application/xmlCP violation in the charged pion energy spectra of the decays [formula omitted]G. FäldtE. ShabalinPhysics Letters B 635 (2006) 295-298. doi:10.1016/j.physletb.2006.03.017journalPhysics Letters BCopyright © 2006 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26936355-620 April 20062006-04-20295-29829529810.1016/j.physletb.2006.03.017http://dx.doi.org/10.1016/j.physletb.2006.03.017doi:10.1016/j.physletb.2006.03.017http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB22880S0370-2693(06)00311-X10.1016/j.physletb.2006.03.017Elsevier B.V.PhenomenologyCP violation in the charged pion energy spectra of the decays K±→π0π0π±G.Fäldta⁎faldt@tsl.uu.seE.Shabalinbshabalin@heron.itep.ruaDivision of Nuclear Physics, Box 535, 751 21 Uppsala, SwedenbInstitute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia⁎Corresponding author.Editor: H. GeorgiAbstractCP violation leads to a difference between the parameters g+ and g− that characterise the energy distributions of the “odd” pion in the decays K±→π0π0π± and K±→π±π±π∓. We argue that for the first decay, the asymmetry Δg=(g+−g−)/(g++g−) is fixed at a value around Δg=2×10−6, whereas for the second decay, the asymmetry Δg may be one order of magnitude larger.PACS13.75.Jz14.20.-c25.10.+sIt is well known that the strength of direct CP violation in the KL→2π decays, as determined by the parameter ε′, is crucially depending on the fact that the QCD penguin (QCDP) and the electroweak penguin (EWP) contributions partially cancel one another [1]. Thus, it is not difficult to understand that before the experimental value ε′/ε=(1.67±0.26)×10−3[2] was available the theoretical predictions for ε′/ε were one order of magnitude smaller than this value [3], or very uncertain, leading to values of this ratio varying all over the range 10−4⩽ε′/ε⩽10−3[4,5]. In the present note we discuss some consequences for the K±→3π decays.In [6–8], it was found that contrary to the case of ε′, in K±→π±π±π∓ decay, the EWP contribution enhance the QCDP contribution. But in order to estimate the magnitude of the CP-violating effect, it was necessary to resort to unreliable theoretical estimates of the QCDP and the EWP contributions (see Ref. [8]).For the K±→π0π0π± decay, the situation is cleaner, because as explained in the present note, the CP-odd asymmetry Δg in this case turns out to be proportional to practically the same combination of QCDP and EWP contributions as in ε′. Consequently, Δg can be estimated reliably using the known value of ε′. For the K±→π±π±π∓ decay, on the other hand, we argue that Δg may be one order of magnitude larger than in the K±→π0π0π± decay. This conclusion differs from those proposed in Refs. [9,10].Our investigation is based on the effective ΔS=1 non-leptonic Lagrangian proposed in Ref. [11],(1)L(ΔS=1)=2GFsinθCcosθC∑iciOi, where the Oi are four-quark operators, defined as(2)O1=s¯LγμdL⋅u¯LγμuL−s¯LγμuL⋅u¯LγμdL,O2=s¯LγμdL⋅u¯LγμuL+s¯LγμuL⋅u¯LγμdL+2s¯LγμdL⋅d¯LγμdL+2s¯LγμdL⋅s¯LγμsL,O3=s¯LγμdL⋅u¯LγμuL+s¯LγμuL⋅u¯LγμdL+2s¯LγμdL⋅d¯LγμdL−3s¯LγμdL⋅s¯LγμsL,O4=s¯LγμdL⋅u¯LγμuL+s¯LγμuL⋅u¯LγμdL−s¯LγμdL⋅d¯LγμdL,O5=s¯LγμλadL(∑q=u,d,sq¯RγμλaqR),O6=s¯LγμdL(∑q=u,d,sq¯RγμqR). For our study of CP violation, we must add two more four-quark operators,(3)O7=32s¯γμ(1+γ5)d(∑q=u,d,seqq¯γμ(1−γ5)q),O8=−12∑q=u,d,seq(s¯LqR)(q¯RdL), where eq is the quark-charge matrix.The operators O5,6 arise from the QCD penguin diagram and the operators O7,8 arise, analogously, from the electroweak penguin diagram. The Wilson coefficients c5–8 contain the imaginary parts necessary for CP violation. The bosonization of the operators O1–8 can be achieved by exploiting the relations between di-quark field operators and pseudoscalar fields as represented in [12], and the reordering relations in colour and spinor spaces as from [13].Representing the K→2π amplitudes in the form(4)M(K10→π+π−)=A0eiδ0−A2eiδ2,M(K10→π0π0)=A0eiδ0+2A2eiδ2, this approach yields(5)A0=κ[c1−c2−c3+329β(Rec˜5+iImc˜5)],(6)A2=κ[c4+i23βΛ2Imc˜7(mK2−mπ2)−1]. Here, δ0 and δ2 are the pion–pion scattering phase shifts in the isospin T=0 and T=2 channels, and the remaining parameters areκ=GFFπsinθCcosθCmK2−mπ22,β=2mπ4Λ2(mu+md)2,c˜5=c5+316c6,c˜7=c7+3c8,Λ≈1GeV. Since c˜7/c˜5∼αem and small, we have neglected the EWP contribution to A0.From data on K→2π rates one can deduce the values of the real parts of the amplitudes A0 and A2[14], i.e.,(7)c4=0.328,(8)c1−c2−c3+329βRec˜5=−10.13. Furthermore, if as suggested by Refs. [11,13], we assume c1−c2−c3=−2.89, then we have in addition 329βRec˜5=−7.24.Using the definition of the parameter ε′,(9)ε′=iei(δ2−δ0)[−ImA0ReA0+ImA2ReA2]⋅|A2A0|, and its experimental value, we deduce(10)−Imc˜5Rec˜5(1−Ω+24.4Imc˜7Imc˜5)=(1.63±0.16)×10−4. The new parameter Ω takes into account effects of isospin violation, coming from the quark mass difference md≠mu and the electromagnetic interaction. As a result hereof, the physical state vector of the isovector I=1 neutral pi-meson acquires an admixture of states with isospin I=0,(11)|πphys0〉=|π0〉+λ|η〉+λ′|η′〉. For a recent review see Ref. [15].As a consequence of mixing there are alternative contributions to the K→π0π0 decays. The weak-interaction Lagrangian of Eq. (1) can in a first step induce the transitions K0→π0η(η′) which, in a second step, are followed by the transitions η(η′)→π0 induced by the isospin mixing of Eq. (11). Thus, in the tree approximation, the isospin decompositions of the K0→π+π− and K0→π0π0 amplitudes change into〈π+π−|Hw|K0〉=(A0−γ)(I=0)−(A2−γ)(I=2)=A0−A2,〈π0π0|Hw|K0〉=(A0−γ)(I=0)+2(A2−γ)(I=2)=A0+2A2−3γ. Here, −γ is the matrix element coming from the mixing. It is the same for both isospin channel amplitudes and 1/3 of the amplitude for the K0→η(η′)→π0π0 transition.Now, in the absence of EWP contributions, the combination(12)−ImA0ReA0+ImA2ReA2 of Eq. (9) transforms into(13)−ImA0ReA0[1−ImγImA0+ReA0ReA2ImγImA0(1+ReγReA2)]≡−ImA0ReA0[1−Ω], where only the most important terms have been retained. Calculations [16] of Ω to leading order in a low-energy expansion suggested Ω=0.25±0.10[17]. However, it was later found that the next-to-leading order corrections reduce the value of Ω to Ω=0.16±0.03[18], and possibly to an even to smaller value, Ωeff=0.060±0.077, providing electromagnetic corrections (without EWP though) are considered as well [19]. Taking this last value of Ωeff as our preferred one we can rewrite Eq. (10) as(14)βImc˜5[1+(26.1±2.1)Imc˜7Imc˜5]=(3.56±0.61)×10−4. Below, we shall see that the numerical result of Eq. (14) leads to a reliable estimate of the asymmetry parameter Δg in the K±→π0π0π± decay.Let us now turn to the K±→3π decays. Applying the same techniques as above and taking into account the appearance of CP-even imaginary parts due the to strong ππ final-state rescattering, we get in leading p2 approximation for the τ and τ′ decay amplitudes:(15)M(K±(k)→π±(p1)π±(p2)π∓(p3))=κ˜[1+ia+12gτY(1+ibτ±idKMτ)+⋯],(16)M(K±(k)→π0(p1)π0(p2)π±(p3))=κ˜2[1+ia+12gτ′Y(1+ibτ′±idKMτ′)+⋯]. The indices τ and τ′ refer to the decay modes of the kaon. The parameters a, bτ and bτ′ arise from the strong pion–pion rescattering and are consequently CP-even. The dKMτ,τ′ are CP-odd imaginary terms produced by the Kobayashi–Maskawa phase. Furthermore, Y is a kinematic factor, Y=(s3−s0)/mπ2, with s3=(k−p3)2 and s0=13mK2+mπ2.In K±→π±π±π∓ decay, the parameter values area=0.12,bτ=0.71,gτ=−3mπ2mK2(1+9c4/c0),c0=c1−c2−c3−c4+329βRec˜5=−10.46,κ˜=GFmK2sinθCcosθCc0/32, and for the CP-odd contribution we get(17)dKMτ=−329βImc˜59c4c0(c0+9c4)[1+3Λ2(c0+9c4)16mK2c4(1+12c4mK2Λ2(c0+9c4))Imc˜7Imc˜5]=−216c4c0(c0+9c4)βImc˜5(1−14.36Imc˜7Imc˜5).In K±→π0π0π± decay, two parameters are different, i.e.,bτ′=0.49,gτ′=6mπ2mK2(1−9c4/2c0), as is the CP-odd contribution(18)dKMτ′=329βImc˜59c4/2c0(c0−9c4/2)[1−3Λ2(c0−9c4/2)8mK2c4(1−c0mK22(c0−9c4/2)(mK2−mπ2))Imc˜7Imc˜5]=16c4c0(c0−9c4/2)βImc˜5(1+27.8Imc˜7Imc˜5).The slope parameters gτ± in τ decay are defined by the equation(19)|M(K±(k)→π±(p1)π±(p2)π∓(p3))|2∼[1+gτ±Y+⋯] with a similar definition for gτ′± in τ′ decay. The CP-asymmetry parameters Δgτ,τ′ in the two decays are defined as(20)Δgτ=gτ+−gτ−gτ++gτ−=adKMτ1+abτ,Δgτ′=gτ′+−gτ′−gτ′++gτ′−=adKMτ′1+abτ′. Discussing first τ′ decay, we realise when comparing Eqs. (18) and (14) that the linear combinations of QCDP and EWP contributions appearing in these expressions are very similar. In fact, at Ω=0.124 the two combinations are identical. Thus, exploiting our knowledge of the experimental value of ε′ we predict for the asymmetry parameter of Eq. (20)(21)Δgτ′=(1.8±0.24)×10−6. At Ωeff=0.060±0.077 from Ref. [19](22)Δgτ′=(1.71±0.29)×10−6. We conclude that due to the close resemblance of the expressions for ε′ and Δgτ′ decay our prediction for Δgτ′ should be quite robust.The CP-asymmetry parameter in τ decay is most easily discussed via the ratio,(23)−ΔgτΔgτ′=2(c0−9c4/2c0+9c4)(1+abτ′1+abτ)(1−14.36Imc˜7/Imc˜51+27.8Imc˜7/Imc˜5), which is obtained by combining Eqs. (17)–(19). Therefore,(a)if the EWP contributions do not play any significant role in direct CP violation, i.e., when Imc˜7/Imc˜5 is negligibly small, then(24)−Δgτ/Δgτ′=3.1or−Δgτ⩾0.56×10−5;(b)if the EWP contribution cancels half of the QCDP contribution in ε′ (see Refs. [20,21]), then(25)−Δgτ=7.8Δgτ′⩾1.3×10−5. The above results are obtained in leading p2 approximation. The role of p4 corrections for Δgτ were studied in Refs. [6,8], and they were found to increase the value of Δgτ by 23%. For Δgτ′ the corresponding investigation has not yet been performed. But one effect can be seen at once. According to Refs. [6,8], the corrections of order p4 increase the rescattering parameter a in Eq. (20) by 30%. Thus, we expect the corrected value of Δgτ′ to lie in the range (1.8–2.5)×10−6.Finally, we remark once more that our numerical results not only differ from those reported in [9], which are −Δgτ=(2.3±0.6)×10−6 and Δgτ′=(1.3±0.4)×10−6, but also from the more recent ones reported in [10], which are −Δgτ=(2.4±1.2)×10−5 and Δgτ′=(1.1±0.7)×10−5. Both investigations were performed within the framework of chiral perturbation theory. In Ref. [10] attempts were made to estimate contributions of order p4, but the predicted value for Δgτ′ has large uncertainties.Our results strongly suggest, that accurate measurements of Δgτ and Δgτ′ should clarify the relative importance of QCDP and EWP mechanisms in direct CP violation.References[1]J.M.FlynnL.RandallPhys. Lett. B2241989221G.BuchallaNucl. Phys. B3371990313E.A.PaschosY.L.WuMod. Phys. Lett. A6199193[2]S.EidelmanPhys. Lett. B5922004140[3]M. Ciuchini, et al., in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second DAPhNE Physics Handbook, INFN-LNF, 1995, p. 27[4]T.HambyeS.PerisE.de RafaelNucl. Phys. B5642001301[5]S.BertoliniNucl. Phys. B514199893[6]E.Shabalinin: Proceedings Les Rencontres de Physique de la Vallée d'Aoste, La Thuile, 2003, p. 417hep-ph/0305320[7]E. Shabalin, in: F. Anulli, et al. (Eds.), Proceedins DAPhNE 2004: Physics at Meson Factories, Frascati, June 2004, INFN-LNF, 2004, p. 89[8]E.P.ShabalinPhys. At. Nucl.68200588[9]L. Maiani, N. Paver, in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second DAPhNE Physics Handbook, INFN-LNF, 1995, p. 51[10]I.ScimemiE.GamizJ.Pradesin: Proceedings 39th Rencontres de Moriond, La Thuile, Aosta Valley, March 2004hep-ph/0405204[11]M.A.ShifmanA.I.VainshteinV.I.ZakharovZh. Eksp. Teor. Fiz.7219771275Sov. Phys. JETP451977670M.A.ShifmanA.I.VainshteinV.I.ZakharovNucl. Phys. B2101977316[12]W.A.BardeenA.J.BurasJ.-M.GerardNucl. Phys. B2931987787[13]L.B.OkunLeptons and Quarks1982North-HollandAmsterdampp. 315–323[14]E.ShabalinNucl. Phys. B409199387[15]T.FeldmannInt. J. Mod. Phys. A152000159[16]J.F.DonoghueE.GolowichB.R.HolsteinPhys. Lett. B1791986361A.J.BurasJ.-M.GerardPhys. Lett. B1921987156S.R.SharpePhys. Lett. B1941987551H.-Y.ChengPhys. Lett. B2011988155M.LusignoliNucl. Phys. B325198933[17]S.BertoliniJ.O.EegM.FabbrichesiRev. Mod. Phys.72200065[18]G.EckerPhys. Lett. B477200088[19]V.CiriglianoEur. Phys. J. C332004369[20]T.HambyeS.PerisE.de RafaelJHEP03052003027[21]J.F.DonoghueE.GolowichPhys. Lett. B4782000172