T-odd Asymmetries
(through September 2022)

T-odd asymmetry is measured using triple-product correlations and assuming the validity of the CPT theorem. Triple-product correlations of the form a • (b × c), where a, b, and c are spins or momenta are odd under time reversal T.

As an example, for the decay mode D⁰ → K⁺ K⁻ π⁺ π⁻ one can form the T-odd correlation C_T = P_K+ • (P_π+ × P_π-) using the momenta of the daughter particles.
The asymmetry for D⁰ is A_T ≡ [Γ(C_T > 0) − Γ(C_T < 0)] / [Γ(C_T > 0) + Γ(C_T < 0)] where the Γ's are partial widths.

The corresponding quantity for D⁰ are C_T = P_K- • (P_π- × P_π+). and A_T ≡ [Γ(−C_T > 0) − Γ(−C_T < 0)] / [Γ(−C_T > 0) + Γ(−C_T < 0)] , These asymmetries can have a nonzero value because of FSI.
On the other hand, one can construct the T-odd asymmetry: A_T-odd ≡ (A_T − A_T)/2 that is nonzero only if CP violation is present.

For hadronic decays of charm mesons, the T-odd method becomes P-odd and allows one to probe CP violation occurring via P-violation. This type of CP violation may arise only in P-odd amplitudes, which are allowed in decays of mesons into final states with at least four spinless particles. All the two- and three-body hadronic decays of charm mesons involve only P-even amplitudes. Thus T-odd asymmetry is a clean and alternative way to search for CP violation in the charm sector.

References:
E. Golowich and G. Valencia, Phys. Rev. D 40, 112 (1989).
I.I. Bigi, Proceedings of KAON2001, 417 (2001).
W. Bensalem and D. London, Phys. Rev. D 64, 116003 (2001).
W. Bensalem et al., Phys. Rev. D 66, 094004 (2002).
W. Bensalem et al., Phys. Lett. B 538, 309 (2002).

T-odd asymmetries in D⁰ decays

Year	Experiment	T-odd asymmetry in the decay mode D0 to K+K-π+π-	AT-odd = (AT − AT)/2
2019	BELLE	J. B. Kim et al. (BELLE Collab.), Phys. Rev. D 99, 011104 (2019).	+0.0052 ± 0.0037 ± 0.0007
2014	LHCb	R. Aaij et al. (LHCb Collab.), JHEP 10, 5 (2014).	+0.0018 ± 0.0029 ± 0.0004
2010	BABAR	P. del Amo Sanchez et al. (BABAR Collab.), Phys. Rev. D81, 111103 (2010).	+0.0010 ± 0.0051 ± 0.0044
2005	FOCUS	J.M. Link et al. (FOCUS Collab.), Phys. Lett. B 622, 239 (2005).	+0.010 ± 0.057 ± 0.037
		HFLAV average	+0.0035 ± 0.0021
Year	Experiment	T-odd asymmetry in the decay mode D0 to K0sπ+π-π0	AT-odd = (AT − AT)/2
2017	BELLE	K. Prasanth et al. (BELLE Collab.), Phys. Rev. D 95, 091101 (2017).	-0.00028 ± 0.00138 (+0.00023 -0.00076)
Year	Experiment	T-odd asymmetry in the decay mode D0 to K0sK0sπ+π-	AT-odd = (AT − AT)/2
2022	BELLE	(BELLE Collab.), arXiv:2207.07555 [PRD].	-0.0195 ± 0.0142 (+0.0014 -0.0012)

T-odd asymmetries in D⁺ decays

Year	Experiment	T-odd asymmetry in the decay mode D+ to K0sK+π+π-	AT-odd = (AT − AT)/2
2011	BABAR	J.P. Lees et al. (BABAR Collab.), Phys. Rev. D 84, 031103 (2011).	-0.0120 ± 0.0100 ± 0.0046
2005	FOCUS	J.M. Link et al. (FOCUS Collab.), Phys. Lett. B 622, 239 (2005).	+0.023 ± 0.062 ± 0.022
		HFLAV average	-0.0110 ± 0.0109

T-odd asymmetries in D_s⁺ decays

Year	Experiment	T-odd asymmetry in the decay mode Ds+ to K0sK+π+π-	AT-odd = (AT − AT)/2
2011	BABAR	J.P. Lees et al. (BABAR Collab.), Phys. Rev. D 84, 031103 (2011).	-0.0136 ± 0.0077 ± 0.0034
2005	FOCUS	J.M. Link et al. (FOCUS Collab.), Phys. Lett. B 622, 239 (2005).	-0.036 ± 0.067 ± 0.023
		HFLAV average	-0.0139 ± 0.0084