THE STANDARD MODEL Stephen West Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX. NExT graduate school lectures 2010-2014. 1 Contents 1 Introduction, Conventions and the fermionic content of theSM 5 1.1 Booksandpapers................................... .... 5 1.2 Overview.......................................... .. 6 1.3 Conventions....................................... ... 8 1.4 LorentzInvariance ................................. ..... 8 1.5 C, P and CP ......................................... 12 1.6 FermionicContentofSM............................... .... 14 2 Abelian Gauge Symmetry 16 2.1 GaugeTransformations.............................. ...... 16 2.2 GaugeFixing ........................................ 18 2.3 Summary-QEDLagrangian............................. .... 20 3 Non-Abelian Gauge Theories 21 3.1 Non-Abeliangaugetransformations . ........ 21 3.2 Non-Abelian Gauge Fields . ... 22 3.3 GaugeFixing ........................................ 25 3.4 Summary: The Lagrangian for a General non-Abelian Gauge Theory .......... 27 3.5 FeynmanRules ...................................... .. 29 4 Quantum Chromodynamics 31 4.1 Colour and the Spin-Statistics Problem . ....... 31 4.2 QCD.............................................. 32 4.3 RunningCoupling..................................... .. 33 4.4 Quark(andGluon)Confinement . ..... 36 4.5 θ ParameterandStrongCPproblem. ... 38 − 2 4.6 Summary .......................................... 40 5 Spontaneous Symmetry Breaking 41 5.1 Massive gauge bosons and Renormalisability . ....... 41 5.2 SpontaneousSymmetryBreaking . ....... 42 5.3 SpontaneousSymmetryBreaking . ....... 44 5.4 GoldstoneBosons................................... .... 47 5.5 TheHiggsMechanism .................................. .. 47 5.6 GaugeFixing ........................................ 50 5.7 Bit more on Renormalisability . ... 54 5.8 Spontaneous Symmetry Breaking in a Non-Abelian Gauge Theory . ........... 56 6 Spontaneous Breaking of SU(2) U(1) U(1) 58 L × Y → EM 6.1 More on renormalisability . ... 61 6.2 Summary .......................................... 63 7 The Electroweak Model of Leptons 64 7.1 Left-andright-handedfermions. ........ 64 7.2 Fermionmasses-Yukawacouplings . ...... 66 7.3 WeakInteractionsofLeptons . ....... 67 7.4 ClassifyingtheFreeParameters . ....... 70 7.5 Summary .......................................... 70 8 Electroweak Interactions of Hadrons 72 8.1 OneGenerationofQuarks ............................. ..... 72 8.2 QuarkMasses...................................... ... 73 8.3 AddingAnotherGeneration .. .. .. ..... 74 8.4 TheGIMMechanism................................... .. 76 3 8.5 AddingaThirdGeneration .............................. ... 78 8.6 Massgeneration-asummary . .. .. ..... 80 8.7 CP Violation ......................................... 82 8.8 Summary .......................................... 86 9 Neutrinos and the MNS matrix 88 9.1 Dirac Neutrino masses and mixings . ..... 88 9.2 Majorana Neutrinos and the see-saw mechanism . .......... 88 9.3 MNS matrix and Counting phases for Dirac and Majorana Neutrinos .......... 91 10 Anomalies 94 10.1Trianglediagrams................................... .... 97 10.2 Chiral Anomalies and Chiral Gauge Theories . ........ 101 10.3 Cancellation of anomalies in the Standard Model. ........ 107 11 Chiral Lagrangian approximation 111 11.1 Explicit Chiral Symmetry Breaking . ...... 118 11.2 ElectroweakInteractions . ........ 120 4 1 Introduction, Conventions and the fermionic content of the SM 1.1 Books and papers F Halzen and A. D. Martin, Quarks and Leptons, Wiley, 1984. • M. E. Peskin and D V Schroeder An Introduction to Quantum Field Theory Addison Wesley • 1995. I J R Aitchison and A J G Hey Gauge theories in particle physics 2nd edition Adam Hilger 1989. • J D Bjorken and S D Drell Relativistic Quantum Mechanics McGraw-Hill 1964. • F Mandl and G Shaw Quantum Field Theory Wiley 1984. • P. Ramond, Journeys beyond the standard model, Perseus Books, 1999. • D. Balin and A. Love, Introduction to Gauge Field Theory, 1986. • Several reviews on Spires, e.g. Teubner, Langacker, Signer, Novaes, Evans. • Also useful S. P. Martin, A Supersymmetry primer • 5 1.2 Overview The Standard Model is a description of the strong (SU(3) ), weak (SU(2) ) and electromagnetic • c L (U(1)em) interactions in terms of “gauge theories”. A gauge theory is one that possesses invariance under a set of “local transformations” i.e. transformations whose parameters are space-time dependent. Electromagnetism is a gauge theory, associated with the group U(1) . • em In this case the gauge transformations are local complex phase transformations of the fields of charged particles. Gauge invariance necessitates the introduction of a massless vector (spin-1) particle, the photon, whose exchange mediates the electromagnetic interactions. In the 1950’s Yang and Mills considered (as a purely mathematical exercise) extending gauge • invariance to include local non-Abelian (i.e. non-commuting) transformations such as SU(2). In this case one needs a set of massless vector fields (three in the case of SU(2)), which were formally called “Yang-Mills” fields, but are now known as “gauge bosons”. In order to apply such gauge theories to the weak interaction, considers particle transforming into • each other under the weak interactions. Doublets of weak isospin. → u ν L and L dL ! eL ! For SU(2) we have three gauge bosons interpreted as the W ± and Z bosons, that mediate • L − − weak interactions in the same way that the photon mediates electromagnetic interactions. However, weak interactions are known to be short ranged, mediated by very massive vector-bosons, • whereas Yang-Mills fields are required to be massless in order to preserve the gauge invariance. Solved by the “Higgs mechanism”. ⇒ Prescription for breaking the gauge symmetry spontaneously. • Start with a theory that possesses the required gauge invariance, but the physical quantum states • of the theory are not invariant under the gauge transformations. The breaking of the invariance arises in the quantisation of the theory, whereas the Lagrangian • only contains terms which are invariant. 6 One of the consequences of this is that the gauge bosons acquire a mass and can thus be applied • to weak interactions. Spontaneous symmetry breaking and the Higgs mechanism has another extremely important con- • sequence. It leads to a renormalisable theory with massive vector bosons. Theory is renormalisable ⇒ Infinities that arise in higher order calculations can be re-absorbed into the parameters of the Lagrangian (as in the case of QED). If we had simply broken the gauge invariance explicitly by adding mass terms for the gauge bosons, • resulting theory would not be renormalisable Could not therefore have been used to carry out perturbative calculations. ⇒ Consequence of the Higgs mechanism is the existence of a scalar (spin-0) particle - the Higgs • boson. Remaining step is to apply the ideas of gauge theories to the strong interactions. • The gauge theory of strong interactions is called “Quantum ChromoDynamics” (QCD), associated with the group SU(3)c. Quarks possess an internal property called “colour” and the gauge transformations are local transformations between quarks of different colours. The gauge bosons of QCD are called “gluons” and they mediate the strong interactions. The union of QCD and the electroweak gauge theory, which describes the weak and electromag- • netic interactions is known as the Standard Model. It has eighteen fundamental parameters, most of which are associated with the masses of the • gauge bosons, the quarks and leptons, and the Higgs. Not all independent and, for example, the ratio of the W and Z boson masses are (correctly) • − − predicted by the model. Since the theory is renormalisable, perturbative calculations can be performed which predict cross- • sections and decay rates both for strongly and weakly interacting processes. Predictions have met with considerable success when confronted with ever stringent experimental data. 7 1.3 Conventions Natural units: ~ = 1, c =1 • energy, momentum, mass, inverse time, inverse length all have same dimensions. ⇒ The metric convention is • g = diag(1, 1, 1, 1) (1.1) µν − − − Space time coords and derivatives • xµ =(x0, xi)=(t, x); x = g xν =(t, x), µ µν − ∂ ∂ ∂ = , ; ∂µ = , . µ ∂t ∇ ∂t −∇ Lorentz invariant inner product of two vectors: • x2 = g xµxν = x xµ = t2 x.x µν µ − 1.4 Lorentz Invariance An arbitrary Lorentz transformation can be written as • µ µ µ ν x x′ =Λ x → ν Lorentz transforms defined by invariance of x2 under Λ Lorentz Group • ∈ g xµxν = g Λρ xµΛσxν , x µν ρσ µ ν ∀ g = g (Λ)ρ (Λ)σ. ⇒ µν ρσ µ ν Klein-Gordon field, φ(x), transforms as • 1 φ(x) φ′(x)= φ(Λ− x) → Transformation should leave KG equation invariant, clearly 1 m2φ2(x) is invariant, what about • 2 the derivatives 1 ν ρ 1 ∂ 1 ∂(Λ− )ρx ∂ 1 ∂µφ(x) ∂µφ(Λ− x)= µ φ(Λ− x)= µ 1 ν ρ φ(Λ− x) → ∂x dx ∂(Λ− )ρx 1 ν ∂ 1 1 ν 1 − − − − = (Λ )µ 1 ν ρ φ(Λ x)=(Λ )µ(∂νφ)(Λ x). ∂(Λ− )ρx 8 Transformation of the kinetic term in KG Lagrangian • 2 µν µν 1 ρ 1 σ 1 (∂ φ) g (∂ φ′(x))(∂ φ′(x)) = g [(Λ− ) (∂ φ)][(Λ− ) (∂ φ)](Λ− x) µ → µ ν µ ρ µ σ ρσ 1 2 1 = g (∂ρφ)(∂σφ)(Λ− x)=(∂ρφ) (Λ− x) Thus the whole Lagrangian transforms as a scalar • 1 (Λ− x) L→L and with the action, S, being an integral over all space of the Lagrangian it is clear that the action is invariant. Now Dirac Fermions Need a representation for the Lorentz group for Dirac spinors • General 4-d Lorentz transformation • Lµν = i(xµ∂ν