CP Asymmetries
(through September 2023)

CP Asymmetries in D⁺ decays

CP Asymmetries in D_s⁺ decays

CP Asymmetries in decays of charm baryons

For charm baryons CP asymmetry has been also accessed through comparison of P asymmetry in decays of, for instance, Λ_c⁺ and Λ_c^-, measured with the weak-asymmetry parameters, respectively α(Λ_c⁺) and α(Λ_c^-). As under P-parity conservation: α(Λ_c⁺) = -α(Λ_c^-), the CP-violating asymmetry is defined as: A^α_CP = [α(Λ_c⁺) + α(Λ_c^-)] /[α(Λ_c⁺) - α(Λ_c^-]. The α(Λ_c) can be measured through an angular analysis exploiting a helicity angle of weakly-decaying baryon in Λ_c decay chain, for instance Λ->pπ^- in Λ_c⁺->Λπ⁺ decays.
This method of accessing CP violation occuring through P-violation can be applied also to other charm baryons, like Ξ_c⁰->Ξ^-π⁺ with Ξ^-->Λπ^-.

Year	Experiment	CP Asymmetry in the decay mode Λc+ to Λ e+ ν	[α(Λc+)+α(Λc-)]/[α(Λc+)-α(Λc-]
2005	CLEO	J.W. Hinson et al. (CLEO Collab.), Phys. Rev. Lett. 94, 191801 (2005).	0.00 ± 0.03 ± 0.01 ± 0.02
Year	Experiment	CP Asymmetry in the decay mode Λc+ to Λπ+	[α(Λc+)+α(Λc-)]/[α(Λc+)-α(Λc-]
2006	FOCUS	J.M. Link et al. (FOCUS Collab.), Phys. Lett. B 34, 165 (2006).	-0.07 ± 0.19 ± 0.12
Year	Experiment	CP Asymmetry in the decay mode Λc+ to ΛK+	[α(Λc+)+α(Λc-)]/[α(Λc+)-α(Λc-]
2022	BELLE	L.K. Li et al. (BELLE Collab.), Science Bulletin 68, 583 (2023).	-0.023 ± 0.086 ± 0.071
Year	Experiment	CP Asymmetry in the decay mode Λc+ to Σ0K+	[α(Λc+)+α(Λc-)]/[α(Λc+)-α(Λc-]
2022	BELLE	L.K. Li et al. (BELLE Collab.), Science Bulletin 68, 583 (2023).	+0.08 ± 0.35 ± 0.14
Year	Experiment	CP Asymmetry in the decay mode Ξc0 to Ξ-π+	[α(Ξc0)+α(Ξc0)]/[α(Ξc0)-α(Ξc0))]
2021	BELLE	Y.B. Li et al. (BELLE Collab.), Phys. Rev. Lett. 127, 121803 (2021).	+0.024 ± 0.052 ± 0.014

ΔA_CP measurements

Probing CP violation through differences between A_CP asymmetries of similar decay channels allows for cancelation of nuisance asymmetries, like charm-production asymmetry and/or charge-track detection asymmetry. It significantly facilitates experimental measurements and increases sensitivity.

Measurements of ΔA_CP for D⁰->h⁺h^- decays, defined as A_CP(D⁰->K⁺K^-)-A_CP(D⁰->π⁺π^-), are discussed here.
Notably, the LHCb measurement in R. Aaij et al. (LHCb Collab.), Phys. Rev. Lett. 122, 211803 (2019) yields an LHCb combination of
ΔA_CP(D⁰->h⁺h^-) = (-15.4 ± 2.9)×10^-4, comprising the discovery of CP violation in charm sector.

ΔA_CP is also measured for three-body Λ_c⁺->ph⁺h^- decays, in a phase-space integrated manner, in R. Aaij et al. (LHCb Collab.), JHEP 03, 182 (2018) giving
ΔA_CP(Λ_c⁺->ph⁺h^-) = (+0.30 ± 0.91 ± 0.61)×10^-2.

Sum rule for D->ππ decays

This sume rule was proposed in Y. Grossman, A.L. Kagan, J. Zupan, Phys. Rev. D 85, 114036 (2012) and for the first time experimentally tested in V. Babu et al. (BELLE Collab.), Phys. Rev. D 97, 011101 (2018).
An isospin decomposition of D->ππ decay amplitudes gives:

where A₁ and A₃ are amplitudes of decays with an isospin change of, respectively, ΔI = 1/2 and ΔI = 3/2 (i.e. transitions to ππ final states with I=0 and I=2). Therefore the following sum holds for decay amplitudes of D->ππ channels:

It translates into the following sum for differences between D and D decay rates:

The above sum being non-zero would mean that CP violation arises from the ΔI = 1/2 amplitude. If the sum is zero, while individual CP asymmetries of D->ππ modes are non-zero, it would suggest New Physics contributing to the ΔI = 3/2 amplitude. As penguin amplitude is, to a good approximation, ΔI = 1/2 within the Standard Model, while D⁺->π⁺π⁰ decay is purely a ΔI = 3/2 transistion, any CP asymmetry observed for this mode would point to New Physics in the ΔI = 3/2 amplitude.

To facilitate an experimental test, one can exploit also the sum of decay rates:

Dividing these two equations gives:

Using the relation between amplitude and branching fraction, |A|²~B/τ_D, and employing A_CP definition, |A|² - |A|² = A_CP(|A|² + |A|²), the above turns into:

where B_+-, B₀₀ and B₊₀ denote branching fractions of D⁰->π⁺π^-, D⁰->π⁰π⁰ and D⁺->π⁺π⁰ decays, respectively, whereas τ_D0 and τ_D+ are lifetimes of D⁰ and D⁺.
The sum R is calculated using HFLAV averages for A_CP given in the tables above, and PDG averages for branching fractions and lifetimes from R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022).

The result is: R = (+0.86 ± 3.09)×10^-3, and is consistent with zero.

More details are given in Sec.10.2 of Y. Amhis et al. (HFLAV), arXiv:2206.07501

 

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