Topics in Standard Model. Alexey Boyarsky Autumn 2013

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1 Topics in Standard Model Alexey Boyarsky Autumn 2013

2 New particles Nuclear physics, two types of nuclear physics phenomena: α- decay and β-decay See Introduction of this article for the history Cosmic rays, first accelerators produced many new particles (positrons, muon, pions,... ) These phenomena did not find their explanation in the framework of QED Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 1

3 Weak & strong interactions α-decay Strong interactions: Discovery of proton (1919) Discovery of neutron (1932) Yukawa theory (1935). Prediction of strongly interacting particle with mass 100 MeV/c 2 Discovery of pion (1947) β-decay Weak interactions: Measurement of continuous spectrum of electrons in β decay (1927) Prediction of neutrino (1930) Discovery of positron (1932) Fermi theory (1934) Discovery of muon (1937) Demonstration that muon interacts too weakly for nuclear forces (1947) See timeline here: hep-ph/ Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 2

4 α-decay and Yukawa theory 1 Electromagnetic interactions are mediated by exchange of photons Yukawa s idea proton and neutron exchange some new massive particle to interact in the nuclei H. Yukawa On the interaction of elementary particles Proc.Phys.Math.Soc.Jap. 17 (1935) 48 Static p n interaction was known to decrease rapidly at the distances 2 fm = cm. The idea of Yukawa there is p n interaction of the form Yukawa 1935 where a 1 fm U(r) = g2 4πr e r/a (1) 0 This presentation closely follows Aitchison & Hey, Sec. 2.2 (see also Sec. 2.1 and ) Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 3

5 α-decay and Yukawa theory Yukawa interaction Electromagnetic interaction U(r) = g2 4πr e r/a short ranged U(r) = e2 long ranged 4πr Poisson equation ( ) 2 1 a U(r) = g 2 δ(r) 2 U(r) = e 2 δ(r) 2 Free-space propagation ( 2 1c 2 2 t 2 1 ) a 2 U(r) = 0 ( 2 1c 2 ) 2 t 2 U(r) = 0 ) De Broglie solution U exp implies that E 2 = p 2 c 2 + c2 2 Mass of quanta : m Y ac ( ip x iet a 2 E 2 = c 2 p 2 Mass of quanta= 0 Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 4

6 Massive Klein-Gordon equation Build the Green s function of the Klein-Gordon equation ( + m 2 )φ(x) = δ (4) (x x ) use m 2 m 2 iɛ to obtain causal Green s function Find its spatial representation. Consider two cases: (x x ) µ = (t, 0) and (x x ) µ = (0, x)). Find behavior of the Green s function when t or x Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 5

7 Virtual quanta Yukawa theory predicted a particle with the mass m Y 100 MeV/c 2 (from a 2 fm), responsible for proton-neutron (nuclear) interactions m n m p 1.3 MeV/c 2 m Y. Therefore an emission process n p + Y is impossible for real particle Y (energy conservation violated) Exchange by the virtual particle mediates the proton-neutron interaction Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 6

8 Weak & strong interactions α-decay Strong interactions: Discovery of proton (1919) Discovery of neutron (1932) Yukawa theory (1935). Prediction of strongly interacting particle with mass 100 MeV/c 2 Discovery of pion (1947) β-decay Weak interactions: Measurement of continuous spectrum of electrons in β decay (1927) Prediction of neutrino (1930) Discovery of positron (1932) Fermi theory (1934) Discovery of muon (1937) Demonstration that muon interacts too weakly for nuclear forces (1947) See timeline here: hep-ph/ Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 7

9 Fermi theory of β-decay 3 Fermi 4-fermion theory: Neutron decay n p + e + ν e Two papers by E. Fermi: An attempt of a theory of beta radiation. 1. (In German) Z.Phys. 88 (1934) DOI: /BF Trends to a Theory of beta Radiation. (In Italian) Nuovo Cim. 11 (1934) 1-19 DOI: /BF Continuum spectrum of electrons (1927) Prediction of neutrino (1930, 1934) Fermi theory (1934) Universality of Fermi interactions (1949) L Fermi = G F 2 [ p(x)γ µ n(x)][ē(x)γ µ ν(x)] (2) New phenomenological constant, G F, Fermi constant. 1 History of β-decay (see [hep-ph/ ], Sec. 1,1); Cheng & Li, Chap. 11, Sec. 11.1) Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 8

10 Fermi theory of β-decay 4 Dimension of the Fermi constant?. [L] = [G F ][( ψψ) 2 ]. Dimension of Lagrangian [L] = M 4. 2 M ψψ [ ψψ] = M 3 = [G F ] = M 2 Mass operator Value determined experimentally to be G F precisely G F ( c) 3 = GeV 2.) 10 5 GeV 2 (More Fermi Lagrangian includes leptonic and hardronic terms: L Fermi = G [ ] F J 2 lepton (x) + J hadron (x) [ ] J lepton (x) + J hadron (x) (3) Lagrangian (3) predicts universality of weak interactions: all processes in Fermi theory can be described in terms of only one constant, G F 2 Dimension [Aµ ] = [ µ ] = M. Therefore [L] = [F 2 µν ] = M 4. Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 9

11 Neutrino-electron scattering Fermi theory predicts lepton-only weak interactions, such as e + ν e e + ν e scattering L νe = 4G F 2 (ēγ λ ν e )( ν e γ λ e) (4) Matrix element for e + ν e e + ν e scattering M 2 = e, ν e L Fermi e, ν e 2 (summed over spins of outgoing particles and averaged over spins of initial particles) spins M 2 G F ū(p e )γ λ u(p in e )ū(p ν )γ λ u(p in ν ) 2 G 2 F (p in e p in ν )(p ν p e ) G 2 F E 4 c.m. (5) Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 10

12 Neutrino-electron scattering center-of-mass energy of the system is E 2 c.m. = 1 4 (pin e + pin ν )2 (p in e pin ν ) for E c.m. m e momentum of one of the incoming particles p c.m. E c.m. for E c.m. m e Cross-section: ν e p in ν e p in e e G F ν e p νe p e e dσ = 1 d 3 p e 4 p c.m. E c.m. (2E e ) d 3 p ν (2E ν ) M 2 δ(e c.m. E e E ν )δ 3 (p e + p ν ) (6) Reproduce this calculation using the Lagrangian (4). Keep in mind that neutrino is a purely massless two-component spinor, i.e. γ µ p µ (1 + γ 5 )ν(p) = 0 where 1 2 (1 + γ 5) is the projector that selects only two (left) component of any 4-component spinor Show that any solution of the Dirac equation for neutrino depending only on t and one spatial coordinate (z) is left-moving (i.e. the wave-function has the general form ψ(t, z) = f(t + z)) Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 11

13 Low energy weak processes Muon (µ ) discovered in cosmic rays (charge of the electron, mass 100 MeV) was first confused with the meson (predicted by Yukawa theory). Originally called µ-meson Later π mesons and other mesons were discovered. They were all similar, but µ was not interacting with nuclear forces as any meson So muon turned out to be just a heavy electron Fermi Lagrangian (3) was extended so that leptonic current J lepton would include muon: J λ lepton = ν eγ µ e + ν µ γ λ µ... describing for example, the muon decay µ e + ν e + ν µ Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 12

14 Low energy weak processes implies the existence of muon neutrino ν µ discovered in 1962 Muon mass m µ 105 MeV. m µ m e. In the rest frame of muon e, ν e, ν µ can be taken as approximately massless. 5 The answer should be constructed as a product of two dimensionful quantities. Can construct the only object with dimension of time, proportional to G 2 F.6 Decay width Γ µ G 2 F m5 µ some powers of π... Find the exact answer: Γ = G2 F m5 µ 192π 3 using the analog of Lagrangian (4) where one ēγ λ ν e term is substituted with µγ λ ν µ term 5 There can be a caveat here! Will see it when analyzing pion decay 6 Actually, we will talk about decay width with the dimension sec 1 or equivalently, GeV. Recall that decay width of 1 GeV corresponds to the lifetime sec! Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 13

15 Low energy weak processes Notice, that similar argument would not work on the neutron s decay: n p + e + ν e as this time m n m p and therefore one cannot simply write Γ n G 2 F m5 n. Rather it should be some combination of m n = MeV and (m n m p ) 1.3 MeV. Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 14

16 Massive intermediate particle We saw in Eq. (5) that matrix element for e + ν e e + ν e scattering grows with energy, M 2 G 2 F E4 c.m.. Unitarity means that this element should be bounded from above: M const Therefore, the Fermi theory would predict meaningless answers for scattering at energies E c.m. G F 300 GeV The situation would be ameliorated if G F were energy dependent and decreasing with energy Promote point-like 4-fermion Fermi interaction to interaction, mediated by a vector boson : L Fermi L W = g(w µ J µ + h.c.) 1957 Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 15

17 Massive intermediate particle Propagator of the massive vector boson: η µν p µp ν M W W µ (p)w ν ( p) = i 2 p 2 MW 2 ; σ E 2 (E 2 M 2 W )2 When mass M W p reduces to Fermi Lagrangian Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 16

18 Vector boson vs. photon Propagator of the massive vector boson: η µν p µp ν M W W µ (p)w ν ( p) = i 2 p 2 MW 2 + iɛ Recall: photon propagator: A µ (p)a ν ( p) = iη µν p 2 +iɛ Try to put M W 0. Will you recover photon-like propagator? No! Trouble with the term in the numerator Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 17