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Charge conjugation and in weak interactions Particle 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 , 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 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 Space-time symmetries ContinuousDiscrete transformations that can Parity Transformations that affects the be regardedSpace-time as a series of infinitely Charge conjugation space-- and time coordinates i.e. small steps.Symmetries transformation of the 4-vector Time reversal Minkowski space. Global Discretesymmetries transformations have only number two elements i.e. two transformations. SU(2)flavour Internal Isospin+Hypercharge SU(3)flavour symmetries i.e. it is the same everywhere in space.

Local gauge U(1) Transformations thatit do is notdifferent in different points in space. Thesymmetries transformation does not dependWeak on charge+weak isospin U(1)xSU(2) affect the space- and time- r coordinates. Colour SU(3) i.e. The transformation depends on r

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 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 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 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. 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 () 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 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 (+1 or -1) must add to give the spin of the electron (+1/2 or -1/2). The helicity (polarization) of the 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 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 , - 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 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.

The electron must have a spin parallel to the muon spin ⇒ configuration (a) with left-handed electrons is strongly preferred ⇒ this is observed experimentally as a forward-backward asymmetry.

Oxana Smirnova & Vincent Hedberg Lund University 261 Charge conjugation and parity in weak interactions Particle Physics Neutral It is possible to produce the neutral kaons K0=ds and K0=sd in πp-collisions. This is a strong interaction process and strangeness has to be conserved: π– + p → K0 + Λ s: 0 0 +1 -1 + π++ p → K0 ++pK s: 00-10+1 The kaons that are produced in this way are pure K0=ds and K0=sd states. However, K0 and K0 can be converted into each other since strangeness is not conserved in weak interactions:

u d s 0 S=+1 K W+ W- K0 S=-1 sd u

Figure 132: Example of a process converting K0 to K0.

Oxana Smirnova & Vincent Hedberg Lund University 262 Charge conjugation and parity in weak interactions Particle Physics The observed physical particles are linear combinations of K0 and K0, since there is no conserved to distinguish them. The phenomenon is called K0–K0 mixing . We know that neither parity nor charge conjugation are conserved in weak decays. The combined operation CP is, however, almost conserved. In this case the CP operators eigenstates can be written as a mixture of K0 and K0:

1 K0 = ------{}K0 + K0 (152) 1 2

1 K0 = ------{}K0 – K0 (153) 2 2 so that

ˆ 0 0 0 0 CPK1 ==K1 and CPK2 –K2 (154) 0 i.e. the CP eigenvalues are cp=+1 for K1 and cp=-1 0 0 for K2 and the K1 can therefore only decay to 0 cp-even states while the K2 only to cp-odd states.

Oxana Smirnova & Vincent Hedberg Lund University 263 Charge conjugation and parity in weak interactions Particle Physics Experimentally observed are two types of neutral 0 τ × –10 kaons: KS (“S” for “short”, lifetime = 0,9 10 s ) 0 τ × –10 and KL (“long”, = 500 10 s ). 0 0 Can the KS be identified with the K1 0 0 CP-eigenstate, and the KL with the K2 ? 0 If CP-invariance holds for neutral kaons, KS should decay only into states with cp=1 such as π 0 2 -states, and KL into states with cp=-1 such as 3π-states:

0 → π+π-, 0 → π0π0 KS KS (155) 2 L − The parity of a two-pion state is PP==π()–1 1 − π0π0 ()2 The C-parity of a state is CC==π0 1 , and of a π+π- state: C1==()– L 1

− i.e. cp= 1 for the π+π- and π0π0 states

− K0 0 i.e. the assumption that S = K1 seem to be correct.

Oxana Smirnova & Vincent Hedberg Lund University 264 Charge conjugation and parity in weak interactions Particle Physics 0 → π+π-π0, 0 → π0π0π0 KL KL (156) − The parity of the 3-π states are -1

− π0π0π0 ()3 The C-parity of is CC==π0 1 − π+π-π0 ()Lππ The C-parity of is CC==π0 –1 1 − i.e. the 3-π final states above have cp=-1

− K0 0 i.e. the assumption that L = K2 seem to be correct. Summary: The neutral Kaon eigenstates in strong interactions are: K0 = ds K0 = sd The neutral Kaon eigenstates in weak interactions (if CP is conserved) are: 1 K0 = K0 = ------{}K0 + K0 s 1 2 1 K0 = K0 = ------{}K0 – K0 L 2 2

Oxana Smirnova & Vincent Hedberg Lund University 265 Charge conjugation and parity in weak interactions Particle Physics CP-violation The CP-violating decay

0 → π+π- KL (157) was first observed in 1964, with a branching ratio of B≈10-3.

π+

t ne ag m 0 Helium bag KL Lead collimator Spark Cherenkov chambers detectors

m a gn et π-

Figure 133: Sketch of the experiment that discovered CP-violation in weak decays.

Oxana Smirnova & Vincent Hedberg Lund University 266 Charge conjugation and parity in weak interactions Particle Physics 0 0 In general, the physical states KS and KL don’t have to correspond to pure CP-eigenstates 0 0 K1 and K2 . Instead 1 K0 = ------{}K0 + εK0 S 2 1 2 1 + ε 1 K0 = ------{}εK0 + K0 L 2 1 2 1 + ε –3 where ε is a small complex parameter: ε = 2×10 0 0 KS contains mostly K1 but has also a small 0 0 0 K2 component while KL consists mostly of K2 0 with a small component of K1 . Mixing occur also for neutral B- (B0 = 0 db, B = bd, Bs=sb and Bs=bs) and for neutral D-mesons (D0=cu and D0=uc). There can be different mechanisms for CP-violation, especially in the B0-B0 systems. Several dedicated experiments have been built to study this system.

Oxana Smirnova & Vincent Hedberg Lund University 267 Charge conjugation and parity in weak interactions Particle Physics Summary • Parity and charge conjugation a) Parity is violated in weak processes. b) Parity violation was first observed in 60Co-decays. c) Muon decays can be used to show that both parity and charged conjugation is violated while the combined CP operation is conserved. • Helicity d) Helicity is the spin quantized along the direction of motion. e) Neutrinos are left-handed and antineutrinos right-handed. f) This was first observed in reactions between electrons and 152Eu .

Oxana Smirnova & Vincent Hedberg Lund University 268 Charge conjugation and parity in weak interactions Particle Physics • V-A interactions g) While neutrinos are always left-handed other fermions are exclusively left-handed only when they are relativistic. • Neutral kaons h) The neutral kaons that are observed 0 0 0 0 experimentally (K S and K L) are due to K -K mixing i) CP-violating decays of neutral kaons have been observed with a small branching ratios.

Oxana Smirnova & Vincent Hedberg Lund University 269