Electroweak Theory & 01.12.2005 Electroweak Theory &
Contents Glashow-Weinberg-Salam-Model Electroweak Theory &
Contents Glashow-Weinberg-Salam-Model Electroweak Theory &
Contents Glashow-Weinberg-Salam-Model Electroweak Theory &
Contents Glashow-Weinberg-Salam-Model Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Quantum Field Theory Quantum electrodynamics is a quantum field theory Quantum field theory describes interaction between particles Combines classical field theory and quantum mechanics Includes field quantization aka 2. quantization Considers explicitly creation and destruction of particles Electroweak Theory &
Field Theory Maxwell Equations Field Quantization General Field Theory Field theory is a mathematical construct describing effects invoked by force and interaction Classical field theory neglegts quantum mechanics. Forces act instantaniously (ED) Quantum field theory incorporates quantum mechanics. Exchange of gauge bosons as force mediator Abelian theory: gauge bosons do not interact among themselves (QED) Non abelian Yang-Mills-theory: gauge bosons can interact (QCD,Weak) Electroweak Theory &
Formalism of Field Theory Field Theory Maxwell Equations Field Quantization All field theories can be described by the Lagrange density formalism For a known Lagrange density of a field theory L = L(φ i, φ i,...) Variation of the action S = d n xl(φ i, φ i,...) yields Euler-Lagrange equation: L L µ = 0 i = 0, 1,... φ i µ φ i System of differential equations aka motion equation of the field theory Boundary conditions need to be fixed for physical systems Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Gauge Theory Quantum field theories describing fundamental forces are gauge theories Gauge theories are field theories depending on gauge invariance Gauge invariance is invariance of i.e. Lagrange density under gauge transformations Local, continuous symmetry transformations of inner degrees of freedom of particles Transformations have properties of a group aka gauge group Electroweak Theory &
Field Theory Maxwell Equations Field Quantization QED is a quantum field theoretical description of ED Follows from ED through quantization of Maxwell equations QED was developed in 40ies as first QFT Consistent quantum theoretical description of fields Creation and destruction of particles explicitly included L = 1 4 F µνf µν + n ψ n (iγ µ D µ m)ψ n D µ = µ + iea µ Developed by Heisenberg, Schrödinger and Pauli 1965 Nobelprize: Feynman, Schwinger and Tomonaga for Renormalization Electroweak Theory &
Introduction: Maxwell Equations Field Theory Maxwell Equations Field Quantization About 1860 Maxwell proposed the laws of electricity and magnetism in 4 short equations: curl B = j + t E div B = 0 curl E = t B div E = ρ Heavyside-Lorentz-Units: = c = ɛ 0 = µ 0 = 1 Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Relativistic Formulation of Maxwell Equations Field Strength Tensor & 4-Current Density 0 E 1 E 2 E 3 (F µν ) = E 1 0 B 3 B 2 E 2 B 3 0 B 1 (j µ ) = E 3 B 2 B 1 0 ( ρ j ) Maxwell Equations: µ F µν = j ν ɛ µνρσ ν F ρσ = 0 Electroweak Theory &
Field Theory Maxwell Equations Field Quantization 4-Potential A µ ( ) φ Combine potentials φ, A to (A µ ) = A F µν = µ A ν ν A µ implies ɛ µνρσ ν F ρσ = 0 A µ is not yet fixed. For A µ (x) A µ (x) + µ Λ(x) F µν remains unchanged This transformation is called a gauge transformation Lorentz-condition µ A µ = 0 yields A µ = j µ Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Field Operator A µ For a free photon field A µ = 0 In quantum theory this is an operator equation Therefore we can rewrite the field operator A µ as d 3 k A µ (x) = (2π) 3 2ω (eikx a µ(k) + e ikx a µ (k)) ( ) ω k = ω = k k kx = k µ x µ = ωt kx Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Lorentz-Condition as Boundary condition a, a are operators in Fock-space with a µ (k) 0 = 0 [a µ (k), a ν(k )] = g µν (2π) 3 2ωδ 3 (k k ) A µ = 0 Maxwell equations only under Lorentz-condition Lorentz-condition in operator form does not consider relativistic invariance Lorentz-condition as boundary condition for states (Gupta,Bleuler) Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Physical States Considering only the part of the states fullfilling Lorentz-condition as physical yields A ( ) µ (x) = d 3 k (2π) 3 2ω e ikx a µ (k) Asking that µ A ( ) µ (x) Phys.State = 0 Therefore Phys.State µ A µ (x) Phys.State = 0 Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Base for Creation and Destruction Operator Chose e 1, e 2 k and e 3 = k/ k orthonormal Define Operators α α 0 (k) = 1 2 (a 0 (k) e 3 a (k)) α 1 (k) = e 1 a (k) α 2 (k) = e 2 a (k) α 3 (k) = 1 2 (a 0 (k) + e 3 a (k)) Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Commutator Relations [α 0 (k), α 0 (k )] = [α 3 (k), α 3 (k )] = 0 [α 0 (k), α 3 (k )] = [α 3 (k), α 0 (k )] = (2π) 3 2ωδ 3 (k k ) [α 1 (k), α 1 (k )] = [α 2 (k), α 2 (k )] = (2π) 3 2ωδ 3 (k k ) All other commutators vanish Electroweak Theory &
Field Theory Maxwell Equations Field Quantization State vectors Boundary condition α 0 (k) Phys.State = 0 State vector α 1 (k 1) α 1 (k 2)..α 2 (k 1)..α 0 (k 1).. 0 Phys. state vector is a linear combination of state vectors Phys.State Phys.State 0 1 2 if ( 1 2 )( 1 2 ) = 0 yielding a Hilbert-space with positiv definite metric Electroweak Theory &
Wick-Theorem Field Theory Maxwell Equations Field Quantization Normal ordered product : A 1 A 2 A 3 : is product over field operators A i with creation operators all to the left For fermionic operators include ( 1) N with N: Number of transitions of fermionic operators Example: : a k1 a k 2 a k3 a k 4 := a k 2 a k 4 a k1 a k3 Time ordered product T (φ(x)φ(y)) =: φ(x)φ(y) : +φ(x)φ(y) φ(x)φ(y) = i F (x y) = d 3 k [e ik(x y) θ(x 0 y 0 ) + e ik(y x) θ(y 0 x 0 )] 2ω k (2π) 3 Express any time ordered product as summation over normal ordered products generated by all possible contractions Electroweak Theory &
Field Theory Maxwell Equations Field Quantization Renormalization Quantization of a field theory may lead to divergencies in the integrals of quantum mechanic amplitudes, rendering the theory useless Divergencies are created by the transition of non interacting theory to interacting theory Elimination of divergencies through renormalization relying on gauge invariance If there is only a finite number of divergencies per order of perturbation theory the theory is renormizable If renormizable the theory can still be used Electroweak Theory &
Proposal of Problems of In analogy to electromagnetic interaction j µ A µ Fermi proposed in 1934 the interaction j µ j µ for the β-decay by replacing the vector current A µ ēγ µ ν e by a leptonic current Considering the process 2 1 + 3 + 4 i.e. (n p + e + ν e ) yields H int = G[ Ψ 1 (x)γ µ Ψ 2 ][ Ψ 3 (x)γ µ Ψ 4 (x)] + h.c. The hermitian conjugate is added to account for the β + -decay Electroweak Theory &
Proposal of Problems of Problems of has a bad high energy behavior Violation of unitarity for high energies Higher order calculations diverge and need to be regulated No systematical regularization Not renormizable Solved by exchange of massive vector bosons instead of contact interactions GWS theory of electroweak interaction Electroweak Theory &
Proposal of Problems of Problems of has a bad high energy behavior Violation of unitarity for high energies Higher order calculations diverge and need to be regulated No systematical regularization Not renormizable Solved by exchange of massive vector bosons instead of contact interactions GWS theory of electroweak interaction Electroweak Theory &
Motivation Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Why combine electromagnetic and weak interaction? Each electromagnetic process is mediated by an uncharged photon Z 0 is uncharged as well Why shouldn t the same process be mediated by a Z 0 boson In the amplitude of each electromagnetic process is a part of the corresponding weak process included Thus it is not farfetched to consider photon and Z 0 as of one family Electroweak Theory &
Historical Overview Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory 1957 First try by Julian Schwinger 1960 His Phd Sheldon Lee Glashow proposes an electroweak gauge theorie without explaining the masses of weak bosons 1964 Abdus Salam proposes Glashows model, apparently not knowing of it 1967 Steven Weinberg proposes an electroweak theory including the Higgs mechanism as explanation of the weak boson masses 1968 Abdus Salam proposes independently the same theory as Weinberg 1979 Glashow,Weinberg and Salam were awarded the Nobel prize for the unification of electromagnetic and weak theory Electroweak Theory &
Weak Isospin T Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Consider a weak isospin T to introduce electroweak multiplets (L) Fermions change into one other by W-boson exchange ( ) ( νe (L) 1 ) Dublet e with T (L) 3 = 2 1 2 Singlet (e (R)) with T 3 = (0) Electroweak Theory &
Isospin Preservation Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Process ν e + X e + Y exchanges a W For isospin preservation W needs T 3 = 1 and T 3 (W + ) = 1 W + andw belong to a T = 1 triplet with W 0 Weak charge g T 3 (W 0 ) = 0 Be careful W 0 Z 0 Introduce singlet state B 0 Weak charge g Photon and Z 0 are mixed states of B 0 andw 0 Electroweak Theory &
Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Weinberg Angle θ w ( ) ( ) ( ) Ψγ cos θw sin θ = w ΨB Ψ Z sin θ w cos θ w Ψ W Geometric rotation, therefore θ w identified as an angle Relation between charge and Weinberg-angle e = gg g 2 + g 2 sin θ w = g g 2 + g 2 cos θ w = g g 2 + g 2 Relation of electric and weak charge: e = g sin θ w Electroweak Theory &
Weak Interaction Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory For sin 2 θ w = 0, 23192 sin θ w 0, 5 e 0, 5 g Comparing coupling constants electromagnetic and weak interaction should be of almost equal strength In reality electromagnetic interaction is much stronger than weak Reason is propagator of exchange particle i q 2 m 2 Photon is massless, W and Z bosons have large masses yielding a small propagator Small propagator explains weakness of weak interaction Electroweak Theory &
Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Magnet above T C Basic properties of a system have a symmetry not shown by groundstate Consider potential V (Φ) = 1 2 µ2 Φ 2 + 1 4 λφ4 Example: Magnetization state of an iron rod above T C with µ 2 0 Free energy symmetric to unmagnetized state Ground state on symmetry axis Electroweak Theory &
Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Magnet below T C Lovering T below T C yields a potential with µ < 0 Thus, the apex goes up, but rotation symmetry is preserved Spontaneously there are 2 equilibrium states for M 0 Equilibrium situation does not reflect original rotation symmetry Spontaneous symmetry breaking yields a phase transition Electroweak Theory &
Higgs-Mechanism Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Higgs proposed a limiting energy for a phase transition for electroweak exchange particles Above the energy exchange particles are massless, below they receive mass For each particle one considers a Higgs-field with a Higgs-boson At the transition point W- and Z-bosons eat their Higgs-boson, acquiering mass Photon stays massless Theory proposes a free Higgs-boson Electroweak Theory &
Goldstone Theorem Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Massless scalars (Goldstone-bosons) arise if symmetry is spontaneously broken Number of scalars equal number of spontaneously broken generators of the group No evidence for existence in electroweak processes For local gauge invariance the Goldstone-boson is needed for longitudinal polarization of massive gauge-bosons Gauge-field eats Goldstone-bosons Degrees of freedom of Goldstone bosons combine with gauge-fields to form massive vector-bosons Electroweak Theory &
Achievements of EW Theory Motivation Historical Overview Weak Isospin T Spontaneus Symmetry Breaking Higgs-Mechanism Achievements of EW Theory Existence of weak current mediated by Z 0 (1973 CERN) Existence of charm-quark (1974 Discovery of J/Ψ particle) t Hooft showed renormizability of GWS-Model (1971) Nobelprize 1999 Existence of massive W- and Z-bosons (1983 CERN) Still missing: Existence of free Higgs-boson Electroweak Theory &
Neutrino-Nucleon-Scattering Deep Inelastic Scattering Cross Sections Measurement of sin 2 θ w Neutrino-Nucleon-Scattering Very good way of measuring θ w Consider reactions: ν µ (k) + N(p) ν µ (k ) + X ν µ (k) + N(p) ν µ (k ) + X First seen by the Gargamelle collaboration Electroweak Theory &
Neutrino-Nucleon-Scattering Deep Inelastic Scattering Cross Sections Measurement of sin 2 θ w Deep Inelastic Scattering Deep inelastic scattering uses the parton model Hadrons are composed of partons For a rate calculation one considers the rate for single partons and sums incoherently over all partons in a nucleon Considering only u- and d quarks leaves following reactions Electroweak Theory &
Neutrino-Nucleon-Scattering Deep Inelastic Scattering Cross Sections Measurement of sin 2 θ w Elemental Neutrino-Parton-Reactions For neutral currents: ν µ ( ν µ ) + u ν µ ( ν µ ) + u ν µ ( ν µ ) + d ν µ ( ν µ ) + d For charged currents: ν µ + d µ + u ν µ + u µ + + d Measurement was obtained on heavy targets averaged over proton and neutron Distribution function N(x) = N u (x) + N d (x) = 2N u d because of isospin invariance Electroweak Theory &
Neutrino-Nucleon-Scattering Deep Inelastic Scattering Cross Sections Measurement of sin 2 θ w Cross Sections averaged over Proton and Neutron σ(ν µ N µ X ) = G 2 π ME dx x N(x) 0 σ(ν µ N ν µ X ) = G 2 1 [ 1 π ME dx x N(x) 2 sin2 θ w + 20 ] 27 sin4 θ w σ( ν µ N ν µ X ) = G 2 σ( ν µ N µ + X ) = G 2 3π ME 0 1 0 3π ME dx x N(x) 1 1 0 dx x N(x) [ 1 2 sin2 θ w + 20 ] 9 sin4 θ w Electroweak Theory &
Neutrino-Nucleon-Scattering Deep Inelastic Scattering Cross Sections Measurement of sin 2 θ w Measurement of sin 2 θ w Relations of neutral to charged current R ν andr ν R ν = σ(νµn νµx ) σ(ν µn µ X ) = 1 2 sin2 θ w + 20 27 sin4 θ w R ν = σ( νµn νµx ) σ( ν µn µ + X ) = 1 2 sin2 θ w + 20 9 sin4 θ w By plotting R ν R ν plane sin 2 θ w should lie on the Weinberg-Nose Measurements of CHDS- and CHARM-collaboration yielded sin 2 θ w = 0, 23 Electroweak Theory &
Neutrino-Nucleon-Scattering Deep Inelastic Scattering Cross Sections Measurement of sin 2 θ w Measurement of sin 2 θ w For precision measurement one has to consider Antiquarks in the nucleon Behavior of sin 2 θ w in higher order of perturbation theory Convention: sin 2 θ w = 1 m2 W m 2 Z Best value of sin 2 θ w = 0, 23192 ± 0, 00023 Electroweak Theory &
Key Ideas References Key Ideas QED and weak theory are the pillars of EW theory In the amplitude of em processes part of a weak process is included Photon and Z 0 are mixed states of B 0 andw 0 Small propagator explains weakness of weak interaction Spontaneous symmetry breaking: Basic properties of a system have a symmetry not shown by groundstate All predictions of EW theory were discovered, except Higgs-boson Neutrino scattering is a good way to obtain θ w Electroweak Theory &
Key Ideas References References A Modern Introduction to Particle Physics by Fayyazuddin and Riazuddin Gauge Theory of Elementary Particle Physics by Ta-Pei Cheng and Ling-Fong Li Introduction to Quantum Field Theory by S.J.Chang Feynman-Graphen und Eichtheorien für Experimentalphysiker by Peter Schmüser Elementarteilchenphysik: Phänomene und Konzepte by Otto Nachtmann Electroweak Theory &