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The 4-Fermi Theory PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 52 Wednesday, April 22, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: The 4-Fermi Theory. CP Violation in the Standard Model. The 4-Fermi Theory Well before the electroweak unification was understood, its effective low-energy description, the 4-Fermi theory, was proven to give a very accurate phenomenological description of the weak inter- actions. Precision measurements at low energies gave indications of how heavy the W and Z bosons should be. They also indicated that the theory should involve vector currents (V ) such as γµ and axial vector currents such as γµγ5 . In fact, the structure of the electroweak theory was deduced from the V − A (pronounced as “V minus A”) structure of the 4-Fermi theory. Writing µ µ 5 µ V − A = γ − γ γ = 2γ PL; (1) 1 5 with PL = 2 (1 − γ ), we see that the V − A structure in the low-energy theory corresponds to a chiral theory in which weak interactions involve only left-handed fermions. ± L± The W couple to the left-handed currents Jµ as e µ + µ − L = p W+Jµ + W−Jµ ; (2) 2 sin θW where + µ µ µ i µ j Jµ = νeLγ eL + νµLγ µL + ντLγ τL + VijuLγ dL; (3) − µ µ µ y i µ j Jµ = eLγ νeL + µLγ νµL + τ Lγ ντL + VijdLγ uL: (4) To derive the 4-Fermi theory, let us start with the lepton sector treating the neutrinos as massless (so we can ignore mixing angles). At tree-level, the interactions among the electron, muon and their PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 neutrinos are µ ν 2 −i gµν − p p ie m2 p µ µ W ν ν (eLγ νeL + µLγ νµL) 2 2 (νeLγ eL + νµLγ µL) : (5) 2 sin θW p − mW We call these charged-current interactions. 2 2 At low energy, p mW , and we can approximate these exchanges with local 4-Fermi interac- tion: 4GF µ µ µ µ L4F = − p (eγ PLνe + µγ PLνµ)(νeγ PLe + νµγ PLµ) ; (6) 2 5 1 5 where we have put in the γ matrices using PL = 2 (1 − γ ) so we can use Dirac spinors to describe the fermions, and 4G e2 g2 2 p F ≡ 2 2 = 2 = 2 : (7) 2 2mW sin θW 2mW v We see that using the 4-Fermi Lagrangian gives a current-current interaction amplitude that is 2 2 identical to Eq. (6) for p mW . Thus, at low energy, the weak theory reduces to a set of 4-Fermi interactions among leptons (and quarks) with a universal strength given by GF . In particular, the muon decay rate is easy to calculate from the 4-Fermi theory. In the limit, mµ me, we find m5 Γ(µ ! eνν) = G2 µ : (8) F 192π3 −5 −2 From the measured muon lifetime, τµ = 2:197µs; this lets us deduce GF = 1:166 × 10 GeV . This determines that the electroweak vacuum expectation value is v = 247 GeV; (9) and constraints one combination of sin θW and mW . Note that, since αe is known and sin θW < 1, we also know that v e mW > 37:4 GeV; (10) 2 sin θW and mW mZ = > mW : (11) cos θW Thus, simply from the muon lifetime, we already knew in the 1960s that the W and Z must be quite heavy. Having an idea where to look helped motivate the design of the Super Proton Synchrotron (SPS) at CERN, with which the W and Z bosons were discovered in 1983. Quarks can be studied with charged-current interactions in the 4-Fermi theory, much like leptons. The only complication is that now flavor mixing is an issue. Including the first two generations, the weak currents are expanded to + µ µ Jµ = ··· + (uL cos θC − cL sin θC ) γ dL + (cL cos θC + uL sin θC ) γ sL; (12) 2 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 − where the ··· are the terms and similarly for Jµ . These mediate processes such as beta decay: + − n ! p e νe. From precision measurements of radioactive decays and from rates for kaon decay, such as + 0 + + K ! π e νe (here, K = su), it was deduced that GF in these processes is consistent with the leptonic measurements, and that sin θC = 0:22. The neutral current interactions, mediated by Z-boson exchange are much harder to measure − directly in the lepton sector. The first observation was in 1973 when νµe elastic scattering was observed. This was a great test of the electroweak theory, consistent with a Z boson, but it only gave a very poor measure of mZ and θW . It was not until the mid 1990s that θW could be measured from this process directly. 2 We now have very precise measurements: MW = 80 GeV, MZ = 91:2 GeV and sin θW = 0:21. Moreover, measuring these quantities in multiple ways has provided important tests of the Standard Model and constraints on beyond the Standard Model physics. 3 The Z boson couples to linear combinations of the Jµ and QED currents. The interactions are e µ+ + µ − e µ Z µ Lint = p W Jµ + W−Jµ + Z Jµ + eAµJEM: (13) 2 sin θW sin θW We have 2 Z 1 3 sin θW EM Jµ = Jµ − Jµ cos θW cos θW 2 X 1 sin θW = γµ T 3 − Q γµ ; (14) cos θ i i cos θ i i i i W W 3 with i including quarks and leptons and T being SU(2) generator in the appropriate representa- Z tion. Note that Jµ only couples fermions to fermions of the same flavor. The full 4-Fermi theory can then be written as GF + − Z 2 L4F = − p J J + (J ) : (15) 2 µ µ µ µ µ There is no JEM − JEM 4-Fermi interaction since the photon is massless and so, unlike the W and Z bosons, its propagator can never be approximated by a constant. One immediate prediction of L4F is that, since the neutral current is flavor diagonal, there will be no flavor-changing neutral current (FCNC) processes, such as s ! ueνe. This is an obvious result the way we have set things up, but it is not at all obvious without an electroweak theory. Indeed, historically, in the 1960s it was not at all clear why there were no FCNCs. In the 1960s the only hadrons known were made up of u, d and s quarks. The charm quark was then predicted to exist based on the absence of neutral currents. When charm was discovered in 1974 the electroweak theory was spectacularly confirmed. To see why charm is required to avoid FCNCs, let us forget about leptons and consider a theory with only two generations of quarks. Then there is only one mixing angle, θC , so we can choose a 3 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 basis so that u and c quarks are flavor and mass eigenstates, while the left-handed d and s quarks are mixed. Then the two left-handed doublets are ! ! u c Q1 = L ;Q2 = L : (16) cos θC dL + sin θC sL cos θC sL − sin θC dL The electromagnetic current is flavor diagonal for any number of quarks, so we will ignore it. The neutral current coming from weak interactions is 3 µ µ Jµ = uγ u + cos θC d + sin θC s γ (cos θC d + sin θC s) µ µ + cγ c + cos θC s − sin θC d γ (cos θC d − sin θC s) = uγµu + cγµc + dγµd + sγµs; (17) where we have dropped the L subscripts for readability. This current is flavor diagonal as expected. Now, suppose there were no charm quark.Then there would be no Q2 and the neutral current µ + − 0 + − would have a non-vanishing cross term cos θC sin θC dγ s, implying ds ! µ µ and K ! µ µ . So Glashow, Iliopoulos and Maiani (GIM) predicted that there must be a charm quark so that the flavor-changing process would cancel. The absence of FCNCs works for any number of generations, and is known as the GIM mechanism. It is general consequence of the T 3 generator of SU(2) commuting with rotations in flavor space, as can be seen in Eq. (15). CP Violation in the Standard Model It is obvious to see that parity is violated in the weak interactions. The left-handed fields couple differently from the right-handed fields. Parity violation is manifest in beta decay, which always produced left-handed electrons. The Universe might still be invariant under reflection accompanied by the interchange of particles and antiparticles. This is CP invariance. We now know that CP invariance is violated by rare processes involving hadrons. We call this weak CP violation. There is another possible form of CP violation, called strong CP violation, which is expected but has not been observed. The non-observation is known as the strong CP problem. Weak CP Violation In QFT I, we looked at how C and P act on fields and spinor bilinears. We can check which terms in the Standard Model Lagrangian can violate CP. One can perform chiral rotations on the left- and right-handed fermions of the Standard Model so that quark masses are diagonal and the mixing is moved to the CKM matrix V . 4 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 The relevant part of the electroweak Lagrangian is e h + y − i Lmix = p uLV W= dL + dLV W= uL 2 sin θW e + µ 1 5 − y µ 1 5 = p Wµ uV γ (1 − γ )d + Wµ dV γ (1 − γ )u ; (18) 2 sin θW 2 2 1 5 where L=R = 2 (1 ± γ ) has been used to remove the projectors on the second line.
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