[Formula: see text] rule, [Formula: see text] and [Formula: see text] in [Formula: see text] and [Formula: see text]...

The experimental value for the isospin amplitude [Formula: see text] in [Formula: see text] decays has been successfully explained within the standard model (SM), both within the large [Formula: see text] approach to QCD and by QCD lattice calculations. On the other hand within the large [Formula: see text] approach the value of [Formula: see text] is by at least [Formula: see text] below the data. While this deficit could be the result of theoretical uncertainties in this approach and could be removed by future precise QCD lattice calculations, it cannot be excluded that the missing piece in [Formula: see text] comes from new physics (NP). We demonstrate that this deficit can be significantly softened by tree-level FCNC transitions mediated by a heavy colourless [Formula: see text] gauge boson with a flavour-violating left-handed coupling [Formula: see text] and an approximately universal flavour diagonal right-handed coupling [Formula: see text] to the quarks. The approximate flavour universality of the latter coupling assures negligible NP contributions to [Formula: see text]. This property, together with the breakdown of the GIM mechanisms at tree level, allows one to enhance significantly the contribution of the leading QCD-penguin operator [Formula: see text] to [Formula: see text]. A large fraction of the missing piece in the [Formula: see text] rule can be explained in this manner for [Formula: see text] in the reach of the LHC, while satisfying the constraints from [Formula: see text], [Formula: see text], [Formula: see text], LEP-II and the LHC. The presence of a small right-handed flavour-violating coupling [Formula: see text] and of enhanced matrix elements of [Formula: see text] left-right operators allows one to satisfy simultaneously the constraints from [Formula: see text] and [Formula: see text], although this requires some fine-tuning. We identify the quartic correlation between [Formula: see text] contributions to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. The tests of this proposal will require much improved evaluations of [Formula: see text] and [Formula: see text] within the SM, of [Formula: see text] as well as precise tree-level determinations of [Formula: see text] and [Formula: see text]. We present correlations between [Formula: see text], [Formula: see text] and [Formula: see text] with and without the [Formula: see text] rule constraint and generalise the whole analysis to [Formula: see text] with colour ([Formula: see text]) and [Formula: see text] with FCNC couplings. In the latter case no improvement on [Formula: see text] can be achieved without destroying the agreement of the SM with the data on [Formula: see text]. Moreover, this scenario is very tightly constrained by [Formula: see text]. On the other hand, in the context of the [Formula: see text] rule [Formula: see text] is even more effective than [Formula: see text]: it provides the missing piece in [Formula: see text] for [Formula: see text]-[Formula: see text].