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1 Theory of Pion Decays 1 1 Theory of Pion Decays There could be no fairer destiny for any physical theory that that it should point the way to a more comprehensive theory in which it lives on as a limiting case Albert Einstein This section will review the underlying theory and consequences of the different pion decay channels. Although this work is mostly interested in the π ! lνl(γ) decays it will would incomplete without a small discussion of muon decays as it is the biggest background in the PEN experiment. Muon decays alone can also be used to search for Physics Beyond the Standard Model. It would also be an injustice if the pion beta decay was not mentioned since this work is the successor to the PiBeta experiment which looked at this particular channel in great detail. + + 1.1 Michel and Electronic Decay: π ! l (γ)νl All decay processes can be understood through Fermi’s gold rules, where the probability of a process to 2 transition from state i to f, Γfi is proportional to j h f jHintjii j ρ where ρ represents the density of states, a 2 kinematics factor that imposes energy and momentum conservation, and j h f jHintji j is related to the inter- action itself. This term can be transformed into the square of a matrix element by normalizing the initial and final state wavefunctions in a manner that adheres to Lorentz invariance The differential decay rate for π ! lν is 1 1 d3 p d3 p dΓ = jMj2 l ν δ4(q − p − p ) 3 3 l ν 2mπ 2El2Eν (2π) (2π) Where mπ is the pion mass, q; pl, and pν are the four momenta of the pion, lepton, and neutrino, and M is the matrix element of the interaction. This matrix element would originate from the Lagrangian (or Hamiltonian, whichever you prefer). The type of interaction which occurs determines the decay rate of a + + 1.1 Michel and Electronic Decay: π ! l (γ)νl 2 particular process. To get a sense of the type of interactionss that could be involved one must go back to the Fermi era where β− decays were first being studied in detail. By constructing the Lagrangian for the beta decay, it is possible to see what type of interactions occur. −G h i F ¯ ¯ λ ¯ ¯ λ p pγλ n eγ νe + nγλ p νe γ e 2 The expresson on the left in side the brackets represent the process n!pe¯νe, neutron beta decay. The second expression is electron capture. This Lagrangian was constructed in analogy to quantum electrodynamics. However, Fermii, Gammow, and Teller [12] argued that a general method to construct the Lagranian is to have a linear combination of all different types of Lorentz invariant terms, and therefore a matrix element in general, would have the following form G Z F ¯ ¯ M = p Σi Ci pOi n lOi ν¯e 2 Where i is summed over all types of operators, scalar, pseudoscalar, vector, axial-vector (pseudo-vector) and tensor. Lee and Yang [16] were the first to propose that parity is not a conserved quantity which was confirmed by a Wu et al. [23]. Since parity conservation was no longer considered to be necessarily conserved, the lagrangian doesn’t need to be invariant under parity transformation. β− decays revealed that νe are left handed andν ¯e are right handed. This meant that the scalar, pseudoscalar and tensor term had to be very small (or zero). So a matrix element with vector and axial vector interaction would be expected and generalized as 1 γ (C + C γ ) 2 λ V A 5 Further From Moller scattering, results had shown that β decays are longitudainally polarized with helicity −(−v)=c for electrons and +(−v)=c for positrons, this means that CV = −CA, which leads to the expected interaction for weak decays as 1 γ (1 − γ ) 2 λ 5 It was pointed out by Feynman and Gell-Mann and independently by Sudarshan and Marshak [22] that proposed that all charged weak processes are described by an effective lagrangian that has a weak current, + + 1.1 Michel and Electronic Decay: π ! l (γ)νl 3 Jµ that coupled to itself, giving a lagrangian of 1 GF µ µ L = − p (Jµ J y +J y Jµ) 2 2 Now the weak current itself is a sum of two portions, Jµ = jlµ + jhµ, a leptonic portion which is the V−A interaction that has been mentioned, and the hadronic portion, which is similar to the V−A interaction, but acting on quarks. For the pion decay, the hadronic portions takes the form jhµ = D¯ Cγµ(1 − γ5)U Once it was realized that an intermediate vector boson was required in order to maintain unitarity the full interaction can be realized igµν M = j j 2 2 πµ lν MW − q Where the pion is the hadron and the lepton will either be the muon or positron, and MW is the mass of the intermediating W boson. For the case of low momentum transer, that is energies much less than mass of the W boson (80 GeV) ,the fermi theory can be recovered and the matrix element becomes iG M = p h0jV − Ajπi u¯lγµ(1 − γ5)vνl 2 Since the first part of the matrix element invovles the pseudoscalar pion to the vacuum using an vector term, this term should vanish, describing a matrix element −iG M = p h0jAjπi u¯lγµ(1 − γ5)vνl 2 The first term is a little difficult to describe, but it should be a Lorentz four vector, and since the only four vector available is the momentum transfer qµ, one can write µ h0jAjπi = i fπq Which produces −iG µ M = p fπq u¯lγµ(1 − γ5)vνl 2 + + 1.1 Michel and Electronic Decay: π ! l (γ)νl 4 After accounting for spin polarizations, the square of the matrix element is 2 2 2 2 jMj = 4G fπ ml pl pν When all is said and done and the phase space factor (density of states) is accounted for (see appendix B), the rate of the decay for a particular channel is G2 f 2m2 Γ = F π l (m2 − m2)2 3 π l 8πmπ Here it seen that the the decay rate depends on the square of the lepton mass, in particular, by looking at the ratio of the decay rates, otherwise known as the branching ratio, is given as Γ(π ! eν) m2 (m2 − m2)2 = e π e ≈ 1:283 × 10−4 2 2 2 2 Γ(π ! µν) mµ (mπ − mµ) Notice, that the ratio of the masses squared appears in the branching ratio. This term is extremely small, ≈ 2:5 × 10−5 This very small term is a manifestation of the V − A nature of the electroweak interaction. It is this term that so highly supresses the π ! eν decay channel. A deeper dive into the weak interaction would show that these types of decays prefer to pick out left-handed helicity states for fermions and right-handed helicity states for anti-fermions. This is apparent in a term that appears in the feynman rules (matrix element calculation) as 1 − γ5, which is the helicity projection operator for massless leptons. Now since the positron and muon are not massless, their helicities are mixed according their masses. This wouldn’t seem like such a problem since both the positron and neutrino are light particles (chirality and helicity become the same in the massless, high energy limit), however, consider a pion decaying at rest. In this case the essentially massless neutrino will decay as a left-handed particle (no problem so far). But, because angular momentum has to be conserved and the pion is a spin 0 particle, then the positron must emerge in the opposite direction of the neutrino with a left-handed helicity state as well. This conflicts with the weak interaction which demands the positron to be right-handed. It can be shown that the probability of picking out a left handed state from this type of interaction is ∼ (1 − vl=c). Now when looking at the ratio again, Γ(π ! eν) 1 − v =c m2 ∼ e ∼ e 2 Γ(π ! µν) 1 − vµ=c mµ + + 1.1 Michel and Electronic Decay: π ! l (γ)νl 5 We see the origin of the term that supresses the signal channel by so much comes from the probability of picking the proper helicity of this interaction. Also note that if the positron were a massless particle, then the velocity emerging from the interaction would be c, and thus the branching ratio would be 0, (ie the π ! eν would never occur).Thus far, only tree level calculations have been discussed. The branching ratio is not complete unless radiative and loop corrections are accounted for. These corrections for a pion decay process will depend on the mass of the resulting lepton and they originate from the emission of a photon via Inner Bremsstrahlung, and the virtual emision and reabsorption of photons. Classically, Bremstrahulung radiation is emitted whenever a charged particle is accelerated or deceleratated (old school physicists call this breaking radiation). Whenever a charged particle scatters in a medium, this means the directon changes, ie the velocity changes. This means that acceleration has occurred and therefore a photon must be emitted. Now a positron cann t radiate a photon unless it exchanges a soft photon with the nucleus that is near. Once the corrections are put into the branching ratio, the branching ratio becomes Γ(π ! eν) m2 (m2 − m2)2 = e π e (1 + δ)(1 + ) = 1:233 × 10−4 2 2 2 2 Γ(π ! µν) mµ (mπ − mµ) where δ = −3α/π ln mµ=me and = −:92α/π.
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