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The Weak Interaction

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The Weak Interaction PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 47 Wednesday, April 15, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: The Weak Interaction. The Weak Interaction The Beta Decay The history of the weak interactions can be traced back to the beta decay process. The energetics of neutron decay indicates that there must be an unseen and electrically neutral particle. The conservation of spin in the neutron decay process requires that it must be a fermion. In 1930 Pauli postulated the existence of this particle and in 1932 Fermi named it as the neutrino (“the little neutral one” in Italian). The neutron decay process is − n ! p + e + νe; (1) where νe is the electron anti-neutrino. The astonishingly long neutron decay time (about 10 minutes) demands that the mediating interaction be very, very weak. We now know that the neutrinos indeed are an almost omnipresent indicator of any weak process. They are part of a family of fermions known as leptons. In Table. 1 we show the lepton family: it contains the trio corresponding to the charged particles: the electron, the muon, and the tau particle, and their respective neutrino cousins. The anti-leptons have the same mass and opposite charge to that of their leptonic counterparts. The Fermi Model of the Weak Interaction All fermions participate in the weak interaction. But many weak interaction processes have a strong or electromagnetic pathway, which dwarfs and thus masks the weak interaction contribution. Electrons repel one another weakly, just as they do electrically, but unless the impact parameter is PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 − l νl ml (MeV) − e νe 0.511 − µ νµ 105.7 − τ ντ 1776.8 Table 1: The three generations (or flavors) of leptons, with the corresponding mass of the charged leptons. We do not yet know the masses of the neutrinos but the existence of mass (as well as their mass differences) is constrained by neutrino oscillation. on the scale of an atomic nucleus, the contribution of the weak interaction is effectively zero for any realistic system. However, processes involving neutrinos have no electromagnetic pathway, as is the case for neutron decay. While the ultimate description of the weak interaction will involve a mediator, as a first attempt, Fermi in 1933 proposed an interaction of the form µ LFermi int = 2GF pγ n eγµ ν ; (2) where the subscript for each spinor refers to the particle type, and GF is known as the Fermi constant. We know that this Lagrangian is incorrect in a number of ways. Protons and neutrons are composite particles; a more accurate calculation would involve a down quark decaying into an up quark. Another effect that is not incorporated in this Lagrangian is that the weak interaction violates parity. Also, Fermi theory does not incorporate a mediator particle. The value of the Fermi constant is approximately −2 GF = (292:8 GeV) : (3) (Note that this coupling constant has a negative mass dimension and thus this theory is not renormalizable.) Against all these inadequacies, this model can compute the decay rate of a station- ary neutron to a moderate accuracy: It gives a neutron lifetime of about 1300 s, which is very close to the true neutron decay time of 882 s. Thus the 4-Fermi theory, which is the low-energy limit of a massive vector gauge theory (we will see the details of this theory later), completely characterizes the most familiar effect of the weak force: beta decay. At high energy, there is no weak force, per se, only an electroweak force, which is spontaneously broken down to the electric and weak forces at low energy. There are many aspects of electroweak physics that only become apparent at high energy, such as the existence of the weak force mediators: the W and Z bosons. 2 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 The Weak Force Mediators There must be a mediator for the weak interaction. Indeed, there three: the W + and the W − (which are antiparticles of one another) and the neutral Z0 (which, like a photon, is its own anti-particle). The masses of the weak mediators are huge. Experimentally measured masses are: - MW = 80:385 ± 0:015 GeV, - MZ = 91:1876 ± 0:0021 GeV. For interactions significantly less energetic than the mediator masses, the virtual mediators have a very short range 1 ' 2:5 × 10−18 m; (4) MW well confined to the interior of an atomic nucleus. Explaining the mechanism for giving mass to these weak mediators turned out to be one of the major physics discoveries of our time. It resulted in the 2013 Nobel Prize in Physics for Englert and Brout, and Higgs. The Fermionic Doublet Under weak interactions, particles can be sorted into doublets. Let us consider the symmetry under SU(2). We begin with a doublet of bi-spinors ! Ψ = u : (5) d Conceptually, these two components are as similar to one another as a spin-up electron is to a spin-down. Within the context of the weak interaction, the particle doublet might be ! ! ν u Ψ = e or ; (6) e− d though we will focus on the lepton doublet for the time being. We may also define an adjoint Ψ = (νe e): (7) Just as we will use e to represent the bi-spinor of an electron, e will be the adjoint of an electron, not anti-particle. Under the assumption that all fermions are massless (which is not true), the free-field doublet Lagrangian may be written as µ L = iΨγ @µΨ: (8) 3 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 To the fermionic doublet let us apply a global SU(2) gauge transformation of the form a a Ψ ! e−igα T Ψ; (9) a a Ψ ! Ψeigα T ; (10) where g is a parameter, which would serve as the weak coupling constant later, αa are parameters a 1 a and T = 2 σ are generators of the gauge transformation. We note that the above transformations must leave ΨΨ = νeνe + ee (11) invariant. The statement that Lagrangians of Ψ are invariant under SU(2) transformations means that as far as the weak interaction is concerned, the two particles in the doublet are interchangeable. That is, turning all electrons to neutrinos and vice versa will not change the weak interaction calculation. The states 1 jeei; jνeνei; p (jνeei + jeνei) (12) 2 should all be energetically identical under weak interaction. Electromagnetism distinguishes very strongly between the two particles, as one (the electron) has charge and the other (the neutrino) does not. However, the strong force will ignore both neutrinos and electrons equally. We find that, in this notation for gauge invariant quantities, the lepton number L (specifically electron number Le here) is conserved. Anti-particles have a lepton number of −1. We have - Electron, neutrino: Le = 1 - Positron, anti-neutrino: Le = −1 The relation of νe to e is the same as the relation of spin up to spin down, and accordingly, we can introduce a quantity called the weak isospin, normally labelled as T3. We have 1 T = + (for the upper component of doublet); (13) 3 2 1 T = − (for the lower component of doublet): (14) 3 2 Helicity In 1956 Chien-Shiung Wu demonstrated that the weak interaction has handedness. In experimental weak decay reactions all neutrinos are created so that they are left handed, with anti-neutrinos produced as right handed. We say that “neutrinos are left handed” but in fact it is nearly impossible to directly measure the spin of a neutrino. We have to infer it from a neutrino’s partners in a reaction. Since lepton number is conserved, electrons and other charged leptons are often produced in weak interactions as well, and they are non-relativistic. 4 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 For a spin-up fermionic particle propagating in the +z-direction we have 0q m 1 E+p B C B 0 C B C u+(p) / Bq C : (15) B E+p C @ m A 0 In Weyl basis, the upper two components correspond to a left-handed helicity, while the bottom two correspond to a right-handed one. For massless particles, there is no distinction between the helicity (the dot product of the spin and the momentum) and the handedness. If a massless neutrino is left handed in one Lorentz frame, it will be left handed in all Lorentz frames. Let us look at the weak channel decay of a pion, which is a spinless composite particle made of a quark and an anti-quark. For π−, which is composed of ud, we have the decay channels − − π ! e + νe; (16) − − π ! µ + νµ: (17) Since pions are spin-0, the spins of the outgoing particles need to cancel each other. If the electron is spin-up, then the anti-neutrino must be spin-down, but with opposite momenta. Thus, we would expect that both should be left-handed or both right-handed. See Fig. 1. Figure 1: Since pions are spin-0, the spins of the outgoing particles need to cancel each other. If the electron is spin-up, then the anti-neutrino must be spin-down, but with opposite momenta. Due to the reasons we do not know yet, the theory of weak interaction presents itself as a chiral theory. That is, the the weak force must make the distinction between left- and right-handedness or particles, and thus parity by itself is not a symmetry of the weak force.
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