$CP$ asymmetry in the angular distribution of $\tau\to K_S\pi\nu_\tau$ decays

In this work, we study the $CP$ asymmetry in the angular distribution of $\tau\to K_S\pi\nu_\tau$ decays, taking into account the known $CP$ violation in $K^0-\bar{K}^0$ mixing. It is pointed out for the first time that, once the well-measured $CP$ violation in the neutral kaon system is invoked, a non-zero $CP$ asymmetry would appear in the angular observable of the decays considered, even within the Standard Model. By employing the reciprocal basis, which is most convenient when a $K_{S(L)}$ is involved in the final state, the $CP$-violating angular observable is derived to be two times the product of the time-dependent $CP$ asymmetry in $K\to \pi^+\pi^-$ and the mean value of the angular distribution in $\tau^\pm\to K^0(\bar{K}^0)\pi^\pm\bar{\nu}_\tau(\nu_\tau)$ decays. Compared with the Belle results measured in four different bins of the $K\pi$ invariant mass, our predictions lie within the margins of these measurements, except for a $1.7~\sigma$ deviation for the lowest mass bin. While being below the current Belle detection sensitivity that is of $\mathcal{O}(10^{-3})$, our predictions are expected to be detectable at the Belle II experiment, where $\sqrt{70}$ times more sensitive results will be obtained with a $50~\text{ab}^{-1}$ data sample.


Introduction
which is defined as the difference of the differential τ − and τ + decay widths weighted by cos α, and can be evaluated in bins of the Kπ invariant mass squared s [21]. Here α is the angle between the directions of K and τ as seen in the Kπ rest frame [15,21,36]. When considering the observabe A CP i , one should keep in mind the following two facts [25]: (i) the τ + (τ − ) decay produces initially a K 0 (K 0 ) state due to the ∆S = ∆Q rule; (ii) the intermediate state K S is not observed directly in experiment, but rather reconstructed in terms of a π + π − final state with M ππ ≈ M K and a time that is close to the K S lifetime. However, since CP is violated in K 0 −K 0 mixing, the final state π + π − can be obtained not only from the short-lived K S but also from the long-lived K L state. Thus, the measured CP asymmetry depends sensitively on the kaon decay time interval over which it is integrated. As emphasized by Grossman and Nir [25] in the study of the decay-rate asymmetry, the contribution from the interference between the amplitudes of intermediate K S and K L is not negligible, but is as important as the pure K S amplitude. In addition, one should also take into account the experiment-dependent effects, due to the efficiency as a function of the kaon decay time as well as the kaon energy in the laboratory frame to account for the time dilation [25].
It is known that, after neglecting the effect generated by the second-order weak interaction, which is estimated to be of O(10 −12 ) [39], there exists direct CP violation neither in the decay rate nor in the angular distribution of τ ± → K 0 (K 0 )π ±ν τ (ν τ ) decays within the SM [15,36]. Once the known CP violation in the neutral kaon system is taken into account, however, a non-zero CP asymmetry is predicted in the decay rates [23][24][25]. Here we shall investigate for the first time whether an observable CP asymmetry in the angular distribution of τ → K S πν τ decays could be generated by the well-measured CP violation in K 0 −K 0 mixing. To this end, we shall employ the reciprocal basis [40][41][42][43][44][45], which is most convenient when a K S(L) is involved in the final state and has been used to reproduce conveniently the decay-rate asymmetry [33].
It is then found that a non-zero CP asymmetry would appear in the angular observable of the decays considered, even within the SM. Furthermore, this observable is derived to be two times the product of the time-dependent CP asymmetry in K → π + π − and the mean value of the angular distribution in τ ± → K 0 (K 0 )π ±ν τ (ν τ ) decays. While the formalism of the former is quite clear [43], a precise description of the latter requires information about the Kπ vector and scalar form factors, including both their moduli and phases. As the form-factor phases fitted via a superposition of Breit-Wigner functions with complex coefficients do not vanish at threshold and violate Watson's final-state interaction theorem before the higher resonances start to play an effect [31][32][33], we cannot rely on the formalism developed in Refs. [46][47][48][49] to study the CP asymmetry in τ → K S πν τ decays. Instead, we shall adopt the thrice-subtracted (for the vector form factor) [50,51] and the coupled-channel (for the scalar form factor) [52][53][54] dispersive representations, which warrant the properties of unitarity and analyticity, and contain a full knowledge of QCD in both the perturbative and non-perturbative regimes.
With all the above points taken into account, we shall then present our predictions for the CP -violating angular observable A CP i (t 1 , t 2 ) defined by Eq. (2.21). It should be emphasized again that our presentation is confined to the SM framework. This is totally different from the studies made in Refs. [19][20][21], which are aimed to probe possible CP -violating (pseudo-)scalar couplings beyond the SM [15,18,[36][37][38]. It is numerically found that our predictions always lie within the margins of the Belle measurements, except for a 1.7 σ deviation for the lowest mass bin [21]. While being below the current Belle detection sensitivity that is of O(10 −3 ), our predictions are expected to be detectable at the Belle II experiment [34], where √ 70 times more sensitive results will be available with a 50 ab −1 data sample.
Our paper is organized as follows. In Sec. 2, we firstly derive the CP -violating angular observable in τ → K S πν τ decays by means of the reciprocal basis. The angular distribution of τ − →K 0 π − ν τ decay is then presented in Sec. 3, and Sec. 4 contains our numerical results and discussions. Finally, we conclude in Sec. 5. For convenience, dispersive representations of the Kπ vector and scalar form factors are given in the appendix.
2 CP -violating angular observable in τ → K S πν τ decay As the τ − (τ + ) decay produces initially aK 0 (K 0 ) state due to the ∆S = ∆Q rule, we have for the SM transition amplitudes the following relation: which is due to the fact that the CKM matrix element V us involved is real and the strong phase must be the same for these two CP -related processes [23]. The relevant Feynman diagrams at the tree level in weak interaction are shown in Fig. 1.
are not the flavour (|K 0 and |K 0 ) but rather the mass (|K S and |K L ) eigenstates, which, in the absence of CP violation in the system, are related to each other via [43,45] where ζ is the spurious phase brought about by the CP transformation, CP|K 0 = e iζ |K 0 [45].
In this case, the double differential decay widths of τ → K S πν τ decays satisfy which, taken together with Eq. (2.1), indicate that there exists CP asymmetry neither in the integrated decay rate nor in the angular distribution of τ → K S πν τ decays within the SM. 1 Once the well-measured CP violation in K 0 −K 0 mixing [1,26] is taken into account, however, the two mass eigenkets |K S,L will be now given by [45] 2 with the normalization |p| 2 + |q| 2 = 1. The corresponding mass eigenbras K S,L | read [45] K S,L | = 1 2 Here we do not consider the contribution from second-order weak interaction, which is estimated to be of O(10 −12 ), and can be therefore neglected safely [39]. 2 The CP T invariance is still assumed here. For the general case in which both CP and CP T are violated in the mixing, the readers are referred to Refs. [43,45].
which form the so-called reciprocal basis that is featured by both the orthornormality and completeness conditions [45]: Notice that the mass eigenbras K S,L | do not coincide with K S,L |, the Hermitian conjugates of the mass eigenkets |K S,L . This is because the 2 × 2 effective Hamiltonian H responsible for the K 0 −K 0 mixing is not a normal matrix, and hence cannot be diagonalized by a unitary transformation but rather by a general similarity transformation [43][44][45]. Consequently, the time-evolution operator for the K 0 −K 0 system is determined by where µ S,L = M S,L − i/2 Γ S,L are the two eigenvalues of the effective Hamiltonian H, with the real and imaginary parts representing their masses and decay widths, respectively.
Experimentally, the intermediate state K S in τ → K S πν τ decays is not directly observed, but rather reconstructed in terms of a π + π − final state with M ππ ≈ M K [19][20][21]46]. When CP violation in the neutral kaon system is invoked, however, the final state π + π − can be obtained not only from the short-lived K S , but also from the long-lived K L state, when the kaon decay time is long enough. Thus, the processes τ ± → [π + π − ]π ±ν τ (ν τ ) proceed actually as follows: the initial states τ ± decay into the intermediate states K S,L π ±ν τ (ν τ ), which after a time t decay into the final state [π + π − ]π ±ν τ (ν τ ). In this context, it is convenient to apply Eq. (2.7) to describe the time evolution of these processes. With the reference to π ±ν τ (ν τ ) suppressed, the complete amplitudes for these two CP -related processes can be written as [33,44] where Eq. (2.5) and the ∆S = ∆Q rule have been used to obtain the second lines. It is obvious from Eqs. (2.8) and (2.9) that the kaon decays are independent of the τ decays, which means that the complete double differential decay widths can be written as with the time-dependent K → π + π − decay widths given by denote respectively the mass difference and the average width of the K 0 −K 0 system, while η +− is defined as the CP -violating amplitude ratio for the π + π − final state, with its modulus |η +− | = (2.232 ± 0.011) × 10 −3 and its phase φ +− = (43.51 ± 0.05) • [26].
Keeping in mind that [26,43] where K is the CP -violating parameter in neutral kaon decays, one can see from Eqs. (2.13) and (2.12) that, for the sum of the two time-dependent decay widths, both the interference (the last) and the pure K L term (the second) are suppressed compared to the pure K S contribution (the first term in the square bracket); for their difference, however, the interference between the amplitudes of K S and K L is found to be as important as the pure K S amplitude [25].
As a consequence, the CP -violating angular observable defined by Eq. (1.1) will depend on the times over which the differential decay rates are integrated. In addition, once the time evolution of the kaons are considered, one has to take into account not only the efficiency as a function of the kaon decay time, but also the kaon energy in the laboratory frame to account for the time dilation [25]. With all these experiment-dependent effects parametrized by a function F (t) [25], and for a decay-time interval [t 1 , t 2 ], we can then define ds d cos α , Γ(Γ) π + π − (t) = Γ(K 0 (K 0 )(t) → π + π − ), and cos α τ ± i denote the differential τ ± decay widths weighted by cos α and evaluated in the i-th bin.
Within the SM, one has A CP τ,i = 0 and cos α τ − i = cos α τ + i due to dΓ τ + dω = dΓ τ − dω , and thus the CP -violating angular observable defined by Eq. (2.16) reduces to which is the key result obtained in this work, and indicates that, once the well-measured CP violation in the neutral kaon system is invoked, a non-zero CP asymmetry would appear in the angular observable of the decays considered, even within the SM.
As indicated by Eq. (2.21), in order to get a prediction of the CP asymmetry A CP i (t 1 , t 2 ), one should firstly determine both cos α τ − i and A CP K (t 1 , t 2 ). The mean value of the angular observable cos α τ − i is related to the so-called forward-backward asymmetry A FB [55], the computation of which will be detailed in the next section. As the observable A CP K (t 1 , t 2 ) is sensitive to the experimental cuts, its prediction can be made only when the kaon decay time interval over which it is integrated as well as the experiment-dependent function F (t) have been determined. Here we shall quote the particularly simple prediction made in Ref. [25], in which the approximations with |η +− | ≈ 2 e( K ) , as well as a double step function [25]  3 Angular distribution in τ − →K 0 π − ν τ decay Within the SM, the effective weak Hamiltonian responsible for the strangeness-changing hadronic τ decays is given by where G F is the Fermi coupling constant, and V us is the CKM matrix element involved in the transitions. The decay amplitude for τ − →K 0 π − ν τ decay can then be written as where S EW = 1.0201(3) encodes the short-distance electroweak radiative correction to the hadronic τ decays [56]. In Eq. (3.2), L µ denotes the leptonic current given by while H µ denotes the hadronic matrix element and can be parametrized as 3

4)
where s = (p K + p π ) 2 , q µ = (p K + p π ) µ , ∆ Kπ = M 2 K − M 2 π , and F + (s) and F 0 (s) are the Kπ vector and scalar form factors associated with the J P = 1 − and J P = 0 + components of the weak charged current, respectively. As mentioned already in Sec. 1, we shall use the dispersive representations rather than the Breit-Wigner parameterizations of these form factors in this work. For convenience, their explicit expressions are presented in the appendix.
Working in the Kπ rest frame, and after integrating over the unobserved neutrino direction, one can then write the double differential decay width of τ − →K 0 π − ν τ decay as [36,57] which involves only the moduli of the two Kπ form factors, and is dominated by the vector one. This implies that there exists no CP violation in the decay rate within the SM [23][24][25].
The Kπ invariant mass distribution of the differential decay width dΓ τ − To obtain further information about the Kπ vector and scalar form factors and, especially, about their relative phase that is of particular interest in relation to the study of CP violation, one must resort to other observables that involve the interference between these two form factors. For this purpose, one introduces the angular observable [34] cos α τ − (s) = which is defined as the differential decay width weighted by cos α. It is interesting to note that the observable cos α τ − (s) is connected with the so-called forward-backward asymmetry [34], with the latter defined by [38,55,58] Being proportional to the factor ∆ Kπ = M 2 K − M 2 π that would vanish in the limit of the exact SU(3) flavour symmetry, the angular observable cos α τ − (s) (or equivalently the forwardbackward asymmetry A τ − FB (s)) also allows us to measure the SU(3) breaking effect in the decays considered [55]. The Kπ invariant mass distribution of cos α τ − (s) is shown in the right plot of Fig. 2 As the CP -violating angular observable A CP i (t 1 , t 2 ) defined by Eq. (2.21) is usually measured in bins of the Kπ invariant mass [19][20][21], one can also make the observable cos α τ − (s) to be bin-dependent, The right plot of Fig. 2 suggests then that, in order to obtain a value of cos α τ − i as large as possible, the Kπ invariant mass bins can be optimally set at the vicinities of the two negative extrema of cos α τ − (s). To see this clearly, we shall make a detailed numerical estimate in the next section.

Numerical results and discussions
Before presenting our numerical results, we firstly collect in Table 1 all the input parameters used throughout this work. For any further details, the readers are referred to the original references. With the time-dependent CP -violating observable A CP K (t 1 , t 2 ) fixed by Eq. (2.22), the computation of the CP -violating angular observable A CP i (t 1 , t 2 ) is then attributed to that of the observable cos α τ − i , which becomes now straightforward. In order to make a direct numerical comparison with the Belle measurements, we choose the same intervals for the four bins of the Kπ invariant mass as in Ref. [21]. Notice that the mass threshold used by the Belle collaboration for the lowest mass bin, s Kπ = 0.625 GeV [21], lies slightly below the theoretical one, s Kπ = M K + M π = 0.637 GeV. As such a numerical difference has only a marginal impact on our prediction, we shall use the latter as input in this work. Our final predictions for the CP -violating angular observable A CP i (t 1 , t 2 ) are shown in Table 2, where, for a comparison, the Belle measurements after background subtraction in each mass bin have also been given. One can see that our predictions always lie within the margins of the Belle results [21], except for a 1.7 σ deviation for the lowest mass bin. It should be pointed out that our predictions, while being below the current Belle detection sensitivity  CP -violating parameters in the neutral kaon system [26] |η +− | × 10 3 φ +− e( K ) × 10 3 2.232 ± 0.011 (43.51 ± 0.05) • 1.66 ± 0.02 times more sensitive results will be obtained following the increase of the integrated luminosity of a 50 ab −1 data sample.
As mentioned already in last section, in order to get a value of A CP i (t 1 , t 2 ) as large as possible for a given time interval, we present two more predictions with the Kπ invariant mass intervals selected at the vicinities of the two negative extrema of cos α τ − : It is interesting to note that the value of this observable in the mass interval [0.70, 0.75] GeV is as large as the SM prediction for the decay-rate asymmetry [25]. Thus, we suggest the experimental τ physics groups at Belle II to measure the CP -violating angular observable in this mass interval.

Conclusion
In this work, inspired by the study of decay-rate asymmetry in τ → K S πν τ decays induced by the known CP violation in K 0 −K 0 mixing [23,25], we have performed an investigation of the same effect on the CP asymmetry in the angular distribution of the same channels within the SM. Our main conclusions are summarized as follows: (i) Once the well-measured CP violation in the neutral kaon system is invoked, a nonzero CP asymmetry would appear in the angular observable of the decays considered, even within the SM. By utilizing the reciprocal basis, which has been used to reproduce conveniently the decay-rate asymmetry [33], this observable is derived to be two times the product of the time-dependent CP asymmetry in K → π + π − and the mean value of the angular distribution in τ ± → K 0 (K 0 )π ±ν τ (ν τ ) decays (see Eq. (2.21)).
(ii) As the relative phase between the Kπ vector and scalar form factors is required for the study of CP violation, but the form-factor phases fitted via a superposition of Breit-Wigner functions do not vanish at threshold and violate Watson's theorem before the higher resonances start to play an effect [31], we did not adopt the Breit-Wigner parameterizations of these form factors. Instead, the thrice-subtracted (for the vector form factor) [50,51] and the coupled-channel (for the scalar form factor) [52][53][54] dispersive representations have been employed, which warrant the properties of unitarity and analyticity, and contain a full knowledge of QCD in both the perturbative and non-perturbative regimes.
(iii) Our predictions for the CP -violating angular observable A CP i (t 1 , t 2 ) always lie within the margins of the Belle measurements [21], except for a 1.7 σ deviation for the lowest mass bin. While being below the current Belle detection sensitivity that is of O(10 −3 ), our predictions are expected to be detectable at the Belle II experiment [34], where √ 70 times more sensitive results will be obtained following the increase of the integrated luminosity of a 50 ab −1 data sample.
(iv) In order to get a value of A CP i (t 1 , t 2 ) as large as possible, two more predictions have been made with the Kπ invariant mass intervals selected at 0.70 GeV < √ s < 0.75 GeV and where one subtraction constant is fixed by the form-factor normalization F + (0) = 1, while the other two λ + and λ + describe the slope and curvature ofF + (s) when performing its Taylor expansion around s = 0, and hence encode the low-energy behaviour ofF + (s). To capture our ignorance of the higher-energy part of the dispersion integral, they will be treated as free parameters, and are determined by fitting to the experimental data [50,51,60]. The form-factor phase δ + (s) in Eq. (A.1) is calculated from the relation where the explicit expression off + (s) has been given by Eq. (4.1) of Ref. [50], which is derived in the context of chiral perturbation theory with resonances (RχT) [61,62], with both K * (892) and K * (1410) included as explicit degrees of freedom [50,51,63,64]. The cut-off s cut is introduced in Eq. (A.1) to quantify the suppression of the higher-energy part of the integral, and the stability of the numerical results has been checked by varying s cut in the range m τ < √ s cut < ∞ [50,51]. Here we shall choose s cut = 4 GeV 2 , because such a choice is, on the one hand, large enough to not spoil the a priori infinite interval of the dispersive integral and, on the other hand, low enough to have a good description of the form-factor phase within the interval considered [65]. Following such a procedure [50,51,65], we show in the left panel of Fig. 3 both the modulus and the phase of the normalized form factorF + (s). form factor is taken from Refs. [50,51], while the scalar form factor is from Ref. [53,60], with the ranges of the modulus obtained by varying the form factor at the Callan-Treiman point [54].
For the Kπ scalar form factor, we shall employ the coupled-channel dispersive representa-tion presented in Ref. [53] and updated later in Refs. [54,66,67], which is obtained by solving the multi-channel Muskelishivili-Omnès problem for three channels (with 1 ≡ Kπ, 2 ≡ Kη, and 3 ≡ Kη ). Explicitly, the scalar form factor for the channel i can be written as [53] F i 0 (s) = where s j and σ j (s) denote respectively the threshold and the two-body phase-space factor for the channel j, and t i→j 0 (s) is the partial wave T -matrix element for the scattering i → j [52,53]. As these form factors are coupled to each other, they can be obtained by solving numerically the coupled dispersion relations arising from Eq. (A.3), taking into account the chiral symmetry constraints at low energies as well as the short-distance dynamical QCD constraints at high energies [52,53]. Here we shall make use of the numerical results obtained from a combined analysis of the τ − → K S π − ν τ and τ − → K − ην τ decays [60]. 4 Again, both the modulus and phase of the reduced form factorF 0 (s) are shown in the right panel of Fig. 3.