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

We calculate the K ⁰ - K⁰, B_d,s⁰ - B_d,s⁰ mixing mass differences ΔM_K, ΔM_d,s and the CP-violating parameter x_L in the littlest Higgs (LH) model. For f/v as low as 5 and the Yukawa < 0.8, the enhancement of ΔM_d amounts to at most 20%. Similar comments apply to ΔM_s and ε_K. The correction to ΔM_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(v²/f²) factors. This contribution is strictly positive. The explicit O(v² f²) 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(v²/f²) 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_K 0.85 box diagrams with two T exchanges have to be included. Although formally O (v⁴ f⁴), 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 (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 W_L^± and W_H^±.