Introduction to particle physics Lecture 6

Transcription

1 Introduction to particle physics Lecture 6 Frank Krauss IPPP Durham U Durham, Epiphany term 2009

2 Outline 1 Fermi s theory, once more 2 From effective to full theory: Weak gauge bosons 3 Massive gauge bosons: Spontaneous symmetry breaking

3 Fermi s theory, once more General idea Task: Formulate a general framework to describe weak interactions. Three classes of weak interactions: purely leptonic, e.g. µ e + ν e + ν µ semi-leptonic, e.g. n p + e + ν e purely hadronic, e.g. K 0 1 π + + π. Will formulate the theory in terms of currents. Example: Current for e ν e - transitions ψ νe Γψ e. (Convention: Read from right to left) As before, the ψ are Dirac-spinors, the four-component objects describing fermions. The Γ encodes details of how the spins behave in the interaction. Remember: Only left-handed fermions take part in the weak interaction, therefore the Γ will project on these chirality states.

4 Lepton currents First: weak interaction of leptons (e ±, µ ±, ν e,µ, ν e,µ ) only. Write down lepton currents for flow of lepton quantum numbers: L W = ψ e Γψ νe + ψ µ Γψ νµ and L W = ψ νe Γψ e + ψ νµ Γψ µ. Then in first order perturbation theory, weak transition amplitudes given by Fermi interaction Pictorially: M (1) G F (L W L W) with G F GeV 2. Currents: L W = ν e e + ν µ µ Amplitude: e νe µ νµ L W = e ν e + µ ν µ νe e νe e

5 An example process: e + ν e e + ν e Relevant Feynman diagrams: e νe e νe e νe e νe e νe Cross section in electron rest frame is tiny: σ lab (ν e + e ν e + e ) = σ 0 Eν m e cm 2 Definition of cross section: dσ dω = N scattered N incident unit surface solid angle. E ν 511keV. (equal to probability to observe one scattered particle per unit solid angle, if target is shot at with one particle per unit area) Typically, unit in particle physics is barn : 1 b = cm 2. Leads to a situation where high fluxes from reactors are related to low cross sections, and large cross sections appear only, if neutrinos have large energies (tricky to produce, so far, only pion beams).

6 Weak interactions of hadrons Try to extend the idea of leptonic currents to hadrons. Problem: Too many hadrons = formulate in terms of quarks. (Ignore effects of hadrons being composed of quarks parametrised by suitable wave functions) Thus: Formulate theory in terms of weak currents J W and J W, composed of lepton and quark currents: J W = L W + H W. Assume only u, d, s quarks (there are more, cf. later lectures): H W = cos θ C ψu Γψ d + sin θ c ψu Γψ s. Again, the Γ map out the spin structure, which as before consists of vector and axial-vector components in equal proportion - only left-handed quarks and right-handed anti-quarks experience the weak interaction, similar to the leptons. The Cabibbo-angle parametrises the ratio of strangeness violating and conserving weak interactions. It is given by sinθ C It can be deduced, e.g. by comparing the branching ratios of the decays τ K + ν τ and τ π + ν τ.

7 Example processes: K π 0 + e + ν e and K 0 π + + π K π 0 + e + ν e Remember flavour content of kaon and pion: K = sū and π 0 = 1 2 ` uū d d. A relevant Feynman diagram: s ν e s ū ū ū Note the two thick blobs correspond to a transition of the hadron state to the quark state. u e Fermi s interaction as four-fermion vertex. u K 0 π + + π Remember flavour content of kaon and pion: K 0 = s d, π + = u d, and π = dū. A relevant Feynman diagram: s ū s d d d d d Note the three thick blobs correspond to a transition of the hadron state to the quark state. u ū u

8 Weak gauge bosons General idea and its problems Fermi s theory is in some contradiction with a previous finding: Interactions are transmitted by intermediate particles. Natural question: Where are the mediators of weak interactions? (The photons of weak interactions) First guess (60 s physics): Since they have not been seen so far (like the photon is seen), they must be too heavy for direct detection: Unlike the massless photon they must be massive! Problem: Massless spin-1 bosons have two (transverse polarisation) degrees of freedom, its propagator behaves like 1/p 2. In contrast, massive (mass M) spin-1 bosons have an additional longitudinal degree of freedom - at high energies the propagator behaves like a constant, 1/M 2. Naively, this leads to a violation of unitarity, and therefore, of causality. In addition (and closely related): Such a theory of massive gauge bosons cannot be renormalised.

9 Gauge invariance Why is this important? The concept of gauge invariance is the most significant guiding principle in modern particle physics: Seemingly all four fundamental interactions can be understood in its terms. Idea: Formulate Lagrangian/Hamiltonian such that it remains invariant under certain (internal) symmetry transformations, which in turn give rise to conserved quantum numbers (charges).

10 A prime example: QED Global phase invariance First example: QED describing interactions between charged particles such that the total electrical charge are always conserved. To represent this consider the Lagrangian L for the electron wave function. It (or, more precisely, the action S = d 4 xl must be invariant under a certain group of transformations, G, modifying the wave function: GL(ψ e ) L(ψ e). In fact the group in question is the group U(1) of all phase transitions of the wave function. This can be represented as G = exp[ iθ] with θ R. Since it acts the same way on all point in space time, a symmetry transformation like this is called global. We know that global changes in phase cannot be observed (because typically squares are taken), but phase differences are observable.

11 Local phase transformations To measure phase differences: Must establish a specific θ = 0. Different such conventions are related by global phase transformations. Clearly the choice, being unobservable, must not matter for physical observables: The theory must be invariant under global phase transformations. What happens if the phase depends on space-time: θ θ(x)? (This is called a local phase transformation.) Simple answer: Then the Lagrangian is not invariant any more. G(x)L(ψ e ) L (ψ e) L(ψ e ).

12 Restoration of local phase invariance But: Can make the Lagrangian invariant by introducing another field, A. G(x)L(ψ e,a) L(ψ e,a ). Properties of this field: Must be massless to allow for infinite range - it must connect different phase conventions all over space. It s a four vector, A µ, identified with the photon field. The photon field must also transform under G(x) such that the combination with changes due to the electron field are compensated. Summary of this construction: Global phase invariance yields conserved charges. Local phase invariance gives rise to the photon field, i.e. interactions. Final remark: A trivial mass term for the photon would look like L m m 2 A 2 and it is not invariant under local phase transformations.

13 Relation to gauge invariance in classical theory Write the photon field as A µ = (Φ, A), where E = Φ t A and B = A gives the relation to the electric and magnetic field. The potentials are invariant under the gauge transformation Φ Φ = Φ + t Λ and A A = A Λ, or, in four-vector notation ( µ = ( t, )) A µ A µ = A µ + µ Λ. Invariance of the Lagrangian L = E 2 B 2 follows trivially. Finally: The Λ(x) here is more or less identical with the θ(x) of the local gauge transformation before.

14 Generalised gauge invariance A classical example Seemingly, gauge invariance an elegant way to produce interactions. Added benefit: protects high-energy behaviour of QED. (renormalisability) Extent this to other interactions, e.g. of nucleons with pions: Pions transform nucleons into nucleons, put p and n into iso-doublet. (isospin: like the spin-up and down states of a fermion) Then: Need gauge transformations acting on the nucleon field N = (p, n), mixing the states. G(x) must have 2 2 matrix form = use Pauli matrices as basis: There are 3 Pauli matrices - each corresponds to a field: 3 ρ s! Due to Gell-Mann-Nishijima formula: ρ s carry isospin = self-interactions! Can show that Lagrangian is invariant under global SU(2): G SU(2) L(N) L(N ). But: ρ s not elementary and π s are the true isospin force carriers.

15 Gauge invariance and weak interactions Realise that weak interactions base on currents corresponding to transitions e ν e, µ ν µ and similar. Natural to group the leptons into weak iso-doublets of the same lepton type/family: l e = (ν e, e ), l µ = (ν µ,µ ). Define weak isospin and corresponding gauge fields W = W +,,0. Demand Lagrangian invariant under weak SU(2) gauge interactions G SU(2) W L(l e,l µ, W) L(l e,l µ, W ). Consequence: Replace Fermi s four-fermion interactions (current-current interactions) with the currents interacting with the gauge bosons, like in QED. Repeat exercise with quarks - replace quark currents with doublets, taking into account the Cabibbo-angle.

16 Feynman rules in weak interactions Charged interactions (similar for quarks and charge-conjugated): ν e e ν µ µ W + W + Neutral interactions (similar for other leptons and quarks): ν e ν e e e Z 0 Z 0 Note: Charged weak interactions change flavour, neutral ones don t. The construction of the theory is a bit more complicated, the Z 0 in fact is a linear combination of the neutral SU(2) gauge boson and a U(1) gauge boson, the photon is the orthogonal linear combination.

17 Problems of weak gauge theory While charged weak interactions (of the e ν e -type) have been known, evidence for neutral ones came in 1973 only. Local gauge invariance dictates massless gauge bosons (like in QED) but experimental evidence showed that they are in fact very massive: m W 80.4 GeV, m Z 91.2 GeV. Neutral currents Gargamelle at CERN (1973): first photo of a neutral current interaction. Neutrinos interact with matter in a 1200 litre bubble chamber. Here: A neutrino interacts with an electron (the horizontal line) and evades unseen.

18 Spontaneous symmetry breaking Basic idea Common phenomenon: Asymmetric solutions to symmetric theory. Example: Magnet (e.g. Heisenberg/Ising model). Theory: local spin-spin interactions with preference for alignment, but direction not fixed. Nevertheless: Preferred direction emerges. Symmetric state (no alignment) is not state of minimal energy. In QFT: Every particle related to a field. Ground state of the field (a.k.a. vacuum ) is state of minimal energy and no particles are present. For most fields, minimal energy equals 0, but cases can be constructed, where E mmin = E 0 = v < 0. Add such a field with E 0 = v < 0 and couple to gauge bosons. Can show that v sets scale for mass of gauge bosons.

19 The Higgs mechanism: Making the gauge bosons massive Add a complex iso-doublet Φ to the theory. (Iso-doublet to trigger interactions with W.) Lagrangian invariant under SU(2): G SU(2) W L( W,Φ) L( W,Φ ) Give it a potential of the form µ 2 Φ 2 + λ Φ 4 = non-trivial minimum at Φ 2 = v 2. Pick a vacuum, for instance R(Φ 1 ) 0 = v. Expand in new fields around this vacuum: Three orbital modes parallel to the minimum (the Goldstone modes θ), one radial mode (the Higgs field η), which feels the potential. The Lagrangian is not trivially SU(2) invariant any longer! Absorb the Goldstones into the gauge bosons (by choosing a gauge ), gives their 3rd polarisation d.o.f. = become massive! One real scalar η as remainder with very fixed interactions.

20 Summary Recapitulated Fermi s theory in form of current-current interactions. Discussed gauge invariance in the case of QED and classical electrodynamics. Generalised the idea to the case of weak isospin, SU(2) W as gauge group of weak interactions. Showed that gauge bosons must be massless. Discussed idea of Higgs mechanism to generate mass in a gauge-invariant way. To read: Coughlan, Dodd & Gripaios, The ideas of particle physics, Sec