Particle-antiparticle mixing, -bar K and the unitarity triangle in the littlest Higgs model

We calculate the K0-K-bar 0, Bd,s0-B-bar d,s0 mixing mass differences ΔMK, ΔMd,s and the CP-violating parameter -bar K in the littlest Higgs (LH) model. For f/v as low as 5 and the Yukawa parameter xL<0.8, the enhancement of ΔMd amounts to at most 20%. Similar comments apply to ΔMs and -bar K. The correction to ΔMK is negligible. The dominant new contribution in this parameter range, calculated here for the first time, comes from the box diagrams with (WL±,WH±) exchanges and ordinary quarks that are only suppressed by the mass of WH± but do not involve explicit O(v2/f2) factors. This contribution is strictly positive. The explicit O(v2/f2) corrections to the SM diagrams with ordinary quarks and two WL± exchanges have to be combined with the box diagrams with a single heavy T quark exchange for the GIM mechanism to work. These O(v2/f2) corrections are found to be of the same order of magnitude as the (WL±,WH±) contribution but only for xL approaching 0.8 they can compete with it. We point out that for xL>0.85 box diagrams with two T exchanges have to be included. Although formally O(v4/f4), this contribution is dominant for xL∼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(v2/f2) 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 WL± and WH±