CP violation in $B^{\pm} \rightarrow \pi^{\pm}\pi^{+}\pi^{-}$ in the region with low invariant mass of one $\pi^{+}\pi^{-}$ pair

G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68

1996

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H.-Y. Cheng and B. Tseng, Phys. Rev.D 58

1998

Earlier work this paper cites.

Y.-H. Chen, H.-Y. Cheng, B. Tseng, and K.-C. Yang, Phys. Rev.D 60

1999

Cited alongside, same era.

H.-Y. Cheng, C.-K. Chua, and C.-W. Hwang, Phys. Rev.D 69

2004

Cited alongside, same era.

J. M. de Miranda (LHCb Collaboration), arXiv:1301.0283 [hep-ex]

Original

Cited in the paper.

For the decay channel B − → π − ​ π + ​ π − B^{-}\to\pi^{-}\pi^{+}\pi^{-} , there are two identical pions with negative charge. When combining the momentum of each π − \pi^{-} meson with that of the π + \pi^{+} meson, we will have two Lorentz invariant mass squares which are usually different in values and are denoted by m π + ​ π − ​ low 2 m_{\pi^{+}\pi^{-}~\text{low}}^{2} and m π + ​ π − ​ high 2 m_{\pi^{+}\pi^{-}~\text{high}}^{2} in Ref. [ 1 ] , respectively. Throughout this paper, we will denote m π + ​ π − ​ low(high) 2 m_{\pi^{+}\pi^{-}~\text{low(high)}}^{2} as s L ⁡ ( H ) s_{L(H)} for simplicity

Cited in the paper.

In fact, a large C ​ P CP asymmetry difference in upper and lower parts of localized region m π + ​ π − 2 < 0.4 ​ GeV 2 m_{\pi^{+}\pi^{-}}^{2}<0.4~\text{GeV}^{2} was also observed in B ± → π + ​ π − ​ K ± B^{\pm}\to\pi^{+}\pi^{-}K^{\pm}

Cited in the paper.

The s s -dependent decay width of Z Z ( Z Z can be X X or Y Y ) takes a general form Γ Z ​ ( s ) = [ g Z ​ ( s ) g Z ] 2 ​ [ λ ⁡ ( s ) / s λ ⁡ ( m Z ) / m Z 2 ] ξ Z ​ [ m Z 2 s ] ​ Γ Z , \Gamma_{Z}(s)=\left[\frac{g_{Z}(s)}{g_{Z}}\right]^{2}\left[\frac{\lambda(\sqrt{s})/s}{\lambda(m_{Z})/m_{Z}^{2}}\right]^{\xi_{Z}}\left[\frac{m_{Z}^{2}}{s}\right]\Gamma_{Z}, where ξ Z = S Z + 1 / 2 \xi_{Z}=S_{Z}+1/2 , with S Z S_{Z} being the spin of Z Z , λ ⁡ ( x ) = [ x 2 − ( m M 1 + m M 2 ) 2 ] ​ [ x 2 − ( m M 1 − m M 2 ) 2 ] \lambda(x)=[x^{2}-(m_{M_{1}}+m_{M_{2}})^{2}][x^{2}-(m_{M_{1}}-m_{M_{2}})^{2}] . In the numerical calculation of this paper, we simply set g Z ​ ( s ) = g Z g_{Z}(s)=g_{Z}

Cited in the paper.

Generally, both the tree and penguin amplitudes of B → X ​ M 3 B\to XM_{3} should be proportional to ε ∗ ⋅ p B \varepsilon^{\ast}\cdot p_{B} . When considering a process in which the vector meson X X predominantly decays into M 1 ​ M 2 M_{1}M_{2} , one should replace the polarization vector ε μ ∗ \varepsilon^{\ast}_{\mu} by g X s X ​ ( p M 1 − p M 2 ) ν ​ [ g μ ​ ν − ( p M 1 + p M 2 ) μ ​ ( p M 1 + p M 2 ) ν s 12 ] . \frac{g_{X}}{s_{X}}(p_{M_{1}}-p_{M_{2}})^{\nu}\left[g_{\mu\nu}-\frac{(p_{M_{1}}+p_{M_{2}})_{\mu}(p_{M_{1}}+p_{M_{2}})_{\nu}}{s_{12}}\right]. Taking the dot product of this term and p B p_{B} , one will get g X ​ ( s 13 − \mathaccentV ​ h ​ a ​ t ​ 05 ​ E ​ s 13 ) / s X g_{X}(s_{13}-\mathaccentV{hat}05E{s}_{13})/s_{X}

Cited in the paper.

However, with this criteria alone, we cannot know the spin of Y Y . For example, if Y Y is a tensor resonance, one can also observe a larger event density at s 12 ∼ m Y 2 s_{12}\sim m_{Y}^{2} than that at s 12 ∼ m X 2 s_{12}\sim m_{X}^{2} when s 13 s_{13} is close to \mathaccentV ​ h ​ a ​ t ​ 05 ​ E ​ s 13 \mathaccentV{hat}05E{s}_{13}

Cited in the paper.

R. Aaij et al. (LHCb Collaboration), LHCb-CONF-2012-028

2012

Later among the works it cites.

J. Beringer et al. (Particle Data Group), Phys. Rev.D 86

2012

Later among the works it cites.