CP-Violation in the decay B^0, (B^0)bar ->pi^+ pi^- gamma

The decay (B^0)bar ->pi^+ pi^- gamma has a bremsstrahlung component determined by the amplitude for (B^0)bar ->pi^+ pi^-, as well as a direct component determined by the penguin interaction V_tb V^*_td c_7 O_7. Interference of these amplitudes produces a photon energy spectrum d Gamma/d x = a/x + b + c_1 x + c_2 x^2 + ... (x = 2 E_gamma/m_B) where the terms c_1,2 contain a dependence on the phase alpha_eff = pi - arg[(V_tb V^*_td)^* Amp((B^0)bar ->pi^+ pi^-)]. We also examine the angular distribution of these decays, and show that in the presence of strong phases, an untagged B^0/(B^0)bar beam can exhibit an asymmetry between the pi^+ and pi^- energy spectra.


Introduction
In this paper, we analyse the reaction B 0 (B 0 ) → π + π − γ, with the aim of finding new ways of probing CP violation in the non-leptonic Hamiltonian, and in particular to test current assumptions about the weak and strong phases in the amplitude for B 0 → π + π − . We will focus on observables that can be measured in an untagged B 0 , B 0 beam, which are complementary to the time-dependent asymmetries in channels such as B 0 , B 0 → π + π − which are currently under study.
The amplitude for the decay of B 0 into two charged pions and a photon can be written as A(B 0 → π + π − γ) = A brems + A dir (1) where A brems is the bremsstahlung amplitude and A dir is the direct emission amplitude. Our main interest will be in the continuum region of π + π − invariant masses (large compared to the ρ-mass), and the possible interference of the two terms in Eq. (1). The bremsstrahlung amplitude is directly proportional to the amplitude for a B 0 decaying into two pions, the modulus of which is determined by the measured branching ratio [1]. Theoretically, the B 0 → π + π − decay amplitude can be written as [2] A where γ = arg(−V ud V * ub /V cd V * cb ), and T and P denote the tree-and penguinamplitudes, which can possess strong phases. (We follow the notation of Ref. [2], in which the phase of Pππ Tππ is estimated to be ≤ 10 • ). The B 0 → π + π − amplitude is obtained by taking the complex conjugate of the CKM factors in Eq. (2), leaving possible strong phases unchanged. In the experiments [1], one measures the time-dependent CP asymmetry where
Here, however, we will be interested in the decay B 0 → π + π − γ for π + π − masses in the continuum region, especially for large s (or low photon energy).
The Hamiltonian H peng leads to where the electric (E dir ) and magnetic (M dir ) amplitudes depend on the two Dalitz plot coordinates: the photon energy ω in the B 0 -meson rest frame and θ, the angle of the π + relative to the photon in the π + π − c.m. frame. As long as the photon polarization is not observed, only the electric component of the direct amplitude interferes with the bremsstrahlung amplitude. This interference is in principle sensitive to the relative phase of T λ u + P λ c and λ t (λ i = V ib V * id ) and therefore could serve as a probe of the phase of A(B 0 → π + π − ).

Differential Decay Rate
In accordance with the Low theorem the bremsstrahlung matrix element is directly proportional to the amplitude of B 0 → π + π − on the mass shell: To obtain the direct amplitude A dir , we observe first that the operator O 7 ∼ dσ µν (1 + γ 5 )F µν b, and the identity σ µν = i 2 ǫ µναβ σ αβ γ 5 enables one to write E dir (ω, cos θ) = M dir (ω, cos θ). One can write a multipole expansion for these direct amplitudes in the form [4] The simplest assumption is the dipole approximation in which E dir is independent of cos θ. In Section 4 we will consider also consequences arising from a quadrupole term.
To get a dimensionless decay distribution we introduce Then the differential branching ratio for the process B 0 → π + π − γ is: where 1,3] and P ππ /T ππ = 0.28, with a negligible strong phase [2]. The bremsstrahlung part of the decay distribution populates preferentially the region of small x and | cos θ| ∼ 1. Far from this region, the spectrum is determined by the direct term |E dir | 2 . In principle, a fit to the Dalitz plot can determine the scale and the shape of the direct amplitude. Fig. 1 shows the two-dimensional decay distribution, for an assumed direct branching ratio Br dir = 10 −6 and two choices of form factor E (1) = const. and E (1) ∼ 1/s. In the dipole approximation the distribution is symmetric with respect to cos θ and identical for B 0 and B 0 decay.

Photon Energy Spectrum
Integrating over the variable cos θ, the branching ratio, for small energies x, can be written in the form: In agreement with the Low theorem, the coefficients a and b are determined entirely by bremsstrahlung whereas c 1 , c 2 depend on the interference of bremsstrahlung and direct emission and therefore contain information about the relative phase of T λ u + P λ c and λ t given by β + γ ef f ≡ π − α ef f . Higher order terms in x (c i , i = 3, 4, . . .) contain pure direct emission, in addition to interference terms involving higher multipoles. Numerical estimates of the expansion parameters are given in table 1.
In figure 2 we show typical photon energy spectra, for direct branching ratios 10 −6 or 10 −7 , and two choices of the form factor in E dir . The phase α is allowed to vary between 60 • and 120 • . As expected, the sensitivity to α depends on the degree of overlap between the bremsstrahlung and direct amplitudes. The expansion in Eq. (17) turns out to be a good description in the region x ≤ 0.25. (Note that the resonant contribution B 0 → ργ → π + π − γ would appear as a spike in dBr dx at .) In the absence of strong phases, the photon energy spectrum (17) holds for B 0 as well as B 0 decay, and the results in table 1 and figure 2 are, in that limit, valid for an untagged B 0 , B 0 beam as well.

Asymmetry in the Angular Distribution
In the dipole approximation, the angular distribution is the same for B 0 and B 0 decay and is symmetric in cos θ (c.f. Eq. (11)). In the presence of an additional quadrupole term in the direct emission amplitude the multipole expansion for B 0 and B 0 differs by a sign, as shown in Eq. (8). For a numerical estimate of E (2) /E (1) , we take [6] Br(B → π + π − γ; E (2) ) The result is plotted in fig. 3, which is obtained from Eq. (11) after integrating over photon energies ω > 50MeV . While the distribution dBr d cos θ is symmetric in a dipole approximation, it develops a forward-backward asymmetry in the presence of a quadrupole term.
Comparing B 0 to B 0 we see that as long as strong phases are absent, so that the decay of an untagged beam would be symmetric in cos θ. This conclusion changes, however, if strong phases are not negligible. Let us associate strong phases δ 0 , δ 1 (s) and δ 2 (s) with the bremsstrahlung, direct dipole, and direct quadrupole terms. Notice that the bremsstrahlung phase δ 0 is the phase of the B 0 → π + π − amplitude, and therefore describes a 2π state with L = 0 and invariant mass m B . By contrast, the phases δ 1 (s) and δ 2 (s) describe L = 1 and L = 2 states of a 2π system with invariant mass s. As a net result, the relative phase of the bremsstrahlung and direct amplitudes in Eq. (11) may be written as π − α ef f + δ str in B 0 decay, and −π + α ef f + δ str in B 0 decay, where δ str denotes some effective combination of δ 0 , δ 1 and δ 2 . Defining we obtain the forward-backward asymmetry in a mixture of B 0 and B 0 : Note, that the forward-backward asymmetry in (untagged) B 0 , B 0 → π + π − γ decay is equivalent to an asymmetry in the energy spectrum of π + and π − in the B meson rest frame. The function f (x) is plotted in figure 4. Thus Asy(x) is a signal of CP -violation, that is present even in an untagged B 0 /B 0 mixture, and requires α ef f = 0 and δ str = 0, in addition to a quadrupole term in the direct electric amplitude.

Summary
We have studied observables in the decay B 0 , B 0 → π + π − γ that do not require tagging or measurement of time-dependence, but which nevertheless probe weak and strong phases appearing in the decay amplitude. Interference of the bremsstrahlung and direct components affects the linear and quadratic terms in the photon energy spectrum Eq. (17), with potential sensitivity to the phase arg[(V tb V * td ) * A(B 0 → π + π − )]. In the presence of non-trivial strong phases, there is a difference in the π + and π − energy spectra even for an untagged B 0 /B 0 beam, or, equivalently, a forward-backward asymmetry of the π + relative to the photon direction.