ar X iv : h ep - p h / 05 06 22 4 v 2 2 9 Ju n 20 05 Strong Phases in the Decays B to ππ

Two sources of strong phases in the decays B to ππ are identified: (1) “quasi-elastic scattering” corresponding to intermediate states like ππ and ρρ, (2) “cc̄” corresponding to intermediate states like DD̄ and D∗D̄∗. Possibilities of using data to identify these two sources are discussed and illustrated. Present data suggests both sources may be significant. The decay of B → ππ may be considered as due to the effective interaction λud(uū− tt̄)b+ λcd(cc̄− tt̄)b, (1) where λu = V ∗ ubVud = e Aλ sinβ sinα λc = V ∗ cbVcd = −Aλ. The decay amplitudes may be written, neglecting the small electroweak penguins and assuming isospin invariance, −A(ππ) = TeT e + PeP , (2) − √ 2A(ππ) = CeCe − PeP , (3) − √ 2A(ππ) = (TeT + CeC )e, (4) where δT , δC , and δP are phases due to the strong final state interaction. In terms of the isospin analysis of the λu terms, T and C may be replaced by A2 and A0 and (δC , δT ) by (δ2, δ0) TeT = e(A2e i(δ2−δ0) + A0), (5) CeC = e(2A2e i(δ2−δ0) − A0). (6) The T term is often referred to as the tree amplitude corresponding to the b quark decay into uūd, while P is called the penguin corresponding to a loop diagram dominated by the virtual t quark. However, as can be seen from Eq (1) there is also a tt̄ loop contributing presumably in a small way to T . This is because we are using what is called the c convention in contrast to the t convention that separates off the t loop[1][2]. Note that P makes a contribution to the I = 0 final state so that the complete amplitude for I = 0 (sometimes given the notation A0) is here a sum of the A0 and P terms. If the final state scattering were purely elastic the phases δ2 and δ0 would be ππ scattering phases in accordance with the Watson theorem. In fact the final state scattering is expected to be very inelastic and described by the N × N S matrix at the center-of-mass energy equal to the B mass and J = 0. The simple Watson theorem can be applied only to final states that are eigenstates of the S matrix. In general for any weak interaction operator O the phase due to the strong interaction for the amplitude Af is determined from ImAf = Im ∑ i 〈f |S|i〉〈i|O|B〉. (7)