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X. Charge Conjugation and Parity in Weak Interactions → Charge conjugation and parity in weak interactions Particle Physics X. Charge conjugation and parity in weak interactions REMINDER: Parity The parity transformation is the transformation by reflection: → xi x'i = –xi A parity operator Pˆ is defined as Pˆ ψ()()xt, = pψ()–x, t where p = +1 Charge conjugation The charge conjugation replaces particles by their antiparticles, reversing charges and magnetic moments ˆ Ψ Ψ C a = c a where c = +1 meaning that from the particle in the initial state we go to the antiparticle in the final state. Oxana Smirnova & Vincent Hedberg Lund University 248 Charge conjugation and parity in weak interactions Particle Physics Reminder Symmetries Continuous Lorentz transformation Space-time Translation in space Symmetries Translation in time Rotation around an axis Continuous transformations that can Space-time be regarded as a series of infinitely small steps. symmetries Discrete Parity Transformations that affects the Space-time Charge conjugation space-- and time coordinates i.e. transformation of the 4-vector Symmetries Time reversal Minkowski space. Discrete transformations have only two elements i.e. two transformations. Baryon number Global Lepton number symmetries Strangeness number Isospin SU(2)flavour Internal The transformation does not depend on Isospin+Hypercharge SU(3)flavour symmetries r i.e. it is the same everywhere in space. Transformations that do not affect the space- and time- Local gauge Electric charge U(1) coordinates. symmetries Weak charge+weak isospin U(1)xSU(2) Colour SU(3) The transformation depends on r i.e. it is different in different points in space. Oxana Smirnova & Vincent Hedberg Lund University 249 Charge conjugation and parity in weak interactions Particle Physics While parity is conserved in strong and electromagnetic interactions, it is violated in weak processes: − 1956: Based on the measurements of Kaon decays, Lee & Yang propose that parity is violated in weak processes: Two known decays of the K+ were: 0 + + + − K+ → π + π and K+ → π + π + π 0 + The intrinsic parity of a pion Pπ= -1, and for the π π + + − and π π π states the parities are 2()L Pππ ==Pπ –1 1 since L = L12 = 0 3()L Pπππ ==Pπ –1 –1 since L = L12 + L3 = 0 Since the two final states have opposite parities, one of the K+ decays must violate parity! Oxana Smirnova & Vincent Hedberg Lund University 250 Charge conjugation and parity in weak interactions Particle Physics − 1957: Wu carries out studies of parity violation in β 60 β 60 ∗ - ν -decay. The Co -decay into Ni +e + e was studied. − The 60Co sample was cooled to 0.01 K to prevent thermal disorder. − The sample was placed in a magnetic field ⇒ the nuclear spins were aligned along the field direction p spin vector J=5 J=4 Mag. Field p Figure 125: Possible β-decays of 60Co: case (a) is preferred. − If parity is conserved, processes (a) and (b) must have equal rates. Electrons were emitted predominantly in the direction opposite the 60Co spin Oxana Smirnova & Vincent Hedberg Lund University 251 Charge conjugation and parity in weak interactions Particle Physics Another case of both parity and C-parity violation was observed in muon decays: µ− → - ν ν e + e + µ µ+ → + ν ν e + e + µ spin vector ± θ e π−θ P ± ± µ p µ ± e e Figure 126: Effect of a parity transformation on the muon decays above The angular distribution of the electrons (positrons) emitted in µ- (µ+) decay are given by ξ 1 ⎛⎞± Γ ()cosθ = ---Γ 1 – ------- cosθ (136) µ± 2 ± ⎝⎠3 here ξ± are constants – “asymmetry parameters”, and Γ± are total decay rates ⇒ inverse lifetimes 1 1 Γ Γ ()θθ ≡ ------- ± = ∫ ± cos dcos τ (137) ± –1 Oxana Smirnova & Vincent Hedberg Lund University 252 Charge conjugation and parity in weak interactions Particle Physics If the process is invariant under charge conjugation (C-invariance) ⇒ Γ Γ ξ ξ + = - + = - (138) (rates and angular distributions are the same for e- and e+) If the process is P-invariant, then angular distributions in forward and backward directions are the same: Γ ()Γθ ()θ ξ ξ µ± cos = µ± –cos + ==- 0 (139) Experimental results: Γ Γ ξ ξ ± + = - + ==– - 1,00 0,04 (140) Both C- and P-invariance are violated! Oxana Smirnova & Vincent Hedberg Lund University 253 Charge conjugation and parity in weak interactions Particle Physics However, the combined operation CP is conserved since that requires Γ ()Γθ ()θ µ+ cos = µ- –cos (141) Γ Γ ξ ξ + = - + = – - (142) which is in agreement with the experiments. Figure 127: P-, C- and CP-transformation of an electron The combined transformation CP is a weaker requirement than the individual transformations P and C and it is conserved. Oxana Smirnova & Vincent Hedberg Lund University 254 Charge conjugation and parity in weak interactions Particle Physics Helicity helicity – the spin is quantized along the particle´s direction of motion instead of along an arbitrary z-direction ˆ sp• Λ = ----------- (143) p ˆ Λψλψ= The eigenvalues of the helicity operator are λ=-s,-s+1,...,+s, ⇒ for spin-1/2 particle it can be either -1/2 or 1/2 Direction of motion spin vectors right-handed left-handed Figure 128: Helicity states of spin-1/2 particle A particle with λ=+1/2 is called right handed. A particle with λ=-1/2 is called left handed. A subscript R or L is used to denote if a state is right - ν or left handed e.g. e R and L Oxana Smirnova & Vincent Hedberg Lund University 255 Charge conjugation and parity in weak interactions Particle Physics 1958: Goldhaber et al. measured the helicity of the neutrino by studying electron capture in europium: - 152 → 152 ∗ ν e + Eu Sm + e (144) 152Sm + γ (145) In this reaction the initial state has zero 152 ∗ ν momentum and Sm and e recoil in opposite directions. Events with the γ emitted in the direction of ∗ motion of the 152Sm were selected so that the overall observed reaction was: - 152 → 152 ν γ e + Eu (J=0) Sm(J=0) + e+ (146) The spin of the neutrino (+1/2 or -1/2) and the photon (+1 or -1) must add to give the spin of the electron (+1/2 or -1/2). The helicity (polarization) of the photons was determined by studying their absorption in magnetized iron. Oxana Smirnova & Vincent Hedberg Lund University 256 Charge conjugation and parity in weak interactions Particle Physics - e 152Eu J=0 = -1/2 se se = +1/2 e- e- ν ν R sν = +1/2 sν = -1/2 L 152Sm J=0 J=0 152Sm γ sγ = -1 sγ = +1 γ Not Observed ! Observed ! Figure 129: From the helicity of the photons it is possible to determine the helicity of the neutrinos. From the measured photon helicity it was concluded that neutrinos must be left-handed. Oxana Smirnova & Vincent Hedberg Lund University 257 Charge conjugation and parity in weak interactions Particle Physics V-A interaction V-A interaction theory was introduced by Fermi as an analytic description of spin dependence of charged current interactions. It denotes “polar Vector - Axial vector” interaction − A Polar vector is one which direction is reversed by parity transformation e.g. momentum p − An Axial vector is one which direction is not changed by parity transformation e.g. spin s or orbital angular momentum Lrp= × − The weak current has both vector and axial components, hence parity is not conserved in weak interactions Main conclusion: if vc≈ , only left-handed ν , - fermions L eL etc. are emitted, and right-handed antifermions. The very existence of preferred states violates both C- and P- invariance Oxana Smirnova & Vincent Hedberg Lund University 258 Charge conjugation and parity in weak interactions Particle Physics Neutrinos (antineutrinos) are always relativistic and hence always left(right)-handed For other fermions, the preferred states are left-handed. Right-handed states are not completely forbidden but suppressed by the factor 2 ⎛⎞v m 1 – -- ≈ --------- (147) ⎝⎠ 2 c 2E Consider the two pion decay modes: π+ → + ν e + e (148) + + π → µ + νµ (149) + + ν Relativistic: e π e Neutrinos are always µ+ π+ ν Non-relativistic: µ lefthanded Figure 130: Helicities of leptons emitted in a pion decay − The π+ has spin-0 and it is at rest ⇒ the spins of the charged lepton and the neutrino must be opposite. Oxana Smirnova & Vincent Hedberg Lund University 259 Charge conjugation and parity in weak interactions Particle Physics − The neutrinos are always left-handed ⇒ the charged leptons have to be left-handed as well. − BUT: the e+ and the µ+ should be right-handed since they are anti-fermions. In these decays the electron will be relativistic but not the muon (due to its large mass). It follows that the pion to muon decay should be allowed but the pion to positron decay should be suppressed. Left-handed + + ν Should be right-handed e π e since it is relativistic + Can be left-handed µ+ π νµ since it is non-relativistic − The suppression factor for positrons is expected to be of the order 10-5. The measured ratio: Γπ()+ → +ν e –4 --------------------------------------e = ()1,230± 0,004 ×10 (150) + + Γπ()→ µ νµ Oxana Smirnova & Vincent Hedberg Lund University 260 Charge conjugation and parity in weak interactions Particle Physics Muons emitted in pion decays are always polarized and this can be used to measure muon decay symmetries by detecting the relativistic electrons in the following decays: - - π → µ + νµ (151) - ν ν e + e + µ The electrons are emitted in decays when both the ν ν µ and the e are emitted in the direction opposite to the e-: νµ νµ e- µ− µ− e- ν ν e e (a) Left-handed electrons (b) Right-handed electrons Favoured Surpressed Figure 131: Muon decays with high energy electron emission.
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