The Theory of Cp-violation – in as Much of a Nutshell as Will Fit on 8 Pages

Do you know that CP violation is intrinsically linked to the scalar sector of the Standard Model and its extensions? If yes, you need read no further — if no, you may turn over the titlepage and start reading now. It is difficult to do justice to a topic as vast and complex as CP-violation in a 30-minutes conference talk — and even more so in a 8-pages contribution to the proceedings. Well, practitioners in teaching & learning do know that nothing is impossible, and so I shall try to stand up to the challenge and concentrate on a less common viewpoint on the subject than is to be found in most textbooks, in the hope the reader may find it as entertaining as enlightening. Everything you ever wanted to know about CP-violation (and more) can be found in Ref. 1). It is actually very surprising that CP should be violated at all. Many gauge-theories preserve C(harge conjugation symmetry) and P(arity) naturally & separately, the probably most prominent ones being (massless) QED and QCD. Even more contrived theories, especially designed to violate parity, like the chiral gauge-theory L = − 1 4 FμνF μν + ψ̄LiσDψL, (1) where only the left-handed (Weyl) fermions ψL interact with gauge-bosons, 2 are still invariant under CP transformations, which implies that CP is a natural symmetry of massless gauge theories. So where does CP-violation come in? The catch is that, as the mass term mψ̄ψ ≡ m(ψ̄LψR + ψ̄RψL) (2) violates gauge-symmetry, it is forbidden in L and hence left-handed fermions must be massless — at obvious variance with experiment. If the theory (1) is to serve as model for parity-violating interactions, it has to be amended in some ingenious way as to give mass to the fermions (and gauge-bosons), but at the same time preserve gauge-invariance. In the Standard Model (SM), this objective is being achieved by adding a scalar (Higgs) sector which generates a nontrivial ground-state (vacuum) of the theory. In general, this vacuum-state is less symmetric than the full theory — a phenomenon usually referred to as spontaneous symmetry breaking (SSB), which in the case of gauge-theories is dubbed Higgs mechanism and allows gauge-bosons (and chiral fermions) to become massive. The Lagrangian of the SM can be written as LSM = Lgauge(ψL, ψR,W, φ) + LHiggs(φ) + LYukawa(ψL, ψR, φ), (3) where the first term on the right-hand side, the equivalent of (1), contains the kinetic terms of the fields involved, i.e. leftand right-handed fermions ψL and ψR, gauge-bosons W and scalar (Higgs) fields φ, as well as their gaugeinteractions. The second term is the potential felt by the scalar fields and is responsible for some of them to acquire a nonzero vacuum expectation value Whereas their right-handed counterparts are “sterile” and hence omitted from the theory. (VEV) which gives rise to SSB. The third term describes interactions between fermionic and scalar fields, which after SSB induce fermion mass terms. In the SM, LHiggs is automatically CP-invariant, 3 which leaves us with LYukawa as the only possible source of CP-violation in the SM. It is given by LYukawa = −λ d ijQ̄ i L · Φd j R − (λ d ij) d̄jRΦ † ·QiL + . . . , (4) where the indices i, j run over the three generations and the dots denote terms with up-type quarks. QiL denotes the SUL(2) quark doublet (u i L, d i L) and Φ the SUL(2) Higgs doublet (φ , φ). The second term on the right-hand side of (4) is the complex conjugate of the first one — as required by the condition that the Lagrangian be a Hermitian operator. So how does LYukawa transform under CP? The P-transformation exchanges L (left) and R (right) indices, the C-transformation exchanges particles (d etc.) and antiparticles (d̄ etc.), so that CP : Q̄iL ·Φd j R → d̄ j RΦ † ·QiL. (5) Comparing with (4), we see that LYukawa is CP-invariant if λ ≡ λ . Hence, a necessary (but not sufficient) condition for CP-violation is that the Yukawa couplings λ are complex. What does all that actually mean? Well, one conclusion is that CPviolation happens in the scalar sector — at least in the SM. What about extensions? The statement stays evidently true for “simple” extensions of the SM with just an enlarged gaugeand scalar-field content (e.g. two Higgs-doublet model), and it also applies to theories where CP is not violated explicitly by complex couplings, but by spontaneous symmetry breaking — which by definition is related to the scalar sector. What about supersymmetry? Again, CP is conserved in theories with unbroken SUSY, for the same reasons as above, but complex couplings occur after SUSY-breaking. Another conclusion is that studying CP-violation means probing the scalar sector — which is also one of the main objectives of the Tevatron and the LHC. In this sense the measurement of CP-asymmetries in K and B decays is complementary to the direct searches for Higgs et al. at high-energy colliders. The reason being that there is only one Higgs-doublet; CP-violation in LHiggs can occur, however, in models with more than one Higgs-doublet. Note that the QCD θ-term θQCDg 2 s/(64π )ǫGμνG a ρσ can be set to 0 if all quarks are massless. What about CP-violation in the SM? Well, after SSB the Yukawa couplings λ induce 3 × 3 mass matrices for u and d-type quarks which are eigenstates under weak interactions. If the theory is to be expressed in terms of states with definite mass, these matrices have to be diagonalized. The resulting transformation from the basis of weak eigenstates to that of mass eigenstates, u (weak) i = U (u) ij u (mass) j , d (weak) i = U (d) ij d (mass) j , (6) has no effect on neutral interactions, ū (weak) i u (weak) i ≡ ū (mass) i u (mass) i , but profoundly changes charged interactions: ū (weak) i d (weak) i → ū (mass) i (U )U d (mass) i . (7) The matrix V ≡ (U )U (d) describes the strength of d-type quarks decaying into u-type quarks and is nothing else but the well-known CKM matrix. As U (u,d) just rotate the quark basis, they are unitary, and so is V . Any 3 × 3 unitary matrix can be parametrised in terms of three angles (the familiar Euler angles of three-dimensional rotations) and six complex phases. In the present case, however, not all six phases are physical: five of them can be “rotated away” by redefining the phases of the quark fields — which leaves three angles and one phase to describe the CKM matrix V . It is this complex phase that is the one and only source of CP-violation in the SM. The fact that V is unitary allows one to express the conditions for CPviolation in the SM in an intuitively appealing form: unitarity means