The Standar Model of Particle Physics Lecture II

Transcription

1 The Standar Model of Particle Physics Lecture II Radiative corrections, renormalization, and physical observables Laura Reina Maria Laach School, Bautzen, September 2011

2 Outline of Lectures II Radiative corrections and Renormalization: renormalization, general structure; SM case: main results. Radiative corrections and physical observables: precision tests: testing consistency and constraining unknowns lepton collider dominated (LEP) obtain best estimate of SM processes for searches of new physics hadron collider dominated (Tevatron, LHC) This last point will be expanded in Lectures III and IV

3 Systematic of renormalization, in a nutshell Simplest case: scalar (gφ 4 ) theory L(φ 0, µ φ 0,m 0,g 0 ) Calculating scattering amplitudes f φ... φ i via perturbative approach introduce divergencies beyond the tree level: ultraviolet (UV): in the p 2 region of momentum loop integrals, ex.: d 4 p p dp p 2 (p 2 m 2 log.divergence ) p infrared (IR): in both loop and real corrections, due to soft (p 2 0) or collinear (p p i 0) radiation/loop-momenta; d 4 p p 0 dp p 2 (p 2 +2p q 1 )(p 2 log.divergence +2p q 2 ) p

4 Actions taken: at a given perturbative order, regularize and extract UV and IR singularities, most common: d 4 p µ 4 d d d p dimensional regularization divergencies extracted as poles in (4 d). µ (renormalization) scale parameter associated to regularization procedure cancel UV singularities by switching from bare to renormalized parameters/fields, fixed by suitable renormalization conditions, L(φ 0, µ φ 0,m 0,g 0 ) L(φ, µ φ,m,g) where m 2 0 = m 2 +δm 2 φ 0 = Z φ φ g 0 = g +δg and m, φ and g (alternatively δm 2, Z φ = 1+δZ φ, δg) are defined by fixing the renormalized proper vertices (or vertex functions) of the theory. cancel IR singularities in the sum of virtual+real corrections (only hard radiation can be resolved).

5 UV systematics: consider the proper vertices of the theory Γ φ...φ Γ φ...φ one-particle irreducible (1PI) diagrams with n external legs. 2-point proper vertex : Γ φφ Γ φ 0φ 0 = i(p 2 m 2 0)+ iσ(p 2 ) Ô = + È Σ=self-energy=sum of 1PI (one-particle-irreducible) diagrams (all orders). Notice relation with all orders propagator: = + È + È È + = = i i p 2 m p 2 m 2 iσ(p 2 i ) 0 p 2 m 2 0 i p 2 m 2 0 +Σ(p2 ) = (Γφ 0φ 0 ) 1 + In terms of bare parameters, Γ φ 0φ 0 is UV divergent: fix δm 2 and δz φ by requiring, e.g., that Γ φφ = Z 1 φ Γφ 0φ 0 has a pole at the physical mass with unit residue, which gives: Σ(m 2 ) = δm 2 and Σ (m 2 ) = δz φ

6 4-point proper vertex: Γ φφφφ Γ φ 0φ 0 φ 0 φ 0 = ig 0 +iγ(p 1,p 2,p 3 ) = Ô + È In terms of bare parameters, Γ φ 0φ 0 φ 0 φ 0 is UV divergent: fix δg by requiring, e.g., that g corresponds to the coupling measured in a specific kinematic realization, which gives: δg = 2gδZ φ Γ(p exp 1,p exp 2,p exp 3 ) where we also use that Γ φφφφ = Z 2 φ Γφ 0φ 0 φ 0 φ 0. All other n-point proper vertices, Γ φ...φ : are obtainable using Γ φφ and Γ φφφφ as building blocks, and finite when expressed in terms of m and g. Notice: parameters are now scale-dependent, m(q 2 ), g(q 2 ). Any physical observables calculated in terms of m and g is finite and well defined, although affected by a systematic (perturbative) uncertainty.

7 Standard Model renormalization: main results The SM Lagrangian is made of renormalizable field structures, L SM = L QCD +L EW = L ferm EW +L gauge EW +L SSB EW +L Y ukawa EW where, L QCD ψ( / m)ψ, ψa/ψ, 1 4 Ga,µν G a µν L ferm EW ψ L ( /)ψ L, ψ L V/ψ L L gauge EW 1 4 Fa,µν F a µν, 1 4 Bµν B µν L SSB EW µ φ µ φ, µ 2 φ 2, φ 4 L Y ukawa EW ψ L Hψ R The systematic procedure outlined in these lectures will apply with extra constraints imposed by the presence of a partially spontaneously broken gauge symmetry.

8 The set of fundamental parameters of the SM Lagrangian is: g s,0, g 0, g 0, µ 0, λ 0, y f,0, V ij 0 here taken as bare parameters. Thanks to relations induced by the symmetries of the theory, e.g. e = gsinθ W = g cosθ W e = gg g2 +g 2 M W = gv 2, M Z = v g 2 +g 2 2 M W M Z = g g2 +g 2 = e g = cosθ W we can trade them for other or better sets of input parameters, for example: g s,0, e 0, M W,0, M Z,0, M H,0, m f,0, V ij 0 and switch to the corresponding set of renormalized or physical parameters upon imposing suitable renormalization conditions. Relations like M W /M Z = cosθ W will automatically be finite but corrections depend on input parameters (e.g. m t, M H ) natural relations. Need to specify renormalization scheme and use consistency.

9 Definitions and renormalization conditions QCD in the absence of a mass scale, use MS scheme or minimal subtraction scheme, i.e. subtract just pole parts of each divergent proper vertex. EW use procedure illustrated in this lecture for a scalar gφ 4 toy model on-shell subtraction scheme. mass/coupling renormalization: M 2 W,0 = M 2 W +δm 2 W,..., m f,0 = m f +δm f, V ij 0 = V ij +δv ij field renormalization: W ± 0 = Z W W ±, ( Z 0 A 0 ) = ( ZZZ )( ZZA ZAZ ZAA Z A )... where, the following renormalization conditions are traditionally adopted: δm 2 W = Re[Σ W T (M 2 W)], δz W = Re[Σ W T (M 2 W)],... and similar ones for other vector+scalar and field renormalization constants the bulk of corrections are in the self-energies!

10 Flavor sector: need to carefully account for the rotation to mass eigenstates (beyond the scope of these lectures). Finally, the QED electric charge renormalization condition is adopted: e defined as measure in the Thomson limit (k 0 scattering of photons, on-shell electrons) α(0) = e2 2π Once expressed in terms of the renormalized parameters and fields, any physical observable is finite and can be calculated at the proper perturbative order in QCD+EW and compared with experimental results. Electroweak precision fits

11 Electroweak precision fits An incredible amount of measurements of electroweak observables have been collected over the past many decades: Sp ps at CERN ( ) GeV p p collider: discovery of W and Z bosons! LEP I at CERN ( ) at energies around s = M Z : e + e Z f f. LEP II at CERN ( ) at energies around s = GeV: e + e WW 4f. SLC at SLAC ( ) at energies up to 100 GeV, polarized beams. Tevatron at Fermilab, RUN I+II ( ) TeV p p collider: top-quark discovery! Precision measurement of M W and m t. LHC at CERN, now running at TeV, will go up to 7 7 TeV: will rediscovery the SM and more!

12 Measurement of M Z and Γ Z at LEP I At the Z pole e + e f f (f e) dominated by Z exchange: dσ f Z dω = 9 4 sγ ee Γ f f/m 2 Z (s M 2 Z )2 +s 2 Γ 2 Z /M2 Z [ (1+cos 2 θ)(1 P e A e )+2cosθA f ( P e +A e ) ] σ had [nb] ALEPH DELPHI L3 OPAL measurements, error bars increased by factor 10 σ from fit QED unfolded Γ Z σ 0 P e : polarization of electron beam Γ ee,γ f f: partial widths for Z e + e f f A f : L-R coupling constant asymmetry A f = (gf L )2 (g f R )2 f (g f L )2 +(g = 2gV gf A f R )2 (g f V )2 +(g f A )2 M Z E cm [GeV] Scanning at the Z pick and fitting to σ f Z yields measurements of M Z, Γ Z and σ had = 12πΓ ee Γ had /(M 2 Z Γ2 Z ).

13 Measurement of M W at LEP II Events per 1 GeV/c ALEPH Preliminary µνqq selection Data (Luminosity = 237 pb -1 ) MC (m W = GeV/c 2 ) Non-WW background s = 191.6,195.5, 199.5,201.6 GeV σ WW (pb) LEP PRELIMINARY 11/07/ YFSWW/RacoonWW no ZWW vertex (Gentle) only ν e exchange (Gentle) M W (GeV/c 2 ) s (GeV) From W invariant-mass reconstruction: t-channel ν-exchange, s-channel γ- and Z-exchange test of 3V-gauge coupling. From rise of the W W cross section near threshold (statistically limited) but till another test of non-abelian interactions.

14 Measurement of m t at the Tevatron New measurement from LHC should improve precision.

15 Strategy Having a variety of measurement for different observables, test the SM by comparing theory and experiment. Pick a set of input parameters, typical choice: α s, α, G F, M Z, M H, m t, m f,... Compute theoretical predictions, including radiative corrections, in a given renormalization scheme treating the best measured parameters as inputs (α, G F, fermion masses except m t and m c ), i.e. as fixed parameters. Perform a best fit to the electroweak data, defined by a χ 2 test χ 2 (α,g F,...) = i (Ôexp i O th i (α,g F,...)) 2 ( Ôexp I ) 2 This results in a best fit of the non-fixed or floating parameters. Compare best-fit values to measurements if available (ex.: M W, m t, α s, not M H!) For the best-fit values of all input parameters, quote the SM theoretical prediction for each observable and compare with the experimental measurements. Tensions may signal new physics...

16 Fine structure constant, α Measured at low energies, α e2 (0) 4π = e 2 0 4π(1 α(0)) = (50) then evolved to M Z : α e (M Z ) = α 1 α(m Z ) α QED and (2-loop) QCD contributions ( α (5) had ). Uncertainties from: h.o. perturbative and nonperturbative corrections, light quark masses (mainly m c ), lack of data below 1.8 GeV, slight disagreement in extraction of α (5) had. Fermi constant, G F (or G µ ) From muon lifetime: ( ) τµ 1 = G2 Fm 5 µ m 2 ( 192π F e 1+ 3 ) [ ( ) 3 m 2 µ 5 m2 µmw π2 α(mµ ) 2 π +O ( α 2)] Measure: G F = (1) 10 5 GeV 2 uncertainty: almost all from residual experimental error.

17 Example of best fit of floating parameters W-Boson Mass [GeV] Top-Quark Mass [GeV] TEVATRON ± LEP ± Average ± χ 2 /DoF: 0.9 / 1 NuTeV ± LEP1/SLD ± LEP1/SLD/m t ± CDF ± 1.00 D ± 1.4 Average ± 0.90 χ 2 /DoF: 6.1 / 10 LEP1/SLD LEP1/SLD/m W /Γ W m W [GeV] July m t [GeV] July 2011 All following plots from: The LHC Electroweak Working Group (

18 Summary of various pulls : theory vs experiment Measurement Fit O meas O fit /σ meas α (5) had (m Z ) ± m Z [GeV] ± Γ Z [GeV] ± σ 0 had [nb] ± R l ± A 0,l fb ± A l (P τ ) ± R b ± R c ± A 0,b fb ± A 0,c fb ± A b ± A c ± A l (SLD) ± sin 2 θ lept eff (Q fb ) ± m W [GeV] ± Γ W [GeV] ± m t [GeV] ± July

19 SM Higgs-boson mass range: constrained by EW precision fits Increasing precision will continue to provide an invaluable tool to test the consistency of the SM and its extensions July 2011 LEP2 and Tevatron LEP1 and SLD 68% CL m W = ±0.023 GeV m t = 173.2±0.90 GeV m W [GeV] m H [GeV] α m t [GeV] M H = GeV M H < 161(185) GeV plus exclusion limits (95% c.l.): M H > GeV (LEP) M H GeV (Tevatron) focus is now on exclusion limits and discovery!

20 Disentangling m t M H and M W M H correlations 200 July 2011 High Q 2 except m t 68% CL 80.5 July 2011 High Q 2 except m W /Γ W 68% CL m t [GeV] 180 m t (Tevatron) m W [GeV] 80.4 m W (LEP2, Tevatron) 160 Excluded m H [GeV] 80.3 Excluded m H [GeV] M W /(GeV) = ln ( (5) ) α h ( MH 100 GeV ) [ ( m t 178 GeV ln 2 ( MH 100 GeV ) ) 2 1 ] A. Ferroglia, G. Ossola, M. Passera, A. Sirlin, PRD 65 (2002) W. Marciano, hep-ph/