PARTICLE PHYSICS Major Option

Transcription

1 PATICE PHYSICS Major Option Michaelmas Term 00 ichard Batley Handout No 8

2 QED Maxwell s equations are invariant under the gauge transformation A A A χ where A ( φ, A) and χ χ ( t, x) is the 4-vector potential is any function of position and time Gauge invariance is related to conservation of electric charge : ρ. J 0 t j j correspond to massless photons : 0 ( ρ,j) The free particle solutions to Maxwell s eqns q 0 which can come in two independent helicity (polarisation) states e.g. A ( x) ε ( q) e iq. x the left-handed and right-handed circular polarisation states ε and ε

3 e.g. for a photon travelling along the z axis, the left- and right-handed states are ε i ( 0,,,0) ε ( 0,, i,0) helicity helicity The equation of motion for a spin / Dirac particle of charge e in an EM field is ( i ea ) ψ mψ 0 γ To keep this equation gauge invariant, the spinor wavefunction ψ must transform as ψ ψ e ie χ ( t, x) ψ a local phase transformation i.e. the phase of ψ changes by an amount which depends on x and t

4 Conversely: if we demand that the free particle Dirac equation iγ ψ mψ 0 be invariant under local phase transformations then: ψ ψ e ie we must introduce a massless gauge boson A which transforms as A χ ( t, x) ψ A A χ and satisfies ( i ea ) ψ mψ 0 γ the interaction between a spin / particle and a photon is completely specified In QFT, the interaction term eγ Aψ generates the Feynman rule for a vertex : (ψ ) ieγ γ ( A )

5 Thus, in QED : ocal phase invariance Gauge invariance Maxwell s equations charge conservation The gauge principle of QED extends to the weak and strong interations also : The EM, weak and strong interactions are all gauge field theories obtained by demanding local phase invariance The interactions of the gauge bosons with other particles, and with themselves, are completely determined

6 QCD equire invariance under local SU() phase transformations: i (x) q q e q where: 8 functions of (x,t) Corresponds to rotating states in colour space about an axis whose direction is different at every spacetime point q ( q, q, q ) r g ( λ, λ, λ ), ( θ ( x), θ ( x),, θ ( )) ( x) 8 b x 8 vector in colour space eight x λ matrices By analogy with QED, local phase invariance must introduce 8 massless gauge bosons (gluons) esulting theory (QCD) is invariant under gauge transformations colour is conserved and interaction of gluons with spin / quarks and with themselves is completely specified -gluon, 4-gluon vertices

7 The eak Interactions The ± bosons interact only with left-handed spinor components : ( 5 ψ γ )ψ In the lepton sector, we have only (ν e ) (ν ) (ν τ ) (e ) / : / : ( ) (τ ) natural to organise leptons into doublets of left-handed particles: ν e e ν ν τ τ / and singlets of right-handed particles: 0 : ( e ) ( ) ( τ ) 0 multiplets of weak isospin ( I I ), ( nothing to do with ordinary spin or flavour isospin... just common SU() mathematics )

8 Similar for the quark sector : (d ) (s ) (b ) u c t (d, s, b are the CKM-rotated weak eigenstates) Form doublets of left-handed quarks: / : / : u d c s t b / and singlets of right-handed quarks: ( u) c) ( (t) I 0 : 0 ( d ) ( s ) ( b )

9 where: and Turn this into a gauge theory by requiring invariance under local SU() phase transformations: i ( x) ψ ψ e : ψ τ 0 ( τ, τ τ ) 0, τ 0 i are the Pauli matrices i 0 ( ( x), ( x), ( )) ( x ) x τ 0 0 Corresponds to invariance under rotations in weak isospin space about a direction varying with position and time equires the introduction of gauge fields,, and the resulting gauge invariance leads to conservation of weak isospin The combinations 0 ( i ) ( i ) form a multiplet with

10 e.g. and can be identified with the charged ± bosons e ν e e ν e : must mediate interactions like e e : 0 corresponds to a neutral gauge boson 0 i.e. SU() gauge invariance requires the existence of weak neutral currents e ν e e ν e Tempting to identify Z But: ± and Z 0 have different masses can t belong to same SU() multiplet 0

11 But: there is another neutral gauge boson, the photon, with the same quantum numbers as the Z 0 could be a mixture of Z 0 and γ unification of EM and weak forces In the Standard Model: cosθ. Z sinθ. where θ is the einberg angle A has to be determined from experiment : sin θ 0. The combination orthogonal to is B sinθ. Z cosθ. In the Standard Model, B is the gauge boson associated with a local U() phase invariance ψ ψ e ig θ (x) where U() group of all x unitary matrices: U ( iφ e ) ψ U U I A

12 Hence, overall, the electroweak sector of the SM is invariant under two independent symmetries: SU() U(),, B mix to form Z 0 and γ Interactions of ±, Z 0 and γ are completely specified, and find g sinθ g cosθ sinθ Z e photon coupling Hence ±, Z 0 and γ interaction strengths are all of similar size in fact ±, Z 0 are stronger: g 4π 9 g Z 4π e 4π 7 eak interactions only appear weak at low energies because of large, Z mass propagator q m m (small)

13 eft- and right-handed components interact with Z 0 with strengths c and c : (e ) (e ) c c (e ) Z 0 (e ) Z 0 c Q sin θ For a particle of charge Q, weak isospin : c I Q sin θ eak bit of Z (i.e. 0 ) couples to left-handed particles only (with strength prop to weak isospin ) EM bit of Z (i.e. B ) couples equally to leftand right-handed (with strength prop to electric charge Q) Overall interaction contains factor c ( 5 ) ( 5 γ c γ ) ( c c ) ( c c ) γ 5 c c A γ V 5

14 Vertex factor for Z 0 is: g Z i γ V ( 5 c c γ ) A Z 0 where c c c c V A c c vector and axial vector coupling constants For sin θ 0. : fermion Q c c c V c A ν, e ν, ν τ 0 0 e,,τ u, c, t d, s, b Note that, for neutrinos: c 0 c cv ca coupling to Z 0 is VA, just as for ± still only left-handed neutrinos