Resolving the ϕ2 (α) ambiguity in B → ρρ

Abstract

I propose an alternative method for measuring the CP violating phase φ 2 (α) without ambiguity in an extended SU(3) flavour symmetry analysis, which can ultimately be achieved by exploiting interference effects between B → AP and B → V V decay channels, where A, V, P indicates an axial-vector, vector and pseudo-scalar meson, respectively. Under certain assumptions on the relevant decays based on current experimental results and minimal theoretical input, I demonstrate with an idealised amplitude model that a programme to extract a single solution for φ 2 in the range [0,π], with the added possibility to simultaneously constrain non-factorisable SU(3)-breaking effects, could be executed to similar precision using Run 3 data at LHCb and the final Belle II sample.

Resolving the ϕ2 (α) ambiguity in $$ {\mathrm{B}}^0\to {\mathrm{a}}_1^{\pm }{\uppi}^{\mp } $$

“…However, armed with knowledge of the B 0 → π + π − π + π − amplitude as suggested in ref. [24], combined with measurements of the B + → K 0 π + π − π + and B 0 → K + π − π + π − amplitudes proposed here, a single solution for φ 2 is also attainable in B → a 1 π.…”

“…The original method creating the possibility to remove the isospin-conserving component of this contamination invokes SU(2) arguments in a triangular analysis with input coming from the three B → ππ or ρρ charge configurations for an 8-fold degeneracy in φ 2 [28]. This can be reduced to two solutions if mixing-induced CP violation can be measured in their respective colour-suppressed channels [21] and reduced further to a single solution in B → ρρ if the complex CP violating parameter of B 0 → ρ 0 ρ 0 can be measured directly in a time-dependent, flavour-tagged amplitude analysis [24]. For flavournon-specific channels such as B 0 → ρ ± π ∓ and a ± 1 π ∓ , the original idea was subsequently extended to isospin pentagonal relations [29].…”

“…For the sake of argument, I now assume that the combined analysis of B 0 → ρ 0 ρ 0 and B 0 → a ± 1 π ∓ suggested in ref. [24] had since been performed at Belle and somehow they managed, albeit unrealistically given their data sample size, to resolve the weak phase solution degeneracy. In this test, the first solution is taken for the phase difference between B 0 → a − 1 π + and B 0 → a + 1 π − and the central values of φ ± 2 are taken from QCD factorisation [36] as they cannot be inferred from the quasi-two-body Belle result.…”

“…However, what these plots really indicate is that if the analysis from ref. [24] had been performed with as little as the full Belle data set, realistically there could already be an only two-fold φ 2 solution degeneracy in the B 0 → a ± 1 π ∓ system at 1σ instead of the 8 solutions it currently has today [20]. Next, I test the impact of the phase difference between B + → K 0 a + 1 and B + → K 0 1A π + for the most probable (figure 3) and mean values (figure 4) of the B + → K 0 1A π + branching fraction.…”

“…Leading-order Feynman diagrams shown producing B 0 → a ± 1 π ∓ decays, though the same quark transition can also produce B 0 → π + π − , ρ ± π ∓ and ρ + ρ − . (a) depicts the dominant In a previous work, I showed how the ratio of time-dependent decay amplitudes, better known as the complex CP violating parameter λ CP ≡ exp (−2iφ 2 )Ā/A, could be measured at amplitude level in B 0 → ρ 0 ρ 0 leading to a unique solution for φ 2 in the B → ρρ system [24]. What was not discussed at the time however, was that due to the procedure in which degenerate solutions are searched for in time-dependent amplitude analyses, λ CP is also resolved for both B 0 → a + 1 π − and a − 1 π + individually with the same significance as it would be in B 0 → ρ 0 ρ 0 .…”