Particle–antiparticle mixing, [formula omitted] and the unitarity triangle in the littlest Higgs model

Abstract

We calculate the K 0 – K ¯ 0 , B d , s 0 – B ¯ d , s 0 mixing mass differences Δ M K , Δ M d , s and the CP-violating parameter ɛ K in the littlest Higgs (LH) model. For f / v as low as 5 and the Yukawa parameter x L < 0.8 , the enhancement of Δ M d amounts to at most 20%. Similar comments apply to Δ M s and ɛ K . The correction to Δ M K is negligible. The dominant new contribution in this parameter range, calculated here for the first time, comes from the box diagrams with ( W L ± , W H ± ) exchanges and ordinary quarks that are only suppressed by the mass of W H ± but do not involve explicit O ( v 2 / f 2 ) factors. This contribution is strictly positive. The explicit O ( v 2 / f 2 ) corrections to the SM diagrams with ordinary quarks and two W L ± exchanges have to be combined with the box diagrams with a single heavy T quark exchange for the GIM mechanism to work. These O ( v 2 / f 2 ) corrections are found to be of the same order of magnitude as the ( W L ± , W H ± ) contribution but only for x L approaching 0.8 they can compete with it. We point out that for x L > 0.85 box diagrams with two T exchanges have to be included. Although formally O ( v 4 / f 4 ) , this contribution is dominant for x L ≈ 1 due to non-decoupling of T that becomes fully effective only at this order. We emphasize, that the concept of the unitarity triangle is still useful in the LH model, in spite of the O ( v 2 / f 2 ) corrections to the CKM unitarity involving only ordinary quarks. We demonstrate the cancellation of the divergences in box diagrams that appear when one uses the unitary gauge for W L ± and W H ± .