6. ELECTROWEAK INTERACTIONS Question: why do mm+-and have same lifetime? Answer: CP-conservation (weak interaction is invariant under combined operation CP, even though both C and P are Charged and neutral currents separately violated). CP operator transforms particles at rest to corresponding antiparticles at Neutral current reactions fi mediated by Z0 boson; charged current reactions fi mediated rest; CP-invariance fi these states have identical properties. In muon decays, P changes q to by W ± boson. Electroweak interaction: unifies weak and electromagnetic interactions, but p - q, while C changes particles to antiparticles. Hence CP-invariance alone fi GG+-()cosqq=-() cos . Substituting G (cosq ), gives GG= (equal lifetimes) and unification only becomes manifest at very high energies; at low energies weak and mm m ± +- electromagnetic interactions still clearly separated. Predicts existence of new spin-0 boson xx+-=- (CP-invariance) in agreement experiment. (Only known violations of CP are very (Higgs boson) associated with origin of particle masses within the model. small effects that occur in the decays of some neutral mesons.) Symmetries of the weak interaction Spin structure of weak interactions Will consider weak interactions alone and deduce general properties valid at all energies based on parity (P) and charge conjugation ()C operators. First direct demonstration of (a) Neutrinos: Use helicity states (spin quantised along direction of motion of particle). 60 60 * - parity violation: study of CoÆ++ Ni e ne in magnetic field and cooled to 0.01K. At Definitions for spin-1/2 particle: nL means left-handed neutrino (spin antiparallel to this temperature, interaction of magnetic moments of nuclei with magnetic field overcomes momentum), eR right-handed electron etc. Fact: only left-handed neutrinos nL and right- tendency to thermal disorder fi nuclear spins align parallel to the field direction fi cobalt- handed antineutrinos nR are observed in nature. Violates C-invariance (neutrinos and 60 nuclei (partially) polarized. Parity violation established by observation of “forward- antineutrinos have identical weak interactions) and P-invariance (nnand have identical backward decay asymmetry”, i.e. fewer electrons emitted in forward hemisphere than in LR weak interactions), but compatible with CP-invariance: backward hemisphere with respect to spins of the decaying nuclei. Follows because parity reverses particle momenta p while leaving orbital angular momenta rpŸ (and by analogy spin angular momenta), unchanged. In rest frame of decaying nuclei, effect is: - 152 152 * Helicity of neutrino first measured in reaction eEuJ+=Æ=+()01Sm () J ne . Consistent with occurrence of left-handed neutrinos only. (Later experiments have shown that only right-handed antineutrinos take part in weak interactions.) Parity invariance requires rates for both processes to be equal fi equal numbers of electrons emitted in forward and backward hemispheres with respect to the nuclear spins – (b) Pions and muons: Spin dependence of V-A form. V denotes proper vector, (direction contradicted by experiment. Charge conjugation operator C changes all particles to reversed by parity: example p) Because parity not conserved in weak interactions fi antiparticles – also not conserved in weak interactions. corresponding weak charged current has another component whose direction is unchanged by a parity transformation – called an axial-vector (A) (example: L = r Ÿp) Important result: in C-violation and P-violation in leptonic decays mnn--Æ++e and mnn++Æ++e of - e m e m the limit that their velocities approach that of light, only left-handed fermions nLL, e etc polarized muons. In rest frame of muon, angular distributions of final leptons are of the form + and right-handed antifermions nRR, e etc are emitted in charged current interactions -+ (“forbidden” helicity states eeRL, etc are suppressed by factors typically of order 1 È x± ˘ GG± (cosq )=-1 cosq 24 2 m 2 ± ÎÍ 3 ˚˙ mc2 E , where m is the appropriate fermion mass.) ++l q = angle between muon spin direction and lepton momentum, x± = asymmetry parameters, Illustrated by pnÆ+l. In rest frame of pion, charged lepton and neutrino recoil in -1 - + opposite directions fi spins must be opposed (angular momentum conservation about decay G± = total decay rates (lifetimes t± ∫G± ). C transforms m decay to m decay fi GG+-= axis). Since neutrino is left-handed fi charged lepton must also be left-handed, in and xx+-= (C-invariance). Parity preserves identity of particles, but reverses their momenta contradiction to the expectations for a relativistic antilepton. For positive muon this is fi q Æ fi GG±±cosqq=- cos while leaving spins unchanged pq- , mm() () (P-invariance). ± unimportant, since it recoils non-relativistically so both helicity states are allowed. However, Substituting for G ± (cosq ) fi x± = 0 (P-invariance). Experimentally, m lifetimes equal to 25- m a positron does recoil relativistically fi this mode is suppressed by 22610().mme p = x . very high precision fi prediction for lifetimes satisfied; but xx-+=- =100.. ± 004 fi both Thus positron decay mode is predicted to be much rarer than the muonic mode: measured C-invariance and P-invariance violated. 6.1 6.2 -4 2 h ratio is ()1..218± 0 014x 10 (agrees with calculation). Helicity argument fi muons emitted Strength of each vertex is given by (dimensionless parameter) apww=ªgc41400 for all in pion decays are polarized (also agrees with experiment). three generations. Thus the weak interaction has a similar intrinsic strength to the electromagnetic interaction. Its apparent weakness in many low-energy reactions, is because W± and Z0 bosons the exchange bosons are heavy (effect of propagator). WW+-and and Z0 all discovered at CERN in 1983 in the reactions Weak interactions of hadrons ± +- -+ 00 Understood in terms of basic processes in which W bosons are emitted or absorbed by their pp+Æ W + X,, pp +Æ W + Xand pp +Æ Z + X - constituent quarks, e.g. neutron decay: dueÆ+ +ne: XX± and 0 are arbitrary hadronic states allowed by conservation laws. Beams produced using a proton-antiproton collider (built specifically for this purpose). At the time it had proton and antiproton beams with maximum energies of 270 GeV each, (total centre-of-mass energy of 540 GeV). Two independent experiments were mounted, one (called UA1) shown schematically in Fig.6.11; example of the modern type of layered detector system. WZ± and 0 boson masses are: 22 MWZ==80.,. 6GeV / cM 91 2 GeV / c lifetimes are about 3x10-25 s. Dominant decays lead to jets of hadrons: -- and pion decay: pmn()du Æ+m : Also leptonic decays: Understood in terms of two ideas: lepton-quark symmetry, and quark mixing. (For simplicity, WW++Æ+llnn, -- Æ+ and ZZ00Æ+ll+-, Æ+nn ll l l just consider two generations of quarks.) Lepton-quark symmetry: first two generations of quarks and first two generations of leptons have identical weak interactions fi can obtain ± 0 In contrast to the zero mass photons and gluons, the WZand are very massive particles W ±-quark vertices from replacements nnmÆÆue,,,-- d Æ c Æ s in basic W ±- -3 e m fi very short ranges, RRWZªª¥210 fm. ± lepton vertices, leaving the coupling constant gW unchanged. Resulting W –quark vertices: As in QED, Feynman diagrams for weak interactions are constructed from fundamental three line vertices. For lepton-quark (charged current) interactions: 6.3 6.4 - - weak mixing angle q is given by ; = coupling constant which fi fundamental processes du+Æ W and s+cÆ W occur with same couplings gW as W cosqWWZ∫ MM gz corresponding leptonic processes, while processes su+Æ W- and dc+Æ W- are characterises strength of neutral current vertices. forbidden, i.e. gg== g. But: many decays forbidden in this simple scheme actually cs ud W 0 occur, albeit at a rate which is suppressed relative to the “allowed” decays. Example: Neutral current interactions accounted for in terms of basic Z -lepton vertices. K --Æ+mn, which requires su+Æ W- vertex that is not present in the above scheme. Corresponding quark vertices can be obtained from the lepton vertices by using lepton-quark m symmetry and quark mixing. Resulting set of vertices is: All suppressed decays can be successfully incorporated into theory by quark mixing: d and s quarks participate in the weak interactions via the linear combinations dd¢ =+cosqqCC s sin and sd¢ =-sinqqCC + s cos qC = Cabibbo angle. Lepton-quark symmetry now applies to the doublets Ê u ˆ Ê cˆ Á ˜ and Á ˜ Ëd¢¯ Ës¢¯ fi neutral current interactions, like electromagnetic interactions, conserve quark numbers Generates new vertices previously forbidden. Example: usW vertex required for NNNu ,,d s ..., in contrast to the charged current interactions that do not conserve them. -- K Æ+mnm arises from interpretation of ud¢ W vertex shown: Finally, in any process in which a photon is exchanged, a Z0 boson can be exchanged as well. At energies that are small compared to the Z0 mass, the Z0 -exchange contributions can be neglected compared to the corresponding photon exchange contributions, and these reactions can be regarded as purely electromagnetic to a high degree of accuracy. However, at very high energy and momentum transfers, Z0 -exchange contributions become comparable with photon exchange, and we are therefore dealing with genuinely electroweak processes which involve both weak and electromagnetic interactions to a comparable degree. Illustrated by cross-section ee+-+Æmm + + -. Assume energy is large enough for the Hypothesis enables theory and experiment to be brought into good agreement by choosing o lepton masses to be neglected, then centre-of-mass energy E is the only quantity in the q C = 13 . Rates for previously “allowed” decays occur at rates which are suppressed by system that has dimensions. Because cross-sections have dimensions of an area, a simple 2 cosqC ª 095 . , while previously “forbidden” decays are now allowed, but with rates dimensional argument gives the electromagnetic cross-section for one-photon exchange to be 2 saª 222()hcE.