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 interactions. 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 ψγµψ and axial vector currents such as ψγµγ5ψ. 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 = √ 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 † 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 √ 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 interaction: 4GF µ µ µ µ L4F = − √ (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 √ 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)

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− 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 = √ 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 = ψ γµ T 3 ψ − Q ψ γµψ , (14) cos θ i i cos θ i i i i W W

3 with ψi 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 = − √ 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 .

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The relevant part of the electroweak Lagrangian is

e h + † − i Lmix = √ uLV W/ dL + dLV W/ uL 2 sin θW e + µ 1 5 − † µ 1 5 = √ Wµ uV γ (1 − γ )d + Wµ dV γ (1 − γ )u , (18) 2 sin θW 2 2

1 5 where ψL/R = 2 (1 ± γ )ψ has been used to remove the projectors on the second line. Under CP, W + and W − switch places since they are each other’s anti-particles. So, e − T µ 1 5 + † T µ 1 5 CP : Lmix → √ Wµ dV γ (1 − γ )u + Wµ u(V ) γ (1 − γ )d . (19) 2 sin θW 2 2

Thus, the Standard Model Lagrangian is invariant under CP if V ∗ = V . That is, if V is real. Thus a non-zero phase in the CKM matrix implies CP violation. We also note that if there were only two generations, then one could remove all the phases completely. Thus, any CP-violating eﬀect in the Standard Model must involve all three generations. There are lots of ways to measure the one CP phase in the Standard Model. That all these measurements are consistent is an important check on the CKM matrix and often provides stringent constraints on beyond the Standard Model physics. Historically, the ﬁrst measurement of CP violation was through decays of neutral kaons. In 1964, Cronin and Fitch found CP violation in kaon decays. For a long time kaon decays and mixing provided venues for CP violation. The advent of B physics opened up a whole new world of CP-violating observables and has provided important checks on the CKM framework and strong constraints on new physics. CP violation has been observed in B0 → K+π−, B0 → π+π−, B+ → ρ0K+ etc. So far, to the extend that we can connect these measurements to the CKM matrix, everything seems perfectly consistent with a single CP phase. However, we cannot rule out CP violation induced through BSM physics.

Strong CP Violation

The strong CP phase has two components. One is the θQCD angle associated with

g2 L = θ s µναβF a F a . (20) CP QCD 32π2 µν αβ

The other is

θF = arg det[YuYd], (21) where Yu and Yd are Yukawa matrices. These two angles rotate into each other under global chiral transformations of the Standard Model. Only the combination

θ = θQCD − θF (22)

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Moving θ to θQCD, we see that it has no eﬀect to any order in perturbation theory, since µναβ a a FµνFαβ is a total derivative. However, it does have an important eﬀect at low energy, where non-perturbative dynamics of QCD translate it into a CP-violating coupling between pions and nucleons. This should lead to an electric dipole moment for the neutron of order (5.2 × 10−16)e· cm θ. Current bounds then force θ ≤ 10−10. The smallness of θ despite the large amount of CP violation in the weak sector is known as the strong CP problem.

Possible solutions to the strong CP problem include: (i.) One of the quarks is massless, mu = 0. (ii.) Axions. We can promote the parameter θ to a dynamical ﬁeld. The excitations of this ﬁeld are known as axions. They could solve the strong CP problem and also provide a viable candidate for dark matter. (iii.) Spontaneous breaking of CP symmetry. CP violation is also necessary to explain the the abundance of matter over anti-matter in the Universe. It turns out that there is not enough CP violation in the Standard Model to explain this abundance. Thus there is a good reason to think that there is more to be learned about CP violation.

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

[2] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013).

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