Lecture 11 Weak interactions, Cabbibo-angle. angle. SS2011: Introduction to Nuclear and Particle Physics, Part 2

Transcription

1 Lecture 11 Weak interactions, Cabbibo-angle angle SS2011: Introduction to Nuclear and Particle Physics, Part 2 1

2 Neutrino-lepton reactions Consider the reaction of neutrino-electron scattering: Feynman diagram with W-boson exchange (OBE) (1) Matrix element (cf. Lecture N 10): (2) q For small q: (3) 2

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5 Trace with g 5 -matrix Consider Tr (Spur) with g 5 -matrix: 5 ( g ) 2 = 1, 5 m { g, g } = 0 a) indeed, for any component for m = n - g m g n = ± 1 Þ Spur( g ab / / ) = 5 Spur g 5 = 0 for m ¹ n - Spur ( g g 5 m g n ) = 0 b) due to the odd number of g s c) total antisymmetric tensor E.g. 5

6 Neutrino-lepton reactions M mn - n m- m-tensor : (12) E mn - e-n e -tensor: (13) due to averaging over spin Consider the different terms in this term is equal zero since whereas - is antisymmetric in m,n is a symmetric tensor, Consider, e.g., a component with which can be 1 or 3 6

7 Neutrino-lepton reactions Finally, (14) Matrix element squared: (15) (16) The matrix element squared doesn t depend on the scattering angle q! 7

8 Neutrino-lepton reactions Differential cross section in the cms: (17) (18) Total (angle integrated) cross section: for s>>m m 2 for s>>m m 2 (19) In the Lab. system (20) 8

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10 Antineutrino-lepton reactions q Use the Crossing symmetry for matrix elements: According to (14) the matrix element for is Thus, the martix element for using the crossing symmetry for (23): (23) (24) q Kinematics in the CMS: (25) For we have (26) 10

11 Antineutrino-lepton reactions The differential cross section in the CMS reads: (27) Angular momentum J=0 J=1 J Z =1 11

12 Antineutrino-lepton reactions The total cross section then reads (28) Factor 1/3 since for the reaction the total angular momentum is J=1 and only the J Z =1 state is realized from 3 possible (J Z =-1,0,1) combinations, whereas for the total angular momentum is J=0. q Neutrino, antineutrino reactions with antileptons (e + ) can be calculated in full analogy to the reactions with leptons (e - ) (29) 12

13 (Anti-)neutrino )neutrino-nucleon nucleon reactions Quark constituent model: the reactions on a nucleon è reactions on (3 valence) quarks One would expect that for valence quarks From experiment è è scattering on other constituents, not only on valence quarks è interaction with the q-qbar sea 13

14 Weak interaction with hadrons q Consider the weak decay of strange baryons: L : uds p : uud (30) total isospin strangeness n : udd - p : ud p 0 1 : 2 ( uu - dd ) èrule: Quantum numbers are changed by the weak hadronic decay : DI=1/2, DS=1 ès-quark transforms to a light quark (u or d) Weak interaction changes the quark flavor! The ratio: - from Clebsch-Gordon-Coefficient - from experiment 14

15 Weak interaction with hadrons q Consider the weak decay of strange mesons: (31) strangeness charge K - : us K + : us p 0 : 1 2 ( uu - dd ) è Weak interaction changes the quark flavor! q Rule: in semileptonic weak decays the strangeness and charge are changed by the same value: DQ=1, DS=1 (32) Experimental proof for the rule (32): 1) Branching ratio for the decay is , here DS=DQ 2) Branching ratio for the decay is < , here DS=-DQ since one has to change two quarks uusàudd - S : dds + S : uus n : udd 15

16 Weak interaction with hadrons Thus, for the weak decay of strange hadrons: q for semileptonic weak decay :DQ=1, DS=1 q for weak hadronic decay : DI=1/2, DS=1 q Changes of the quark flavor can be interpreted as an emission of a W-boson DQ=1, DS=1 DQ=1, S=0 E.g.: From experiment we know that the weak decay processes with changes of strangeness DS=1 are supressed by a factor of 20 compared to the non-strange hadronic decay, 0 - i.e. L pp is supressed compared to n pe è Cabbibo model v e 16

17 Weak interaction with hadrons, Cabbibo model ècabbibo model (1963): Hamiltonian for strong+weak interaction: H = H strong + H week H strong æ H = ç è 0 s 0 H s ö ø H week æ 0 = ç è H W H 0 W ö ø H æ H µ ç è H s W H H W s ö ø (33) èeigenstates (in flavor space) of the strong hamiltonian H S have to be rotated by an angle q C in order to be eigenstates of the total strong+weak interaction hamiltonian H. The rotation is done by the unitary matrix U: U i.e. the physical states (d,s ) are a superposition of d and s quarks: (34) (35) q C - Cabbibo angle 17

18 Weak interaction with hadrons, Cabbibo model Angle q C - Cabbibo angle - is a measure of the amplitude that one flavor of quark (either down or strange) will change into another flavor (up) under the action of the weak force. Thus, the weak current for the d àu+w is u(d ) (36) 18

19 Weak interaction with hadrons, Cabbibo model Cabbibo angle q C from the weak meson decay: (37) From experiment: 19