Weak Interactions: towards the Standard Model of Physics

Transcription

1 Weak Interactions: towards the Standard Model of Physics Weak interactions From β-decay to Neutral currents Weak interactions: are very different world CP-violation: power of logics and audacity Some experimental consequences K - oscillations Setting the stage for the Standard Model 1

2 The Fundamental Forces Presently: we see four forces in nature Force Strength* Theory Mediator Strong 1 Chromodynamics (QCD) Gluon Electromagnetic 1 - Electrodynamics (QED) Photon Weak 1-13 (Flavordynamics) W, Z Glashow-Weinberg-Salam Gravitational 1-4 General Theory of Relativity Graviton Strength: to be taken as an indication; depends on force, energy, distance (and maybe on time!)

3 Effective range of Fundamental Forces Strong interaction Potential 1/r Electrom. interaction Weak interaction M(gluon)=; but range limited by Gluon-gluon selfinteraction Range limited due to heavy IVB Potential 1/r 3

4 Quantum Electrodynamics (QED) Oldest, simplest, most successful of the dynamical theories All electromagnetic processes are ultimately reducible to the process represented by the diagram below - Convention for interpreting the diagram time flows horizontally the charged particle enters emits (or absorbs) a photon the charged particle exists charged particle could be - charged lepton - a quark 4

5 Quantum Chromodynamics (QCD) o o o in QCD: color plays the role of charge fundamental vertex q q + gluon g e e + γ Analogous to Bottom diagram p. 66 o o bound state of qq force between two quarks is mediated by the exchange of gluons 5

6 Quantum Chromo Dynamics QCD: Fundamental differences to QED QED: one type of charge, i.e. one number (+, -); photon is neutral QCD: three kinds of color: red, green, blue - Fundamental process q q + : color of quark (not its flavor may change e.g.: blue up-quark red up-quark color is conserved gluon carries away the difference gluons are bicolored with one positive and negative unit (e.g.: one unit of blueness and minus one unit of redness) 3 x 3 = 9 possibilities experimentally only 8 different gluons observed; ninth gluon would be color singlet and therefore observable not observed, i.e. does not exist g 6

7 β Decay: experimental facts Fundamental reaction n p + e + v n ( u, d, d) p ( u, u, d) + e + In the quark picture: - This particular weak interaction process changes quark flavor However, the following weak interaction process preserves quark flavor ν µ p ν + + µ - Neutrino-proton scattering: hint of two different types of processes (mediators) Observation of kinematics of β-decay led to neutrino hypothesis - Is the neutrino really massless? The happy end to a 4-year struggle will be discussed later Observation of (maximal) parity violation in β-decay (first observed in the decay of 6 Co) p v 7

8 Neutron decay through weak interactions β-decay of neutron : n p + e - + anti ν e apart from strong interaction contamination from the spectator quarks is identical in structure to µ decay, and clearly related to π- decay 8

9 Neutron decay through weak interactions Measurements of neutron decay parameters and their correlations are a very active area of research These measurements probe Standard Model Parameters, like quark coupling; they are also sensitive to Physics Beyond Standard Model, e.g. Supersymmetry or other new particles (Axions) 9

10 β Decay: attempting to measure the mass of neutrinos 1

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12 Neutrinos: There is much more to it µ e + γ not observed, although charge and Lepton numbers are conserved Why? famous rule of thumb (Feynman): - whatever is not expressively forbidden, is mandatory (in physics) Absence of µ e + γ conservation of muon-ness - but µ e + ν + ν Perhaps: two kinds of neutrinos? - one associated with e ν e (L e = +1) - one associated with µ ν µ (L µ = +1) 1

13 Neutrinos: There is much more to it Answer: conservation of electron number and muon number n p + + e + ν e π µ + ν µ ; π + µ + + ν µ µ e _ + ν e + ν µ ; µ + e + + ν e + ν µ Lepton quantum number is attached to leptons and is conserved (however: see later: neutrino oscillations) 13

14 Neutral K-Meson System revisited: Oscillations K and K can be produced in strong interaction processes + - K + p K + n, K + n K + p ;.. - Kaons are produced in states of unique strangeness - K (S= -1) is antiparticle of K (S=+1) Neutral kaons are unstable and decay through weak interaction - Experimentally observed: two different decay times Possible, if these states consist of a superposition of two distinct states with different lifetimes - a short-lived one, originally labeled K 1 - a long-lived one, originally labeled K K and K are eigenstates of the Strong Hamiltonian, but not eigenstates of the Weak Hamiltonian K 1 and K are eigenstates of the weak Hamiltonian Eigenstates of weak Hamiltonian are different from those of st.h. Weak interaction does not conserve strangeness: K + -> π +π +

15 Neutral K-System K s are pseudoscalars P K = K P K = K C K = K C K = K CP K = K CP K = K The normalized eigenstates of CP are (assuming CP is good symmetry) ( )( ) ( ) ( ) 1 1 K K K = K K + 1 = K CP K1 = K1 CP K = K If CP is conserved in weak interactions - K 1 can decay only in CP = +1 state - K can decay only in CP = -1 state Kaons typically decay into (P π = - π C π = π ) π state (CP = +1) 3 π state (CP = -1) Conclusion: K 1 π, K 3π 15

16 K 1, K π decay is much faster (larger phase space, more energy available) Start with K -beam - component will decay quickly, leaving more s K 1 K ( 1 )( K K ) 1964: Cronin, Fitch and collaborators observe CP violation K - beam: by letting the K 1 component decay can produce arbitrarily pure K - beam; K is a CP=-1 state; can only decay into CP=-1 (3 pions), if CP is conserved Observation: 7 3π-decays and 45 π-decays = Long-lived K is NOT perfect eigenstate of CP, contains a small admixture of K 1 1 K 16

17 Source of inspiration: K-Meson System K ( strangeness S= +1) can turn into antiparticle K K, because both particle can decay into π + + π - (or π + π ) through (second order) weak interaction (W.I. does not conserve strangeness Feynman diagrams in modern formulation K, K as observed in the laboratory are superpositions of K 1 and K 17

18 Oscillations in the neutral K-System Inverting the eigenstates of CP ( )( ) ( ) ( ) 1 1 K = K K K = K + one obtains K o 1 K = ( 1 )( ) ( ) ( ) K + K, K = 1 K K Can be seen as : the strong interaction eigenstates, K and K are superpositions of the weak eigenstates K 1 and K At time of production the K and K states correspond to the superposition of K 1 and K states; K 1 component will decay much faster, getting enriched in the component; BUT is an equal admixture of K and K 1 An initially pure K or K state will evolve into a mixture of these two states! K K K Strangeness oscillation K 1 18

19 Oscillations in the neutral K-System: time evolution The states and are stable with respect to strong interactions, but decay weakly; decays phenomenologically described with an effective Hamiltonian in the time-dependent Schrödinger equation Hamiltonian involves the masses and lifetimes of the two states and After a few pages of matrix manipulation and algebra one obtains for the time-dependent probabilities of the and states: 19 K K ) ( ) ( t H t t i eff ψ ψ = K L K S K K + + = + t m e e e t K P t t t L S L S cos 4 1 ), ( ) 1 1 ( 1 τ τ τ τ S L t t t m m m t m e e e t K P L S L S = + = + cos 4 ), ( ) 1 1 ( 1 τ τ τ τ κ

20 Oscillations in the neutral K-System: time evolution If the states K and K had the same masses, the beam intensities would exhibit the sum of two exponential decays, corresponding to the two characteristic decay times Experimentally observed are oscillations between the two states, implying a finite mass difference, for which one obtains m = m L m S 3.5 * 1-1 MeV/ c (recall: mass K = MeV/ c ) This mass difference can be converted into an upper limit on the K K mass difference of < 1-18 m K implying a very stringent on the validity of CPT Similar physics also observed in the B B system - Studied at Beauty Factories - Babar at SLAC (terminated) - Belle and follow-experiment Belle II HEPHY is leading institute in Belle and Belle II

21 One further new effect: Strangeness changing weak interactions This description is incomplete Strangeness changing weak interactions s quark up quark: involves conversion of a strange quark into an up quark; examples : Λ p + + π ; Ω Λ + K 1

22 Strangeness changing weak interactions Cabibbo Mechanism In 1963: only u, d, s quarks were known N. Cabbibo (1963) - Starting from the fact that Strangeness is NOT conserved in W.I. - Concept in analogy to the K K system Concept: a redefinition of the quark doublet eigenstates of the Weak Hamiltonian u u d d with the newly defined state d = d cosθ c + s sinθ c This state does not have a unique strangeness quantum number Angle θ c parameterizes the level of mixing; data consistent with θ c =13.15 With this new definition (almost) all experimental data are reproduced

23 From Cabibbo to GIM Cabbibo (1963): d -> u + W - vertex carries a factor cosθ c ; s -> u + W - carries factor sinθ c Cabbibo s suggestion was very successful, explaining a number of decay rates, except: - Within this description K would decay into μ + μ - with a rate far larger than experimentally measured Solution proposed in 197 by Glashow, Iliopoulos, Maiani (GIM); - In the GIM mechanism a fourth quark, charm was introduced, four years before first experimental indications - Approximate cancellation between the two diagrams resulting in a rate consistent with experiment 3

24 Charged weak interactions of quarks Physics interpretation of the Cabbibo- GIM suggestion - For the weak interactions the proper weak states of the quarks are not the d and the s quark but linear combinations d = d cosθ c + s sinθ c ; s = - d sinθ c + s cosθ c Written in matrix form u d s ' ' c cosϑ c = sinϑc sinϑ The W s couple to ; in exactly the same way as they couple ' ' d s to the lepton pairs At the time this seemed a bit extravagant to introduce a new quark to solve the rate problem; Even more extravagant was the generalization by Kobayashi, Maskawa to explain CP-violation within the Standard Model by introducing a third quark generation in BUT: their intuition was correct, as we know today! 4 c cosϑ c d s

25 Conclusions so far Eigenstates of the Weak Hamiltonian are different to those of the Strong Hamiltonian Eigenstates of the Weak Hamiltonian do not have unique Strangeness Weak Interactions may change quark flavor (e.g. Neutron decay) Transposed to the quark level (anticipating): - Quark Doublets of the Weak Hamiltonian are mixed states of these doublets u ' d c s ' and also b t 5

26 From Fermi to Glashow-Salam-Weinberg: from Weak to Electroweak Interactions Fermi theory of weak interactions: - Fermions involved have point (contact) interaction; no mediating particle required - Excellent approximation for low-energy phenomena, such as β-decay - Failure at high energy: ν e + e - ν e + e - has total cross section σ(ν e e) = G s/π; G Fermi coupling constant; s (total cms energy) cross section depends on coupling G and energy ~ s (phase space) Problem: - This cross section exceeds Unitarity limit : total scattered intensity is larger than incident intensity for cms momenta >~ 3 GeV/c - Fundamentally: the Fermi contact interaction has divergent higher order terms - Introducing the concept of an exchange interaction with a heavy, mass M, particle (gauge boson) being exchanged spreads the interaction over range ~ 1/M and saturates cross section at G M / π

27 Unifying Interactions Maxwell (1865): electricity and magnetism described by one single theory involving a vector field, the electromagnetic field, interacting between electric charges ad currents Maxwell equations involve arbitrary constants: e.g. velocity of light c and electric charge e, to be determined experimentally Einstein: dream of unified field theory of all four intersections Unifying electromagnetic and weak interactions into electroweak : - Fundamental idea: symmetry between electromagnetic and weak interactions manifest at large momentum transfers (q >>1 4 GeV ) - At low energies: symmetry is broken First ideas for unified treatment of electromagnetic and weak interactions : - Schwinger (1957): first conceptual work - Glashow (1961)... 7

28 From Fermi to Glashow-Salam-Weinberg: from Weak to Electroweak Interactions Remember Yukawa s problem with the strong interactions - Estimate of the mass of the exchanges particle (the pion) based on the range of the strong force: approximately size of nucleon Problem for weak interactions: no equivalent way to estimate range of weak interaction (no weakly bound states) - Exchange particle was given the (not very imaginative) name: Intermediate Vector Boson (IVB) - Estimate of mass initially based on educated guesses, i.e. limits obtained by experiments searching for existence of IVBs Emergence of the electroweak theory, formulated by Glashow, Weinberg and Salam (end of 196 s) - Three IVB s: W +, W -, Z Note the presence of charged and neutral mediators! Neutral mediator is a novel feature of this theory 8

29 Electroweak unification Glashow (1961): aim to unify electromagnetic and weak interactions in a single theoretical description, building on work by Schwinger Several difficulties had to be overcome... - Enormous difference in strength of em and weak interactions - Glashow (and others) recognized that this could be solved, if the weak interactions were mediated by extremely massive particle - Glashow did not know how to introduce a heavy mediator: Quoting Glashow: Is is a stumbling block, we must overlook - Difference in electromagnetic (only vector coupling γ μ ) and weak vertex factors γ μ (1- γ 5 ) - γ μ represents the vector coupling, as in QED; γ μ γ 5 axial coupling which violates parity - This problem was formally treated with the spinor concept, as sketched on following transparencies 9

30 From weak to electroweak interaction Quarks and leptons are arranged in doublets - Lepton generations - Quark generations (weak interaction generation) Reminiscent of spin, isospin structure for hadrons, represented by SU() transformation properties - Strong interaction isospin used to classify hadrons; good symmetry, if electromagnetic interaction is ignored ( turned off ) Weak isospin classifies leptons and quarks - Up and down states of leptons and quarks are the same, if electromagnetic interaction is turned off Weak Hypercharge : Y= (Q-J 3 ); Q = J 3 + Y/; 3 ' ' ' ; ; b t s c d u τ ν µ ν ν τ µ ; ; e e

31 Weak Hypercharge and left-handed and right-handed particles Example : (ν, e - ) doublet: Y(ν) = (-1/) = -1; Y(e - ) = (-1+1/) = -1 (u,d) quark doublet : Y(u) = (/3-1/) = 1/3; Y(d) = (-1/3+1/) = 1/3 weak hypercharge quantum number is the same for both members of a doublet In the Standard Model only left-handed leptons and quarks have doublet structure - Isospin doublets Right-handed quarks and right-handed charged leptons are - Isospin I= singlets Weak interactions have doublet (SU L () symmetry) and singlet (U Y (1)) structure Electromagnetic interaction is U(1) invariant -> can be regarded as a particular combination of weak isospin and weak hypercharge 31

32 Weak currents This symmetry is reflected in the structure of the Weak Lagrangian The process is described by the current Similarly, is described by the current Written more compactly (we want to exhibit spinor SU L () symmetry..) with the corresponding currents and a weak hypercharge current The same framework applies to all three lepton doublets and weak interaction quark doublets 3 + W e e ν L L e j µ µ γ =ν + + W e e ν L e L j ν γ µ µ = + L e L e = ν χ ; 1 ; 1 3 τ τ τ L L j χ τ γ χ µ µ ± ± = L L L L R R em Y e e e e j j j ν γ ν γ γ µ µ µ µ µ µ = = 3

33 Weak Interactions mediated by heavy exchange particle Basic concept, using framework of gauge theories gauge potentials quantization gauge particles, carrier of the corresponding force - Mediators of weak interactions are heavy particles (in analogy to photon exchange in QED) - In today s framework: W +, W -, Z with masses at ~ 9 GeV/c ( Intermediate Vector Bosons, IVB) Coupling to the constituents (leptons and quarks) analogous to QED, but with modifications corresponding to exchange of the IVB s In analogy to QED: need to consider the coupling and the propagator - Propagator for photons in QED: i gµν / q - QED vertex factor: ig γ µ e g e = 4πα γ μ vector coupling - Propagator in Weak Int., exchange of mass M i( gµν qµ qν / M c ) /( q M c ) - WI vertex factor: ig w γ µ 5 (1 γ ) g w = 4παw ; g w is weak coupling - one important addition: γ μ γ 5 33 axial vector coupling; parity violating!

34 Electroweak Theory Developed by S. Weinberg and A. Salam, applying new ideas about gauge theories originally developed by S. Glashow. All three received Nobel Prize in 1979 Weinberg was actually studying Strong Interactions, until he realized that his concept corresponded to Weak Interactions! Theory has 4 mediators of electroweak force ( gauge bosons ) - Photon γ (m ) - three heavy IVB s required to make theory consistent EW Theory predicts - M Z = 9 GeV/c M W = M Z cos θ W ; - θ W = 8.75 ; M W = 8 GeV/c ; - sinθ w is parameter in theory which must be be measured experimentally Actual measured values M W ± = ±.5 GeV M Z = ±.1GeV 34

35 Weak Interactions The newly developed electroweak theory predicts two kinds of weak interactions: charged and neutral Neutral weak interactions (actually discovered much later than charged WI) - fundamental vertex - f (fermion) can be any lepton or any quark - Z (one of the four bosons responsible for the electroweak interaction) mediates a new kind of weak force, resulting in processes such as Example: neutrino-electron scattering ν µ + e ν µ + e 35

36 Neutral Weak Interactions Middle p. 73 neutrino proton scattering ν µ p ν + + µ p two spectator quarks go along, bound by Strong Interaction to d quark Note: any process mediated by photon could also be mediated by Z example: electron-electron scattering very minute corrections to Coulomb law observed in e + + e - -> μ + + μ - 36

37 First Observation of Weak Neutral Current Interaction (CERN 1973) Muon antineutrino electron scattering producing energetic electron and antineutrino First event: beginning of several months of intensive further searches and a violent scientific controversy with an US Collaboration which originally reported no evidence for this type of interactions Joke: the US physicists had identified Alternating neutral currents The confirmation of neutral current reactions provided a major additional contribution to the emergent idea of electroweak theory Unified physics theory confirmed a finding of historic importance (New 37 York Times, 1973)

38 Charged Weak Interactions Electromagnetic, strong and neutral weak interactions: - Quark or lepton not changed: same quark (lepton) in same out - Strong interaction only changes color of quark, not its flavor New effect in charged weak interactions - Charged weak interactions are the only interactions that change the flavor true decay of a particle Fundamental charged lepton vertex o negative lepton converts into corresponding neutrino with emission of W - decay of muon Neutrino-muon scattering (difficult to study ) µ + ν e + e ν µ µ e + ν µ + ν e 38

39 Example: Muon decay Calculations proceed analogously to QED calculations - Feynman rules are the same as for QED, apart from the modifications in vertex factor and propagator Muon decay: µ e + ν µ + ν e with the matrix element (compare the QED calculation for e + µ e + µ) gw µ 5 M = u p (1 γ ) u p 8( M W c) [ ] [ ( ) ( )] 5 u( p ) γ (1 γ ) u( p) 3 γ 1 4 µ Decay rate Γ(Fermi s Golden Rule ) - Γ ~ M * Phase factor - With the result for total decay rate - Note that only the ratio g W /M W enters mµ gw Γ = M W 4 m µ c 1h(8π ) 3 39

40 Charged weak interactions of quarks Leptonic weak vertices connect members of the same generation (e - ν e emitting W -, μ - μ - emitting Z ): electron never goes to muon, etc. Quarks: fundamental charged quark vertex quark (-⅓, d, s, b) quark (⅔, u, c, t) Surprise: flavor is NOT conserved in charged weak interactions! (far end of the W-line can couple to lepton ( semileptonic process) or to other quarks (purely hadronic interaction) d +ν e u + e (not observed in Nature because of quark confinement) π e + ν e 4

41 Charged weak interactions of quarks: an interplay between theory and experiment Coupling of the leptons to W+ and W- to takes place strictly within one generation: ν e ν µ ν τ ; ; e µ τ example: e - ->ν e + W - ; μ - -> ν μ +W - ; τ - -> ν τ + W - are possible, but e - -> ν μ +W - is not possible Coupling of the quarks to W+ and W- is more complicated, as was realized through a close interplay between experiment and theory - Weak interactions do not strictly respect always the separate generations - N.b. : in neutron decay d -> u + W - (quark generation respected) - However: in Λ p + e + ν e s -> u + W - ; this transition occurs; otherwise, the lightest strange particle (K - ) and lightest B-meson would be stable. Strangeness-changing weak interactions needed an explanation: If they would not exist, our world would be very different 41

42 Ultima ratio: power of logic 1973: Kobayashi and Maskawa Structure of Weak charged current was established GIM mechanism successful: mixing of quarks good strategy Kobayashi, Maskawa : aim to treat CP violation in Standard Model: - Discover that treatment of CP violation within the SM framework requires a complex number in the rotation matrix; non-trivial only in a 3*3 matrix 3 generation of quarks and leptons: implemented by Kobayashi and Maskawa in 1973 before the discovery of c-, b-, t-quark! Requires 3x3 Matrix, which can be parameterized with three parameters, which are obtained from measurments - one parameter is a phase, which expresses CP violation in the Standard Model These measurements are a very active research area CP violation Nobel Prize 8 for Kobayashi and Maskawa 4

43 Quark Mixing in the Weak Interaction Sector Mixing of the mass eigenstates is expressed with the matrix V CKM ' d Vud Vus Vub d ' s = Vcd Vcs Vcb s ' b Vtd Vts Vtb b iδ c1c13 s1c13 s13 e iδ iδ VCKM = s1c3 c1 s3s13e c1c3 s1s3s13e s3c13 iδ iδ s1s3 c1c3s13e c1 s3 s1c3s13e c3c13 c ij cos θ ij; s ij...sin θ ij ; if θ 3 = θ 13 = Cabibbo-GIM picture Elements of this matrix reflect the coupling of the W boson to all the quark pairs: e.g. W tb ; cb ; cs ; etc. V ud :coupling of u to d: d u + W - Transition rate q i q j is proportional to [V ij ] Dominant transitions in W-decay are to members of the same doublet 43 diagonal matrix elements close to unity; off-diagonal small

44 CKM Matrix Weak eigenstates d, s, b are linear combinations of the mass eigenstates d, s, b CKM Matrix V CKM = 44

45 Quark Mixing: Cabibbo-Kobayashi-Maskawa matrix Mixing of the mass eigenstates is expressed with the matrix V CKM Alternate representation V V V ud us ub V V V cd cs cb V V V td ts tb 1 λ λ Aλ 3 (ρ iη) = λ 1 λ Aλ Aλ 3 (1 ρ iη) Aλ 1 Matrix can be parametrized with three parameters (three generalized Cabibbo angles ), to be determined by experiment and one phase In the Standard Model, CP-violation can be explained by a nontrivial complex phase iη in this matrix This would not be possible with only two generations 45

46 Quark Mixing: Cabibbo-Kobayashi-Maskawa matrix Matrix desribes a mixture of states whose total number does not change; so the matrix has to be unitary: V ud V us V ub V V + = V + V = 1 * V * * cd V td V ud V us V ub * V * * cs V ts V cd V cs V cb * V * * cb V = tb V td V ts V tb this yields the relation V * ub V ud + V * cb V cd + V * tb V td = which can be graphically represented in the form of one of six socalled unitarity triangles 46

47 The unitarity triangle Im ρ + iη α V tb * V td V cb * V cd γ V ub * V ud 1 β Re normalized to 1

48 The unitarity triangle: measurements Measurements have confirmed with high precision the unitarity of the triangle: the red ellipse is the 1 σ error contour; Next steps: increase the precision by factor ten to search for significant deviations: physics beyond Standard Model should be reflected also in deviations from the SM description of CP violation

49 1978-1?: The Inexorable Rise of the Standard Model (SM) Today s physics world as described by the Standard Model, is made of - four interactions: (gravitation), weak, electromagnetic, strong - 1x3 quarks and 1 leptons - 4 IVB s for ew force - 8 gluons, mediating the strong force - Higgs particle (explained later) There are 61 particles - is this really the hallmark of a fundamental description? The obvious question: why 3 generations of quarks - perhaps: matter-antimatter symmetry? After the pp collider, the Discovery Machine the LEP e + e - collider was the next logical step to perform precise tests of the Standard Model and LHC to complete (?) the Standard Model This will be the story of the next lectures. Thank you. 49