Γ VV MVV, GF M MVV VV µ MVV ➁ Introduce the notion of -degenerate solutions for the iggs couplings to SM particles Dixon - Li arxiv: , Caola - We dene an off-shell production cross-section for all channels as follows: σ prop ij all = 1 s π σij s s Γ tot s When the cross-section ij refers to an off-shell iggs boson the choice of the QCD scales should be made according to the virtuality and not to a fixed value. Therefore, for the PDFs and σij +X one should select µ µ F = µ R = z s/4 z z s being the invariant mass of the detectable final state. Let us consider the case of a light iggs boson; here, the common belief was that the product of on-shell production cross-section say in gluon-gluon fusion and branching ratios 1 = = s M π δ s M 1 + Γ M + PV M Γ s M the first term puts you on-shell and the second one gives you the off-shell tail is the iggs propagator, there is no space for anything else in QFT e.g. Breit-Wigner distributions.

10 OFF SELL IV pp gg e + e γ 10 Me γ > 0.1 Me + e γ Me + γ > 0.1 Me + e γ 10 3 Me + e > 0.1 Me+ e γ σ [ fb] Madgraph VBF Signal Phantom VBF Bkg Phantom VBF Bkg + Sig Me + e γ [ GeV]

11 reproduces the correct result to great accuracy. The expectation is based on the well-known result Γ M where PV denotes the principal value understood as a distribution. Furthermore s is the iggs virtuality and M and Γ should be understood as M = µ and Γ = γ and not as the corresponding on-shell values. In more simple terms, ➂ ➃ MelnikovarXiv: Observe that the enhanced tail is obviously γ -independent and that this could be exploited to constrain the iggs width model-independently Use a matrix element method MEM to construct a kinematic discriminant to sharpen the constraint Campbell, Ellis and Williams arxiv: CMS preliminary s -1 = 8 TeV, L = 19.7 fb Events/bin Data gg+vv ZZ Γ = 5 Γ, µ = 1 SM gg+vv ZZ SM qq ZZ Z+X We dene an off-shell production cross-section for all channels as follows: σ prop ij all = 1 s π σij s s Γ tot s When the cross-section ij refers to an off-shell iggs boson the choice of the QCD scales should be made according to the virtuality and not to a fixed value. Therefore, for the PDFs and σij +X one should select µ µ F = µ R = z s/4 z z s being the invariant mass of the detectable final state m 4l GeV Let us consider the case of a light iggs boson; here, the common belief was that the product of on-shell production cross-section say in gluon-gluon fusion and branching ratios = 1 s M + Γ M = π M Γ δ s M + PV the first term puts you on-shell and the second one gives you the off-shell tail 1 s M is the iggs propagator, there is no space for anything else in QFT e.g. Breit-Wigner distributions. A short istory of beyond ZWA don t try fixing something that is already broken in the first place ➀ There is an enhanced iggs tail Kauer - Passarino arxiv: : away from the narrow peak the propagator and the off-shell width behave like 1 Γ VV MVV, GF M MVV VV µ MVV ➁ Introduce the notion of -degenerate solutions for the iggs couplings to SM particles Dixon - Li arxiv: , Caola -

12 σ i f = σ BR = σprod i Γ f γ σ i f g i gf CMS preliminary s γ -1 = 8 TeV, L = 19.7 fb g i,f = ξ g SM γ = ξ 4 γ SM i,f a consistent BSM interpretation? On the whole, we have a constraint in the multidimensional κ-space κ g = κ g κ t,κ b κ κ = κ κ j, j On-shell -degeneracy arxiv: , , The generalization is an -degeneracy Events/bin Data gg+vv ZZ Γ = 5 Γ, µ = 1 SM gg+vv ZZ SM qq ZZ Z+X g ii i κ jj j g i g f = γ m 4l GeV Only on the assumption of degeneracy one can Simplified version prove that off-shell effects measure γ Γ gg Γ SM gg µ = κ t Γ tt ggµ +κ b Γbb ggµ +κ t κ b Γ tb ggµ Γ tt ggµ +Γ bb ggµ +Γ tb ggµ original κ-language a combination of on-shell effects measuring g i g f /γ and off-shell effects measuring g i g f gives information on γ without prejudices

13 MOU PO EFT The only limit to our realization of tomorrow will be our doubts of today Memo: Skip meetings Main Theorem: EFT is a realization of κ -language Definition: κ -language is BSM MI approach Chapter IV Renorm. dim. reg. QFT role of Λ top? - down Strategy: ow to interpret κx? 1 measure κ Γgg Γ SM gg m = κ t Γ tt ggm+κb Γbb ggm+κtκb Γ tb ggm Γ tt ggm+γ bb ggm+γ tb ggm find Oi κx epistemological stop, true ESM believers stop here Corollary: κ -language requires insertion of O d operators in SM loops Chapter II Nature of O d Chapter III Ontology of EFT 3 find {LBSM} LESM = LSM + n>4 that produces Oi Nn i=1 ai n Od=n Λn 4 i is there a QFT behind degeneracy?

14 annotated DIAGRAMMATICA Og 3 g6 t, b W/φ/X ± W/φ W/φ Figure 3: Example of one-loop SM diagrams with O-insertions, contributing to the amplitude for γγ ց κ f, A NF mix under ren. with Og g6 Og 3 g6 ց κ W, A NF ր A NF OΦW W Figure 4: Example of one-loop O-diagrams, contributing to the amplitude for γγ

15 Caveat Note that for Λ 5 TeV we have 1/ GFΛ g /4π, No hierarchy assumed g 4+k 4+k 4+k = 1/ G F Λ k N defines LO A = n=n n l=0 k=1 gn g 4+k 4+k 4+k l A nlk i.e. the contributions of d = 6 operators are loop effects. For higher scales, loop contributions tend to be more important PTG PTG - operators versus LG - operators, cf. Einhorn, Wudka,... It can be argued that at LO the basis operator should be chosen from among the PTG operators take O 6 LG, contract two lines, is ren of some O4 O 4 a SM vertex with O 6 PTG required... same order 1/Λ expansion power-counting S PTG: T - generated in at least one extension of SM LG low-energy analytic structure

16 PROPOSITION: There are two ways of formulating EFT a mass-dependent schemes or Wilsonian EFT b mass-independent schemes or Continuum EFT CEFT only a is conceptually consistent with the image of an EFT as a low-energy approximation to a high-energy theory however, inclusion of NLO corrections is only meaningful in b since we cannot regularize with a cutoff and NLO requires regularization There is an additional problem, CEFT requires evolving our theory to lower scales until we get below the heavy-mass scale where we use L = L SM + dl, dl encoding matching corrections at the boundary. Therefore, CEFT does not integrate out heavy degrees of freedom but removes them compensating for by an appropriate matching calculation Not quite the same as it is usually discussed no theory approaing the boundary from above... cf. low-energy SM, weak effects on g etc.

17 dimφ = d/ 1 dimo d = N φ dimφ + N der For d 3 there is a finite number of relevant + marginal operators For d 1 there is a finite number of irrelevant operators Sounds good for finite dependence on high-energy theory Footnotes Annotations This assumes that high-energy theory is weakly coupled Dimensional arguments work for LO EFT In NLO EFT scaling may break down, implying appeal to a particular renormalization scheme Ren. group should only be applied to EFTs that are nearly massless Decoupling theorem fails for CEFT, but, arguably this does not prevent them from supporting a well dened seme, but decoupling must be inserted in the form of mating calculations whi we don't have... Match Feynman diagrams EFT with corresponding 1lightPI PI diagrams high-energy theory and discover that Taylor-expanding is not always a good idea

18 aving said that... no space left for annotations MOU PO EFT FP-sector: handle with care Oops!... 4f O 4f O needed for b b M W Renormalization [ g = g ren 1+ g ren 16π etc. γγ not finite [ = MW ren 1+ 1 dz g +g 6 dz 6 1 ] ε gren 16π dz MW +g 6 dz 6 1 M W ε Wilson coefficients W i W i Z wc ij Make finite all Green s functions = j Zij wc Wj ren = δ ij + g ren wc 1 dz 16π ij ε Schemes: remember β QED g 1 ] β QED in large m e -limit Don t forget background ε = ε γ lnπ + ln µ R

19 Appendix C. Dimension-Six Basis Operators for the SM. Einhorn, Wudka O is PTG O is LG Grzadkowski, Iskrzynski, Misiak, Rosiek QG Q G QW Q W X 3 LG ϕ 6 and ϕ 4 D PTG ψ ϕ 3 PTG f ABC G Aν µ GBρ ν GCµ ρ Qϕ ϕ ϕ 3 Qeϕ ϕ ϕ lperϕ ABC Aν f G µ GBρ ν GCµ ρ Qϕ ϕ ϕ ϕ ϕ Quϕ ϕ ϕ qpur ϕ ε IJK Wµ Iν Wν Jρ Wρ Kµ QϕD ϕ D µ ϕ ϕ Dµϕ Qdϕ ϕ ϕ qpdrϕ Iν εijk W µ WJρ ν WKµ ρ X ϕ LG ψ Xϕ LG ψ ϕ D PTG QϕG ϕ ϕg A µν GAµν QeW lpσ µν erτ I ϕwµν I Q 1 ϕl Q ϕ G ϕ ϕ G A µν GAµν QeB lpσ µν erϕbµν Q 3 ϕl ϕ i Dµϕ lpγ µ lr ϕ i D I µ ϕ lpτ I γ µ lr QϕW ϕ ϕw I µνw Iµν QuG qpσ µν T A ur ϕg A µν Qϕe ϕ i Dµϕēpγ µ er Q ϕ W ϕ ϕ W µν I WIµν QuW qpσ µν urτ I ϕw µν I Q 1 ϕq QϕB ϕ ϕbµνb µν QuB qpσ µν ur ϕbµν Q 3 ϕq ϕ i Dµϕ qpγ µ qr ϕ i D I µ ϕ qpτ I γ µ qr Q ϕ B ϕ ϕ BµνB µν QdG qpσ µν T A drϕg A µν Qϕu ϕ i Dµϕūpγ µ ur QϕWB ϕ τ I ϕw I µν Bµν QdW qpσ µν drτ I ϕw I µν Qϕd ϕ i Dµϕ dpγ µ dr Q ϕ WB ϕ τ I ϕ W I µν Bµν QdB qpσ µν drϕbµν Qϕud i ϕ Dµϕūpγ µ dr Table C.1: Dimension-six operators other than the four-fermion ones. These tables are taken from [5], by permission of the authors. Effective Lagrangians cannot be blithely used without acknowledging implications of their choice ex: non gauge-invariant, intended to be used in U-gauge ex: WW is virtual W + something else, depending on the operator basis

20 Tadpoles β γ W/Z g γ Z f µ -decay

21 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R γ W/Z g γ Z f µ -decay

22 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R Z φ = 1 + g δz 4 16π φ + g 6 δz 6 φ γ W/Z g γ Z f µ -decay

23 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R Z φ = 1 + g δz 4 16π φ + g 6 δz 6 φ Self-energies UV O 4,O 6 -finite γ W/Z g γ Z f µ -decay

24 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R Z φ = 1 + g δz 4 16π φ + g 6 δz 6 φ Self-energies UV O 4,O 6 -finite µ-decay γ W/Z g γ Z f µ -decay

25 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R Z φ = 1 + g δz 4 16π φ + g 6 δz 6 φ Self-energies UV O 4,O 6 -finite µ-decay g g R γ W/Z g γ Z f µ -decay

26 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R Z φ = 1 + g δz 4 16π φ + g 6 δz 6 φ Self-energies UV O 4,O 6 -finite µ-decay g g R Finite ren. γ W/Z g γ Z f µ -decay

27 Tadpoles β Φ = Z 1/ Φ R etc. φ Φ R Z φ = 1 + g δz 4 16π φ + g 6 δz 6 φ Self-energies UV O 4,O 6 -finite µ-decay g g R Finite ren. [ MR = ] M W 1 + g R 16π Re ΣWW δz M etc Propagators finite and µ R -independent µ R γ W/Z g γ f Z µ -decay

28 EXAMPLE UV -propagator 1 = Z s + Z m M 1 π 4 i Σ Z = 1 + g R 16π δz 4 + g 6 δz 6 [ δz 4 1 = δz 6 = 3 mb + m ] t 3 M etc 1 ε 8 16 cθ 5 s θ 14sθ 14s θ c θ c θ [ 5 cθ m b + m t M 1 m M ] a φ

29 EXAMPLE finite ren. m = M [ 1 + g ] R 16π dm 4 + g 6 dm 6 M 16 dm4 = M W cθ M t 3 MW M 4M t B 0 M ; Mt,Mt 3 M b 3 MW M 4M b B 0 M ; M b,m b 9 M 4 18 MW B 0 M ; M,M 1 M 4 64 MW 4M 1M W B 0 M ; M W,M W 1 M 4 18 MW 4 M cθ + 1 M ] W cθ 4 B 0 M ; M Z,M Z

30 v = iggs virtuality requires Z, Z g, Z g, Z gs

31 requires Z, Z g, Z g, Z gs It is O 4 -finite but not O 6 -finite v = iggs virtuality

32 requires Z, Z g, Z g, Z gs It is O 4 -finite but not O 6 -finite involves a φd, a φ, a tφ, a bφ, a φw, a φg, a tg, a bg, v = iggs virtuality a tg = W 1 a bg = W a φg = W 3 a bφ a φd a ΦW a φ = W 4 a tφ 1 4 a φd + a ΦW + a φ = W 5

33 requires Z, Z g, Z g, Z gs It is O 4 -finite but not O 6 -finite involves a φd, a φ, a tφ, a bφ, a φw, a φg, a tg, a bg, v = iggs virtuality a tg = W 1 a bg = W a φg = W 3 a bφ a φd a ΦW a φ = W 4 a tφ 1 4 a φd + a ΦW + a φ = W 5 requires extra renormalization W i = j Zij mix Wj R µ R Zij mix = δ ij + gg S 16π δzmix ij δz mix 31 = 1 M tb M W 1 ε

34 Define building blocks 8π i g S M W M q A LO q = 4Mq v C 0 v,0,0; M q,m q,m q 3π i g S M W M q A nf q = 8M 4 q C 0 v,0,0; M q,m q,m q + v [1 B 0 v ; M q,m q ] 4M q

35 Define process dependent κ-factors [ 1 M b κ b = 1 + g 6 W R M 1 ] W4 R W [ 1 M t κ t = 1 + g 6 W1 R M 1 ] W5 R W

36 Define process dependent κ-factors [ 1 M b κ b = 1 + g 6 W R M 1 ] W4 R W [ 1 M t κ t = 1 + g 6 W1 R M 1 ] W5 R W Obtain the 4+6 amplitude A 4+6 = g [ + g 6 g q=b,t κ q A LO q + i g 6 g S W R 1 Anf t + W R Anf b M W3 R M W ]

37 Define process dependent κ-factors [ 1 M b κ b = 1 + g 6 W R M 1 ] W4 R W [ 1 M t κ t = 1 + g 6 W1 R M 1 ] W5 R W Obtain the 4+6 amplitude A 4+6 = g [ + g 6 g Derive true relation q=b,t κ q A LO q + i g 6 g S W R 1 Anf t + W R Anf b M W3 R M W ] A 4+6 gg = g g v A 4 gg

38 Define process dependent κ-factors [ 1 M b κ b = 1 + g 6 W R M 1 ] W4 R W [ 1 M t κ t = 1 + g 6 W1 R M 1 ] W5 R W Obtain the 4+6 amplitude A 4+6 = g [ + g 6 g Derive true relation q=b,t κ q A LO q + i g 6 g S W R 1 Anf t + W R Anf b M W3 R M W ] A 4+6 gg = g g v A 4 gg Effective running scaling g g i is not a κ constant parameter unless O 6 = 0 and κ b = κ t

39 Non-factorizable not included κt = Γgg κb = 1.5 κt = virtuality [ GeV]

40 Non-factorizable not included Γγγ κw = κt = κt = 1.5 κw = virtuality [ GeV]

41 SCALE dependence no subtraction point

42 SCALE dependence no subtraction point Consider γγ Zij mix = δ ij + g [ R 16π δzij mix 1 ε + ij ln M ] µ R W 1 = a γγ = s θ c θ a ΦWB + cθ a φb + sθ a φw MW 11 = 1 [ 8sθ sθ cθ MW + 4sθ cθ 5 4 M ]

43 SCALE dependence no subtraction point Consider γγ Zij mix = δ ij + g [ R 16π δzij mix 1 ε + ij ln M ] µ R W 1 = a γγ = s θ c θ a ΦWB + cθ a φb + sθ a φw MW 11 = 1 [ 8sθ sθ cθ MW + 4sθ cθ 5 4 M ] etc

44 Symphony No. 8 in B minor S p What do we lose without matching? p1 toy model: S dark iggs field L = L SM 1 µs µ S 1 M S S + µ S Φ ΦS [ Ieff DR = 3 4 g M 1 M W Λ s 1 3M ε ] s i 0 ln + finite part µ R I full = 3 g M µ S [1 1 M W MS 4 s 1 1 MS ] s ln s i 0 s + O MS MS MS 4 full starts at Oµ S /M S eff starts at Os/Λ large mass expansion of full follows from Mellin-Barnes expansion and not from Taylor expansion

45 Background? Consider uu ZZ

46 Background? Consider uu ZZ The following Wilson coefficients appear: W 1 = a γγ = s θ c θ a ΦWB + c θ a φb + s θ a φw W = a ZZ = s θ c θ a ΦWB + sθ a φb + cθ a φw W 3 = a γz = s θ c θ a φw a φb + cθ sθ a ΦWB W 4 = a φd W 5 = a 3 φq + a1 φq a φu W 6 = a 3 φq + a1 φq + a φu

47 Background? Consider uu ZZ The following Wilson coefficients appear: W 1 = a γγ = s θ c θ a ΦWB + c θ a φb + s θ a φw W = a ZZ = s θ c θ a ΦWB + sθ a φb + cθ a φw W 3 = a γz = s θ c θ a φw a φb + cθ sθ a ΦWB W 4 = a φd W 5 = a 3 φq W 6 = a 3 φq + a1 φq + a1 φq a φu + a φu Define A LO = M4 Z t + M4 Z u t u u t 4 M Z s tu

48 Obtain the result uu ZZ spin A 4+6 [ = g 4 A LO F LO s θ + g 6 ] 6 F i s θ W i i=1

49 Obtain the result uu ZZ spin A 4+6 [ = g 4 A LO F LO s θ + g 6 ] 6 F i s θ W i i=1 Background changes!

50 Obtain the result uu ZZ spin A 4+6 [ = g 4 A LO F LO s θ + g 6 ] 6 F i s θ W i i=1 Background changes! Note that F LO 0.57 F F 3.31 F F 4.46 F 4.46 F

51 CONCLUSIONS FUTURE [from arxiv: ] Moriod EW L =L 4 + n> 4 Nn i i =1 an Λ n i 4 O d =n i e ing rov guag p n m -la si NLO κ -language is NOT a simple scaling ThiNLO κ wit

52 Thanks for your attention

53 Backup Slides

54 Large Scale φj, χ Lχ,φ + Lφ renormalization group µ = M particle mass MATCING Lφ + δlφ renormalization group Low Energy φj Figure 4: The general form of a matching calculation. terms. In this region, the physics is described by a set fields, χ, describing the heaviest particles, of mass M, and a set of light particle fields, φ, describing all the lighter particles. The Lagrangian has the form Lχ,φ + Lφ, 3.15 where Lφ contains all the terms that depend only on the light fields, and Lχ, φ is everything else. You then evolve the theory down to lower scales. As long as no particle masses are encountered, this evolution is described by the renormalization group. owever, when µ goes below the mass, M, of the heavy particles, you should change the effective theory to a new theory without the heavy particles. In the process, the parameters of the theory change, and new, nonrenormalizable interactions may be introduced. Thus the Lagrangian of the effective theory below M has the form Lφ + δlφ,

55 Increasing COMPLEXITY γγ ➀ 3 LO amplitudes A LO A LO t,a LO b,alo W, 3κ 3κ-factors ➁ 6 Wilson coefficients & non-factorizable amplitudes

56 Increasing COMPLEXITY γγ ➀ 3 LO amplitudes A LO A LO t,a LO b,alo W, 3κ 3κ-factors ➁ 6 Wilson coefficients & non-factorizable amplitudes ZZ ➀ 1 LO amplitude ➁ 6 NLO amplitudes, 6κ-factors δ µν A NLO i,d + pµ pν 1 A NLO i,p i=t,b,b i=t,b,b ➁ 16 Wilson coefficients & non-factorizable amplitudes

57 Increasing COMPLEXITY γγ ➀ 3 LO amplitudes A LO A LO t,a LO b,alo W, 3κ 3κ-factors ➁ 6 Wilson coefficients & non-factorizable amplitudes ZZ etc. ➀ 1 LO amplitude ➁ 6 NLO amplitudes, 6κ-factors δ µν A NLO i,d + pµ pν 1 A NLO i,p i=t,b,b i=t,b,b ➁ 16 Wilson coefficients & non-factorizable amplitudes

58 g finite renormalization g exp = G [ 1 + G 16π dg 4 + g 6 dg 6] G = 4 G F M W dg 4,6 from µ -decay

59 g finite renormalization g exp = G [ 1 + G 16π dg 4 + g 6 dg 6] G = 4 G F M W dg 4,6 from µ -decay Involving Σ WW 0 easy

60 g finite renormalization g exp = G [ 1 + G 16π dg 4 + g 6 dg 6] G = 4 G F M W dg 4,6 from µ -decay Involving Σ WW 0 easy and vertices & boxes not easy with O 6 O 6 O 6 -insertions

61 g exp wave function renormalization 1 1 δz 16π δz 4 = B f M t MW B0 M f ; Mt,Mt + 3 M b MW B0 M f ; M b,m b M ; M W,M W 1/ 1 cθ B0 M f ; M Z,M Z 3 M M 4M t t MW B p 0 M ; Mt,Mt + 3 M M 4M b b MW B p 0 M ; M b,m b 1 M 4 4 MW 4M + 1M W B p 0 M ; M W,M W + 1 M 4 8 MW 4 M cθ + 1 M Z cθ B p 0 M ; M Z,M Z + 9 M 4 8 MW B p 0 M ; M,M etc.

62 Fine points on PTG versus LG O 6 operators Proposition: if we assume that the high-energy theory is ➀ weakly-coupled and ➁ renormalizable

63 Fine points on PTG versus LG O 6 operators Proposition: if we assume that the high-energy theory is ➀ weakly-coupled and ➁ renormalizable it follows that the PTG/LG classification of arxiv: used here is correct.

64 Fine points on PTG versus LG O 6 operators Proposition: if we assume that the high-energy theory is ➀ weakly-coupled and ➁ renormalizable it follows that the PTG/LG classification of arxiv: used here is correct. If we do not assume the above but work always in some EFT context i.e.. also the next high-energy theory is EFT, possibly involving some strongly interacting theory then classification changes, see Eqs. A1-A of arxiv: v

65 STU: combination of Wilson coefficients W 1 = a γγ = s θ c θ a ΦWB + c θ a φb + s θ a φw W = a ZZ = s θ c θ a ΦWB + sθ a φb + cθ a φw W 3 = a γz = s θ c θ a φw a φb + cθ sθ a ΦWB W 4 = a φd W 6 = a bwb W 8 = a twb W 10 = a bφ W 5 = a φ W 7 = a bbw W 9 = a tbw W 11 = a tφ a qw = s θ a qwb + c θ a qbw a qb = s θ a qbw c θ a qwb

66 W 1 = a φba W 14 = a φta W 13 = a φbv W 15 = a φtv a φbv = a 3 φq a φb a 1 φq a φba = a 3 φq + a φb a 1 φq a φtv = a 3 φq a φt a 1 φq a φta = a 3 φq + a φt a 1 φq

67 STU: building blocks γ γ Σ γγ s = Π γγ ss Π γγ s = g sθ 16π Π4 γγ s + g g 6 16 π 11 Π 6 γγ i sw i i=1 Π 4 γγ 0 = 3a0 f 1 [ MW + 1 4a f ] 0 Mb 16a f 9 0 Mt

68 Π 6 γ γ = 8sθ + s4 θ a0 M f W M MW a0 M f s θ [ s θ Π 6 γ γ 0 = { s θ c θ 9 Π 6 γ γ 3 0 = { s θ c θ c θ Π 6 γ γ 4 0 = c θ 1 cθ a0 M f Z a0 f Mt 1 1 s θ a0 M f b ] 34 s θ [ a0 f ] Mt + 4a f 0 M b a0 f c θ [ 4a0 f ]} Mt + a f 0 M b { 3 af 0 Π 6 γ γ 6 0 = M ] b MW s θ [a0 M f b s θ a0 M f W M W } M W + 1 [ 16a f ]} 18 0 Mt + 4a f 0 M b 1 Π 6 γ γ 8 c 0 = 4 θ M [ s b θ s θ MW a0 f ] Mt + 1 Π 6 γ γ 9 0 = 8s θ c M [ t θ MW a0 f ] Mt + 1

69 STU: building blocks Z γ Σ Zγ s = Π Zγ ss Π Zγ s = Π 4 Zγ 0 = 1 6 g 16π s θ c θ Π 4 Zγ s + g g 6 16 π sθ 3 4sθ a0 f MW 9 a0 f M b Π 6 Zγ i sw i g 6 W 3 i=1 3 8sθ 1 sθ a0 f Mt

70 Π 6 Zγ 1 0 = s [ θ 1 c θ c 4 θ Π 6 Zγ 0 = s θ c θ [ s θ af 0 Π 6 Zγ 3 0 = cθ 4 a0 M f W c 4 9 θ a0 f Mt a0 M f b 1 ] 33 1s 9 θ + 70s4 θ 3 cθ a0 M f W 3 9 s θ af 0 Mt M b ] 8 35c 9 θ s θ + 140s4 θ 1 M 4 MW a0 M f 1 4 Π 6 Zγ 4 0 = 1 [ 1 s θ c θ Π 6 Zγ 6 0 = 1 4c θ + 1 9s 3 θ + 6s4 θ a0 M f W M Z 3 4s 9 θ + 16s4 θ a0 f 1 Mt 3 1s 9 θ + 8s4 θ a0 M f b 1 cθ a0 f 19 56sθ + 36s4 θ a0 M f W s 18 θ + 16s4 θ a0 f Mt 3 1s θ + 8s4 θ M b M W a0 M f b 1 ] 1 + 4s 4 7 θ [ ] 1 4cθ a0 M f b 1 Π 6 Zγ 7 0 = M [ ] b MW sθ a0 M f b + 1 Π 6 Zγ 8 0 = 1 4c θ M t M W [ 5 34cθ + 3c4 θ a0 f ] Mt 1 Π 6 Zγ 9 0 = 1 s M [ t θ MW 7 16sθ a0 f ] Mt + 1

71 Π 6 Zγ 13 0 = s θ 3 c θ Π 6 Zγ 15 0 = 4 s θ 3 c θ M b M W M t M W [ ] a0 M f b + 1 [ a0 f ] Mt + 1

72 STU: building blocks Z Z Σ ZZ s = S ZZ + Π ZZ s + Os S ZZ = Π ZZ = g 16π c θ g 16π c θ S 4 ZZ + g g 6 16 π Π 4 ZZ + g g 6 16 π 15 S 6 ZZ i W i i=1 15 Π 6 ZZ i W i i=1

73 S 4 ZZ = Π 4 ZZ = MZ 1 3 M + 1 M 4 1 MZ B0 M f Z ; M,M Z 1 [ ] 7 16c 18 θ 64c θ s θ Mt cθ 3c θ s θ MZ B0 M f Z ; Mt,Mt 1 [ ] 5 + 4c 18 θ 8c θ s θ MZ 17 8cθ + 16c θ s θ Mb B0 M f Z ; M b,m b 1 [ ] 1 0c 1 θ + 36c θ s θ MZ sθ MZ c6 θ B0 M f Z ; M W,M W 1 M 4 1 Z M M Z + M4 B p 0 0; M,M Z + M 3 Z + MZ 4 M 3 M 8 M + 1 M 4 8 Z MZ 1 M 4 Z M 6 Z 1 5 M 1 18 MZ 4 M 1 M 3 M a0 f Z B p 0 0; M,MZ cθ + 16c θ s θ Mb Bp 0 M Z 4 + c 7 θ 5c θ s θ MZ 7 16cθ 64c θ s θ Mt B p 0 0; Mt,M t 0; M b,m b M 4 4 Z M M Z + M4 B0 s 0; M,MZ MZ M a M 0 f Z M Z c 7 θ 5c θ s θ a f 0 M ] [5M Z c θ 4 5 3sθ MZ c6 θ B p 0 0; M W,M W MZ 1 + M M Z a f 0 M

74

75 γ bare propagator 1 γ = s g Σ γγ s = The life and death of µ R 16π Σ γγs D 4 + g 6 D 6 1 ε + x X {X } = {s, m, m 0, m, m t, m b} L 4 i + g 6 L 6 i ln x µ R + Σ rest γγ

76 γ bare propagator 1 γ = s g Σ γγ s = The life and death of µ R 16π Σ γγs D 4 + g 6 D 6 1 ε + x X {X } = {s, m, m 0, m, m t, m b} L 4 i + g 6 L 6 i ln x µ R + Σ rest γγ γ renormalized propagator 1 γ = Z γ s g ren 16π Σ γγs = s g 16π Σren γγ s

77 The life and death of µ R Σ ren γγ s = L 4 i + g 6 L 6 i ln x x X µ R + Σ rest γγ finite renormalization Σ ren γγ s = Π ren γγ s s [ ] Π ren γγ s Π ren γγ 0 µ R = 0

78 The life and death of µ R Σ ren γγ s = L 4 i + g 6 L 6 i ln x x X µ R + Σ rest γγ finite renormalization Σ ren γγ s = Π ren γγ s s [ ] Π ren γγ s Π ren γγ 0 µ R = 0 including O 6 contribution. There is no µ R problem when a subtraction point is available. O 6

79 O 6 O 4 fieldparameter redefinition L = µ K µ K µ K K 1 λ K K 1 M 0 φ 0 M φ + φ + g a φ K Λ K 3 g a φ Λ K K K K g a φd K Λ µ K K1 = + M g + i φ 0 K = i φ

80 Requires µ = β λ g M λ = 1 4 g M M M [ 1 a φd 4a φ M ] g Λ [ 1 + a φd 4a φ + 4a φ M ] g Λ M etc. with non-trivial effects on the S-matrix