Electroweak unification

Transcription

1 Electroweak unification Electroweak unification τ Experimental facts τ SU(2) L U(1) Y gauge theory τ Charged current interaction τ Neutral current interaction τ Gauge self-interactions III/1

2 Experimental facts III/2

3 SU(2) L U(1) Y L 0 < iξ φ j j λ λ ξ j σ < Pauli matrices ξ ξ ξ 1 1 ξ exp{ exp{i i σ θ σ θ / 2} 2}exp{ exp{ iy α 1 / 2} 2} ξ 1 θ θ ξ exp{, σ, iσ 2} / 2}exp{ exp{,, iyα α2} / 2} ξ ξ exp{ iy α / 2} ξ ξ exp{, iy ξ exp{ α 2} ξ ξ exp{, α 2} ξ 1 1 α / 2} ξ ξ exp{ ξ 3 αtransforms 2} ξ ξ as ξexp{, 2 α 2} ξ III/3

4 III/4 SU(2) L U(1) Y ) ( 1 k j k j ijk i i i O W g W W δ λ λ λ λ, )} ( 2 / exp{ ) ( ; ) ( 2 / ) ( x i x U x W x W σ σ λ λ θ θ θ θ ) (, ) ( x x α α < θ < θ ) ( )] ( 2) / '( ) ( [ ) ( x x B y ig x igw x D ξ ξ λ λ λ λ,, ) ( ) ( ) (1/ ) ( ) ( ) ( ) ( ) ( ') (1/ ) ( ) ( ) ( )} ( 2) / )exp{ ( ( ) ( x U x U g x U x W x U x W x g x B x B x D x y i x U x D λ λ λ λ λ λ λ λ α ξ α ξ = x,α=α(x) Wλ(x) σ 2 Wλ (x) ; U x exp{ σ 2 (x)}

5 Charged currents III/5

6 Neutral currents III/6

7 Neutral currents 3 2 v < T, 2Q sin π a < T f f j W f 3 f, T 3 f is the weak isospin III/7

8 Charged & neutral currents s π < sin π c < W π cosπ W III/8

9 Gauge self-interactions III/9

10 Gauge self-interactions e, e cot π W e 2 sin 2 π W e 2, e 2 cotπ W, e 2 cot 2 π W III/10

11 W pair production at LEP2 III/11

12 W pair production at LEP2 III/12

13 Z pair production at LEP2 NB! No e e, Z/φ ZZ vertex in the Standard Model III/13

14 Electroweak unification: summary III/14

15 Higgs mechanism Higgs mechanism τ Spontaneous symmetry breaking τ Higgs mechanism τ The Higgs boson τ Fermion masses τ Fermion mixing III/15

16 Spontaneous symmetry breaking We have been able to generate electromagnetism and the strong interactions through local gauge invariance. We would like to do the same for the weak interactions, but we immediately face a problem. Unlike the photon and the gluons, the W and Z have non-zero masses. A mass term in the Lagrangian, ½ m 2 A λ A λ, would not be gauge invariant. Suppose we forget about gauge invariance and just put it in anyway. Then theory becomes non-renormalizable, i.e. we obtain divergances that we cannot get ride off. There is a more subtle way to generate a mass for the W and Z, called spontaneous symmetry breaking. Let us first consider a real scalar field ε with an arbitrary potential. L = ½ ( λ ε) 2 V(ε) We require the L to be invariant under ε ε. if we expand V, it will have only even values of ε: Then, L = ½ ( λ ε) 2 ½ λ 2 ε 2 - ¼κ ε 4... III/16

17 Spontaneous symmetry breaking But what happens if λ 2 is allowed to be negative L = ½ ( λ ε) 2, ½ λ 2 ε 2, ¼κ ε 4... The sign of the mass term has changed giving an imaginary mass. This makes no sense. The potential looks like in the figure. The minimum is now at ε =,λ 2 /κ. Feynman diagrams represent a perturbation serie. The serie would not converge if we expand about ε = 0, since this is a local maximum. We need to expand about the global minimum, say,λ 2 /κ. ε(x λ )=,λ 2 /κ γ(x λ ), and then γ = 0 corresponds to the minimum. Then L = ½[ λ (,λ 2 /κ γ(x λ ))] 2, ½λ 2 (,λ 2 /κ γ(x λ )) 2, ¼κ(,λ 2 /κ γ(x λ )) 4 L = ½( λ γ(x λ )) 2 ¼λ 4 /κ λ 2 γ 2 (x λ ),,λ 2 κγ 3 (x λ ), ¼κγ 4 (x λ ) = constant m =,2λ 2 III/17

18 Spontaneous symmetry breaking The γ and ε fields are the same fields, but we now see that the γ has a mass of,2λ 2, and does not have reflection symmetry γ,γ. This symmetry has been spontaneously broken. A more intersting case occurs, when we consider a complex field, ε = (1/ 2)(ι 1 iι 2 ) L = ( λ ε)*( λ ε), λ 2 ε*ε, κ(ε*ε) 2 L = ½( λ ι 1 ) 2 ½( λ ι 2 ) 2, ½λ 2 (ι 12 ι 22 ), ¼κ(ι 12 ι 22 ) 2 The minimum is now a circle: ι 12 ι 22 =, λ 1 /κ We choose to expand about ι 1 =, λ 1 /κ, ι 2 = 0. ε(x λ ) = (1/ 2) [, λ 1 /κ γ(x λ ) iω(x λ )] We get: L = ½( λ γ(x λ )) 2 ½( λ ω(x λ )) 2 ¼λ 4 /κ λ 1 γ 1 (x λ ),λ 1 κ [γ 3 (x λ ) γ(x λ )ω 2 (x λ )], ¼κ[γ 4 (x λ ) ω 4 (x λ ) 2γ 2 (x λ )ω 2 (x λ )] The γ field gets a mass,2λ 1, as before, but the ω field is massless. This always happens whenever a continuous symmetry is spontaneously broken. The massless particle is called a Goldstone boson. III/18

19 Spontaneous symmetry breaking Let us consider a complex scalar field which is invariant under a local gauge transformation ε exp(i (x λ )) ε λ D λ λ,iqa λ A λ A λ (1/q) λ (x λ ) L = ( λ iqa λ )ε*( λ, iqa λ )ε, λ 2 ε*ε, κ(ε*ε) 2, ¼F λµ F λµ As before ε = (1/ 2)(ι 1 iι 2 ), we choose ι 1 (x λ )=,λ 1 /κ γ(x λ ); ι 2 (x λ ) = ω(x λ ) and we get L = ½( λ γ) 2 ½( λ ω) 2 ¼λ 3 /κ λ 2 γ 2, ¼F λµ F λµ,½q 2 (,λ 1 /κ) 2 A λ A λ q(,λ 1 /κ)( λ ω)a λ (+ interaction terms in γ, ω and A.) Note: (a) The A field has aquired a mass q(,λ 1 /κ) (b) There is still a Goldstone boson, ω. (c) There is a strange term q(,λ 1 /κ)( λ ω)a λ ω A This indicates that we haven t chosen the correct fields. III/19

20 Higgs mechanism We can use our freedom to choose a gauge so that ε(x λ ) (1/ 2)[,λ 1 /κ h(x λ )] exp{iπ(x λ )/(,λ 1 /κ)} (h(x λ ) and π(x λ ) are real) A λ A λ (1/q(,λ 1 /κ)) λ π(x λ ) L = ½( λ h(x λ )) 2 λ 2 h 2 (x λ ) ¼λ 3 /κ, ¼F λµ F λµ, ½q 2 (,λ 1 /κ) A λ A λ (,λ 1 /κ)q 2 h A λ A λ ½q 2 h 2 A λ A λ,,λ 1 /κ h 3, ¼κh 4 A h A h etc.. A h A We have 1 massive scalar h and 1 massive vector A. Note that we started with two scalars (γ and ω) or (h and π) and one massless vector A. This is 4 spin degrees of freedom, (2x1 1x2). We end with one scalar and one massive vector. This is still 4 spin degrees of freedom (1x1 1x3). The gauge field has eaten up the Goldstone boson. This is the Higgs mechanism, and h is called the Higgs boson. III/20

21 Higgs mechanism µ vacuum expectation value III/21

22 Higgs mechanism III/22

23 Bosonic dof s III/23

24 The Higgs boson should exist!!! LHC: discovery of a Higgs-like particle !!! (direct) (indirect) m H = GeV (average ATLAS & CMS) Up to now all measurements of the newly discovered particle consistent with the Standard Model Higgs boson! Tevatron direct search: M H < 158 GeV or M H > 175 GeV III/24

25 Higgs-weak boson couplings III/25

26 Fermion masses III/26

27 Fermion generations Phenomenology 2016 III/27

28 Fermion generations III/28

29 Fermion mixing III/29

30 Quark mixing III/30

31 Quark mixing CKM = Cabibbo-Kobayashi-Maskawa As observed for K 0, D 0 and B 0 mesons: III/31

32 Electroweak precision measurements Electroweak precision measurement τ LEP and its experiments τ Electroweak radiative corrections τ Z precision measurements τ Global electroweak fit III/32

33 The LEP collider ALEPH OPAL DELPHI L3 An electron,positron collider at CERN, European laboratory for particle physics, in a tunnel situated m underground. Four experiments focused on studying the Standard Model (electroweak measurements ~ W & Z, QCD measurement & Higgs searches) LEP1, s m Z ( ) LEP2, s = GeV ( ) III/33

34 Electron positron annihilation III/34

35 Z physics at LEP III/35

36 Luminosity determination III/36

37 Standard Model couplings a f v f III/37

38 Radiative corrections QED ISR QCD FSR III/38

39 Radiative corrections = ( )= III/39

40 Initial state radiation III/40

41 Differential cross section III/41

42 Total cross section & partial widths = III/42

43 Cross sections III/43

44 Lepton forward-backward asymmetries III/44

45 Neutral current lepton universality III/45

46 Global electroweak fit Inputs to Measurements of W, Z, quark, properties + excluding coupling m H constants from LEP/CERN, SLD/SLAC (e + e - Z 0 ) Tevatron/Fermilab, LHC/CERN & lower se + e - colliders. Results & plots: J. Haller et al. (Gfitter group), arxiv: remarkable consistency, the Standard Model works well!! III/46

47 Global electroweak fit including m H III/47

48 Global electroweak fit consistency between m W & m t from radiative corrections and direct measurements. a measure of Standard Model success: m t from radiative corrections vs m t direct measurement. III/48

49 Global electroweak fit 131 prefered Prefered mminimum H mass of fit: m H 114 = 90 GeV 21 (=,18 lower GeV limit from direct search) direct measurement (ATLAS+CMS) m H = GeV Overall SM fit: β 2 = 18.6 for n dof = 15 P-value = 0.23 NB! minimum of electroweak fit strongly correlated with m W & m (March 2018) t SM still very reasonable!! III/49

50 Standard Model parameters III/50