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Can the observed CP asymmetry in tau -->K pi nu_tau be due to Non-Standard Tensor interactions?

An intriguing opposite sign of the CP violating asymmetry was recently measured in the tau decay modes tau ->Ks pi nutau by the BaBar collaboration, than that expected within the Standard Model. If this result is confirmed with higher precision, the observed decay rate asymmetry ACP, can only arise from some Non-Standard Interactions(NSI) occurring possibly in both the hadronic as well as in the leptonic sectors. We illustrate that while a simple charged scalar interaction cannot yield this rate asymmetry, it will be possible to generate this in the presence of a tensor interaction. Parameterizing the strength and weak phase of this NSI contribution, the observed branching ratio and the decay rate CP asymmetry for the particular mode tau ->Ks pi nutau, are used to determine the CP violating weak phase and the coupling of a tensorial interaction that can give a consistent sign and magnitude of the asymmetry.


I. INTRODUCTION
While the Kobayashi Maskawa ansatz for CP violation within the Standard Model(SM) [1] in the quark sector has been clearly verified by the plethora of data from the B factories, this is unable to account for the observed Baryon Asymmetry of the Universe (BAU). Hence one needs to look for other sources of CP violation, including searches in the leptonic sector.
Apart from the CP phases that may arise in the neutrino mixing matrix, the decays of the charged tau leptons may allow us to explore non-standard CP violating interactions.
In fact, CP violation through decays of tau leptons had been studied in a series of papers by Tsai [2], around the time of the preliminary design for a tau-charm factory. Various experimental groups have been involved in exploring CP violation in tau decays in the last decade or more. In 2002, the CLEO collaboration [3] and more recently the Belle Collaboration [4] studied the angular distribution of the decay products in τ ± → K 0 s π ± (−) ν τ in search of CP violation, however, neither study revealed any CP asymmetry. The BaBar collaboration [5] for the first time reported a measurement of the rate asymmetry A CP in this decay mode to be: A Exp. CP = (−0.36 ± 0.23 ± 0.11)%. (1) On the theoretical side, for τ ± → K 0 s π ± (−) ν τ → [π + π − ] K π ± (−) ν τ , Bigi and Sanda [6] predicted the CP asymmetry to be, where the CP violation arises from the known K 0 −K 0 mixing. Recently, Grossman and Nir [7] comparing the rate asymmetries for decays to neutral kaons, of the taus with that of D mesons, pointed out that since τ + (τ − ) decays initially to a K 0 (K 0 ) whereas D + (D − ) decays initially toK 0 (K 0 ), the time integrated decay rate CP asymmetry (arising from oscillations of the neutral kaons) of τ decays must have a sign opposite to that of D decays. Further, they emphasized that the decay asymmetry is affected by the reconstruction efficiency as a function of the K s → π + π − decay time. Using the parameterization of Ref. [7], BaBar has obtained a multiplicative correction factor for the decay-rate asymmetry and predicts the SM decay-rate asymmetry to be, A SM CP = (+0.36 ± 0.01) %.
The observation of a CP asymmetry in tau decays to K s having the same sign as that in D decays and moreover of the same magnitude but opposite in sign to the corrected SM expectation, implies that this asymmetry cannot be accounted for by the CP violation in K 0 −K 0 mixing. Apart from this mixing contribution to CPV, within the SM, since there is only a single amplitude with the W boson mediating the decay process, the observed CP asymmetry cannot be explained. Hence, this may be a signal of physics beyond the standard model, if confirmed to a higher statistical significance.
It should be pointed out that the BaBar collaboration has accounted for the modification of the decay rate asymmetry due to different nuclear interaction cross-sections of the K 0 andK 0 mesons with the detector material through a correction, calculated on an event by event basis. Also, BaBar [5] claims that using a control sample in data and Monte Carlo simulations, they have verified that no significant decay rate asymmetry is induced by their detector or the selection criteria. Further, the standard model asymmetry is identical for decays with any number of π 0 mesons and hence can be searched for, in all the modes We note that all the experiments CLEO, Belle, BaBar assume that the CP asymmetry is conserved at the tau production vertex.
In the presence of an additional new physics (NP) amplitude with a complex coupling, along with the strong phases from the K − π scattering, which can be large particularly in this resonance dominated region, the observed CP asymmetry may be attainable. This Non Standard Interaction (NSI) could possibly affect both the hadronic and leptonic currents.
The Feynman diagrams for the SM decay mode of tau via W exchange and via exchange of some exotic particle X, in the presence of NP are shown in Figure (1).
Naively one would expect a charged scalar boson exchange to provide the required additional diagram, where, with a complex weak coupling and the difference in the strong phases of the scalar and vector hadronic form factors, a CP violating asymmetry could arise. However, such an asymmetry appears only in the difference of the τ ± → K S π ± ν τ (ν τ ) decay angular distributions, but vanishes in the integrated difference of the decay rates for τ + and τ − , measured by BaBar. In fact, CLEO had used their non-observation of an asymmetry in the distribution, to set a bound [3]  was compatible with zero. The new physics amplitude that can account for a decay rate asymmetry therefore has to appear from a different kind of interaction and we investigate whether a tensor interaction can produce the asymmetry reported by BaBar and in fact, use the measured branching ratio and asymmetry to constrain the parameters of this kind of new interaction.
In section II, we evaluate the decay rate asymmetry in the presence of any generic NP amplitude. The decay rate for the SM case is calculated in section III, while in section IV the total rate is evaluated in the presence of the new additional tensor interaction term.
The observables, the branching ratio and the decay rate asymmetry are used to estimate the parameters of this new interaction in section V and in section VI we conclude.

II. BRANCHING RATIO AND RATE ASYMMETRY IN τ → Kπν τ IN PRESENCE OF A GENERIC NEW PHYSICS AMPLITUDE
The amplitude for the decay of τ + → K s π +ν τ in the presence of NSI can be written as where A SM and A N SI are the magnitudes of the SM and NP amplitudes respectively, while φ is the weak phase of the NP contribution (since V us is real, there is no weak phase in the SM contribution, except that coming from the neutral K meson mixing, which is accounted for, separately in the theoretical SM expectation). The relative strong phase of the (Kπ) system, δ, between the NP amplitude with respect to the SM contribution is a function of the Kπ invariant mass-squared. The amplitude for the anti-process τ − → K s π − ν τ has the opposite weak phase but the same strong phase.
In presence of a NSI whose interference with SM is non-vanishing even after the angular integrations 1 , the general expression for the differential decay rate of τ → Kπν may be written as, where Q is the sum of the hadron momenta. The differential decay rate for the anti-process is, The branching ratio for τ → K s πν is the ratio of the average of the width of τ + → K s π +ν and τ − → K s π − ν to the total width of τ (Γ total ). Hence, Now, where r(Q 2 ) is the ratio of the amplitude of the NSI contribution to the SM contribution. Therefore, Similarly the difference, Hence, the integrated rate asymmetry, As pointed out in Refs. [12] and [13], this CP asymmetry being linear in the new physics amplitude has a higher sensitivity to it, than effects like lepton flavour violation and electric dipole moments etc., which depend quadratically on the NP amplitude, rendering this observation to play an important role in uncovering physics beyond the SM. Using eqn. (12) and eqn. (15) and the PDG [14] value of the Branching Ratio and the rate asymmetry measured by BaBar, the weak phase and the magnitude of the new physics contribution can be estimated.

III. DECAY RATE OF τ → Kπν IN THE STANDARD MODEL
The tau leptonic and the hadronic decay amplitude can be factorized into a purely leptonic part including the tau and the neutrino and a hadronic part, where the hadronic system is created from the QCD vacuum via the charged weak current.
Hence, the differential decay rate of the process τ (p τ ) → K(p K ) + π(p π ) + ν(p ντ ) [15] may be written as, where L µν is the leptonic term: and the hadronic term: is given in terms of the hadronic vector current A transition from vacuum to two pseudoscalar mesons can occur only via scalar and vector currents as for the Kπ system the allowed values of J P are 0 + and 1 − . The hadronic vector current in eqn. (19) is parameterized in terms of the scalar and the vector form factors as, where Q µ = p µ k + p µ π . The decay rate involves only the mod-squared of the vector and scalar form factors but no scalar-vector interference term. In the hadronic-rest frame where p k + p π = 0, it takes the following form, where S EW = 1.02 [16] is the electroweak correction factor and P (Q 2 ) ≡ | p k |, is the momentum of the kaon in the Kπ rest frame, which is a function of the Kπ invariant mass squared Q 2 , and may be expressed as: After integrating out the neutrino momentum the phase space in the Kπ rest frame is where β is the direction of the kaon with respect to the tau direction, denoted byn τ , viewed from the hadronic rest frame, i.e; cosβ =p K .n τ wherep K = p K | p K | . Hence, the differential decay rate takes the form, The hadronic current is dominated by many resonances, the vector ones: K * (892),K * (1410) and K * (1680) and the scalars: K * 0 (800) and K * 0 (1430). The form factors can be parameterized in terms of Briet-Wigner forms with energy dependent widths. Hence the vector form factor may be written as [17]: where β and χ are the complex coefficients for the relative contributions of of K * (1410) and K * (1680) resonances respectively with respect to the dominant K * (892) contribution, and BW R (s) is a relativistic Breit-Wigner function corresponding to R being, K * (892), K * (1410) or K * (1680) for the vector case.
For each of the resonances, the Breit-Wigner function has the following form: where Here, Γ R (s) is the s-dependent total width of the resonance and Γ 0R (s) is the resonance width at its peak. The orbital angular momentum l = 1 if the Kπ system is in a p-wave or a the vector state and l = 0 for the s-wave or scalar state.
The scalar form factor F S has K * 0 (800) and K * 0 (1430) contributions and has the similar form: where, κ and γ are the complex constants that describe the relative contributions of the K * 0 (800) and K * 0 (1430) resonances respectively. The Belle collaboration had performed fits to the K s π − invariant mass spectrum (Q 2 ) distribution and had listed the values of the complex constants β,κ, χ and γ in Ref. [17]. Their fitted results (as well as those of BaBar reported in [18]) demonstrated that a K * (892) alone is not enough to describe the K s π mass spectrum but rather the distribution shows the clear evidence for a scalar contribution in the low invariant mass and another component at large Q 2 . The fits were best explained with either K * (892) + K * (1410) + K * (800) or K * (892) + K * (1430) + K * (800). Therefore we have used these two possibilities for our study and have excluded K * (1680) in our study as its inclusion worsen the fit quality in the Belle analysis. We would also like to point out that the Belle fit results are also consistent with a theoretical description using Chiral Perturbation Theory, described in Ref. [19] and that of Ref. [20] which is based on analycity and K − π scattering results.

IV. TOTAL RATE IN PRESENCE OF A NEW TENSOR INTERACTION
We now propose an additional tensor contribution to the amplitude. We explore if the interaction of the SM with the new tensorial interaction can account for a non-vanishing CP asymmetry in the decay mode τ → Kπν τ . This tensor interaction could arise in various NP models, however, our approach is to study just the effect of this new structure and keep the analysis as model independent as possible. Interference of the SM with this tensor amplitude must give a non vanishing CP asymmetry for this particular decay mode, moreover the sign and magnitude the CP asymmetry must be consistent with the observed result.
The effective Hamiltonian due to this new operator is written as where G ′ is a complex coupling, The new physics amplitude is then given by where σ µν = i 2 (γ µ γ ν − γ ν γ µ ) and the Hadronic current is given by with F T being the form factor due to the tensor interaction. In A T , we include only left handed neutrinos, as the similar term with a right handed neutrino, in the interference of the SM and NP contributions at the decay rate level, will be suppressed by the neutrino mass. We note here that the A T is a special case of the generalised A N SI mentioned in section III. More specifically now, the effective amplitude A is now given by, where and The observables used in this study are obtained after the integration over the angular variables of the different contributions in eqn.(33). In the interference term of the new tensor contribution with the SM contribution (eqn.(34)), the term that arises from the symmetric (Q µ ) part of the standard current vanishes after this angular integration, and hence only the interference of the tensor with the vector form factor appears. The full differential decay rate may hence be written as, and For the conjugate tau decay mode, only the interference term in eq. (39), will differ, having the opposite weak phase φ.
As mentioned above, the interference of the scalar contribution of the SM and the antisymmetric tensor contribution vanishes. This is similar to the vanishing of the scalar and the vector interference contribution in the SM itself. Note that the scalar term is even under parity, while both vector and tensor are odd under parity, resulting in only the vector-tensor interference being even under parity and hence surviving after the full (parity even) phase space integration. In other words, once the angular integration is performed, terms that are odd in cosβ vanish, however the parity even interference of the vector and tensor terms contribute to the decay rate even after this integration.
We wish to point out that after completion of this work, we became aware of some earlier papers where tensor interactions had been introduced in semileptonic tau decays, namely Refs. [21], [22] and [23]. We notice several differences in our approach and these earlier papers. Firstly, Ref. [21], claims that the tensor amplitude does not interfere with the SM amplitude, however, as shown in our calculations in this section, this is not true. In fact, it is exactly this interference that can possibly account for the CP violating rate asymmetry.
Ref. [22] generates a very tiny CP asymmetry (≈ 10 −12 ) by second order weak interactions and [23] has given a numerical estimate of such an asymmetry, but in the context of a SUSY model, unlike our analytical formulae for a generic tensor interaction. Moreover, since this paper preceeded the BaBar rate asymmetry measurement, they have not used the observables to constrain the NP parameters, which we attempt in the following section.

V. ESTIMATION OF THE NEW PHYSICS PARAMETER FROM OBSERV-ABLES
In order to estimate the parameters of the new tensor interaction term, we need to numerically compute the total decay rates for τ + → K s π +ν and τ − → K s π − ν, using eqns. (36)-(39).
We can write the vector and the scalar form factors in terms of the magnitude and the strong phases as, Having expressed the form factors in terms of the combinations of Briet Wigner forms of the various resonances, given in eqns. (25) and (28), the strong phases of the scalar and the vector form factors can be simply extracted from these complex forms. We have used this vector form factor strong phase (Q 2 dependent) for our results below. As mentioned in section III, Belle proposed two fit models for the Q 2 distribution of their data, which had comparable χ 2 values, both having the vector K * (892) resonance where the data peaks, as well as the scalar K * (800). In the region around 1.4GeV , since the data lied much higher than the fitted curve, inclusion of either K * (1410) or K * (1430) resulted in a significant improvement in the goodness of fit. Hence, in our analysis, we consider both the possibilities: K * (892) + K * (1410) + K * (800) and K * (892) + K * (1430) + K * (800), considering these two cases one at a time and naming them as Case I and Case II respectively.
Note that a relative orbital angular momentum l = 2 of the Kπ system would get contribution from a symmetric 2 + state, however, since our NP amplitude consists of an antisymmetric tensor contribution, such a resonant 2 contribution to the Kπ is not feasible, we take the tensor form factor to be real and hence δ appearing in eqn. (15) in presence of the new tensor interaction will be: δ = δ v , since, δ T = 0. A similar tensor form factor had been introduced in the analysis of K e3 and K µ3 data and in fact PDG [14] gives the constraint on the ratio of |f T /f V | for these decays. However, with no such existing analysis from experiments nor any theoretical lattice estimates of the tensor hadronic form factor in the Q 2 range relevant for the tau decay being considered here, we assume the tensor form factor to be a constant for simplicity, and determine the product of this constant and its coupling strength from the tau decay observables.
In the presence of this tensorial NSI, the experimentally measured CP asymmetry A Exp CP will be a result of the combination of the CP asymmetry arising from the K 0 -K 0 mixing and direct CP asymmetry appearing in the tau decay. If the CP asymmetry is a consequence only of the mixing contribution, then this asymmetry depending on the integrated decay times may be expressed as, The K s produced in the tau decay is observed through the (π + π − ) final state with m ππ = m K . The τ + decays to K 0 at the time t 1 of tau decay (τ − decays toK 0 at t 1 ) and the time difference between the tau decay and the Kaon decay is of the order of the K s lifetime (τ S ).
Hence, in presence of both direct and indirect CP violation, we may express the observed decay rate asymmetries as Interestingly, as shown below, this asymmetry can be factored into A K CP and A τ CP defined in eqs. (15) and (41), where A τ CP is the CP violation due to the tensorial interaction. Defining, the difference of the decay rates in the numerator of A Exp CP in eqn.(42) can be written as, Similarly, the sum, Therefore the observed asymmetry and the branching ratio for this decay mode can be written as, and Using the time dependence of the widths of K 0 andK 0 to ππ for t 1 ≤ τ S and τ S ≤ t 2 ≤ τ L we can show that, where p, q and ǫ are the standard K mixing parameters and we ignore terms of order ǫ 2 in the evaluation of the time integrals of the time dependent decay rates of K 0 andK 0 to ππ, as was done in reference [7]. Hence, the branching ratio for tau decay to Kπν may be written in the following form,  , [24]), we compute the Q 2 integrated results, Γ τ ± for both the decay modes τ ± → K s π ± (−) ν τ within the kinematic limits (m K + m π ) 2 and m 2 τ . This results in the average effective decay width dependent on the unknown parameters of the new interaction: the product of coupling constant R T (ratio wrt SM) and the tensor form factor F T which is assumed to be a constant in this work and the CP violating weak phase φ; numerically the effective widths in MeV are given by, In the above equation, the first number is the integrated width for SM, the second is the width corresponding to the mod-squared of the tensor contribution, while the last two terms are from the interference of the SM and tensor parts and hence dependent on the strong phase (Q 2 dependent, which has been integrated out). The last three terms have been computed in terms of the unknown parameters of NP, the weak phase φ and the product of tensor coupling and form-factor. From eqn. (15) we compute the second observable, the direct CP asymmetry, again in terms of the NP parameters to be, where, the difference of the widths for τ + and τ − are computed, using the difference of Γ 3 andΓ 3 (after integrating the expressions in eqn.(39) corresponding to those for τ + and τ − ).
Using, the average width, (Γ τ + + Γ τ − )/2 evaluated from eqn.  show the ratio of the contribution of the tensor mod-squared term, the interference terms involving the cosine of strong and weak phases and that involving the sine of the phases, respectively, wrt the SM contribution. The SM part uses the vector and scalar form factors corresponding to Case I described in the text. decay rate is computed to be, From eqn. (15) we get Similar to the first case, using the above two equations we get two feasible solutions for R T |F T | and cosφ shown in Table II below, where again, only the first solution is meaningful.  In future, once the hadronic form factor for the tensor contribution is estimated theoretically, hopefully from lattice calculations or a fresh analysis of the larger data sample that may be available 3 is performed by the experimental groups, including the fits with a new tensorial contribution to the amplitude, then, with some handle on the tensor form factor (including its possible Q 2 dependence), the coupling strength as well as the weak phase of NP can be pinned down further. Note that the Q 2 dependence of the mod-squared of the tensor amplitude is quite different from that of the other terms, which will enable its extraction from data.

VI. CONCLUSIONS:
CP violation in the quark sector, observed through decays and mixing of K and B mesons, is consistent with its parametrization within the standard model. However, it fails to explain the large baryon asymmetry of the universe and necessitates searches for CP violation beyond the standard model. Leptonic decays along with semileptonic decays of hadrons may offer a clean environment for searches of CP violating new physics beyond the standard model.
The recent observation of a CP rate asymmetry A CP by BaBar [5] in the tau decay mode τ ± → K s π ± (−) ν τ seems to hint at some new physics, with the observed decay rate asymmetry being approximately 2.8 standard deviations away from the standard model predictions of an asymmetry that arises from K 0 −K 0 mixing. The presence of various resonances in the vicinity of the decay hadrons invariant mass, facilitates the availability of strong phases, while complex couplings in a new physics amplitude could provide the weak phase, enabling the possibility of a direct CP asymmetry. A charged scalar contribution can provide a CP violating asymmetry in the angular distribution, but fails to produce an integrated rate asymmetry. However this is achievable with a generic non-standard tensorial interaction.
We calculated the effective decay rate in the presence of the additional tensor interaction and in fact used the observed branching ratio and CP asymmetry to obtain the parameters of the new physics, the weak phase φ and product of tensorial coupling and form factor.