Direct CP violation in τ ± → K ± ρ 0 (ω) ν τ → K ± π + π − ν τ
aa r X i v : . [ h e p - ph ] A ug Direct CP violation in τ ± → K ± ρ ( ω ) ν τ → K ± π + π − ν τ Chao Wang ∗ , Xin-Heng Guo † , Ying Liu ‡ , Rui-Cheng Li § College of Nuclear Science and Technology,Beijing Normal University, Beijing 100875, China
We study the direct CP violation in the τ ± → K ± ρ ( ω ) ν τ → K ± π + π − ν τ decay process inthe Standard Model. An interesting mechanism involving the charge symmetry violating mixingbetween ρ and ω is applied to enlarge the CP asymmetry. With this mechanism, the maximumdifferential and localized integrated CP asymmetries can reach − (5 . +2 . − . ) × − and 6 . +2 . − . × − , respectively, which still leave plenty room for CP-violating New Physics to be discoveredthrough this process. PACS numbers: 11.30.Er, 13.30.Eg, 13.35.-r, 12.39.-x
I. INTRODUCTION
CP violation was first observed in the neutral kaon system fifty years ago [1]. The asymmetry of CP in the K mesonsystem can be explained by a weak complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the StandardModel (SM) [2, 3]. However, the fundamental origin of CP violation is still an open problem and it is not clear if theCKM mechanism is the only source for CP violation. New Physics (NP) may exist [4–6] and cause CP violation. Toverify the origin of CP violation and look for NP, one needs to collect more information about CP violation in as manyprocesses as possible. One such possible process is the τ decay. τ is the only lepton which is heavy enough to decayinto hadrons and the pure leptonic and semileptonic character of τ decays provides a clean laboratory to test thestructure of the weak currents and the universality of their couplings to the gauge bosons [5]. More importantly, withthe establishment of the high-luminosity Super τ -Charm factories, more τ leptons will be produced and its propertieswill be measured to a very high precision [4]. After the CLEO-c experiment ceased data collection in March 2008, theBESIII experiment began to collect data, and the luminosity reached 10 cm − s − in 2013 [7]. Future high luminositySuper τ -Charm factories are also being considered in Russia and Italy and may reach the luminosity of 10 cm − s − [8–11]. Moreover, Super B -Factories (with the luminosity of 10 cm − s − ) will produce about 10 τ pairs per yearat the Υ(4S) peak [12, 13]. The large statistics collected have considerably improved the statistical accuracy of the τ measurements and brought a new level of systematic understanding, allowing us to make sensible tests of the τ properties, provide more information about CP asymmetries in τ decay processes and seek for the fundamental originof CP violation.Experimental searches for CP violating asymmetries in τ lepton semileptonic decays have been carried out. The ∗ Email: [email protected] † Corresponding author, Email: [email protected] ‡ Email: [email protected] § Email: [email protected] missing evidence for a nonzero CP asymmetry was interpreted in terms of a coupling Λ in the decay τ ± → π ± π ν τ [14]. Recently, the τ ± → K S π ± ν τ rate asymmetry was measured to be of order O (10 − ) by Belle [15] and BaBar[16]. In order to improve our understanding of CP violation in τ decays, more efforts should be put on the theoreticalside and it is important to study the possibility of finding CP signals in τ decays. In the framework of the SM, thedirect CP asymmetries come about due to a relative weak (CP-odd) and a relative strong (CP-even) phase. Thismechanism is forbidden in τ decays in the leading order of the Fermi coupling constant G F [17]. Explicit studies ofthe decay modes τ ± → K ± π + π − ν τ [18, 19], τ ± → π ± K + K − ν τ [18], τ ± → (3 π ) ± ν τ [20, 21] and τ ± → (4 π ) ± ν τ [20]show that sizeable CP-violating effects could be generated in some models of CP violation involving several Higgsdoublets or left-right symmetry. In order to be sure that any eventual observation of CP violation in τ decays has itsorigin beyond the SM, it is essential to study the magnitude of CP violation within the SM.Usually, vanishingly small CP violation in τ decays is predicted in the SM. For example, the CP violation in the τ ± → K ± π ν τ mode is estimated to be of order O (10 − ) when one takes higher order electroweak corrections intoaccount [22]. Note that for the decay τ ± → K S π ± ν τ , the SM predicts a CP violating asymmetry of 3 . × − dueto the K − ¯ K mixing amplitude [23]. In order to obtain a larger CP asymmetry in the SM, one needs to appeal tosome phenomenological mechanisms. The charge symmetry violating mixing between ρ and ω ( ρ - ω mixing) has beenapplied in hadron decays for this purpose in the past few years. ρ - ω mixing has the dual advantages that the strongphase difference is large and well known [24, 25]. From a series of studies on CP violation, it has already been foundthat this mechanism can provide a very large strong phase difference (usually 90 degrees) when the mass of the decayproduct of ρ ( ω ) , π + π − , is in the vicinity of the ω resonance in some decay channels of heavy hadrons including B ,Λ b , and D [24–28]. We will apply this mechanism to the τ lepton decay in the present paper.We will consider the decay process τ ± → K ± π + π − ν τ . The CP violation of this process was analyzed theoreticallywith NP effects in the past [18, 19]. Now, we investigate the CP violation in this decay mode in the framework ofthe SM. The interference between the leading order diagram in G F [Fig. 1(a)] and the second order weak diagrams[Fig. 1(b) and (c)] generates a small CP violation phase [22]. ρ - ω mixing has been applied for getting a large strongphase when the invariant mass of the π + π − pair is near the ω resonance. Hence one can expect that there could bea bigger CP violating asymmetry in the τ ± → K ± ρ ( ω ) ν τ → K ± π + π − ν τ process. Actually, it will be shown fromour explicit calculations that ρ - ω mixing does enlarge the differential CP violating asymmetry by a maximum of fourorders of magnitude and the localized integrated CP asymmetry by a maximum of three orders of magnitude. Eventhough, there is still a large window for studying effects of nonstandard sources of CP violation in experiments.The remainder of this paper is organized as follows. In Sec. II, we first present the formalism for the CP asymmetryin τ ± → K ± ρ ( ω ) ν τ → K ± π + π − ν τ via ρ - ω mixing. Then we give the derivation details of the leading order and thesecond order weak process matrix elements and apply ρ - ω mixing to generate a large CP asymmetry. In Sec. III, withthe expression of meson wave functions and form factors and several parameters we calculate numerical results of thedifferential and localized integrated CP asymmetries. Our conclusion is included in Sec. IV. II. CP VIOLATION IN τ ± → K ± ρ ( ω ) ν τ → K ± π + π − ν τ A decay process described by some amplitudes may have CP-even and -odd relative phases. Within the SM, theCP-odd relative phase is always a weak phase difference which is directly determined by the CKM matrix. On the τ − ν τ W − ρ K − τ − ν τ W − ρ K − ( a ) τ − ν τ W − K − ρ ( c )( b ) W − W − ¯ us ¯ ds ¯ u i d ¯ u ¯ ud i s ¯ uuu ¯ u π + π − π + π − π + π − FIG. 1. The leading order [(a)] and higher order diagrams [(b) and (c)] in G F contributing to the decay τ − → K − ρ ν τ → K − π + π − ν τ . Gluons in (a) are soft ones representing nonperturbative QCD interaction. u i = u, c, t in (b) and d i = d, s, b in (c).contrary, the CP-even relative phase is usually a strong phase difference due to some complicated phenomenologicalmechanism. Letting M and ¯ M be the amplitudes for τ − → K − ρ ( ω ) ν τ → K − π + π − ν τ and its CP conjugate one,respectively, we define the two amplitudes as follows: M = g r e i φ + g r e i φ , (1)¯ M = g ∗ r e i φ + g ∗ r e i φ , (2)where g and g represent CP-odd complex terms which involve coupling constants and CKM matrix elements, r e i φ and r e i φ terms are even under the CP transformation. Then, one has | M | − | ¯ M | = 4 r r Im ( g ∗ g ) sin( φ − φ )= 4 r r | g || g | sin[ Arg( g /g )] sin( φ − φ ) , (3)from which, we can see explicitly that both the CP-odd phase difference Arg( g /g ) and the CP-even phase difference φ − φ are needed to produce CP violation. It will be shown below that the CP-odd phase difference arises fromthe second order weak processes and the CP-even phase difference is determined by the decay widths of intermediateresonances and ρ - ω mixing in the τ − → K ± ρ ( ω ) ν τ → K ± π + π − ν τ decay mode. A. General formalism for CP asymmetry
The hadronic τ decay amplitude can be factorized into a purely leptonic part including τ lepton and neutrino anda hadronic part, where the hadronic system is created from the vacuum via the charged weak current. Thus, theamplitude of τ − decaying into the K − π + π − ν τ final state through K − ρ ν τ with the invariant mass of the π + π − pairnear the ρ resonance can be written in the following general form: M ρ = G F √ g ρππ s ρ L µ H ρµ , (4)where g ρππ is the effective coupling for ρ → ππ , H ρµ is the hadronic matrix element creating ρ K − , L µ is the leptontransition matrix element which can be written as ¯ u ν τ γ µ (1 − γ ) u τ with u ν τ and u τ being the Dirac spinors of ν τ and τ , respectively, and s ρ is the propagator of the ρ meson, s ρ = 1 s − m ρ + i m ρ Γ ρ , (5)where √ s is the invariant mass of the π + π − pair, and m ρ and Γ ρ are the mass and width of the ρ meson, respectively.It should be noted that we assume that the ρ meson is on-shell since the invariant mass of the π + π − pair is near themass of the ρ meson.Because of the absence of the CP-odd phase, the CP asymmetry is zero in the leading order in G F in the SM in the τ decay. In order to have a nonzero CP violating asymmetry, the second order weak terms corresponding to Fig. 1(b)and (c) (with u i = u, c, t and d i = d, s, b ), which provide a CP-odd phase difference, should be taken into account[22]. The leading order amplitude is denoted by M ρ corresponding to Fig. 1(a) and the second order weak terms aredenoted by M ρ and M ρ corresponding to Fig. 1(b) and (c), respectively.As mentioned before, in order to obtain a large CP violation, we intend to apply the ρ - ω mixing mechanism, whichleads 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 − pair is near the ω resonance mass: M = M ρ + M ρ − ω , (6)with M ρ − ω = G F √ g ρππ s ρ L µ H ωµ s ω ˜Π ρω , (7)where ˜Π ρω is the effective ρ - ω mixing amplitude, s ω is the propagator of the ω meson, and H ωµ includes three ωK annihilation terms H ωµ , H ωµ and H ωµ corresponding to Fig. 1(a), (b) and (c), but through the ω intermediateresonance, respectively. We also assume that the ω meson is on-shell. It should be noted that the ρ → ω → π + π − process has been neglected since it is of the second order of isospin violation. The direct coupling ω → π + π − hasbeen effectively absorbed into ˜Π ρω [29]. This leads to the explicit s dependence of ˜Π ρω . Making the expansion˜Π ρω ( s ) = ˜Π ρω ( m ω ) + ( s − m ω ) ˜Π ′ ρω ( m ω ), the ρ - ω mixing parameters were fitted by Gardner and O’Connell [30]: Re ˜Π ρω ( m ω ) = − ± M eV , Im ˜Π ρω ( m ω ) = − ± M eV , (8)˜Π ′ ρω ( m ω ) = 0 . ± . . We define M = M ρ + M ρ − ω , M = M ρ + M ρ − ω and M = M ρ + M ρ − ω , where M ρ − ω , M ρ − ω and M ρ − ω correspondto H ωµ , H ωµ and H ωµ , respectively. The CP violation can arise from the interference between M and M , M . Itshould be noted that M and M are the second order in G F . Therefore, to the G F order, the square of the total τ − ν τ W − K − V τ − ν τ K − VW − = + V − i τ − ν τ V K − W − A − i + τ − ν τ W − P − i K − V ( a ) ( b ) ( c ) ( d ) FIG. 2. The Feynman diagrams with the intermediate virtual mesons that connect the weak current and the strongvertex in the decay τ − → V K − ν τ [ V is ρ (or ω )]. (a) represents the total effective strong vertex; (b), (c) and (d)correspond to the V i (vector), A i (axial-vector) and P i (pseudoscalar) intermediate meson processes, respectively.amplitude M = M + M + M can be written as | M | = ( M + M + M ) † ( M + M + M ) = M † M + ( M † M + M M † ) + ( M † M + M M † ) . (9)Then, the differential CP asymmetry which is defined as A CP = | M | − ¯ | M | | M | + ¯ | M | , (10)can be written as the following to the order G F : A CP = | M | − ¯ | M | | M | + ¯ | M | , (11)where the M † M + M M † and M † M + M M † terms are negligible in the denominator since they do not contributeto the second order in G F . When we take ρ - ω mixing into account, the three terms in Eq. (9) can be rewritten in thefollowing forms: M M † = ( G F / √ g ρππ s ρ L µν (cid:0) H ρ ρµν + H ρ ωµν ˜Π ∗ ρω s ∗ ω + H ω ρµν ˜Π ρω s ω + H ω ωµν ˜Π ρω s ω (cid:1) ,M M † = ( G F / √ g ρππ s ρ L µν (cid:0) H ρ ρµν + H ρ ωµν ˜Π ∗ ρω s ∗ ω + H ω ρµν ˜Π ρω s ω + H ω ωµν ˜Π ρω s ω (cid:1) ,M M † = ( G F / √ g ρππ s ρ L µν (cid:0) H ρ ρµν + H ρ ωµν ˜Π ∗ ρω s ∗ ω + H ω ρµν ˜Π ρω s ω + H ω ωµν ˜Π ρω s ω (cid:1) , (12)where L µν = L µ ( L ν ) † and for example, H ρ ρµν = H ρµ ( H ρν ) † . B. Derivation details of matrix elements
The transition from the vacuum to the pseudoscalar meson K − and the vector one ρ ( ω ) occurs via weak vectorand axial-vector current. Based on Lorentz invariance and parity and time-reversal invariance, one can decomposethe hadronic matrix element in terms of four form factors in the leading order in G F [31]: H ρ ( ω ) µ = − i V ∗ us < ρ ( ω ) K − | sγ µ (1 − γ ) u | > = V ∗ us (cid:2) − gε µναβ ǫ ∗ ν p α p β − i f ǫ ∗ µ − i( a p µ + a p µ )( ǫ ∗ · Q ) (cid:3) , (13)where V us is the CKM matrix element, ¯ s and u are quark field operators, p and p are momenta of ρ (or ω ) and K − ,respectively, Q = p + p is the momentum transfer to the hadronic system, g is the vector current form factor, f , a and a are axial-vector current form factors, and ǫ µ denotes the polarization vector of ρ (or ω ) which satisfies p · ǫ = 0and P λ =0 , ± ǫ ∗ µ ( q, λ ) ǫ ν ( q, λ ) = − g µν + q µ q ν /m V , where λ = ± , m V is the mass of the vector meson V ( V = ρ or ω ). The form factors are functions of Q only.They are difficult to be related directly to experimental measurements but can be dealt with in phenomenologicalmodels. We will calculate the form factors with the meson dominance model [31]. The pseudoscalar and vectormeson annihilation process in the leading order in G F is generated by the strong interaction. In the meson dominancemodel it is assumed that intermediate mesons connect the weak current and the strong vertex shown in the Feynmandiagrams in Fig. 2. Using the Feynman rules for these diagrams, the following expressions for the form factors areobtained [31]: f = − ( Q + m V − m K ) X i h A i t A i V K D A i ( Q ) , g = X i h V i t V i V K D V i ( Q ) ,a = X i h P i t P i V K D P i ( Q ) + X i h A i t A i V K D A i ( Q ) , a = X i h P i t P i V K D P i ( Q ) + X i h A i t A i V K D A i ( Q ) , (14)where V i , A i and P i denote vector, axial-vector and pseudoscalar intermediate meson resonances, respectively, h M i ( M i = A i , V i or P i ) denotes the weak coupling of the M i intermediate meson, t M i V K is its strong coupling to the
V K final state, m K is the mass of the K meson, and D M i ≡ Q − m M i + im M i Γ M i where m M i (Γ M i ) is the mass(width) of the corresponding intermediate meson. The details about the intermediate mesons and their weak couplingsand strong vertex coupling constants will be given in Section III. From Eqs. (12) and (13) it can be found that inthe leading order in G F the CP-odd phase is absent, and the CP-even phase is determined by the decay widths ofintermediate resonances when ρ - ω mixing is not considered.Next, we proceed to evaluate M and M based on the perturbation method. Note that M M † + M † M in Eq. (9)is proportional to | V ud i | | V us | and will not contribute to CP violation. Hence we only have to consider M . In theframework of perturbation method, it can be evaluated in a similar way to B decays [32]. Since the τ mass is muchsmaller than the W -boson mass M W , the momenta of all the particles involved in the τ decay are much smaller than M W . As a result, we can approximate the denominator of the W -boson propagator ( p + p ) − M W by − M W in thenumerator of the W -boson propagator. The wave functions including spin factors of pseudoscalar and vector mesonsare taken as [33] Ψ V ( x, p ) = − I √ √ m V φ V ( x )( m V + p/ ) ǫ/, (15)Ψ P ( x, p ) = − I √ √ m P φ P ( x )( m P + p/ ) γ , (16)where I = 3 is an identity in color space, m P and m V are the masses of the pseudoscalar and vector mesons,respectively, p represents the momentum of the meson P or V , x is the longitudinal momentum fraction of theconstituent quark, and the non-perturbution effects are included in the distribution amplitudes φ V ( x ) and φ P ( x ),which satisfy R φ V ( P ) ( x )d x = f V ( P ) / (2 √ f V ( P ) is the decay constant of V ( P ). According to the Feynmandiagram (b) in Fig. 1, the hadronic matrix elements H ρ (or H ω ) can be expressed as H ρ ( ω ) µ = +( − ) G F √ π ) I X i V ∗ u i s V u i d V ∗ ud q m u m d m s Z d x d yφ ∗ ρ ( ω ) ( x ) φ ∗ K ( y ) · √ m ρ ( ω ) ( m ρ ( ω ) − p/ ) ǫ/ ∗ γ α (1 − γ ) 12 √ m K ( m K − p/ ) γ γ µ (1 − γ ) I u i γ α (1 − γ ) , (17)where V u i s and V u i d are the CKM matrix elements, +( − ) corresponds to ρ ( ω ) and we define I u i = i( p/ u i + m u i ) / ( p u i − m u i ) with m u i and p u i being the current quark mass and the momentum of the intermediate quark u i , respectively.We will neglect the difference between the masses of ρ and ω mesons in the following, i.e., we take m ρ = m ω .Using the unitarity of the CKM matrix, we have X i V ∗ u i s V u i d I u i = V ∗ ud V us ( I u − I c ) + V ∗ td V ts ( I t − I c ) ≈ V ∗ ud V us ( I u − I c ) − V ∗ td V ts I c (18)where the last line is obtained using the fact that m t is much larger than masses of other quarks involved in thisprocess. We note that only V ∗ td V ts provides a weak CP-violation phase, so it is unnecessary to consider the contributionof the first term. As a consequence, the CP asymmetry only depends on V ∗ td V ts I c . We define A ≡ Z d x d yφ ∗ ρ ( ω ) ( y ) φ ∗ K ( x ) 1 xQ + (1 − x ) m ρ + ( x − x ) m K − m c , = Z d xφ ∗ K ( x ) f ρ √ xQ + (1 − x ) m ρ + ( x − x ) m K − m c ] , (19) B ≡ Z d x d yφ ∗ ρ ( ω ) ( y ) φ ∗ K ( x ) xxQ + (1 − x ) m ρ + ( x − x ) m K − m c = Z d xφ ∗ K ( x ) xf ρ √ xQ + (1 − x ) m ρ + ( x − x ) m K − m c ] . (20)Inserting Eqs. (18), (19) and (20) into Eq. (17) and only considering the CP asymmetry term, H ρ ( ω ) µ can be simplifiedas H ρ ( ω ) µ = 6 √ π ) p m u m d m s G F V ∗ ts V td V ∗ ud m k · n − A ε µναβ ǫ ∗ ν p α p β − i ǫ ∗ µ h A ( Q − m ρ − m K ) + B m K i + i A p µ ( Q · ǫ ∗ ) + i2 B p µ ( Q · ǫ ∗ ) o . (21)We can see that the weak phase appears but the strong phase is absent in this amplitude if ρ - ω mixing is not included.Now we take ρ - ω mixing into account and show how ρ - ω mixing enlarges the CP violation. In the meson dominancemodel, the form factors of the annihilation process are dominated by strong interaction. So, we adopt the same formfactors in the Kρ and Kω annihilation processes. According to Eq. (13), we have H ρµ = H ωµ . H ρ ( ω ) µ is dependenton the hadronic wave functions. Since the wave functions of mesons are determined by strong interaction, whichpreserves isospin, we assume that the ρ and ω mesons have the same hadronic wave functions. Therefore, fromEq. (17), we have H ρµ = − H ωµ . Then, the first two equations of Eq. (12) can be written as M M † = ( G F / √ g ρππ s ρ L µν H ρ ρµν (cid:0) ∗ ρω s ∗ ω + ˜Π ρω s ω + ˜Π ρω s ω (cid:1) M M † = ( G F / √ g ρππ s ρ L µν H ρ ρµν (cid:0) − ˜Π ∗ ρω s ∗ ω + ˜Π ρω s ω − ˜Π ρω s ω (cid:1) . (22)We can see explicitly that ρ - ω mixing provides additional complex terms to A CP . As will be shown later, thesecomplex terms enlarge the CP-even phase, which leads to a bigger CP asymmetry.Finally, we will calculate L µν H ρ ρµν and L µν H ρ ρµν . For simplicity, we will consider the unpolarized τ decay process.The unpolarized leptonic scattering tensor is L µν = X λ , λ tr (cid:2) ¯ u ν τ ( p , λ ) γ µ (1 − γ ) u τ ( p , λ )¯ u τ γ ν ( p , λ )(1 − γ ) u ν τ ( p , λ ) (cid:3) = 4 (cid:2) − g µν ( p · p ) + p µ · p ν + p ν · p µ + i ε µναβ p α p β (cid:3) , (23)where p and p represent the momenta of ν τ and τ , respectively, and λ and λ represent the helicities of ν τ and τ ,respectively. We also sum over the spins of hadrons. Then, from Eqs. (13) and (23), one has L µν H ρ ρµν = 4 | V us | (cid:2) ( − x − m ρ x − m K x )( p · p ) + 2 x ( p · p )( p · p ) + 2 x ( p · p )( p · p ) − ( x + + x − )( p · p )( p · p ) + ( x + + x − + 2 gf ∗ + 2 g ∗ f )( p · p )( p · p )+ ( x + + x − − gf ∗ − g ∗ f )( p · p )( p · p ) (cid:3) , (24)where x = − g ( Q + m ρ + m K − m ρ Q − m K Q − m ρ m K ) − f ,x = g m K + a h − Q + ( p · Q ) m ρ i + f m ρ + p · p m ρ f a ∗ + p · p m ρ f ∗ a ,x = g m ρ + a h − Q + ( p · Q ) m ρ i − ( a f ∗ + a ∗ f ) ,x + = − g ( Q − m K − m ρ ) + a a ∗ h − Q + ( p · Q ) m ρ i − a f ∗ + p · p m ρ f a ∗ ,x − = − g ( Q − m K − m ρ ) + a ∗ a h − Q + ( p · Q ) m ρ i − a ∗ f + p · p m ρ f ∗ a . (25)From Eqs. (21) and (23), one has L µν H ρ ρµν = 6 √ π ) p m u m d m s G F V ts V ∗ td V ud V ∗ us m K · (cid:2) ( − x ′ − x ′ m ρ − x ′ m K )( p · p ) + 2 x ′ ( p · p )( p · p ) + 2 x ′ ( p · p )( p · p )+ ( x ′ + + x ′− + 2 f B + 2 gλ )( p · p )( p · p ) + ( x ′ + + x ′− − f ∗ B − g ∗ λ )( p · p )( p · p ) − ( x ′ + + x ′− )( p · p )( p · p ) (cid:3) , (26)where x ′ = − gA ( Q + m ρ + m K − m ρ Q − m K Q − m ρ m K ) − f λ,x ′ = − gA m K + gB ( Q − m ρ − m K ) − a A h − Q + ( p · Q ) m ρ i − A f p · p m ρ + a λ p · p m ρ ,x ′ = − gA m ρ − a B h − Q + ( p · Q ) m ρ i + 2 B f + a λ,x ′ + = gA ( Q − m K − m ρ ) − gA m K − B f p · p m ρ − a λ,x ′− = gA ( Q − m K − m ρ ) − a A h − Q + ( p · Q ) m ρ i − A f + a λ p · p m ρ ,λ = A ( Q − m K − m ρ ) + B m K . C. Hadronic rest-frame
In the previous subsections, we have given the the general expression of CP asymmetry and derivations of matrixelements. For simplicity, we choose to work in a special reference frame and express the products among vectors p , p , p , p , Q and ǫ in terms of the square of momentum transfer Q , the invariant mass of the π + π − pair √ s , and adistribution angle θ in this subsection. We note that it is convenient to express the momenta of hadrons and leptonsand calculate various components of the matrix elements in the hadronic rest-frame [34]. This frame is defined inFig. 3. The z axis is chosen to be in the direction of motion of the ρ (or ω ) meson. The three-momentum of K ischosen to be p = − p . The ( x, z ) plane is aligned with the ρ and ν τ movement plane, with n ⊥ = ( p × p ) / | p × p | (the normal to the ρ and ν τ movement plane) pointing along the y axis. The distribution angle θ is the one betweenthe motion direction of ρ (or ω ) and the neutrino. Then, the momenta of hadrons and leptons in this hadronic restframe are given as follows: p µ = ( E , , , P ) ,p µ = ( E , , , − P ) ,p µ = ( K, K sin θ, , K cos θ ) ,p µ = ( E , K sin θ, , K cos θ ) ,Q µ = ( E + E , , , E − K, , , , (27)0 x yz ρ K ± τ ± ν τ θ FIG. 3. The hadronic rest-frame. The z axis is chosen to be in the direction of the motion of the ρ (or ω ) meson.The three-momentum of K is chosen to be p = − p . The ( x, z ) plane is aligned with the ρ and ν τ movementplane, with n ⊥ = ( p × p ) / | p × p | (the normal to the ρ and ν τ movement plane) pointing along the y axis.The distribution angle θ is the one between the motion direction of ρ (or ω ) and the neutrino.and the polarization vectors of ρ (or ω ) in this hadronic rest frame are ǫ λ = ± = (0 , , ± i , ,ǫ λ =0 = 1 √ m ρ ( P, , , E ) , (28)with E = Q + m K − m ρ p Q , E = Q − m K − m ρ p Q ,P = q m K + m ρ − m K Q − m K m ρ − Q m ρ p Q ,K = m τ − Q p Q , E = m τ + Q p Q . (29)The above expressions for various hadron and lepton momentum vectors allow us to determine simple expressionsfor matrix elements which involve products including p · p , p · p , p · p , p · p , p · p , p · p , p · Q , p · Q , p · Q , p · Q , and Q · ǫ in the term of Q , √ s and θ . We will integrate over the angle θ since we will not consider the angledistribution. Furthermore, by integrating A CP in the region Ω in which Q and s vary in some areas, we obtain the1localized integrated CP asymmetry which takes the following form: A Ω CP = R Ω d Q d s ( | M | − ¯ | M | ) R Ω d Q d s ( | M | + ¯ | M | ) . (30) III. NUMERICAL RESULTS
From the above discussions, the CP violating asymmetries depend on the values of Q and s . In this section wegive the explicit expressions of meson wave functions and form factors, and values of several parameters in order tocalculate the CP violating asymmetries. We find that significant cancellation occurs as one performs the integrationover Q . To show more details about this cancellation, we calculate both the differential and the integrated CPasymmetries. We also compare CP asymmetries with and without ρ - ω mixing. A. Models for form factors and meson wave functions
The hadronic τ decay is dominated by the meson annihilation diagram, Fig. 1(a). As mentioned before, thevector and pseudoscalar meson annihilation form factors in this decay mode are difficult to be related directly toexperimental measurements. One therefore needs to adopt phenomenological models. Following Ref. [31] we use themeson dominance model in our calculation. In this model it is assumed that the vector form factor g is dominatedby the K ∗ (892) and K ∗ (1410) vector mesons and f and a ± are dominated by the exchange of the K − pseudoscalarmeson and the K (1270) and K (1400) axial-vector mesons [31]. The expressions for the form factors are given inEq. (14). In Ref. [31], the values of weak couplings and strong vertex couplings were extracted from experiments andfixed by the SU(3) flavor symmetry. We display these values in Table I.TABLE I. The values of h M i , t M i V K , m M i and Γ M i in the numerical calculations.Pseudoscalar Axial Vector VectorIntermediate mesons K − K (1270) K (1400) K ∗ (892) K ∗ (1680) h M i (10 MeV ) 0.159 ± − ±
25 170 ±
130 188 ± ± t M i V K (10 − MeV − ) − ±
30 MeV − . ± ± ± − . ± m M i (MeV) 494 ± ± ± ± ± M i (MeV) 0 90 ±
20 174 ±
13 50.8 ± ± K meson wave function of the Brodsky-Huang-Lepage prescription which have the following form [35]:Φ K ( x, k ⊥ ) = A K (1 − x ) exp h − b K (cid:16) k ⊥ + m ′ s x + k ⊥ + m ′ u − x (cid:17)i , (31)where k ⊥ is the transverse momentum of the constituents of K , m ′ u and m ′ s are the constituent quark masses of u and s , respectively. Integrating Φ K ( x, k ⊥ ) over k ⊥ one has the following distribution amplitude: φ K ( x ) = A K π b K x (1 − x )(1 − x ) exp h − b K (cid:16) m ′ s x + m ′ u − x (cid:17)i . (32)2In the following numerical calculations we use the parameters A K = 232 GeV − , b K = 0 .
61 GeV − , m ′ u = 350 MeV, m ′ s = 550 MeV and f ρ = 221 MeV [35]. B. Numerical results for the CP asymmetries
We are now ready to evaluate numerical results of CP asymmetries. We take the meson masses m ρ = 770 MeVand m K = 493 MeV, the lepton mass m τ = 1776 MeV, the current quark masses m u = 2 . m d = 4 . m s = 95 MeV and m c = 1275 MeV [36]. The CKM matrix, which elements are determined from experiments, can beexpressed in terms of the Wolfenstein parameters A , ρ , λ and η [36]: − λ λ Aλ ( ρ − i η ) − λ − λ Aλ Aλ (1 − ρ − i η ) − Aλ , (33)where O ( λ ) corrections are neglected. The latest values for the parameters in the CKM matrix are [36]: λ = 0 . ± . , A = 0 . +0 . − . , ¯ ρ = 0 . +0 . − . , ¯ η = 0 . +0 . − . , (34)with ¯ ρ = ρ (1 − λ ) , ¯ η = η (1 − η ) . (35) −12 √ s (GeV) p Q (GeV) A C P FIG. 4. The differential CP asymmetry as a function of √ s and p Q . The numerical results correspond to centralvalues of the parameters involved in the calculation.In our numerical calculations, the most uncertain factors come from the CKM matrix elements and the form factorsin the leading order in G F . In fact, the uncertainties due to the CKM matrix elements are mostly from η since λ is3 −13 √ s (GeV) A s C P (a) 0.76 0.78 0.800.511.522.5 x 10 −13 √ s (GeV) A s C P (b) 0.76 0.78 0.800.511.522.533.54 x 10 −12 √ s (GeV) A s C P (c) 0.76 0.78 0.8012345 x 10 −12 √ s (GeV) A s C P (d)0.76 0.78 0.8−4−3.5−3−2.5−2−1.5−1−0.500.5 x 10 −13 √ s (GeV) A s C P (e) 0.76 0.78 0.8−12−10−8−6−4−202 x 10 −13 √ s (GeV) A s C P (f) 0.76 0.78 0.8−2−1.5−1−0.500.51 x 10 −12 √ s (GeV) A s C P (g) 0.76 0.78 0.8−7−6−5−4−3−2−10 x 10 −13 √ s (GeV) A s C P (h) FIG. 5. The localized integrated CP asymmetry A sCP as a function of √ s . (a) For integrating over Q in p Q =(1.30 GeV, 1.35 GeV): the dash-dotted line corresponds to the CP asymmetry including ρ - ω mixing and thesolid line corresponds to the CP asymmetry without ρ - ω mixing; (b), (c), (d), (e), (f), (g) and (h) correspond to theintegration intervals (1.35 GeV, 1.40 GeV), (1.40 GeV, 1.45 GeV), (1.45 GeV, 1.50 GeV), (1.50 GeV, 1.55 GeV),(1.55 GeV, 1.60 GeV), (1.60 GeV, 1.65 GeV) and (1.65 GeV, 1.70 GeV), respectively. We take central values of theparameters involved in the calculation.well determined and the CP violating asymmetries are independent of ρ . Hence in the following we take the centralvalue of λ , 0.225. In the meson dominance model, the uncertainties arising from form factors are dominated by thoseof the strong and weak coupling constants of the K (1400) meson due to the poor quality of measurements. Thevalues of ρ - ω mixing paraments also bring some uncertainties.In order to find the details about the dependence of the CP violating asymmetries on Q and s , we study thedifferential CP asymmetries. Since CP asymmetries are calculated around the ω (782) resonance region, we take therange of √ s as 760 MeV ≤ √ s ≤
800 MeV. From Eqs. (27) and (29), we obtain ( m ρ + m K ) < Q < m τ . Hence we takethe range of p Q from ( m ρ + m K ) = 1270 MeV to m τ =1770 MeV. The differential CP asymmetry A CP dependingon Q and s is displayed in Fig. 4, where we take central values of the parameters involved in the calculation. Wecan see that A CP varies from around 10 − to around 10 − . The maximum differential CP violating asymmetrycan reach − (5 . +2 . − . ) × − , where the errors come from the uncertainties of the CKM matrix elements, the ρ - ω mixing parameters and the form factors in the leading order in G F . As we expect, there is a peak for the CP violating4parameter A CP when the invariant mass of the π + π − pair is in the vicinity of the ω resonance for a certain p Q . ρ - ω mixing enlarges A CP by four orders of magnitude in some regions. Furthermore, we also find that the sign of A CP changes frequently in some regions of Q . This behaviour can be easily understood if one notes that the denominatorof A (and B ), which is defined in Eq. (19) [and (20)], changes its sign when p Q crosses the pole. This will lead tocancellations when one performs the integration over Q in some regions. These cancellations are found be be quiteobvious around the peak when √ s = 784 MeV. In the experimental study of three-body decays of the B meson, onedivides the Dalitz plot of candidates into bins with equal population using an adaptive binning algorithm and the CPviolating parameter is calculated from the number of B event candidates in each bin [37]. In the following, we willcalculate the localized integrated CP violating parameter in the τ lepton decay, which may be measured in the futureexperiments. We will also compare CP asymmetries with and without ρ - ω mixing in the following.Firstly, we preform the integration over p Q while keeping √ s fixed. We divide the integration region into eightequal intervals: (1.30 GeV, 1.35 GeV), (1.35 GeV, 1.40 GeV), (1.40 GeV, 1.45 GeV), (1.45 GeV, 1.50 GeV), (1.50 GeV,1.55 GeV), (1.55 GeV, 1.60 GeV), (1.60 GeV, 1.65 GeV) and (1.65 GeV, 1.70 GeV). In each interval, we integrate over p Q and calculate the CP asymmetries with and without ρ - ω mixing. The results are denoted by A sCP and shownin Fig. 5. Our numerical results show that the ρ - ω mixing mechanism enlarges the A sCP by about 1-2 order when √ s is around 0 .
784 MeV. All the three NP models in Ref. [19] predict that the s -depending differential CP asymmetriesare to be order of O (10 − ) when √ s varies in the region (0.760 MeV, 0.800 MeV). It can be seen from Fig. 5 that themaximum values of our results including ρ - ω mixing are about nine orders smaller than these predictions. Since the ρ - ω mixing mechanism provides an extra strong phase which enlarges the CP asymmetry, we suggest to measure thelocalized CP asymmetry in the region where the invariant mass of the π + π − pair is around 780 MeV in this decayprocess. TABLE II. The localized integrated asymmetries (in units of 10 − ) with (without) ρ - ω mixing. The central values of the numerical results correspond to central values of theparameters involved in the calculation and the errors are from the uncertainties of theCKM matrix elements, the ρ - ω mixing parameters, and the form factors in the leadingorder in G F . p Q (GeV) A Ω CP p Q (GeV) A Ω CP (1.30, 1.35) 3 . +1 . − . ( − . +0 . − . ) (1.50, 1.55) − . +3 . − . (0 . +0 . − . )(1.35, 1.40) 9 . +3 . − . (0 . +0 . − . ) (1.55, 1.60) − . +1 . − . (0 . +0 . − . )(1.40, 1.45) 63 +24 − (0 . +0 . − . ) (1.60, 1.65) − . +1 . − . ( − +47 − )(1.45, 1.50) 51 +42 − ( − . +0 . − . ) (1.65, 1.70) − . +2 . − . ( − . +0 . − . )Finally, we integrate A CP over both p Q and √ s and obtain the localized integrated asymmetries A Ω CP . Consideringthe significant region of ρ - ω mixing shown in Fig. 5, we choose the integration interval of √ s to be from 0.775 MeVto 0.795 MeV. The numerical results of the localized integrated asymmetries with (without) ρ - ω mixing are shown inTable. ?? , where the central values of the numerical results correspond to central values of the parameters involvedin the calculation and the errors are again from the uncertainties of the CKM matrix elements, the ρ - ω mixingparameters, and the form factors in the leading order in G F . We can see in most of the intervals ρ - ω mixing enlargesthe localized integrated asymmetries. The maximum increase is three orders of magnitude. These predictions lead to5a new upper limit of the CP asymmetries based on the SM in this decay channel.We have calculated the differential and integrated CP asymmetries within the SM in the τ ± → K ± π + π − ν τ decaytaking ρ - ω mixing into account. It is worth noting that even the maximum localized integrated CP asymmetry,6 . +2 . − . × − , is at least six orders smaller than that predicted based on NP in Ref. [19]. If any CP violation biggerthan our predicted values is observed in experiments, one may consider the possibility of NP. IV. CONCLUSION
In the framework of the SM, CP violation in the τ lepton decay process arises from a nontrivial phase in the CKMmatrix and is predicted to be zero in the leading order in G F . However, Delepine pointed out that the CP-odd phasecan arise from the second order weak process in the τ ± → K ± π ν τ decay mode [22]. Since ρ - ω mixing can providevery large CP asymmetries in some decay channels of heavy hadrons, we have tried to enlarge the CP asymmetry inthe τ − → K − ρ ( ω ) ν τ → K − π + π − ν τ decay via this mechanism.We have first studied the differential CP asymmetry depending on p Q and √ s . The numerical results show thatit varies from around 10 − to around 10 − and the maximum CP violating asymmetry can reach − (5 . +2 . − . ) × − .We have found that there is a peak for the CP violating parameter A CP when the invariant mass of the π + π − pair is inthe vicinity of the ω resonance. The advantage of ρ - ω mixing is that it makes the strong phase difference between thehadronic matrix elements of the leading order and the second order in G F larger at the ω resonance. Consequently,the CP violating asymmetry reaches the maximum value when the invariant mass of the π + π − pair in the decayproduct is in the vicinity of the ω resonance. We have also found that A CP changes its sign when p Q varies. Then,we have calculated the localized integrated CP violating parameter in the τ lepton decay, which may be measured inthe future experiments.After integrating over p Q in serval intervals, we have shown that the ρ - ω mixing mechanism enlarges the √ s dependent CP asymmetry by about 1-2 order when √ s is around 0 .