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

We calculate the K0-K¯0, Bd,s0-B¯d,s0 mixing mass differences ΔMK, ΔMd,s and the CP-violating parameter ɛK 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 ɛK. 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±.