Particle-Antiparticle Mixing, \epsilon_K and the Unitarity Triangle in the Littlest Higgs Model

We calculate the K^{0}-\bar{K}^{0}, B_{d,s}^{0}-\bar{B}_{d,s}^{0} mixing mass differences \Delta M_K, \Delta M_{d,s} and the CP-violating parameter \epsilon_{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 \Delta M_{d} amounts to at most 20%. Similar comments apply to \Delta M_s and \epsilon_{K}. The correction to \Delta 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^\pm,W_H^\pm) exchanges and ordinary quarks that are only suppressed by the mass of W_H^\pm 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^\pm 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^\pm,W_H^\pm) 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 \approx 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^\pm and W_H^\pm.