We study the direct CP violation in the τ±→K±ρ0(ω)ντ→K±π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow K^\pm \rho ^0 (\omega )\nu _\tau \rightarrow K^\pm \pi ^+\pi ^-\nu _\tau $$\end{document} decay process in the standard model. An interesting mechanism involving the charge symmetry violating mixing between ρ0\documentclass[12pt]{minimal}
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\begin{document}$$\rho ^0$$\end{document} and ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} is applied to enlarge the CP asymmetry. We find that the CP-violating asymmetry can be enhanced greatly via this ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document}–ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing mechanism when the invariant mass of the π+π-\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^+\pi ^-$$\end{document} pair is in the vicinity of the ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} resonance. With this mechanism, the maximum differential and localized integrated CP asymmetries can reach -(5.6-1.7+2.9)×10-12\documentclass[12pt]{minimal}
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\begin{document}$$-(5.6^{+2.9}_{-1.7})\times 10^{-12}$$\end{document} and 6.3-3.3+2.4×10-11\documentclass[12pt]{minimal}
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\begin{document}$$6.3^{+2.4}_{-3.3}\times 10^{-11}$$\end{document}, respectively, which is still negligible.
volume-issue-count12issue-article-count53issue-toc-levels0issue-typeRegularissue-online-date-year2014issue-online-date-month12issue-online-date-day13issue-pricelist-year2014issue-copyright-holderSIF and Springer-Verlag Berlin Heidelbergissue-copyright-year2014article-contains-esmNoarticle-numbering-styleContentOnlyarticle-toc-levels0article-registration-date-year2014article-registration-date-month10article-registration-date-day21article-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessIntroduction
CP violation was first observed in the neutral kaon system 50 years ago [1]. The violation of CP in the K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document} meson system can be explained by a weak complex phase in the Cabibbo–Kobayashi–Maskawa (CKM) matrix in the standard model (SM) [2, 3]. However, the fundamental origin of CP violation is still an open problem and it is not clear if the CKM mechanism is the only source for CP violation. New physics (NP) may exist [4–6] and cause CP violation. To verify the origin of CP violation and look for NP, one needs to collect more information as regards CP violation in as many processes as possible. One such possible process is the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decay. τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} is the only lepton which is heavy enough to decay into hadrons and the pure leptonic and semileptonic character of τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decays provides a clean laboratory to test the structure of the weak currents and the universality of their couplings to the gauge bosons [5]. More importantly, with the establishment of the high-luminosity Super τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-Charm factories, more τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} leptons will be produced and its properties will be measured to a very high precision [4]. After the CLEO-c experiment ceased data collection in March 2008, the BESIII experiment began to collect data, and the luminosity reached 1032cm-2s-1\documentclass[12pt]{minimal}
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\begin{document}$$10^{32}\,\mathrm {cm^{-2}\,s^{-1}}$$\end{document} in 2013 [7]. Future high-luminosity Super τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-Charm factories are also being considered in Russia and Italy and may reach the luminosity of 1035cm-2s-1\documentclass[12pt]{minimal}
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\begin{document}$$10^{35}\,\mathrm {cm^{-2}\,s^{-1}}$$\end{document} [8–11]. Moreover, Super B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}-Factories (with the luminosity of 1036cm-2s-1\documentclass[12pt]{minimal}
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\begin{document}$$10^{36}\,\mathrm {cm^{-2}\,s^{-1}}$$\end{document}) will produce about 1010\documentclass[12pt]{minimal}
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\begin{document}$$10^{10}$$\end{document}τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} pairs per year at the Υ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\Upsilon }$$\end{document}(4S) peak [12, 13]. The large statistics collected have considerably improved the statistical accuracy of the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} measurements and brought a new level of systematic understanding, allowing us to make sensible tests of the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} properties, provide more information as regards CP asymmetries in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decay processes, and seek the fundamental origin of CP violation.
Experimental searches for CP-violating asymmetries in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} lepton semileptonic decays have been carried out. The missing evidence for a nonzero CP asymmetry was interpreted in terms of a coupling Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document} in the decay τ±→π±π0ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow \pi ^\pm \pi ^0 \nu _\tau $$\end{document} [14]. Recently, the τ±→KSπ±ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow K_S \pi ^\pm \nu _\tau $$\end{document} rate asymmetry was measured to be of order O(10-3)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O} (10^{-3})$$\end{document} by Belle [15] and BaBar [16]. In order to improve our understanding of CP violation in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decays, more efforts should be made on the theoretical side. Explicit studies of the decay modes τ±→K±π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow K^\pm \pi ^+\pi ^-\nu _\tau $$\end{document} [17, 18], τ±→π±K+K-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow \pi ^\pm K^+K^-\nu _\tau $$\end{document} [17], τ±→(3π)±ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow (3\pi )^\pm \nu _\tau $$\end{document} [19, 20], and τ±→(4π)±ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow (4\pi )^\pm \nu _\tau $$\end{document} [19] show that sizeable CP-violating effects could be generated in some models of CP violation involving several Higgs doublets or left–right symmetry.
In the framework of the SM, the direct CP asymmetries come about due to a relative weak (CP-odd) and a relative strong (CP-even) phase. This mechanism is forbidden in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decays in the leading order of the Fermi coupling constant GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document} [21]. The CP violation in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decays is usually predicted vanishingly small in the SM. For example, the CP violation in the τ±→K±π0ντ\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O} (10^{-12})$$\end{document} when one takes higher-order electroweak corrections into account [22]. (Note that for the decay τ±→KSπ±ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow K_S \pi ^\pm \nu _\tau $$\end{document}, the SM predicts a CP-violating asymmetry of 3.3×10-3\documentclass[12pt]{minimal}
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\begin{document}$$3.3\times 10^{-3}$$\end{document} due to the K0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{K}_0$$\end{document} mixing amplitude [23].) In order to obtain a larger CP violation in the SM, one needs to appeal to some phenomenological mechanisms. The charge symmetry violating mixing between ρ0\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} (ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing) has been applied in hadron decays for this purpose in the past few years. ρ\documentclass[12pt]{minimal}
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\begin{document}$$ \omega $$\end{document} mixing has the dual advantages that the strong phase difference is large and well known [24, 25]. From a series of studies on CP violation, it has already been found that this mechanism can provide a very large strong phase difference (usually 90 degrees) when the mass of the decay product of ρ0\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}) , π+π-\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^+\pi ^-$$\end{document}, is in the vicinity of the ω\documentclass[12pt]{minimal}
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\begin{document}$$D$$\end{document} [24–29]. We will apply this mechanism to the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} lepton decay in the present paper.
We will consider the decay process τ±→K±π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow K^\pm \pi ^+ \pi ^- \nu _\tau $$\end{document}. The interference between the leading-order diagram in GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document} (Fig. 1a) and the second-order weak diagrams (Fig. 1b, c) generates a small CP-violation phase [22]. ρ\documentclass[12pt]{minimal}
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\begin{document}$$ \omega $$\end{document} mixing has been applied for getting a large strong phase when the invariant mass of the π+π-\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^+\pi ^-$$\end{document} pair is near the ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} resonance. Hence one can expect that there could be a bigger CP-violating asymmetry in the τ±→K±ρ0(ω)ντ→K±π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^\pm \rightarrow K^\pm \rho ^0 (\omega )\nu _\tau \rightarrow K^\pm \pi ^+\pi ^-\nu _\tau $$\end{document} process. Actually, it will be shown from our explicit calculations that ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing leads to an additional strong phase and enlarges the differential CP-violating asymmetry by a maximum of four orders of magnitude and the localized integrated CP asymmetry by a maximum of three orders of magnitude. Even so, the direct CP asymmetry in the SM is still negligible.
The leading-order (a) and higher-order diagrams (b, c) in GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document} contributing to the decay τ-→K-ρ0ντ→K-π+π-ντ\documentclass[12pt]{minimal}
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The remainder of this paper is organized as follows. In Sect. 2, we first present the formalism for the CP asymmetry in τ±→K±ρ0(ω)ντ→K±π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing. Then we give the derivation details of the leading-order and the second-order weak process matrix elements and apply ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing to generate a large CP asymmetry. In Sect. 3, with the expression of meson wave functions and form factors and several parameters we calculate numerical results of the differential and localized integrated CP asymmetries. Our conclusion is included in Sect. 4.
A decay process described by some amplitudes may have CP-even and -odd relative phases. Within the SM, the CP-odd relative phase is always a weak phase difference which is directly determined by the CKM matrix. On the contrary, the CP-even relative phase is usually a strong phase difference due to some complicated phenomenological mechanism. Letting M\documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document} and M¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{M}$$\end{document} be the amplitudes for τ-→K-ρ0(ω)ντ→K-π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^-\rightarrow K^- \rho ^0 (\omega )\nu _{\tau }\rightarrow K^- \pi ^+\pi ^-\nu _{\tau }$$\end{document} and its CP conjugate one, respectively, we define the two amplitudes as follows:M=g1r1eiϕ1+g2r2eiϕ2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} M&= g_1r_1{e}^{{i}\phi _1}+g_2r_2{e}^{{i}\phi _2},\end{aligned}$$\end{document}M¯=g1∗r1eiϕ1+g2∗r2eiϕ2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{M}&= g_1^*r_1{e}^{{i}\phi _1}+g_2^*r_2{e}^{{i}\phi _2}, \end{aligned}$$\end{document}where g1\documentclass[12pt]{minimal}
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\begin{document}$$g_2$$\end{document} represent CP-odd complex terms which involve coupling constants and CKM matrix elements, and r1eiϕ1\documentclass[12pt]{minimal}
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\begin{document}$$r_1e^{\mathrm {i}\phi _1}$$\end{document} and r2eiϕ2\documentclass[12pt]{minimal}
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\begin{document}$$r_2e^{\mathrm {i}\phi _2}$$\end{document} terms are even under the CP transformation. Then one has|M|2-|M¯|2=4r1r2Im(g1∗g2)sin(ϕ1-ϕ2)=4r1r2|g1||g2|sin[Arg(g2/g1)]sin(ϕ1-ϕ2),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |M|^2-|\bar{M}|^2&= 4r_1r_2\,\mathfrak {I}\mathfrak {m}(g_1^*g_2)\,\sin (\phi _1-\phi _2)\nonumber \\&= 4r_1r_2|g_1||g_2|\! \,\sin [\mathrm {Arg}(g_2/g_1)]\,\sin (\phi _1-\phi _2), \end{aligned}$$\end{document}from which we can see explicitly that both the CP-odd phase difference Arg(g2/g1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Arg}(g_2/g_1)$$\end{document} and the CP-even phase difference ϕ1-ϕ2\documentclass[12pt]{minimal}
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\begin{document}$$\phi _1-\phi _2$$\end{document} are needed to produce CP violation. It will be shown below that the CP-odd phase difference arises from the second-order weak processes and the CP-even phase difference is determined by the decay widths of intermediate resonances and ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing in the τ-→K±ρ0(ω)ντ→K±π+π-ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^-\rightarrow K^\pm \rho ^0 (\omega )\nu _{\tau }\rightarrow K^\pm \pi ^+\pi ^-\nu _{\tau }$$\end{document} decay mode.
General formalism for CP asymmetry
The hadronic τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decay amplitude can be factorized into a purely leptonic part including τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} lepton and neutrino and a hadronic part, where the hadronic system is created from the vacuum via the charged weak current. Thus, the amplitude of τ-\documentclass[12pt]{minimal}
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\begin{document}$$K^-\pi ^+\pi ^-\nu _{\tau }$$\end{document} final state through K-ρ0ντ\documentclass[12pt]{minimal}
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\begin{document}$$\rho ^0$$\end{document} resonance can be written in the following general form:Mρ=GF2gρππsρLμHμρ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} M^\rho&= \frac{G_{F}}{\sqrt{2}}g_{\rho \pi \pi }s_{\rho }L^{\mu }H^\rho _{\mu }, \end{aligned}$$\end{document}where gρππ\documentclass[12pt]{minimal}
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\begin{document}$$H^\rho _{\mu }$$\end{document} is the hadronic matrix element for ρ0K-\documentclass[12pt]{minimal}
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\begin{document}$$L^{\mu }$$\end{document} is the lepton transition matrix element which can be written as u¯ντγμ(1-γ5)uτ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decay. In order to have a nonzero CP-violating asymmetry, the second-order weak terms corresponding to Fig. 1b, c (with ui=u,c,t\documentclass[12pt]{minimal}
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\begin{document}$$d_i=d, s, b$$\end{document}), which provide a CP-odd phase difference, should be taken into account [22]. The leading-order amplitude is denoted by M0ρ\documentclass[12pt]{minimal}
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\begin{document}$$M^\rho _{2}$$\end{document} corresponding to Fig. 1b, c, respectively.
As mentioned before, in order to obtain a large CP violation, we intend to apply the ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing mechanism, which leads to large strong phase differences in heavy hadron decays. In this scenario, to the first order of isospin violation, we have the following total amplitude when the invariant mass of the π+π-\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} M=M^\rho +M^{\rho -\omega }, \end{aligned}$$\end{document}withMρ-ω=GF2gρππsρLμHμωsωΠ~ρω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} M^{\rho -\omega }=\frac{G_{F}}{\sqrt{2}}g_{\rho \pi \pi }s_{\rho }L^{\mu }H^{\omega }_{\mu }s_\omega \tilde{\varPi }_{\rho \omega }, \end{aligned}$$\end{document}where Π~ρω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} intermediate resonance, respectively. We also assume that the ω\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\varPi }_{\rho \omega }$$\end{document} [30, 31]. This leads to the explicit s\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\varPi }_{\rho \omega }(s)=\tilde{\varPi }_{\rho \omega }(m^2_\omega )+(s-m_\omega )\tilde{\varPi }_{\rho \omega }^\prime (m^2_\omega )$$\end{document}, the ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing parameters were fitted by Gardner and O’Connell [32]:ReΠ~ρω(mω2)=-3500±300MeV2,ImΠ~ρω(mω2)=-300±300MeV2,Π~ρω′(mω2)=0.03±0.04.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathfrak {R} \mathfrak {e}\tilde{\varPi }_{\rho \omega }(m^2_{\omega })=-3500\pm 300\ \mathrm{MeV}^2,\nonumber \\&\mathfrak {I} \mathfrak {m}\tilde{\varPi }_{\rho \omega }(m^2_{\omega })=-300\pm 300\ \mathrm{MeV}^2,\\&\nonumber \tilde{\varPi }_{\rho \omega }^\prime (m^2_{\omega })=0.03\pm \ 0.04. \end{aligned}$$\end{document}We define M0=M0ρ+M0ρ-ω\documentclass[12pt]{minimal}
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\begin{document}$$M_0=M^\rho _0+M^{\rho -\omega }_0$$\end{document}, M1=M1ρ+M1ρ-ω\documentclass[12pt]{minimal}
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\begin{document}$$M_1=M_1^\rho +M_1^{\rho -\omega }$$\end{document}, and M2=M2ρ+M2ρ-ω\documentclass[12pt]{minimal}
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\begin{document}$$M_2=M_2^\rho +M_2^{\rho -\omega }$$\end{document}, where M0ρ-ω\documentclass[12pt]{minimal}
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\begin{document}$$H_{\mu }^{2\omega }$$\end{document}, respectively. The CP violation can arise from the interference between M0\documentclass[12pt]{minimal}
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\begin{document}$$M_{2}$$\end{document}. It should be noted that M1\documentclass[12pt]{minimal}
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\begin{document}$$G_{F}$$\end{document}. Therefore, to the GF3\documentclass[12pt]{minimal}
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\begin{document}$$G_F^3$$\end{document} order, the square of the total amplitude M=M0+M1+M2\documentclass[12pt]{minimal}
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\begin{document}$$M=M_0+M_1+M_2$$\end{document} can be written as|M|2=(M0+M1+M2)†(M0+M1+M2)=M0†M0+(M0†M1+M0M1†)+(M0†M2+M0M2†).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |M|^2&= (M_0+M_1+M_2)^{\dagger }(M_0+M_1+M_2) \nonumber \\&= M_0^{\dagger }M_0+(M_0^{\dagger } M_1 +M_0 M_1^{\dagger }) \nonumber \\&\quad +\,(M_0^{\dagger } M_2+M_0 M_2^{\dagger }). \end{aligned}$$\end{document}Then the differential CP asymmetry, which is defined asACP=|M|2-|M|¯2|M|2+|M|¯2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} A_\mathrm{CP}=\frac{|M|^2-\bar{|M|}^2}{|M|^2+\bar{|M|}^2}, \end{aligned}$$\end{document}can be written as follows to the order GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document}:ACP=|M|2-|M|¯2|M0|2+|M0|¯2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} A_\mathrm{CP}=\frac{|M|^2-\bar{|M|}^2}{|M_0|^2+\bar{|M_0|}^2}, \end{aligned}$$\end{document}where the M0†M1+M0M1†\documentclass[12pt]{minimal}
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\begin{document}$$M_0^{\dagger } M_1 +M_0 M_1^{\dagger }$$\end{document} and M0†M2+M0M2†\documentclass[12pt]{minimal}
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\begin{document}$$M_0^{\dagger } M_2 +M_0 M_2^{\dagger }$$\end{document} terms are negligible in the denominator since they do not contribute to the second order in GF\documentclass[12pt]{minimal}
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\begin{document}$$G_{F}$$\end{document}. When we take ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing into account, to the leading order in isospin violation, the three terms in Eq. (9) can be rewritten in the following forms:M0M0†=(GF/2)2gρππ2sρ2Lμν(Hμν0ρ0ρ+Hμν0ρ0ωΠ~ρω∗sω∗+Hμν0ω0ρΠ~ρωsω),M0M1†=(GF/2)2gρππ2sρ2Lμν(Hμν0ρ1ρ+Hμν0ρ1ωΠ~ρω∗sω∗+Hμν0ω1ρΠ~ρωsω),M0M2†=(GF/2)2gρππ2sρ2Lμν(Hμν0ρ2ρ+Hμν0ρ2ωΠ~ρω∗sω∗+Hμν0ω2ρΠ~ρωsω),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} M_0M_0^\dagger&= (G_F/\sqrt{2})^2g_{\rho \pi \pi }^2s_\rho ^2 L^{\mu \nu }\big (H_{\mu \nu }^{0\rho 0\rho }+H_{\mu \nu }^{0\rho 0\omega }\tilde{\varPi }^*_{\rho \omega }s_\omega ^*\nonumber \\&\quad +\,H_{\mu \nu }^{0\omega 0\rho }\tilde{\varPi }_{\rho \omega }s_\omega \big ),\nonumber \\ M_0M_1^\dagger&= (G_F/\sqrt{2})^2g_{\rho \pi \pi }^2s_\rho ^2 L^{\mu \nu }\big (H_{\mu \nu }^{0\rho 1\rho }+H_{\mu \nu }^{0\rho 1\omega }\tilde{\varPi }^*_{\rho \omega }s_\omega ^*\nonumber \\&\quad +\,H_{\mu \nu }^{0\omega 1\rho }\tilde{\varPi }_{\rho \omega }s_\omega \big ),\nonumber \\ M_0M_2^\dagger&= (G_F/\sqrt{2})^2g_{\rho \pi \pi }^2s_\rho ^2 L^{\mu \nu }\big (H_{\mu \nu }^{0\rho 2\rho }+H_{\mu \nu }^{0\rho 2\omega }\tilde{\varPi }^*_{\rho \omega }s_\omega ^*\nonumber \\&\quad +\,H_{\mu \nu }^{0\omega 2\rho }\tilde{\varPi }_{\rho \omega }s_\omega \big ), \end{aligned}$$\end{document}where Lμν=Lμ(Lν)†\documentclass[12pt]{minimal}
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\begin{document}$$L^{\mu \nu }=L^\mu (L^\nu )^\dagger $$\end{document} and, for example, Hμν0ρ0ρ=Hμ0ρ(Hν0ρ)†\documentclass[12pt]{minimal}
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\begin{document}$$H^{0\rho 0\rho }_{\mu \nu }=H^{0\rho }_\mu (H^{0\rho }_\nu )^\dagger $$\end{document}.
Derivation details of matrix elements
The transition from the vacuum to the pseudoscalar meson K-\documentclass[12pt]{minimal}
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\begin{document}$$(\omega )$$\end{document} occurs via weak vector and axial-vector current. Based on Lorentz invariance and parity and time-reversal invariance, one can decompose the hadronic matrix element in terms of four form factors in the leading order in GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document} [33]:Hμ0ρ(ω)=-iVus∗⟨ρ0(ω)K-|s¯γμ(1-γ5)u|0⟩=Vus∗[-gεμναβϵ∗νp1αp2β-ifϵμ∗-i(a1p1μ+a2p2μ)(ϵ∗·Q)],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} H_{\mu }^{0\rho (\omega )}&= -\mathrm {i}V_{us}^*\langle \rho ^{0}(\omega )K^{-}|\overline{s}\gamma _{\mu }(1-\gamma _{5})u|0\rangle \nonumber \\&= V_{us}^*[-g\varepsilon _{\mu \nu \alpha \beta }\epsilon ^{*\nu } p_1^{\alpha }p_2^{\beta }-\mathrm {i}f\epsilon ^*_\mu \nonumber \\&\quad -\,\mathrm {i}(a_1p_{1\mu }+a_2p_{2\mu })(\epsilon ^*\cdot Q)], \end{aligned}$$\end{document}where Vus\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} only. They are difficult to relate directly to experimental measurements but can be dealt with in phenomenological models. We will calculate the form factors with the meson dominance model [33]. The pseudoscalar and vector meson annihilation process in the leading order in GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document} is generated by the strong interaction. In the meson dominance model it is assumed that intermediate mesons connect the weak current and the strong vertex shown in the Feynman diagrams in Fig. 2. Using the Feynman rules for these diagrams, the following expressions for the form factors are obtained [33]:f=-12(Q2+mV2-mK2)∑ihAitAiVKDAi(Q2),g=12∑ihVitViVKDVi(Q2),a1=52∑ihPitPiVKDPi(Q2)+12∑ihAitAiVKDAi(Q2),a2=32∑ihPitPiVKDPi(Q2)+12∑ihAitAiVKDAi(Q2),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\begin{aligned}&f=-\frac{1}{2}(Q^2+m_V^2-m_K^2)\sum _i\frac{h_{A_i}t_{A_iVK}}{D_{A_i}(Q^2)},\\&g=\frac{1}{2}\sum _i\frac{h_{V_i}t_{V_iVK}}{D_{V_i}(Q^2)},\\&a_1=\frac{5}{2}\sum _i\frac{h_{P_i}t_{P_iVK}}{D_{P_i}(Q^2)}+\frac{1}{2}\sum _i\frac{h_{A_i}t_{A_iVK}}{D_{A_i}(Q^2)},\\&a_2=\frac{3}{2}\sum _i\frac{h_{P_i}t_{P_iVK}}{D_{P_i}(Q^2)}+\frac{1}{2}\sum _i\frac{h_{A_i}t_{A_iVK}}{D_{A_i}(Q^2)}, \end{aligned} \end{aligned}$$\end{document}where Vi\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _{M_i}$$\end{document}) is the mass (width) of the corresponding intermediate meson. The details as regards the intermediate mesons and their weak couplings and strong vertex coupling constants will be given in Sect. 3. From Eqs. (12) and (13) it can be found that in the leading order in GF\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document})]. a This panel represents the total effective strong vertex; b–d correspond to the Vi\documentclass[12pt]{minimal}
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\begin{document}$$|V_{ud_i}|^2|V_{us}|^2$$\end{document} and will not contribute to CP violation. Hence we only have to consider M1\documentclass[12pt]{minimal}
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\begin{document}$$x$$\end{document} is the longitudinal momentum fraction of the constituent quark, and the non-perturbative effects are included in the distribution amplitudes ϕV(x)\documentclass[12pt]{minimal}
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\begin{document}$$V(P)$$\end{document}. According to the Feynman diagram (b) in Fig. 1, the hadronic matrix elements H1ρ\documentclass[12pt]{minimal}
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\begin{document}$$H^{1\omega }$$\end{document}) can be expressed aswhere Vuis\documentclass[12pt]{minimal}
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\begin{document}$$u_i$$\end{document}, respectively. We will neglect the difference between the masses of ρ0\documentclass[12pt]{minimal}
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\begin{document}$$m_\rho =m_\omega $$\end{document}.
Using the unitarity of the CKM matrix, we have∑iVuis∗VuidIui=Vud∗Vus(Iu-Ic)+Vtd∗Vts(It-Ic)≈Vud∗Vus(Iu-Ic)-Vtd∗VtsIc,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{i}V^*_{u_i s}V_{u_id}I_{u_i}&= V^*_{ud}V_{us}(I_u-I_c)+V^*_{td}V_{ts}(I_t-I_c)\nonumber \\&\approx V^*_{ud}V_{us}(I_u-I_c)-V^*_{td}V_{ts}I_c, \end{aligned}$$\end{document}where the last line is obtained using the fact that mt\documentclass[12pt]{minimal}
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\begin{document}$$m_t$$\end{document} is much larger than masses of other quarks involved in this process. We note that only Vtd∗Vts\documentclass[12pt]{minimal}
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\begin{document}$$V_{td}^*V_{ts}$$\end{document} provides a weak CP-violation phase, so it is unnecessary to consider the contribution of the first term. As a consequence, the CP asymmetry only depends on Vtd∗VtsIc\documentclass[12pt]{minimal}
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\begin{document}$$V^*_{td}V_{ts}I_c$$\end{document}. We defineA1≡∫01dxdyϕK∗(x)ϕρ(ω)∗(y)xQ2+(1-x)mρ2+(x2-x)mK2-mc2,=∫01dxfρϕK∗(x)26[xQ2+(1-x)mρ2+(x2-x)mK2-mc2],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} A_1&\equiv \int ^1_0\mathrm {d}x\mathrm {d}y\frac{{\phi ^*_K(x)\phi ^*_{\rho (\omega )}}(y)}{xQ^2+(1-x)m_\rho ^2+(x^2-x)m_K^2-m^2_{c}},\nonumber \\&= \int ^1_0\mathrm {d}x\frac{f_\rho \phi ^*_K(x)}{2\sqrt{6}[xQ^2+(1-x)m_\rho ^2 +(x^2-x)m_K^2-m^2_{c}]},\nonumber \\&\end{aligned}$$\end{document}B1≡∫01dxdyxϕK∗(x)ϕρ(ω)∗(y)xQ2+(1-x)mρ2+(x2-x)mK2-mc2=∫01dxxfρϕK∗(x)26[xQ2+(1-x)mρ2+(x2-x)mK2-mc2].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_1&\equiv \int ^1_0\mathrm {d}x\mathrm {d}y\frac{x\phi ^*_K(x)\phi ^*_{\rho (\omega )}(y)}{xQ^2+(1-x)m_\rho ^2+(x^2-x)m_K^2-m^2_{c}}\nonumber \\&= \int ^1_0\mathrm {d}x\frac{xf_\rho \phi ^*_K(x)}{2\sqrt{6}[xQ^2+(1-x)m_\rho ^2 +(x^2-x)m_K^2-m^2_{c}]}.\nonumber \\ \end{aligned}$$\end{document}Inserting Eqs. (18), (19), and (20) into Eq. (17) and only considering the CP asymmetry term, Hμ1ρ(ω)\documentclass[12pt]{minimal}
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\begin{document}$$H^{1\rho (\omega )}_{\mu }$$\end{document} can be simplified asHμ1ρ(ω)=62(2π)3mumd2msGFVts∗VtdVud∗mk×{-A1εμναβϵ∗νp1αp2β-iϵ∗μ[12A1(Q2-mρ2-mK2)+B1mK2]+iA1p1μ(Q·ϵ∗)+i2B1p2μ(Q·ϵ∗)}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&H^{1\rho (\omega )}_{\mu }= \frac{6\sqrt{2}(2\pi )^3\sqrt{m_u m_d^2 m_s}G_F V_{ts}^*V_{td}V_{ud}^*}{m_k}\nonumber \\&\quad {}\times \Bigg \{{}-A_1\varepsilon _{\mu \nu \alpha \beta }\epsilon ^{*\nu }p_1^{\alpha }p_{2}^{\beta }-\mathrm {i}\epsilon ^{*\mu }\Bigg [\frac{1}{2}A_1(Q^2-m_\rho ^2-m_K^2)\nonumber \\&\quad {}+B_1m^2_K\Bigg ]+\mathrm {i}A_1p_1^{\mu }(Q\cdot \epsilon ^*)+\mathrm {i}2B_1p_2^{\mu }(Q\cdot \epsilon ^*)\Bigg \}. \end{aligned}$$\end{document}We can see that the weak phase appears but the strong phase is absent in this amplitude if ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing is not included.
Now we take ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing into account and show how ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing enlarges the CP violation. In the meson dominance model, the form factors of the annihilation process are dominated by strong interaction. So, we adopt the same form factors in the Kρ\documentclass[12pt]{minimal}
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\begin{document}$$K\omega $$\end{document} annihilation processes. According to Eq. (13), we have Hμ0ρ=Hμ0ω\documentclass[12pt]{minimal}
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\begin{document}$$H^{1\rho }_{\mu }=-H^{1\omega }_{\mu }$$\end{document}. Then the first two equations of Eq. (12) can be written asM0M0†=(GF/2)2gρππ2sρ2LμνHμν0ρ0ρ(1+Π~ρω∗sω∗+Π~ρωsω),M0M1†=(GF/2)2gρππ2sρ2LμνHμν0ρ1ρ(1-Π~ρω∗sω∗+Π~ρωsω).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&M_0M_0^\dagger \nonumber \\&\quad =(G_F /\sqrt{2})^2 g_{\rho \pi \pi }^2 s_\rho ^2 L^{\mu \nu }H^{0\rho 0\rho }_{\mu \nu }\big (1+\tilde{\varPi }^*_{\rho \omega }s^*_\omega +\tilde{\varPi }_{\rho \omega }s_\omega \big ),\nonumber \\&M_0M_1^\dagger \nonumber \\&\quad =(G_F/\sqrt{2})^2g_{\rho \pi \pi }^2s_\rho ^2 L^{\mu \nu }H^{0\rho 1\rho }_{\mu \nu }\big (1-\tilde{\varPi }^*_{\rho \omega }s^*_\omega +\tilde{\varPi }_{\rho \omega }s_\omega \big ). \end{aligned}$$\end{document}We can see explicitly that ρ\documentclass[12pt]{minimal}
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\begin{document}$$A_\mathrm{CP}$$\end{document}. As will be shown later, these complex terms enlarge the CP-even phase, which leads to a bigger CP asymmetry.
In the previous subsections, we have given the general expression of CP asymmetry and derivations of the matrix elements. For simplicity, we choose to work in a special reference frame and express the products among vectors p1\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} in this subsection. We note that it is convenient to express the momenta of hadrons and leptons and calculate various components of the matrix elements in the hadronic rest frame [36]. This frame is defined in Fig. 3. The z\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}) and the neutrino. Then the momenta of hadrons and leptons in this hadronic rest frame are given as follows:p1μ=(E1,0,0,P),p2μ=(E2,0,0,-P),p3μ=(K,Ksinθ,0,Kcosθ),p4μ=(E4,Ksinθ,0,Kcosθ),Qμ=(E1+E2,0,0,0)=(E4-K,0,0,0),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} p_1^\mu&=(E_1,\ 0,\ 0,\ P),\\ p_2^\mu&=(E_2,\ 0,\ 0,\ -P),\\ p_3^\mu&=(K,\ K\sin \theta ,\ 0,\ K\cos \theta ),\\ p_4^\mu&=(E_4,\ K\sin \theta ,\ 0,\ K\cos \theta ),\\ Q^\mu&=(E_1+E_2,\ 0,\ 0,\ 0)\\&=(E_4-K,\ 0,\ 0,\ 0), \end{aligned} \end{aligned}$$\end{document}and the polarization vectors of ρ0\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}) in this hadronic rest frame areϵλ=±1=(0,1,±i,0),ϵλ=0=1mρ(P,0,0,E1),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \epsilon ^{\lambda =\pm 1}&=(0,\ 1,\ \pm \mathrm {i}\ ,0),\\ \epsilon ^{\lambda =0}&=\frac{1}{\sqrt{m_\rho }}(P,\ 0,\ 0,\ E_1), \end{aligned} \end{aligned}$$\end{document}withE1=Q2+mK2-mρ22Q2,E2=Q2-mK2-mρ22Q2,P=mK4+mρ4-2mK2Q2-2mK2mρ2-2Q2mρ22Q2,K=mτ2-Q22Q2,E4=mτ2+Q22Q2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&E_1=\frac{Q^2+m_K^2-m^2_\rho }{2\sqrt{Q^2}},\qquad E_2=\frac{Q^2-m_K^2-m^2_\rho }{2\sqrt{Q^2}},\nonumber \\&P=\frac{\sqrt{m_K^4+m^4_\rho -2m_K^2Q^2-2m_K^2m^2_\rho -2Q^2m^2_\rho }}{2\sqrt{Q^2}},\\&K=\frac{m_\tau ^2-Q^2}{2\sqrt{Q^2}},\qquad \qquad E_4=\frac{m_\tau ^2+Q^2}{2\sqrt{Q^2}}.\nonumber \end{aligned}$$\end{document}The above expressions for various hadron and lepton momentum vectors allow us to determine simple expressions for matrix elements which involve products including p1·p2\documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document} vary in some areas, we obtain the localized integrated CP asymmetry, which takes the following form:ACPΩ=∫ΩdQ2ds(|M|2-|M|¯2)∫ΩdQ2ds(|M0|2+|M0|¯2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} A_\mathrm{CP}^\Omega =\frac{\int _\varOmega \mathrm {d}Q^2\mathrm {d}s(|M|^2-\bar{|M|}^2)}{\int _\varOmega \mathrm {d}Q^2\mathrm {d}s(|M_0|^2+\bar{|M_0|}^2)}. \end{aligned}$$\end{document}
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\begin{document}$$\omega $$\end{document}) and the neutrino
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From the above discussions, the CP-violating asymmetries depend on the values of Q2\documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document}. In this section we give the explicit expressions of meson wave functions and form factors, and values of several parameters in order to calculate the CP-violating asymmetries. We find that significant cancelation occurs as one performs the integration over Q2\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} decay is dominated by the meson annihilation diagram, Fig. 1a. As mentioned before, the vector and pseudoscalar meson annihilation form factors in this decay mode are difficult to be related directly to experimental measurements. One therefore needs to adopt phenomenological models. Following Ref. [33] we use the meson dominance model in our calculation. In this model it is assumed that the vector form factor g\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} and a±\documentclass[12pt]{minimal}
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\begin{document}$$a_\pm $$\end{document} are dominated by the exchange of the K-\documentclass[12pt]{minimal}
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\begin{document}$$K_1(1400)$$\end{document} axial-vector mesons [33]. The expressions for the form factors are given in Eq. (14). In Ref. [33], the values of weak couplings and strong vertex couplings were extracted from experiments and fixed by the SU(3) flavor symmetry. We display these values in Table 1.
The values of hMi\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _{M_i}$$\end{document} in the numerical calculations
We use the K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document} meson wave function of the Brodsky–Huang–Lepage prescription which have the following form [37]:ΦK(x,k⊥)=AK(1-2x)2exp[-bK2(k⊥2+ms′2x+k⊥2+mu′21-x)],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\Phi _K(x,\ {\varvec{k}}_\perp )\nonumber \\&\quad =A_K (1-2x)^2\,\mathrm {exp}\Big [-b_K^2\Big (\frac{{\varvec{k}}^2_\perp +m_s^ {\prime 2}}{x}+\frac{{\varvec{k}}^2_\perp +m_u ^ {\prime 2}}{1-x}\Big )\Big ],\nonumber \\ \end{aligned}$$\end{document}where k⊥\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}, mu′\documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document}, respectively. Integrating ΦK(x,k⊥)\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _K(x,\ {\varvec{k}}_\perp )$$\end{document} over k⊥\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{k}}_\perp $$\end{document} one has the following distribution amplitude:ϕK(x)=AK16π2bK2x(1-x)(1-2x)2exp[-bK2(ms′2x+mu′21-x)].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\phi _K(x)\nonumber \\&\quad =\frac{A_K}{16 \pi ^2 b_K^2} x(1-x)(1-2x)^2\mathrm {exp}\Big [-b_K^2\Big (\frac{m_s^ {\prime 2}}{x}+\frac{m_u ^ {\prime 2}}{1-x}\Big )\Big ].\nonumber \\ \end{aligned}$$\end{document}In the following numerical calculations we use the parameters AK=232\documentclass[12pt]{minimal}
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\begin{document}$$f_\rho =221$$\end{document} MeV [37].
Numerical results for the CP asymmetries
We are now ready to evaluate numerical results of CP asymmetries. We take the meson masses mρ=770\documentclass[12pt]{minimal}
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\begin{document}$$m_c=1275$$\end{document} MeV [38]. The CKM matrix, of which the elements are determined from experiments, can be expressed in terms of the Wolfenstein parameters A\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} [38]:1-12λ2λAλ3(ρ-iη)-λ1-12λ2Aλ2Aλ3(1-ρ-iη)-Aλ21,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left( \begin{array}{c@{\quad }c@{\quad }c} 1-\frac{1}{2}\lambda ^2 &{} \lambda &{}A\lambda ^3(\rho -\mathrm {i}\eta ) \\ -\lambda &{} 1-\frac{1}{2}\lambda ^2 &{}A\lambda ^2 \\ A\lambda ^3(1-\rho -\mathrm {i}\eta ) &{} -A\lambda ^2 &{}1\\ \end{array} \right) , \end{aligned}$$\end{document}where O(λ4)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O} (\lambda ^{4})$$\end{document} corrections are neglected. The latest values for the parameters in the CKM matrix are [38]λ=0.22535±0.00065,A=0.811-0.012+0.022,ρ¯=0.131-0.013+0.0026,η¯=0.345-0.014+0.013,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\begin{aligned}&\lambda =0.22535\pm 0.00065,\qquad \ A=0.811^{+0.022}_{-0.012},\\&\bar{\rho }=0.131^{+0.0026}_{-0.013},\qquad \qquad \quad \bar{\eta }=0.345^{+0.013}_{-0.014}, \end{aligned} \end{aligned}$$\end{document}withρ¯=ρ1-λ22,η¯=η1-η22.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{\rho }=\rho \left( 1-\frac{\lambda ^2}{2}\right) ,\qquad \bar{\eta }=\eta \left( 1-\frac{\eta ^2}{2}\right) . \end{aligned}$$\end{document}In our numerical calculations, the most uncertain factors come from the CKM matrix elements and the form factors in the leading-order weak process. In fact, the uncertainties due to the CKM matrix elements are mostly from η\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} is well determined and the CP-violating asymmetries are independent of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document}. Hence in the following we take the central value of λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, 0.225. In the meson dominance model, the uncertainties arising from form factors are dominated by those of the strong and weak coupling constants of the K1(1400)\documentclass[12pt]{minimal}
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\begin{document}$$K_1(1400)$$\end{document} meson due to the poor quality of measurements. The values of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing parameters also bring about some uncertainties.
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\begin{document}$$s$$\end{document}, we study the differential CP asymmetries. Since CP asymmetries are calculated around the ω(782)\documentclass[12pt]{minimal}
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\begin{document}$$\le \sqrt{s}\le $$\end{document}800 MeV. From Eqs. (28) and (30), we obtain (mρ+mK)2<Q2<mτ2\documentclass[12pt]{minimal}
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\begin{document}$$(m_\rho +m_K)^2<Q^2<m_\tau ^2$$\end{document}. Hence we take the range of Q2\documentclass[12pt]{minimal}
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\begin{document}$$m_\tau =$$\end{document}1770 MeV. The differential CP asymmetry ACP\documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document} is displayed in Fig. 4, where we take central values of the parameters involved in the calculation. We can see that ACP\documentclass[12pt]{minimal}
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\begin{document}$$-(5.6^{+2.9}_{-1.7})\times 10^{-12}$$\end{document}, where the errors come from the uncertainties of the CKM matrix elements, the ρ\documentclass[12pt]{minimal}
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\begin{document}$$A_\mathrm{CP}$$\end{document} by four orders of magnitude in some regions. Furthermore, we also find that the sign of ACP\documentclass[12pt]{minimal}
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\begin{document}$$A_1$$\end{document} (and B1\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{Q^2}$$\end{document} crosses the pole. This will lead to cancelations when one performs the integration over Q2\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} in some regions. These cancelations are found to be quite obvious around the peak when s=784MeV\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing in the following.
The differential CP asymmetry as a function of s\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{Q^2}$$\end{document}. The numerical results correspond to central values of the parameters involved in the calculation
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\begin{document}$$\sqrt{s}$$\end{document} fixed. We divide the integration region into eight equal intervals: (1.30, 1.35 GeV), (1.35, 1.40 GeV), (1.40, 1.45 GeV), (1.45, 1.50 GeV), (1.50, 1.55 GeV), (1.55, 1.60 GeV), (1.60, 1.65 GeV), and (1.65, 1.70 GeV). In each interval, we integrate over Q2\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{Q^2}$$\end{document} and calculate the CP asymmetries with and without ρ\documentclass[12pt]{minimal}
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\begin{document}$$A^s_\mathrm{CP}$$\end{document} and shown in Fig. 5. Our numerical results show that the ρ\documentclass[12pt]{minimal}
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\begin{document}$$A^s_\mathrm{CP}$$\end{document} by about one–two orders of magnitude when s\documentclass[12pt]{minimal}
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\begin{document}$$0.784\,\mathrm {MeV}$$\end{document}.
The localized integrated CP asymmetry ACPs\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{Q^2}=$$\end{document}(1.30GeV\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing and the solid line corresponds to the CP asymmetry without ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing; b–h correspond to the integration intervals (1.35, 1.40 GeV), (1.40, 1.45 GeV), (1.45, 1.50 GeV), (1.50, 1.55 GeV), (1.55, 1.60 GeV), (1.60, 1.65 GeV), and (1.65, 1.70 GeV), respectively. We take central values of the parameters involved in the calculation
Finally, we integrate ACP\documentclass[12pt]{minimal}
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\begin{document}$$A_\mathrm{CP}$$\end{document} over both Q2\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}$$\end{document} and obtain the localized integrated asymmetries ACPΩ\documentclass[12pt]{minimal}
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\begin{document}$$A^{\Omega }_\mathrm{CP}$$\end{document}. Considering the significant region of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing shown in Fig. 5, we choose the integration interval of s\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}$$\end{document} to be from 0.775 MeV to 0.795 MeV. The numerical results of the localized integrated asymmetries with (without) ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing are shown in Table 2, where the central values of the numerical results correspond to central values of the parameters involved in the calculation and the errors are again from the uncertainties of the CKM matrix elements, the ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing parameters, and the form factors in the leading-order weak process. We can see in most of the intervals ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing enlarges the localized integrated asymmetries. The maximum increase is three orders of magnitude. These predictions lead to a new upper limit of the CP asymmetries based on the SM in this decay channel. Even though the strong phase is enhanced by ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing, the direct CP asymmetry is still negligible. This is just because the small weak relative phase, which comes from the interference between the leading-order and the second-order weak diagrams, is of the third order in GF\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document}.
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\begin{document}$$\omega $$\end{document} mixing. The central values of the numerical results correspond to central values of the parameters involved in the calculation and the errors are from the uncertainties of the CKM matrix elements, the ρ\documentclass[12pt]{minimal}
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In the framework of the SM, CP violation in the τ\documentclass[12pt]{minimal}
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\begin{document}$$G_F$$\end{document}. However, Delepine pointed out that the CP-odd phase can arise from the second-order weak process in the τ±→K±π0ντ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^-{\rightarrow } K^- \rho ^0 (\omega )\nu _{\tau }{\rightarrow } K^- \pi ^+\pi ^-\nu _{\tau }$$\end{document} decay via this mechanism.
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\begin{document}$$\omega $$\end{document} resonance. Consequently, the CP-violating asymmetry reaches the maximum value when the invariant mass of the π+π-\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing mechanism in present paper can also be considered in the (3π)±ντ\documentclass[12pt]{minimal}
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\begin{document}$$(3\pi )^\pm \nu _\tau $$\end{document} and (4π)±ντ\documentclass[12pt]{minimal}
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\begin{document}$$(4\pi )^\pm \nu _\tau $$\end{document} final states in the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} lepton decay.
Finally, we also have performed integration over both Q2\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{Q^2}$$\end{document} and s\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}$$\end{document} to obtain ACPΩ\documentclass[12pt]{minimal}
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\begin{document}$$A^{\varOmega }_\mathrm{CP}$$\end{document}. The maximum value of ACPΩ\documentclass[12pt]{minimal}
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\begin{document}$$A^{\varOmega }_\mathrm{CP}$$\end{document} turns out to be 6.3-3.3+2.4×10-11\documentclass[12pt]{minimal}
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\begin{document}$$6.3^{+2.4}_{-3.3}\times 10^{-11}$$\end{document}. This value is the largest CP asymmetry in this decay channel within the SM predicted at present. Even though the relatively strong phase is largely enhanced by ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing, the direct CP asymmetry is still negligible. Any measurement of CP violation in this decay channel requires NP, though its effect may be enhanced through the ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} mixing mechanism discussed in this paper.
ReferencesJ.H. Christenson et al., Phys. Rev. Lett. 13, 138 (1964)N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963)KobayashiMMaskawaTProg. Theor. Phys.19734965210.1143/PTP.49.6521973PThPh..49..652KHuangTLuWTaoZPhys. Rev. D199755164310.1103/PhysRevD.55.16431997PhRvD..55.1643HA. Pich, Prog. Part. Nucl. Phys. 75, 41–85 (2014). arXiv:1310.7922 [hep-ph]D. Kimura et al., (2009). arXiv:0905.1802 [hep-ph]Q. Qin, Workshop on Tau Charm at High Luminosity, Elba (Italy), May 2013P. Zweber et al., (BESIII Collaboration), AIP Conf. Proc. 1182, 406 (2009)E. Levichev, Workshop on Tau Charm at High Luminosity, Elba (Italy), May 2013BondarAEPhys. Atom. Nucl.2013761072108510.1134/S10637788130900322013PAN....76.1072BM. Zobov (INFN LNF), (2011). arXiv:1106.5329H.B. Li, Workshop for Super Tau-Charm Factory in Hefei, Hefei (China), June 2013M. Baszczyk et al. (SuperB Collaboration), (2013). arXiv:1306.5655 [physics.ins-det]P. Avery et al., (CLEO Collaboration), Phys. Rev. D 64, 092005 (2001)M. Bischofberger et al., (Belle Collaboration), Phys. Rev. Lett. 107, 131801 (2011)J.P. Lees et al., (BABAR Collaboration), Phys. Rev. D 85, 031102 (2012)U. Kilian, J.G. Körner, K. Schilcher, Y.L. Wu, Z. Phys, C 62, 413 (1994)KiersKPhys. Rev. D20087811300810.1103/PhysRevD.78.1130082008PhRvD..78k3008KDattaAPhys. Rev. D20077507400710.1103/PhysRevD.75.0740072007PhRvD..75g4007DChoiSYHagiwaraKTanabashiMPhys. Rev. D199552161410.1103/PhysRevD.52.16141995PhRvD..52.1614CDattaAPhys. Rev. D20077507400710.1103/PhysRevD.75.0740072007PhRvD..75g4007DD. Delepine, G. López Castro, G. López Castro, L.-T. López Lozano, Phys. Rev. D 72, 033009 (2005)BigiIISandaAIPhys. Lett. B20056254710.1016/j.physletb.2005.08.0332005PhLB..625...47BEnomotoRTanabashiMPhys. Lett. B199638641310.1016/0370-2693(96)00979-31996PhLB..386..413EGardnerSO’ConnellHBThomasAWPhys. Rev. Lett.199880183410.1103/PhysRevLett.80.18341998PhRvL..80.1834GX.-H. Guo, A.W. Thomas, Phys. Rev. D 58, 096013 (1998)X.-H. Guo, A.W. Thomas, Phys. Rev. D 61, 116009 (2000)GuoX-HLeitnerOThomasAWPhys. Rev. D20016305601210.1103/PhysRevD.63.0560122001PhRvD..63e6012GO. Leitner, X.-H. Guo, A.W. Thomas, Eur. Phys. J. C 31, 215 (2003)O’ConnellHBThomasAWWilliamsAGNucl. Phys. A199762355910.1016/S0375-9474(97)88425-41997NuPhA.623..559OMaltmanKO’ConnellHBWilliamsAGPhys. Lett. B19963761910.1016/0370-2693(96)00293-61996PhLB..376...19MGardnerSO’ConnellHBPhys. Rev. D199857271610.1103/PhysRevD.57.27161998PhRvD..57.2716GA. Flores-Tlalpa, G. López Castro, Phys. Rev. D 77, 113011 (2008)DuDYangDZhuGPhys. Rev. D19996005401510.1103/PhysRevD.60.0540151999PhRvD..60e4015DFalkAFNucl. Phys. B1992378799410.1016/0550-3213(92)90004-U1992NuPhB.378...79FJ.H. Kühn, E. Mirkes, Z. Phys. C 56, 661–671 (1992)GuoX-HHuangTPhys. Rev. D199143293110.1103/PhysRevD.43.29311991PhRvD..43.2931GJ. Beringer et al., (Particle Data Group), Phys. Rev. D 86, 010001 (2012)