Exploring new physics contributions to CP violation in τ-→K-π0ντ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^- \rightarrow K^-\pi ^0\nu _{\tau } $$\end{document}

A general analysis of possible violation of CP in processes like τ→Kπν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \rightarrow K\pi \nu $$\end{document}, for unpolarized τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is presented. In this paper, we derive the new contributions to the effective Hamiltonian governs |ΔS|=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \Delta S \vert =1$$\end{document} semileptonic tau decays in the framework of two Higgs doublet model with generic Yukawa structure and Leptoquarks models. Within these models, we list all operators, in the effective Hamiltonian and provide analytical expression for their corresponding Wilson coefficients. Moreover, we analyze the role of the different contributions, originating from the scalar, vecor and tensor hadronic currents, in generating direct CP asymmetry in the decay rate of τ-→K-π0ντ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^-\rightarrow K^-\pi ^0\nu _\tau $$\end{document}. We show that non vanishing direct CP asymmetry in the decay rate of τ-→K-π0ντ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^-\rightarrow K^-\pi ^0\nu _\tau $$\end{document} can be generated due to the presence of both, the weak phase in the Wilson coefficient corresponding to the tensor operator and the strong phase difference resulting from the interference between the form factors expressing the matrix elements of the vector and tensor hadronic currents. After taking into account all relevant constraints, we find that the generated direct CP asymmetry is of order 10-8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-8}$$\end{document} which is several orders of magnitude larger than the standard model prediction. We show also that, in two Higgs doublet model with generic Yukawa structure , direct local or non integrated CP violation can be as large as 0.3 % not far from experimental possibilities. This kind of asymmetry can be generated due to the interference between vector and scalar contributions with different weak phases which is not the case in the SM.


Introduction
Probably the most evident fail of the Standard Model (SM) is the absence of a mechanism to explain baryogenesis, even if all CP violation (CPV) processes measured by now are consistent with SM predictions [1][2][3][4][5][6][7][8][9][10]. At the moment CPV has been observed only in non leptonic decays of kaons, B and B s and recently in D [11]. In the leptonic sector, the quantum mixing of neutrinos yields a source for generating a complex phase [12,13]. This phase is necessary for having CPV that can be measured in ν μ −ν e andν μ −ν e oscillations which are experimentally accessible with the Tokai-to-Kamioka (T2K) experiment [14]. Recently, in Ref. [15], the T2K collaboration has reported a measurement that shows an indication of CPV in the neutrino sector at 3σ confidence level. Particularly, at this confidence level, the reported measurement are in favor of large enhancement of the neutrino oscillation probability. The measurement also excludes values of a complex phase that can lead to a large enhancement of the observed anti-neutrino oscillation probability at 3σ confidence level. Decays involving leptons like K 0 L → π − l + ν l , π + π − e + e − , where CPV has been measured can be understood as CPV in the meson sector. SM predictions for direct CPV in the leptonic sector tell us that it should be very small so its observation would be a clear signal of new physics.
Decays like τ → K πν involve at least two kinds of CPV contributions: direct CPV and the 'known' CPV if neutral kaons are involved. Direct CPV is the same for τ − →K 0 π − ν τ , τ − → K − π 0 ν τ and so on, because the transition τ → sūν τ is the same. Notice that if additional neutral pions are present the conclusions are identical.
In Ref. [21], it was shown that angular integrated direct CPV asymmetry can not be produced by a simple new scalar interactions, even if they provide new weak phases. As it is well known, to have direct CPV one needs two interfering contributions with different weak and strong phases. In these decays once the angular integral is done the interfering signal vanish. However a direct CPV signal may remain if the angular integration is partial or no integration is done (local CPV) at all.
Moreover, as shown in Ref. [21], another possibility is to have nonstandard tensor interactions with non vanishing weak phases. In this case the interfering vector and tensor interactions remain after the full angular integration and a CPV signal is obtained. However in a recent study, it was shown that this interference is severely suppressed due to the bounds from the neutron electric dipole moment and D −D mixing constrains [22].
An estimation of the direct CPV in τ − → K − π 0 ν τ , within SM framework showed that the calculated asymmetry is negligibly small of order 10 −12 [27]. This result motivated further studies of CP violation in this decay mode within the framework of supersymmetric extension of the SM [29,30]. In minimal supersymmetric extension of the SM with R parity conservation, direct CP asymmetry of order O(10 −7 ) can be generated through the interference between the vector and tensor interactions [29]. On the other hand, within supersymmetric extensions of SM with allowed R parity violating terms, no direct CP asymmetry in the decay rate can be generated at tree-level due to the absence of tensor interactions [30]. In Refs. [26,28,31], it was pointed out that CP violation in τ − → K − π 0 ν τ can arise in multi Higgs models with complex couplings in the quark sector due to the interference of the vector and scalar quark currents.
The aim of this paper is to analyze how to generate a C P violation in the semileptonic | S| = 1 tau decays in the integrated and nonintegrated C P asymmetries. We shall apply these considerations to specific models of New Physics (NP) such as 2HDM III and in the SM extensions with scalar leptoquarks. With the presence of new weak phases and new tensor operator, we analyze the direct CP violating effects in the decay rate of τ − → K − π 0 ν τ . This paper is organized as follows. In Sect. 2, we present the effective Hamiltonian describing semileptonic | S| = 1 tau decays, τ − → sūν τ transition, in the presence of NP beyond SM. Based on this Hamiltonian, we derive the general expression of the differential decay width of the decay process τ − → K − π 0 ν τ . Switching off NP contributions to the differential decay width, we show in Sects. 3 and 4 that no direct CP asymmetry in the decay rate of τ − → K − π 0 ν τ can be generated in the SM at tree-level.
In Sect. 3.1, we derive the analytic expressions of the Wilson coefficients up to one loop level originating from the charged Higgs mediation in 2HDM III. Leptoquarks contributions to these processes are presented in Sect. 3.2. In Sect. 3.1 also, we give our estimation of the direct CP asymmetry in the decay rate of τ − → K − π 0 ν τ . In Sect. 4 we give our prediction for the local CP violation in the same decay mode. Finally, in Sect. 5, we give our conclusion.
2 Effective Hamiltonian and the differential decay width of | S| = 1 τ decays In the presence of NP beyond SM, the effective Hamiltonian governs | S| = 1 τ decays transition, taking into consideration the parity conservation in the k → π matrix elements, can be expressed as where V us is the us Cabibbo-Kobayashi-Maskawa (CKM) matrix element and Q i represent the four-fermion local operators at low energy scale μ m τ where with σ μν = i 2 [γ μ , γ ν ]. The Wilson coefficients, C i , corresponding to the operators Q i can be expressed as where C SM i and C N P i represent SM and NP contributions to the Wilson coefficients respectively. In order to proceed to write the amplitude we need to calculate the matrix elements of the operators in the effective Hamiltonian. For this, we need to assign the momenta of the particles involved in the decay process. We express the momenta as The matrix elements of the hadronic currents in the Q i operators, in Eq. (4), are usually parameterized in terms of particles momenta and form factors. Due to parity conservation in the K → π matrix elements we need only to calculate the matrix element of the vector, scalar and tensor quarks currents only. The matrix element of the vector quark current can be expressed as and here s is the invariant mass defined as s = ( p K + p π ) 2 of the π K system and we have defined 2 The matrix element of the scalar quark current can be obtained from Eq. (7) by taking the divergence in the usual form and hence we get m s,u denote s, u current quark masses. Finally, the matrix element of the tensor quarks current, K − π 0 |sσ μν u|0 , can be expressed as [22] For specific models, like the ones under investigation in this study, the Wilson coefficients C i can be expressed in terms of only three independent coefficients. Particularly, in these models, we have C A = −C V and C P = C S and hence we are left with only three independent Wilson coefficients, namely C V , C S and C T . This indicates that, the set of the operators in Eq. (4), within these models, can be rewritten in terms of just three independent operators as ν τ γ μ Lτ sγ μ u , ν τ Rτ su and ν τ σ μν Rτ sσ μν u where L , R = 1 ∓ γ 5 .
Non vanishing direct CP asymmetry in the decay rate requires the presence of two types of phases, the weak CP violating phases and the strong CP conserving phases. The weak CP violating phases can be generated in the Wilson coefficients upon existence of complex couplings. On the other hand, the strong CP conserving phases originate from the phases in the form factors expressing the matrix elements of the hadronic currents.
Breit-Wigner forms are used to parameterize the contributions of the different resonances dominating the scalar and vector hadronic currents. As a consequence, form factors originating from these currents can be expressed as a summation of Breit-Wigner forms. Previous studies of CP asymmetries in τ → K πν τ decays, for instances Refs. [21,29], adopted the assumption that the form factor B T (s) has no strong phases. However, this assumption is incorrect as argued in Ref. [22]. As shown in Refs. [37,38], spin-1 resonances can be described equivalently by vector or antisymmetric tensor fields . Hence the same resonances K * (892) and the K * (1410) that dominate the Breit-Wigner forms in f + (s) will appear in B T (s) as well [22]. It should be noted that, this conclusion can be derived by analyzing the unitarity relation for the form factors as shown in details in Ref. [22]. Thus, we conclude that B T (s) has a strong phase that should be taken into account in the calculations of the CP asymmetry.
In the SM, at tree-level, the Wilson coefficients C i reduces to This accounts for the fact that τ − → K − π 0 ν τ , at tree-level, can be generated as a result of exchanging single W − boson which contributes only to Q V operator. Consequently, the quantities S(s) and T (s) in Eq. (12) reduce to and upon substation in Eq. (11) we get The decay rate of the process τ → K − π 0 ν τ in the SM, SM , can be then obtained upon integrating the previous equation with respect to the kinematic variable s. Thus we get The CP asymmetry in total decay rate of τ − → K − π 0 ν τ is given by: . (17) Clearly, from the expression of SM , direct CP asymmetry in the decay rate will vanish due to the absence of the weak phase, C SM V is real, and also due to the remark that the form factors f + (s) and f 0 (s) do not interfere and hence the relative strong phase essential for CP asymmetry vanishes. Thus, to generate no vanishing CP asymmetry in the decay rate within SM, it is essential to consider higher order terms contributing to the amplitude as done in Ref. [27]. These terms can be generated from diagrams with exchanging two W bosons. Thus, the generated asymmetry is expected to be very small. As shown in Ref. [27], the resulting CP asymmetry is suppressed by the CKM factor V td 10 −3 and also by a higher order suppression factor g 2 /4π M 2 W 10 −8 . As a consequence, the resulting CP rate asymmetry is expected to be negligible. In fact the asymmetry is of order 10 −12 as estimated in Ref. [27].
Integrated direct or local CP violation has been discussed above. However having somehow the ability to measure angular distribution one can extract the local CPV signal [31] and | f 0 |. The CPV signal is produced due to the interference between the vector and the scalar contributions, as long as they contribute with different weak phases. Additional CPV observables are provided by triple products, but in that case polarized taus are needed [25,26,28]. Without considering the exotic tensor interactions, one finds that the local distribution is given as where the negative sign in − corresponds to the QED charge of τ − lepton, x = cos θ and |q| = m τ (1 − s/m 2 τ )/2 is the momentum of the K − π system (and of the neutrino) in the τ center of mass. Similarly |p √ s is the kaon (and of the pion) momentum in the K − π center of mass. The angle θ is the scattering angle of kaon with respect to the incoming τ , or equivalently the angle between − → p π and − → p ν , in the hadronic rest frame. The Local CPV is defined as Similarly to the sign convention in − , the positive sign in + corresponds to the QED charge of τ + . Finally, it should be remarked that the local CPV, defined in Eq. (19), is different than the forward-backward asymmetry, A F B (s), that can be obtained upon integrating d 2 dsdx over x. Explicitly, it is defined as [39] It is clear that within SM at tree-level, from Eq. (19), the local CPV is vanishing due to the remark that C SM S = 0. This is not the case of the forward-backward asymmetry A F B (s) that has non-vanishing term independent of C SM S and C SM T [39]. Thus, non-vanishing values of the local CPV can be used as a probe of physics beyond the SM as we will discuss in Sect. 4.

Integrated CP violation asymmetries
We investigate now the direct CP asymmetry for a general new physics model that can contribute to the effective Hamiltonian in Eq. (3) with Wilson coefficients denoted by C N P i . As can be seen from Eqs. (11,12), the hadronic form factor f 0 (s), in S(s), does not interfere with the form factor f + (s). The absence of this interference leads to the absence of the strong phase difference between their contributions to the decay rate. This phase difference is essential for generating non vanishing direct CP asymmetry. As a consequence, and for having non-vanishing direct CP asymmetry in the decay rate, we are left only with the interference between B T (s), in T (s), and f 0 (s) as a possible source for the required strong phase difference. However, this interference was estimated to be small due to Watson's final-state-interaction theorem [22,40]. Assuming that N P contributions are obtained via integrating out heavy particles, above the electroweak breaking scale, we can set C N P V = 0. In this case, the direct CP asymmetry in total decay rate of τ − → K − π 0 ν τ is given by: where δ + (s), δ T (s) are the phases of f + (s) and B T (s).
Estimation of the C P asymmetry in the previous equation requires information on the form factor f + (s) and B T (s).
Empirically, τ → π K S ν τ spectrum can give information on the form factor f + (s) which is mainly dominated by the K * (892) resonance [41]. As discussed in Refs. [22,37,38], spin-1 resonances can be described equivalently by vector or antisymmetric tensor fields. Thus, both of f + (s) and B T (s) can receive contributions from the same resonances, mainly K * (892) resonance. Considering the moduli of the form factors f + (s) and B T (s), inelastic corrections turns to be negligible and the elastic solution of the unitarity relation for the form factors can be used [22] f in terms of the Omnès factor [42] In the previous equation, the phase shift δ(s) can be approximated by a BW phase [22] with parameters as determined in [41]. As shown in Ref. [22], the modulus of f + (s), given in Eq. (45), can reproduce well the experimental fit below the K * (1410) resonance done in Ref. [41].
We turn now to the phase of the form factor f + (s), namely δ + (s). Experiments are only sensitive to the modulus of the form factor. Thus δ + (s) cannot be directly estimated from experiments. Rather, its extraction can be done with the help of a fit function that preserves the analytic structure of the form factor [22]. As a consequence, the fit function used in [41] cannot be helpful for extracting δ + (s). However, as argued in Ref. [22], still this fit can be useful in extracting δ + (s). This can be achieved through comparing the phase from the experimental fit [41] with δ + (s) computed using a BW approximation for the K * (892). To account for the inelastic contribution δ inel + (s) to δ + (s), the authors of Ref. [22] have added the BW phase for K * (1410) → K * (892)π with a coefficient that allows for a similar phase motion in the vicinity of the K * (1410). This leads to the band shown in Fig. 2, in Ref. [22], which represents a simple estimation of inelastic effects. The result is consistent with more refined estimates along the lines of [43][44][45][46][47]. Up on the assumption that the inelastic contributions in δ T (s) are of similar size with an opposite sign, one can take δ + (s)−δ T (s) ∼ 2δ inel + (s) [22].
With the estimation of the form factors following the discussion above and using BR(τ → K − π 0 ν τ ) = (4.33 ± 0.15) × 10 −3 [48], f + (0)|V us | = 0.2165(4) [48], B T (0)/ f + (0) = 0.676 (27) from lattice QCD [49], particle masses and couplings from [48], we find that the C P asymmetry can be estimated as An investigation of the limits on Im C T for general NP models follow from electric dipole moment (EDM) of the neutron and D-D mixing was carried in Ref. [22]. In the following we present the derivation of such limits as carried out in Ref. [22]. At an energy scale v, the decay processes τ − → K − π 0 ν τ receive contribution from a tensor operator. The operator is originating from the SU (3) × SU (2) × U (1) gauge-invariant Lagrangian that can be expressed as [22] L T = C abcdL i La σ μν e Rb i jq j here a, b, c, d are generation indices and i, j are SU (2) L indices. In the preceding equation q L and L L stand for the quark and lepton SU (2) L doublets while u R and e R represent the charged up-quark and lepton SU (2) L singlets respectively. The tensor operator Q T listed in Eq. (4) is generated from [22] L T = C 3321 (ν τ σ μν Rτ )(sσ μν Ru) where R = (1 + γ 5 )/2 and other terms that involve the charm and top quark have been omitted. Comparing Eq. (4) and Eq. (26) one finds that The renormalization group evolution [50] of the operator (τ σ μν Rτ )(ūσ μν Ru), second term in the square bracket in Eq. (26), can produce via insertion an up-quark EDM d u (μ) [22] Upon solving the RG following [51][52][53] one obtains [22] The last equation above together with the 90% C.L. bound d n = g u T (μ)d u (μ) < 2.9 × 10 −26 e cm [54,55] can be used to obtain a strong bound on Im C T . Thus, for a value 100 GeV and using and the recent lattice result [56] g u T (μ = μ τ = 2 GeV) = −0.233 (28) one finds the constraint [22] |Im C T (μ τ )| 10 −5 , It should be noted that, the above constraint is based on the assumption that there are no other contributions to d n that can cancel the effect of C T . Applying the derived constraint on Im C T (μ τ ), we get a model independent prediction of the direct C P asymmetry as Clearly, from the above discussion, NP contributions to the phases of C N P 5 are needed to have non-vanishing direct C P asymmetry. These contributions will be evaluated below for several NP models.

CP asymmetry of
Two Higgs doublet models(2HDM) are simple extensions of the SM in which the scalar sector of the SM is enlarged to contain new scalars [57][58][59][60][61][62]. Based on the couplings of the new scalars to quarks and leptons we can classify these models to several types such as type I, II or III [63]. In the two Higgs doublet model with generic Yukawa structure or simply type III (2HDM III), the couplings of the new scalars to quarks and leptons can be complex [64][65][66]. As a consequence, these couplings can serve as the source of the weak CP violating phases essential for generating non vanishing CP asymmetries. The effect of these new weak phases on the CP asymmetry in D meson sector have been investigated in Refs. [67][68][69]. The resultant direct CP asymmetry in this model can be enhanced several orders of magnitudes larger than SM predictions. For instances, a large direct CP asymmetry of order 10 −3 can be achieved in the decay mode D 0 → K + π − after taking into account all constraints on the parameter space of the model [69]. This value is 6 orders of magnitude larger than the standard model prediction. The 2HDM III is also motivated by its ability to explain some anomalies in B decays. As shown in Ref. [66], a 2HDM of type II (like the MSSM at tree-level) cannot explain the deviations from the SM in tauonic B decays, while 2HDM III can account for B → Dτ ν and B → D * τ ν simultaneously [65,70].
The scalar sector of the 2HDM III consists of two Higgs doublets. The mass eigenstates constructed from these doublets are H 0 (heavy CP-even Higgs), h 0 (light CP-even Higgs) and A 0 (CP-odd Higgs) and H ± . In 2HDM III, the charged Higgs couplings to quarks and leptons can be expressed as [64,65] where Here v u and v d stand for the vacuum expectations values of the neutral component of the Higgs doublets, tan β = v u /v d and finally V is the CKM matrix. Using the Feynman-rules given in Eq. (32), we can proceed to derive the relevant Wilson coefficients to our investigation. Working in the unitary gauge and after integrating out the charged Higgs in the dia-grams in Fig. 1, we find that the relevant Wilson coefficients at μ = m H scale can be expressed as where v denotes the Higgs vacuum expectation value which It should be noted that an investigation of the dependency of the matrix elements of H → γ γ through one W ± loop in SM on the choice of the gauge was done in Ref. [71]. The results showed that the calculated matrix elements using R ζ gauge and the unitary gauge are explicitly verified to be different. However, in a subsequent study, it was found that the unitary gauge is consistent and equivalent to the R ζ gauge at the level of β-functions [72]. At higher loops, their is a possibility that the observed consistency between the two gauges may breaks down. In this case, their is a demand of a better understanding of the quantum internal structure of spontaneously broken gauge theories [72]. In Ref. [73], the mismatch which was pointed out by S. L. Wu and T. T. Wu group is resolved by a dedicated calculation of the unitary gauge. In a recent study in Ref. [74], it was shown that the unitary gauge is the default gauge for massive gauge bosons.
In order to estimate the contributions of the charged Higgs to the amplitude of the decay process under consideration we need to discuss the constraints imposed on the couplings u,d i j appear in the expressions of C H ± i above. The couplings d 12 , and d 32 are stringently constrained from flavor changing neutral current processes, in the down quark sector, due to the tree-level neutral Higgs exchange [65,66]. On the other hand, the coupling d 22 can be strongly constrained upon applying the naturalness criterion of 't Hooft to the quark masses that reads [66] |v u(d) Clearly, from this bound, d 22 is severely constrained by the smallness of the s quark mass. As a result, we can safely neglect the contributions of the couplings d i j to C H ± 1 , C H ± 4 .
Other terms, in these Wilson coefficients are real and thus are not relevant for generating C P asymmetries. Thus to a good approximation we can write , , It should be noted that, in the above expressions, we neglected non relevant real terms. In addition we neglected terms proportional to u 11 which is severely constrained from the bound in Eq. (36) due to the smallness of the up quark mass.
Recently, a lower bound m H ± 600 GeV, independent of tan β, has been obtained in 2HDM II after taking into account all relevant results from direct charged and neutral Higgs boson searches at LEP and the LHC, as well as the most recent constraints from flavour physics [57]. This bound should be also respected in 2HDM III [65]. Thus, for m H ± = 600 GeV and tan β = 50 we find that Clearly, to a good approximation, we can neglect contribution of C H ± V to the amplitude as it is very small. The previous equation shows that C H ± S is much larger than C H ± T . However, as we discussed before, only C H ± T can generate non vanishing direct CP asymmetry.
The 't Hooft naturalness criterion leading to the bounds in Eq. (36) implies different bounds for u i j and u ji due to the presence of the CKM element in the first line of Eq. (36) corresponding to the case i < j. Thus one expects different bounds for u 12 and u 21 as can be seen from Eq. (16) in Ref. [66]. The authors of Ref. [66] set a bound on u 21 after considering all constraints listed in their Tables VII and VIII. This bound can be read from the first matrix in Eq. (75) as | u 21 |≤ 3.0 × 10 −2 . However, the strong bound | u 12 u * 21 | < 2 × 10 −8 from considering D −D mixing [66] implies that | u 12,21 |< √ 2 × 10 −4 . This can be understood as in the absence of a symmetry that protect | u 12 | from being much smaller than | u 21 | and using the same merit of the 't Hooft naturalness criterion it is unnatural to have | u 12 | much suppressed than | u 21 |. Thus we obtain Knowing that Re(V ts ) O(10 −2 ) and the bound 2.7 × 10 −3 ≤| u 31 |≤ 2.0 × 10 −2 from the process B → τ ν [66], it is clearly trivial that the B → τ ν process sets more severe bound on u 31 compared to the model-independent EDM bound. Consequently, one expects a strong bound I m C H ± T < 10 −6 and after using CKM matrix elements from Ref. [75], we obtain from Eq. (24) Clearly, the charged Higgs contributions can enhance the direct CP asymmetry several orders of magnitude larger than the standard model prediction. However, the estimated CP asymmetry is still so small to be detected by current or near future experiments.
Their Yukawa couplings with SM fermions are given by Here, τ i (i = 1, 2, 3) are the Pauli matrices,Q L i = (ū i ,d i ) L and L L i = (ν i ,ē i ) L and for a spinor field ψ, the charge conjugation ψ c is defined as ψ c R,L = iγ 0 γ 2ψ T R,L . The coupling constants κ L ,R i j and ξ L ,R i j can be in general complex numbers. The Yukawa couplings, in Eq. (41), lead to the Lagrangians , (42) which is relevant to τ → K − π 0 ν decay with i = 1, 2, 3. These Lagrangians result in diagrams similar to the one mediated by the charged Higgs in which the Higgs boson is replaced with the scalar leptoquarks S 1/2 and S 0 . After integrating out the leptoquarks, we can obtain the Wilson coefficients C I,I I i , corresponding to the operators Q i in Eq. (4), as We now discuss the constraints imposed on the leptoquark couplings. For model I I , the couplings |ξ L 2i ξ L * 13 |, appears in C I I V , can be severely constrained from the process K → πνν [83]. From the results of the analysis carried in Ref. [83], the bound reads |ξ L 2i ξ L * 13 | < 2×10 −5 ×(M S 0 /[100 GeV]) 2 . Thus, in model I I , we have C V = C SM V + C I I V C SM V . Turning now to the couplings ξ L 2i ξ R * 13 and setting i = 3, we find that possible constraints can be derived from the observable τ → K ν τ . The expected bound on ξ L 23 ξ R * 13 ≡ ξ L sν τ ξ R * uτ , which corresponds to the Wilson coefficients of the four fermion operator that contributes to both τ → K ν τ and τ − → K − π 0 ν τ , read Re(ξ L 23 ξ R It should be noted that a strong constraint on |I m(C I I T )| can be obtained from the EDM of the neutron noticing that the coupling ξ L 23 ξ R * 13 can be linked to C 3321 ≡ C ν τ τ su , defined in Eq. (26), and using the bound given in Eq. (30). As a consequence, similarly to the obtained value in the charged Higgs model. Regarding model I , we find that the predicted CP asymmetry will be the same as in model I I . This can be explained as similar constraints can be imposed on the couplings κ R 23 κ L * 13 from the observable τ − → K − ν τ [86] and from the electric dipole moment (EDM) of the neutron in Eq. (30).
In the scalar leptoquark models discussed above, the amplitudes of the processes τ − → K − π 0 ν e and τ − → K − π 0 ν μ receive contributions from the couplings ξ L 21 ξ R * 13 ≡ ξ L sν e ξ R * uτ and ξ L 22 ξ R * 13 ≡ ξ L sν μ ξ R * uτ respectively. These couplings can be also constrained using the observable τ → K ν [87]. However, the obtained bounds turns to be one order of magnitude weaker than the bound on ξ L 23 ξ R * 13 ≡ ξ L sν τ ξ R * uτ . Moreover, the couplings ξ L 21 ξ R * 13 ≡ ξ L sν e ξ R * uτ and ξ L 22 ξ R * 13 ≡ ξ L sν μ ξ R * uτ cannot be linked to C 3321 ≡ C ν τ τ su , defined in Eq. (26), and thus evade the strong bound given in Eq. (30). Consequently, we expect a somehow large branching ratios and CP asymmetries for the channels τ − → K − π 0 ν e and τ − → K − π 0 ν μ compared to their corresponding ones of τ − → K − π 0 ν τ . Since these couplings contribute to C S,T one expects that the effect on the branching ratios is not much while a large effect is expected to be seen in the CP asymmetries. With the only constraints from τ → K ν and in the absence of much stronger constraints, the predicted direct CP asymmetry of τ − → K − π 0 ν e and τ − → K − π 0 ν μ can be of order 10 −3 .
Finally, we discuss the possibility of having constraint on the I m C T , in the leptoquark models discussed in this work, from the process B → τ ν. Recall that, the process B → τ ν can be generated through b → uτ ν transition. In model I I , the transition originates at tree-level from the exchanging of the scalar leptoquark S 0 . The relevant product of the leptoquark couplings to the amplitude of B → τ ν decay is ξ L * 3i ξ R 13 . Consequently, the constraints from B → τ ν process are expected to be imposed on ξ L * 3i ξ R 13 which is irrelevant to τ → K − π 0 ν process that receives contributions from ξ L 2i ξ R * 13 as we have shown above. Regarding model I , first line in Eq. (42), the b → uτ ν transition can be generated at tree-level from the exchanging of the scalar leptoquark S 1/2 . In this case, the relevant leptoquark couplings to the amplitude of B → τ ν decay are κ R * 33 κ L 1i . Again, the expected constraints from B → τ ν process on κ R * 33 κ L 1i are irrelevant to τ → K − π 0 ν process that receives contributions from κ R 23 κ L * 1i .

Nonintegrated CP violation asymmetries
Searches for local CPV signal have been done by Cleo [16] and Belle [17] collaborations to obtain bounds on new generic scalar mediators like the ones considered here. Belle collaboration is one order of magnitude more precise. They parameterized new physics with η S C S by doing the replacement f 0 → [1 − η S s/(m τ (m s − m u ))] f 0 . No local CPV signal was found and a bound of |Im (η S )| |Im (C S )| < 0.013 was obtained. Recall that within SM, at tree-level, C SM S = 0 which is not the case in some beyond SM physics that have been investigated in the previous section. In the case of 2HDM III, and using Eq. (39), this bound results in the con-straints |Im ( u 31 )| < 0.4. On the other hand and in the case of scalar Leptoquark, the bound |Im (η S )| |Im (C S )| < 0.013 will lead to a weaker bound on |Im (C I,I I S )| compared to that one obtained from the electric dipole moment (EDM) of the neutron in Eq. (30) as, from Eqs. (30,43), we have |Im (C I,I I S )| 4 × 10 −5 . We proceed now to calculate the local CPV given in Eq. (19) within 2HDM III discussed above. Including charged Higgs contributions to the Wilson coefficients C V and C S in Eq. (19) can be done via the replacement . For a value of I m( u 31 ) ≤ 2 × 10 −2 that satisfies the strongest bound from B → τ ν process we find that I m(C S ) 3.6 × 10 −3 × s.
We turn now to the form factors f + (s) and f 0 (s) required to the calculation of the local CPV. Recall that a discussion about the form factor f + (s) has been presented in the preceding section. So we are left here to discuss the scalar form factor f 0 . While the phase of f 0 was measured by the LASS collaboration [88,89] its magnitude can be obtained by using dispersion relations [43] but uncertainties in the input information avoid a precise prediction. In fact decays as the one we are interested in here should provide experimental information about f 0 , once the angular distribution can be measured [31]. For our purpose of an estimation of the local CPV the form factor f 0 can be parameterized, in terms of a superposition of Breit-Wigner resonances following Refs. [23,41] as with p = |p K |, χ = 2.28 and γ = 1.92e 4.03·i . Breit-Wigner parameters are m K * 0 (700) = 878(23) +64 −51 , K * 0 (700) = 499(52) +55 −87 , m K * 0 (1430) = 1425 and K * 0 (1430) = 270. The spin of the resonance is J and all masses and widths are expressed in MeV [8][9][10]. With all this in hand, we present a typical graphical representation of the local CPV in Fig. 2. Clearly, from Fig. 2, values as large as 0.3 % can be obtained. Due to the strong constraints

Conclusion
In this paper we have derived the contributions to the effective Hamiltonian governing the semileptonic | S| = 1 tau decays in 2HDM III and models with scalar Leptoquarks. We have discussed the imposed constraints on the elements in the parameter space of the model relevant to the decay channel τ − → K − π 0 ν τ . In addition, we have analyzed the role of the different contributions, originating from the scalar, vector and tensor hadronic currents, in generating direct CP asymmetry in the decay rate of τ − → K − π 0 ν τ .
We have shown that non vanishing direct CP asymmetry in the decay rate of τ − → K − π 0 ν τ can be generated in the model due to the presence of the weak phase in the Wilson coefficient C N P T and due to the strong phase difference resulting from the interference between the form factors B T (s), and f + (s). After taking into account the relevant constraints we found that the resultant CP asymmetry can be of order 10 −8 in both models. The asymmetry is so tiny to be probed even in near future experiments.
In this work we have also studied another observable related to the CP violation, namely the local CPV. This kind of asymmetry can be generated if there is interference between vector and scalar contributions, as long as they contribute with different weak phases which is not the case in the SM. In the 2HDM III we studied in this work, we have found that, direct local CPV can be as large as 0.3 % not far from experimental possibilities.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .

Appendix A
The quantities Z and γ can be expressed as Here, a AB V denotes the contribution from Feynman diagram where the gauge boson V = Z , γ connect the two  Finally, the loop function C 0 (m 2 1 , κ 2 , m 2 2 , 2 , m 2 , n 2 ) can be defined as C 0 ( p 2 1 , (p 1 + p 2 ) 2 , p 2 2 , 2 , m 2 , n 2 ) where p 2 1 = m 2 1 , p 2 2 = m 2 2 and ( p 1 + p 2 ) 2 = κ 2 . After Feynman parameterization and shifting the loop momentum k to absorb the terms linear in k one can proceed to obtain the final result of the integration.