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

We calculate the K-0-(K) over bar (0), B-d,s(0)-(B) over bar (0)(d,s) mixing mass differences Delta M-K, Delta M-d,M-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(+/-), W-H(+/-)) exchanges and ordinary quarks that are only suppressed by the mass of WH +/- 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) approximate to 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(+/-). (c) 2005 Elsevier B.V. All rights reserved.