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Triumph of the Symmetry Content Triumph of the Symmetry The (electro{weak) standard model and the discovery of the W §{ and Z{Bosons Hendrik van Hees Content Relativistic quantum theory and gauge symmetry ² Phenomenology of weak interactions ² The (uni¯ed) theory of electromagnetic and weak interactions ² The discovery of W § and Z ² Status quo of experimental standard{model tests ² Relativistic quantum theory #2 Problems in \¯rst quantization": ² No single particle wave function for free particles with ² energy bounded from below: E = p~2 + m2 ² § and \conserved current" with positivpe de¯nite \charge" ² No probability interpretation for single particle wave functions ) Reason: Uncertainty relation ¢x¢p 1=2 ² In principle particle can be sharply lo¸calized (small ¢x) but then ¢p > m Particles can be produced and annihilated need many particle theory! ² ) The way out: \second quantization" = \quantum ¯eld theory" ² Reinterpret \negative energy solutions" of relativ. wave equations as anti{particles ² micro{causality ² [O^ (t ; ~x ); O^ (t ; ~x )]¡ = 0 for t t < ~x ~x 1 1 1 2 2 2 j 1 ¡ 2j j 1 ¡ 2j Measurements of observables cannot influence each other if this would need faster than light travel of signals! existence of a lowest energy state (vacuum) ² Hamiltonian bounded from below Pauli 1940 Spin{statistics theorem: Particles with integer (half integer) spin must be ² bosons (fermions) Field{Quantization #3 Starting point: Hamilton's principle ² 4 S = d xL (Á; @¹Á) Z Lagrangian L : Polynomial of ¯elds Á(x) and its four{dim. gradient @ Á(x) ² ¹ Lorentz invariance L scalar ¯eld ² , Locality L depends only on one space{time point x ² , Classical Equations of motion ² @L @L ±S = 0 @¹ ) @Á ¡ @(@¹Á) Quantization #4 Hamilton formalism: Canonical ¯eld momenta ² @L ¦ = @(@tÁ) Quantization: Classical Fields Á Field operators Á^ ² ! Equal time commutators (bosons) or anti{comutators (fermions): ² Á^(t; ~x); ¦(^ t; ~y) = i±(3)(~x ~y) for bosons ¡ ¡ 8hÁ^(t; ~x); ¦(^ t; ~y)i = i±(3)(~x ~y) for fermions < + ¡ h i Example: Free scalar ¯eld: L = @ Á¤@¹Á m2Á¤Á ² : ¹ ¡ Equations of motion: (@2 ¢)Á(¤) = 0 (Klein{Gordon equation) ² t ¡ Quantization: Field operators in momentum basis: ² d3p~ 1 Á^(x) = [a^(p~) exp( i!t + ip~~x) + ^by(p~) exp(i!t ip~~x)] (2¼)3 2! ¡ ¡ Z a^(p~) annihilates particle with momentum p~ ² ^by(p~) creates anti{particle with momentum p~ Symmetries #5 Noether's theorem: If the action is invariant under an in¯nitesimal transformation: ² Á0(x0) = Á(x) + ±Á(x), x0 = x + ±x then there exists a current j¹ which is conserved: @ j0 + ~ ~j = @ j¹ = 0 t r ¹ Space{time symmetries: ² Symmetry Conserved quantity Translations in time Energy Translations in space Momentum Rotations Angular Momentum Quantization: Need to chose ordering of ¯eld operators ² Physical input: Vacuum should be state of 0 energy and momentum ² 3 y ^y ^ 0 d p~ !(p~) 2 a^ (p~)a^(p~) + b (p~)b(p~) 31 E = BR 6density of particles density of anti{particles7C p~ B 4 5C à ! B | {z } | {z } C B 3 y ^y ^ C B d p~ p~ 2 a^ (p~)a^(p~) + b (p~)b(p~) 3 C B C B density of particles density of anti{particles C B R 6 7 C @ 4 5 A | {z } | {z } Interacting particles: Scattering #6 Scattering cross section: dσ = number of scattered particles per solid angle per time ² d­ incomming particle flux To calculate: Transition amplitude T = f T^ i ² fi D ¯ ¯ E S(cattering){Matrix S = ± + i(2¼)4±(P ¯ P¯) f T^ i ² fi fi f ¡¯ ¯i S = f; t i; t D ¯ ¯ E ² fi h ! 1j ! 1i ¯ ¯ i; t = S^ i; t ¯ ¯ j ! 1i j ! ¡1i S can be calculated only in perturbation theory ² fi S^ is unitary Overall normalization of probability time{independent! ² ) Feynman{diagrams #7 Logics of Model Building: ² (1) Find out symmetries conservation of quantities in scattering experiments , (2) Write down Lagrangian obeying the symmetries (3) Calculate cross sections, life times and check with experiment Invention by R.P. Feynman (1948) diagram rules ² From given L one derives diagram rules for perturbation series of scattering processes ² Example: QED ² Internal lines: Propagators External lines: Initial and ¯nal states ¹ ¹ p º ¹ ¹º ² = iGγ (p) p (²¹)¤ = iGe(p) e+ in ¯nal state or e¡ in initial state = ieγ¹ e+ in initial state or e¡ in ¯nal state Compton{scattering #8 Compton{scattering: Lowest order perturbation theory (Klein{Nishina cross section) Experimental values: Hofstadter, R. and McIntyre, J.A., Phys. Rev. 76, 1269 (1949); Evans, R.D., Handbuch der Physik 34, Ed. S. Flugge,Ä Springer, Berlin (1958) Curves: Klein{Nishina cross section QED-Lagrangian #9 Four{vector potential A : massless vector ¯eld ² ¹ couples to conserved electromagnetic current ² 1 L = (@ A @ A )(@¹Aº @º A¹) j A¹ ¡4 ¹ º ¡ º ¹ ¡ ¡ ¹ F¹º Equations of motion Maxwell equations ² ) | {z } ¹ ½ ¹ © (j ) = ; (A ) = ; E~ = @tA~ ©; B~ = A~ Ã~j! ÃA~! ¡ ¡ r r £ invariant under gauge transformations ² A0 = A + @  ©0 = © + @ Â; A~0 = A~  ¹ ¹ ¹ , t ¡ r Electrons and positrons: Dirac particles current j¹ = eùγ¹Ã ² ) ¡ 1 L = F F ¹º + ù(i@ + eA )γ¹Ã ¡4 ¹º ¹ ¹ Gauge invariance #10 Lagrangian invariant under local transformations ² Ã0(x) = exp[ ieÂ(x)]Ã(x); ù0(x) = exp[+ieÂ(x)]ù(x); A0 = A + @ Â(x) ¡ ¹ ¹ ¹ Vector ¯eld makes global phase{invariance of à local! ² Yang and Mills 1956 Physics invariant under local changes conventions (gauges) for ² \charge spaces" For each local gauge invariance one vector ¯eld = \gauge ¯eld" ² ) Veltmann and 't Hooft 1971 (Nobelpreis 1999): Gauge theories renormalizable, i.e., have ² unitary S{matrix and a ¯nite number of coupling{constants Now: Standardmodel of elementary particle physics is a gauge theory of the gauge group ² SU(3) SU(2) U(1) strong £ weak £ electro Geometry of gauge invariance #11 Ã21 Ã12 º ±x2 Ã(x) x Ã1(x + ±x1) ¹ ±x1 Physicist at x + ±x de¯nes \iso{spin" di®erent from physicist at x ² 1 In¯nitesimal gauge transformation dependent on x necessary to compare wave functions ² at di®erent space{time points ¹ a a ¹D Ã1(x + ±x1) = Ã(x) + ±x1 [@¹Ã(x) + iA¹(x)T ]Ã(x) := Ã(x) + ±x1 ¹Ã(x) De¯nition of wave function depends on path of the \transport" from one space{time ² point to another: à (x) à (x) = ig±x¹±xº [@ Ac @ Ac gf cbaAaAc ]T cÃ(x) := ig±x¹±xº ¹º (x)Ã(x) 12 ¡ 21 1 2 º ¹ ¡ ¹ º ¡ º ¹ 1 2 F T b; T c = if cbaT c ² ¡ For non{commutative groups: F ¹º depends on coupling g! ² £ ¤ A Brief History of weak interactions #12 210 ¯ 210 1927 Ellis and 83 Bi 84 Po: Violation of energy con- Wooster servation?! 1930 Wolfgang Pauli Postulate of existence of Neutrinos (what we call º¹e nowadays) 1933 Enrico Fermi Four{fermion coupling theory for weak interactions 1953 Reines et al. First direct experimental proof for exis- tence of neutrinos 1956 Yang, Lee Solution of the \#{¿"{puzzle: \#" and \¿" are one and the same particle, namely what we call K+ nowadays; Weak interaction violates parity conservation 1957 Wu et al. Direct experimental proof of parity non{ conservation with polarized 60Co 1957 Salam, Feynman Maximal violation of parity conservation, et al. V A{structure ¡ 1962 Ledermann et al Two di®erent sorts of neutrinos, discov- ery of º¹ 1963 Cabibbo Explanation for strangeness changing weak decays, \saving" universality of weak coupling constant quark mixing ! 1973 Hasert et al Discovery of neutral currents in reactions ¡ ¡ like º¹¹ + e º¹¹ + e , Neutral currents never change!quark flavour Fermi's theory of weak interactions #13 Idea: Weak interaction involves always 4 fermions ² Analogy: Successful QED Photon couples to current, is created and absorbed in ² reactions ) Instead of elementary photon direct coupling of fermion currents ² ) L = G(pγ¹ ¹n)(eγ¹ º) int ¡ ¹ Successful description of ¯{decay data, but not complete (Gamow{Teller transitions) ² More Fermion currents needed, if one likes to stay with four{fermion couplings ) Bilinear forms of fermions: Systematics under parity transformations: scalar, pseudo{ ² scalar, vector, axial vector, tensor \¿ +" ¼+¼+¼¡, \#+" ¼+¼0; life time and mass identical, but cannot be identical if ² parity!is conserved ! Lee and Yang: Parity conservation is violated by weak interactions ² The Wu experiment: Proof of P {violation #14 Anti-neutrino: only right{handed ² helicity = ~sp=~ ( ~s p~ ) = 1 ² j jj j angular momentum conservation: ² (e) (º¹) sz = sz momentum conservation p~ = p~ ² ) e ¡ º¹ p^ = s^ ² e ¡ Co The electro{weak standard model #15 Extension of Fermi{theory: Only left{handed particles (right{handed anti{particles) in- ² teract weakly: G L = [º¹γ¹(1 γ )e][n¹γ (1 γ )p] ¡ 2 ¡ 5 ¹ ¡ 5 Big theoretical flaws: non{renormalizable and non{unitary S{matrix ² Idea: Massive gauge bosons give at low energies four{fermion interactions like in Fermi{ ² model masses for vector bosons 80 90 GeV ) ¼ ¡ Two charged heavy vector bosons: W § ² One neutral heavy vector boson W 0 but: Massterms for vector{mesons destroy gauge invariance ² Neutral current coupling depends on electric charge ² Gauge theory: SU(2) gauge group: weak isospin (left{handed particles only!) ² Explanation: Neutral W 0 mixes with another gauge ¯eld B which couples to \weak hyper ² charge" Mass eigenstates are the massive Z 0 and the photon A ) Correct gauge group: SU (2) U (1): ² W £ Y four gauge bosons spontaneously broken to Uem(1) Three massive vector bosons (charged W { and neutral Z{boson), one massless (photon) ² The Higgs{mechanism and masses #16 Remaining problems: Howto preserve gauge invariance and make W {bosons massive? ² Solution: Higgs (1961), Glashow, Salam, and Weinberg ( 1967) ² Spontaneous breakdown of gauge{symmetry ¼ Potential symmetric un- ² der rotations in Á1{Á2{ plane Stable ground state not V ² symmetric degenera- Á0 ) Á2 tion of ground state Á = exp[iÂ~(x)T~]Á0(x)) Á1 ² Gauge{transformations local Can gauge the \polar" degrees of freedom away Â(x) = 0 ² ) Three SU (2){bosons become massive ² W Photon stays massless ² 1979 Nobel prize for Glashow, Salam, and Weinberg ² The discovery of the W { and Z{bosons #17 1983: Discovery of the W { and Z{bosons by C.
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