Topics in Standard Model. Alexey Boyarsky Autumn 2013

Topics in Standard Model Alexey Boyarsky Autumn 2013

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

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/0001283 Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 2

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 = 2 10 13 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 2.3 2.5) Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 3

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

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

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

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/0001283 Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 7

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) 161-177 DOI: 10.1007/BF01351864 Trends to a Theory of beta Radiation. (In Italian) Nuovo Cim. 11 (1934) 1-19 DOI: 10.1007/BF02959820 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/0001283], Sec. 1,1); Cheng & Li, Chap. 11, Sec. 11.1) Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 8

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 = 1.166 10 5 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

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

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

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

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 6.6 10 25 sec! Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 13

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 = 939.5 MeV and (m n m p ) 1.3 MeV. Alexey Boyarsky STRUCTURE OF THE STANDARD MODEL 14

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

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

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