Lecture 12 Weak Decays of Hadrons

Transcription

1 Lecture 12 Weak Decays of Hadrons π + and K + decays Semileptonic decays Hyperon decays Heavy quark decays Rare decays The Cabibbo-Kobayashi-Maskawa Matrix 1

2 Charged Pion Decay π + decay by annihilation of the u d into a W + boson π decay by annihilation of the dū into a W boson: u µ + (p 3 ) f π W d ν µ (p 4 ) The dominant decay mode is to a muon and a neutrino π + µ + ν µ π µ ν µ Why not to an electron and a neutrino? 2

3 Charged Pion Lifetime The matrix element for the weak decay is: M = G ( F f π q µ ū µ γ µ1 ) 2 2 (1 γ5 )u νµ where f π is the charged pion decay constant (probability that quark-antiquark annihilate inside pion) The matrix element squared in the rest frame of the pion is: M 2 = 4G 2 Ffπm 2 2 µ[p 3.p 4 ] ( Γ π = 1 τ π = G2 F 8π f2 πm π m 2 µ 1 m2 µ m 2 π Charged pion mass, lifetime and decay constant: m π + = 139.6MeV τ π + = 26ns f π = 131MeV ) 2 3

4 Helicity Suppression in π + Decays The pion has J=0, so the µ + and ν have the same helicities: The decay to an electron and a neutrino is helicity suppressed R = Γ(π eν) Γ(π µν) = m2 e m 2 µ 1 = (1 m 2 µ/m π) Experimental result proves V A theory of weak interactions: R = (1.267 ± 0.023)

5 Charged Kaon Decay The main decay modes of the charged Kaon are: A leptonic decay (similar to π + µ + ν µ ) B(K + µ + ν µ ) = 63.4% A semileptonic decay (with equal amounts of l = e, µ) B(K + π 0 l + ν l ) = 8.1% Weak hadronic decays to pions B(K + π + π 0 ) = 21.1% B(K + π + π + π ) = 5.6% Charged Kaon mass, lifetime and decay constant: m K + = 494MeV τ K + = 12ns f K = 160MeV f K f π is example of breaking of SU(3) flavour symmetry 5

6 Neutral Kaon Decays For neutral Kaons the decay eigenstates K S and K L are not equal to the flavour eigenstates K 0 and K 0 (more in Lecture 13) The main decay modes are: B(K L π l + ν l ) = 67.5% (l = e, µ) B(K L π + π π 0 ) = 12.6% B(K L π 0 π 0 π 0 ) = 19.6% B(K S π + π ) = 69.2% B(K S π 0 π 0 ) = 30.7% Neutral Kaon mass and lifetimes: m K 0 = 498MeV τ S = 0.09ns τ L = 51ns There are two very different lifetimes (short and long)! 6

7 The Cabibbo Angle The couplings of the W boson to u d and u s quarks are described by the Cabibbo angle θ C Weak coupling becomes G F G F V ud or G F V us d s = V ud V cd V us V cs d s = cosθ C sinθ C sinθ C cosθ C d s Interpret this as a rotation matrix between the flavour eigenstates d, s and the weak eigenstates d, s The Cabibbo angle is measured to be: θ C = 12.7 sinθ C = cosθ C =

8 Semileptonic Decays & Selection Rules The matrix element for semileptonic Kaon decays is: M f + [ (pk + p π ) µ lγ µ (1 + γ 5 )ν ] + f [ ml l(1 + γ 5 )ν ] where f + and f are semileptonic decay form factors which describe the hadronic transitions K π The f term multiplying m l is negligible for electrons Semileptonic kaon decays obey the selection rules: I = I 3 = 1 2 Q = S = 1 Hadronic kaon decays obey the selection rules: I = 1/2, 3/2 (1/2 preferred) I 3 = 1 2 S = 1 8

9 Hyperon Decays Baryons containing strange quarks are known as hyperons With one exception they all have weak decays: Hyperon Quark Content Decay Modes Lifetime Λ uds pπ, nπ ns Σ + uus pπ 0, nπ ns Σ 0 uds Λγ s Σ dds nπ 0.15ns Ξ 0 uss Λπ ns Ξ dds Λπ 0.16ns Ω sss ΛK, Ξ 0 π 0.08ns Lifetimes s Decay Lengths 1cm are observable 9

10 Heavy Quark Decays Charm quark decays are mainly c s (a few percent c d) Examples D Klν, D Kπ, D Kππ Lifetimes τ D + = 1.04ps, τ D 0 = 0.41ps Bottom quark decays are mainly b c (a few percent b u) Examples B Dlν, B Dπ, B J/ψK 0 S Lifetimes τ B + = 1.64ps, τ B 0 = 1.53ps The proper decay lengths of b and c hadrons are µm The top quark decays almost completely t b Its lifetime is too short to form hadrons! 10

11 Semileptonic & Rare b Decays Inclusive and exclusive semileptonic decays (BaBar/Belle) B(b clν l ) = (10.75 ± 0.15)% B(B Dlν l ) = (2.2 ± 0.1)% B(B D lν l ) = (5.6 ± 0.5)% B(b ulν l ) = (1.3±0.1) 10 3 B(B πlν l ) = (1.4±0.1) 10 4 Determine CKM couplings V cb and V ub Flavour-changing neutral currents (CLEO/BaBar/Belle) B(b sγ) = (3.5 ± 0.3) 10 4 B(B K γ) = (4.5 ± 0.2) 10 5 These are second-order weak decays ( penguin loops) Set limits on many New Physics models 11

12 Leptonic Decays of Heavy Quarks Leptonic D decays measured by CLEO-c experiment (2008) B(D + µ + ν µ ) = (4.4 ± 0.6) 10 4 B(D s + µ + ν µ ) = (6.2 ± 0.6) 10 3 B(D s + τ + ν τ ) = (6.6 ± 0.6) 10 2 Leptonic B decay measured by BaBar/Belle experiments (2006) B(B + τ + ν τ ) = (1.4 ± 0.4) 10 4 Determine decay constants f D, f Ds, f B Set limits on possible charged Higgs couplings 12

13 Cabibbo-Kobayashi-Maskawa Matrix Kobayashi & Maskawa awarded Nobel prize in October 2008! By extension from the Cabibbo angle, the full description of weak decays of quarks needs the 3 3 CKM matrix: d V ud V us V ub d s = V cd V cs V cb s b V td V ts V tb b The CKM matrix is unitary, and its elements satisfy: Vij 2 = 1 Vij 2 = 1 i V ij V ik = 0 i j V ij V kj = 0 The CKM matrix can be written in terms of just four parameters j 13

14 CKM Parametrizations With three angles s i = sin θ i, c i = cos θ i, and a complex phase δ: c 1 s 1 c 3 s 1 s 3 s 1 c 3 c 1 c 2 c 3 s 2 s 3 e iδ c 1 c 2 s 3 + s 2 c 3 e iδ s 1 s 2 c 1 s 2 c 3 c 2 s 3 e iδ c 1 s 2 s 3 + c 2 c 3 e iδ Wolfenstein parametrisation is expansion in powers of λ = sinθ C : 1 λ 2 /2 λ Aλ 3 (ρ iη) λ 1 λ 2 /2 Aλ 2 Aλ 3 (1 ρ iη) Aλ

15 Measurements of CKM Elements V ud = from pion and nuclear β decays V us = from Kaon and Hyperon decays V cs = 0.97 ± 0.12 from D Klν semileptonic decays V cd = ± from neutrino production of charm V cb = ± from b clν semileptonic decays V ub = ± from b ulν semileptonic decays V td and V ts are measured in B meson mixing (Lecture 13) V tb 1 is measured in top decays at the Tevatron Wolfenstein parameters: λ = ± A = 0.80±0.03 ρ = 0.19±0.08 η = 0.36±