Theoretical overview of kaon decays

Kaon decays are an important testing ground of the electroweak flavour theory. They can provide new signals of CP violation and, perhaps, a window into physics beyond the Standard Model. At the same time, they exhibit an interesting interplay of long-distance QCD effects in flavour-changing transitions. A brief overview is presented, focusing on a few selected topics of particular interest. A more detailed and comprehensive review can be found in arXiv:1107.6001.


Effective Field Theory
Kaon physics has been at the origin of many fundamental ingredients of the Standard Model (SM), such as flavour quantum numbers, parity violation, meson-antimeson mixing, quark mixing, CP violation and the GIM mechanism [1]. Rare kaon decays provide sensitivity to short-distance scales (c, t, W ± , Z) and offer the exciting possibility of unravelling new physics beyond the SM. Searching for forbidden lepton-flavour-violating processes (K L → e ± µ ∓ , K L → e ± e ± µ ∓ µ ∓ , K + → π + µ ± e ∓ . . . ) beyond the 10 −10 level, one is actually exploring energy scales above the 10 TeV region. The study of allowed decay modes provides, at the same time, very interesting tests of the SM itself, improving our understanding of the interplay among electromagnetic, weak and strong interactions. In addition, new signals of CP violation, which would help to elucidate the source of CP-violating phenomena, can be looked for.
Owing to the presence of very different mass scales (m π < m K M W ), the QCD corrections are amplified by large logarithms. The short-distance logarithmic corrections can be summed up, using the operator product expansion (OPE) and the renormalization group, all the way down from M W to scales µ < m c [2]. One gets in this way an effective Lagrangian, defined in the three-flavour theory, which is a sum of local four-fermion operators Q i , constructed with the light degrees of freedom (u, d, s; e, µ, ν ), modulated by Wilson coefficients C i (µ) which are functions of the heavy (Z, W, t, b, c, τ ) masses: The CP-violating decay amplitudes are proportional to the components y i (µ). The overall renormalization scale µ separates the short-(M > µ) and long-distance (m < µ) contributions, which are contained in C i (µ) and Q i , respectively. The Wilson coefficients are fully known at the next-to-leading order (NLO) [3,4]; this includes all corrections of O(α n s t n ) and O(α n+1 s t n ), where t ≡ log (M 1 /M 2 ) refers to the logarithm of any ratio of heavy mass scales (M 1,2 ≥ µ). In order to calculate the kaon decay amplitudes, we also need to know the non-perturbative matrix elements of the operators Q i between the initial and final hadronic states.    At low energies, one can use symmetry considerations to define another effective field theory in terms of the QCD Goldstone bosons (π, K, η). Chiral Perturbation Theory (χPT) [5,6] describes the pseudoscalar-octet dynamics through a perturbative expansion in powers of momenta and quark masses over the chiral symmetry-breaking scale Λ χ ∼ 1 GeV. Chiral symmetry fixes the allowed operators, while all short-distance information is encoded in their low-energy couplings (LECs) [7,8]. At LO the most general effective Lagrangian, with the same SU (3) L ⊗ SU (3) R transformation properties as the short-distance Lagrangian (1), contains three terms [1]: where U ≡ exp(i λ φ/F ) parameterizes the Goldstone fields, L µ = iF 2 U † D µ U represents the octet of V − A currents, λ ≡ (λ 6 − iλ 7 )/2 projects onto thes →d transition, Q = The χPT framework determines the most general form of the K decay amplitudes, compatible with chiral symmetry, in terms of the LECs multiplying the relevant chiral operators. These LECs, which encode the short-distance dynamics, can be determined phenomenologically and/or calculated in the limit of a large number of QCD colours N C . Chiral loops generate nonpolynomial contributions, with logarithms and threshold factors as required by unitarity. Fig. 1 shows schematically the procedure used to evolve down from M W to m K . While the OPE resums the short-distance logarithmic corrections log (M/µ), the χPT loops take care of the large infrared logarithms log (µ/m π ) associated with unitarity corrections (final-state interactions).

Leptonic and Semileptonic Decays
In (semi)leptonic decays strong interactions only appear through the hadronic matrix elements of the left-handed current, which can be precisely studied within χPT and with lattice simulations.

Non-leptonic Decays
The In the absence of QCD, the SM (W exchange) prediction g 8 = g 27 = 3 5 would disagree with (6). The short-distance QCD corrections show the needed qualitative trend to understand the data. The matching of the effective descriptions L ∆S=1 eff (short-distance) and L ∆S=1 2 (χPT) can be done in the large-N C limit; including the large short-distance logarithmic corrections ∼ 1 N C log (M W /µ), this gives the results g ∞ 8 = 1.13 ± 0.18 and g ∞ 27 = 0.46 ± 0.01 [1], which show the relevance of missing NLO corrections in 1/N C .
A dynamical understanding of the ∆I = 1 2 enhancement was achieved long time ago [29], through a combined expansion in powers of momenta (χPT) and 1/N C . With one virtual W ± field emitted and reabsorbed, and to LO in the chiral expansion (two L µ insertions at most), there are three possible chiral invariant configurations which give rise to the effective Lagrangian The operators Q L ] ij = δ i1 δ j2 V ud + δ i1 δ j3 V us . The underlying functional integral over quark and gluon fields L sources [29].
which gives rise to the effective couplings in (7) is represented diagrammatically in Fig. 4. When further restricted to ∆S = 1 transitions, the effective Lagrangian (7) reduces to (2) with [29] The topology which leads to the a-type coupling is O(N 2 C ), while those generating b and c are O(N C ). In the large-N C limit the coupling a can be calculated because the four-quark operators factorize into QCD currents with well-known χPT realizations. This factorization is only broken by (at least two) gluonic exchanges which are of NNLO in the 1/N C expansion. The c-type configuration corresponds to the so-called penguin-like diagrams which can also be calculated at LO in terms of known phenomenological parameters. Taking into account the factor F 2 ∼ O(N C ), included in the definition of the currents L µ , one easily finds [29]: where L 5 is an O(p 4 ) coupling of the strong χPT Lagrangian. To O(1/N C ) the scale dependence in C 6 (µ) cancels with the one in the quark condensate 0|qq|0 (µ), while C 4 is scale-independent. The non-leading topology b cannot be evaluated in a model-independent way. The important observation made in Ref. [29] is that it contributes with opposite signs to g 8 and g 27 . Taking Eq. (9) into account, the experimental value of A 2 , i.e. |g 27 | ≈ 0.29, implies [29]: Thus, the measured ratio |g 8 /g 27 | requires a large and negative value of b, generating a significant cancellation in g 27 and a sizeable enhancement of g 8 . This is precisely what was previously predicted through model-dependent calculations [30][31][32][33] and confirmed through a rigorous inclusive NLO analysis of the two-point correlators associated with the short-distance Lagrangians L ∆S=1,2 eff [29,30,34]. Recently, the predicted cancellation in g 27 has been observed by a lattice calculation of A 2 which finds b/a ≈ −0.7 [35]; the corresponding enhancement of g 8 is also seen in A 0 , although the present lattice results are still obtained at unphysical kinematics.
A similar cancellation is observed in lattice simulations of the ∆S = 2 transition amplitude [36,37]. In the chiral limit, the so-called B K parameter which regulates the K 0 -K 0 hadronic matrix element is given by B K = 3 4 (a + b) [29], which behaves as g 27 . This result agrees with explicit model calculations [30-33, 38, 39] and QCD sum-rule determinations [40,41].

Direct CP Violation: ε /ε
The measured CP-violating ratio [42][43][44][45] demonstrates the existence of direct CP violation in the K → 2π decay amplitudes. When CP violation is turned on, the amplitudes A I acquire imaginary parts. To first order in CP violation, where the strong phases χ I can be identified with the S-wave ππ scattering phase shifts at √ s = m K , up to isospin-breaking effects [27,28]. The phase φ ε = χ 2 − χ 0 + π/2 = (42.5 ± 0.9) • is very close to the so-called superweak phase, The CP-conserving amplitudes ReA I can be set to their experimentally determined values, avoiding in this way the large uncertainties associated with the hadronic matrix elements of the four-quark operators in L ∆S=1 eff . Thus, one only needs a first-principle calculation of the CP-odd amplitudes ImA 0 and ImA 2 ; the first one is completely dominated by the strong penguin operator Q 6 , while the leading contribution to the second one comes from the electromagnetic penguin Q 8 . Fortunately, those are precisely the only operators that are well approximated through a large-N C estimate of LECs, because their anomalous dimensions are leading in 1/N C . Owing to the large ratio ReA 0 /ReA 2 , isospin violation plays also an important role in ε /ε [28]. The one-loop χPT enhancement of the isoscalar amplitude [46,47] destroys an accidental LO cancellation of the two terms in (12) [48][49][50], bringing the SM prediction of ε /ε in good agreement with the experimental measurement in Eq. (11) [1,46,47]: