The impact of a_0^0(980)-f_0(980) mixing on the localized CP violations of the B^-\rightarrow K^- π^+π^- decay
Jing-Juan Qi, Zhen-Yang Wang, Chao Wang, Zhen-Hua Zhang, Xin-Heng Guo
aa r X i v : . [ h e p - ph ] F e b The impact of a (980) − f (980) mixing on the localized CP violations of the B − → K − π + π − decay Jing-Juan Qi ∗ Junior College, Zhejiang Wanli University, Zhejiang 315101, China
Zhen-Yang Wang † Physics Department, Ningbo University, Zhejiang 315211, China
Chao Wang ‡ Center for Ecological and Environmental Sciences,Key Laboratory for Space Bioscience and Biotechnology,Northwestern Polytechnical University, Xi’an 710072, China
Zhen-Hua Zhang § School of Nuclear and Technology, University of South China, Hengyang, Hunan 421001, China
Xin-Heng Guo ¶ College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China (Dated: February 5, 2021)
Abstract
In the framework of the QCD factorization approach, we study the localized CP violations of the B − → K − π + π − decay with and without a (980) − f (980) mixing mechanism, respectively, and find that the localized CP violationcan be enhanced by this mixing effect when the mass of the π + π − pair is in the vicinity of the f (980) resonance.The corresponding theoretical prediction results are A CP ( B − → Kf → K − π + π − ) = [0 . , . and A CP ( B − → K − f ( a ) → K − π + π − ) = [0 . , . , respectively. Meanwhile, we also calculate the branching fraction of the B − → K − f (980) → K − π + π − decay, which is consistent with the experimental results. We suggest that a (980) − f (980) mixing mechanism should be considered when studying the CP violation of the B or D mesons decaystheoretically and experimentally. PACS numbers: *************** ∗ e-mail: [email protected] † Corresponding author,e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] ¶ Corresponding author, e-mail: [email protected] . INTRODUCTION CP violation plays an important role for the test of the Standard Model (SM) and extractions of theCabibbo-Kobayashi-Maskawa (CKM) matrix. The processes of nonleptonic decays of B mesons provideus with opportunities for exploring CP violation. In SM, CP violation depends on the weak complexphase in the CKM matrix [1, 2]. The main uncertainties of CP violation come from the insufficientunderstanding of strong interaction associated with the nonperturbative QCD. In the past few years, alarge amount of experimental data have been collected for CP violation of two body decays of the B meson by B factories, BABAR, Belle, and LHC experiments. The large CP violations have been foundby the LHCb Collaboration in the three-body decay channels of B ± → π ± π + π − and B ± → K ± π + π − [3, 4]. Hence, the exploration of the theoretical mechanism for CP violation becomes interesting in thetwo- and three-body decays of the B meson.The nature of the light scalar mesons has attracted much attention for decades since its discovery [5–11].Because of sharing the same quantum numbers, light scalar mesons play an important role to understandthe QCD vacuum. The a (980) − f (980) mixing mechanism has been a hot research topic because of itspotential to help understand the structure of scalar mesons. In late 1970s, the a (980) − f (980) mixingeffect was first suggested theoretically [12]. a (980) and f (980) have the same spin parity quantumnumbers but different isospins. Because of the isospin breaking effect, when they decay into K ¯ K thereexists a difference of 8 MeV between the charged and neutral kaon thresholds. Up to now, a (980) and f (980) mixing has been studied extensively in various processes and with respect to its differentaspects [13–33]. The signal of this effect was observed for the first time by the BESIII Collaboration inthe J/ψ → φf (980) → φa (980) → φηπ and χ c → a (980) π → f (980) π → π + π − π decays [34].Inspired by the fact that ρ − ω mixing (also due to isospin breaking effect) can induce large CP violationswhen the invariant mass of the ππ pair is in the ρ − ω mixing effective area [35–37], we intend to studythe a (980) − f (980) mixing effect on the localized CP violations in three-body decays of the B meson.In this paper, we will investigate the localized CP violation by a (980) − f (980) mixing and thebranching fraction of the B − → Kf → K − π + π − decay in the QCDF approach. The remainder of thispaper is organized as follows. In Sect. II , we present the formalism for B decays in the QCDF approach.In Sect. III , we present the a (980) − f (980) mixing mechanism, calculations of the localized CP violationand the branching fraction of the B − → Kf → K − π + π − decay. The numerical results are given in Sect. IV and we summarize and discuss our work in Sect V .2 I. B DECAYS IN THE QCD FACTORIZATION APPROACH
In the framework of the QCD factorization approach [38, 39], one can obtain the matrix element B decaying to two mesons M and M by matching the effective weak Hamiltonian onto a transition operator,which is summarized as follow ( λ ( D ) p = V pb V ∗ pD with D = d or s ) h M M |H eff | B i = X p = u,c λ ( D ) p h M M |T pA + T pB | B i , (1)where T pA and T pB describe the contributions from non-annihilation and annihilation topology amplitudes,respectively, which can be expressed in terms of the parameters a pi and b pi , respectively, both of which aredefined in detail in Ref. [38].Concretely, T pA contains the contributions from naive factorization, vertex correction, penguin ampli-tude and spectator scattering and can be expressed as T pA = δ pu α ( M M ) A ([¯ q s u ][¯ uD ]) + δ pu α ( M M ) A ([¯ q s D ][¯ uu ])+ α p ( M M ) X q A ([¯ q s D ][¯ qq ]) + α p ( M M ) X q A ([¯ q s q ][¯ qD ])+ α p ,EW ( M M ) X q e q A ([¯ q s D ][¯ qq ]) + α p ,EW ( M M ) X q e q A ([¯ q s q ][¯ qD ]) , (2)where the sums extend over q = u, d, s , and ¯ q s (= ¯ u, ¯ d or ¯ s ) denotes the spectator antiquark. The coeffi-cients α pi ( M M ) and α pi,EW ( M M ) contain all dynamical information and can be expressed in terms ofthe coefficients a pi .As for the power-suppressed annihilation part, we can parameterize it into the following form: T pB = δ pu b ( M M ) X q ′ B ([¯ uq ′ ][¯ q ′ u ][ ¯ Db ]) + δ pu b ( M M ) X q ′ B ([¯ uq ′ ][¯ q ′ D ][¯ ub ])+ b p ( M M ) X q,q ′ B ([¯ qq ′ ][¯ q ′ D ][¯ qb ]) + b p ( M M ) X q,q ′ B ([¯ qq ′ ][¯ q ′ q ][ ¯ Db ]+ b p ,EW ( M M ) X q,q ′ e q B ([¯ qq ′ ][¯ q ′ D ][¯ qb ]) + b p ,EW ( M M ) X q,q ′ e q B ([¯ qq ′ ][¯ q ′ q ][ ¯ Db ]) , (3)where q, q ′ = u, d, s and the sums extend over q, q ′ . The sum over q ′ arises because a quark-antiquark pairmust be created via g → ¯ q ′ q ′ after the spectator quark is annihilated. III. a (980) − f (980) MIXING MECHANISM, CALCULATION OF CP VIOLATION ANDBRANCHING FRACTIONA. a (980) − f (980) mixing mechanism In the condition of turning on the a (980) − f (980) mixing mechanism, we can get the propagatormatrix of a (980) and f (980) by summing up all the contributions of a (980) → f (980) → · · · → a (980) f (980) → a (980) → · · · → f (980) , respectively, which are expressed as [33] P a ( s ) P a f ( s ) P f a ( s ) P f ( s ) = 1 D f ( s ) D a ( s ) − | Λ( s ) | D a ( s ) Λ( s )Λ( s ) D f ( s ) , (4)where P a ( s ) and P f ( s ) are the propagators of a and f , respectively, P a f ( s ) , P f a ( s ) and Λ( s ) arisedue to the a (980) − f (980) mixing effect, and D a ( s ) and D f ( s ) are the denominators for the propagatorsof a and f when the a (980) − f (980) mixing effect is absent, respectively, which can be expressed asfollows in the Flatt ´e parametrization: D a ( s ) = m a − s − i √ s [Γ a ηπ ( s ) + Γ a K ¯ K ( s )] ,D f ( s ) = m f − s − i √ s [Γ f ππ ( s ) + Γ f K ¯ K ( s )] , (5)where m a and m f are the masses of the a and f mesons, with the decay width Γ abc can being presentedas Γ abc ( s ) = g abc π √ s ρ bc ( s ) with ρ bc ( s ) = r [1 − ( m b − m c ) s ][1 + ( m b − m c ) s ] . (6)It was pointed out that the contribution from the amplitude of a (980) − f (980) mixing is conver-gent and can be written as an expansion in the K ¯ K phase space when only K ¯ K loop contributions areconsidered [12, 40], Λ( s ) K ¯ K = g a K + K − g f K + K − π (cid:26) i (cid:20) ρ K + K − ( s ) − ρ K ¯ K ( s ) (cid:21) − O ( ρ K + K − ( s ) − ρ K ¯ K ( s )) (cid:27) , (7)where g a K + K − and g f K + K − are the effective coupling constants. Since the mixing mainly comes fromthe K ¯ K loops, we can adopt Λ( s ) ≈ Λ K ¯ K ( s ) . B. Decay amplitudes, localizd CP violation and branching fraction
With the a (980) − f (980) mixing being considered, the process of the B − → K − π + π − decay is shownin Fig. 1 and the amplitude can be expressed as M = h K − π + π − |H T | B − i + h K − π + π − |H P | B − i , (8)in which H T and H P are the tree and penguin operators, respectively, and we have h K − π + π − |H T | B − i = g f ππ T f D f + g f ππ T a Λ D a D f − Λ , h K − π + π − |H P | B − i = g f ππ P f D f + g f ππ P a Λ D a D f − Λ , (9)4 − K − a (980) π + π − K , K + ¯ K , K − K − B − f (980) f (980) π + π − + FIG. 1: The Feynman diagram for the B − → K − π + π − decay with the a (980) − f (980) mixing mechanism. where T a ( f ) and P a ( f ) represent the tree and penguin diagram amplitudes for B → Ka ( f ) decay,respectively. Substituting Eq. (9) into Eq. (8), the total amplitude of the decay B − → K − f ( a ) → K − π + π − can be written as M ( B − → K − π + π − ) = g f ππ D f M ( B − → K − f ) + g f ππ Λ D a D f − Λ M ( B − → K − a ) . (10)In the QCD factorization approach, we derive the amplitudes of the B − → K − f and B − → K − a decays, which are M ( B − → K − f ) = − G F √ X p = u,c λ ( s ) p (cid:26) ( δ pu a + a p − r Kχ a p + a p − r Kχ a p ) f u K ( m B − m f ) f K F Bf u ( m K ) − ( δ pu a + 2 a p + 2 a p + 12 a p + 12 a p ) Kf u ( m B − m K ) ¯ f f u F BK ( m f ) − ( a p + a p + a p − r fχ a p − a p − a p − a p + 12 r fχ a p ) Kf s ( m B − m K ) ¯ f f s F BK ( m f )+ ( δ pu b + b p + b ,EW ) Kf u f B ¯ f f u f K + ( δ p,u b + b p − b ,EW ) Kf s f B ¯ f f s f K (cid:27) , (11)5nd M ( B − → K − a ) = − G F √ X p = u,c λ ( s ) p (cid:26) ( δ pu a + a p − r Kχ a p + a p − r Kχ a p ) a K ( m B − m a ) F Ba ( m K ) f K − ( δ pu a + 32 a p + 32 a p ) Ka ( m B − m K ) F B → K ( m a ) f a + ( δ pu b + b p + b p ,EW ) a K f B f a f K (cid:27) , (12)respectively, where G F represents the Fermi constant, f B , f K , ¯ f f and f a are the decay constants of the B , K , f , and a , ¯ f s = f s m π m b ( µ )( m u ( µ )+ m s ( µ )) (where µ is the scale parameter), F Bf u ( m K ) , F BK ( m f ) and F Ba ( m K ) are the form factors for the B to f , K and a transitions, respectively.By integrating the numerator and denominator of the differential CP asymmetry parameter, one canobtain the localized integrated CP asymmetry, which can be measured by experiments and takes thefollowing form in the region R : A RCP = R R dsds ′ ( | M | − | ¯ M | ) R R dsds ′ ( | M | + | ¯ M | ) , (13)where s and s ′ are the invariant masses squared of ππ or Kπ pair in our case, and ¯ M is the decay amplitudeof the CP -conjugate process.Since the decay process B − → K − π + π − has a three-body final state, the branching fraction of thisdecay can be expressed as [41] B = τ B (2 π ) m B Z ds | p ∗ || p | Z d Ω ∗ Z d Ω |M| , (14)in which Ω ∗ and Ω are the solid angles for the final π in the ππ rest frame and for the final K in the B meson rest frame, respectively, | p ∗ | and | p | are the norms of the three-momenta of final-state π in the ππ rest frame, and K in the B rest frame, respectively, which take the following forms: | p ∗ | = p λ ( s, m π , m π )2 √ s , | p | = q λ ( m B , m K , s )2 m B , (15)where λ ( a, b, c ) is the K ¨ a ll ´e n function and with the form λ ( a, b, c ) = a + b + c − ab + ac + bc ) . IV. NUMERICAL RESULTS
When dealing with the contributions from the hard spectator and the weak annihilation, we encounterthe singularity problem of infrared divergence X = R dx/ (1 − x ) . One can adopt the method in Refs.65, 38, 39] to parameterize the endpoint divergence as X H,A = (1 + ρ H,A e iφ H,A ) ln m B Λ h , with Λ h being atypical scale of order 0.5 GeV , ρ H,A an unknown real parameter and φ H,A the free strong phase in therange [0 , π ] . For convenience, we use the notations ρ = ρ H,A and φ = φ H,A . In our calculations, we adopt ρ ∈ [0 , and φ ∈ [0 , π ] for the two-body B − → K − f and B − → K − a decays. The first term of Eq.(10) is the amplitude of the B − → K − π + π − decay without the effect of the a (980) − f (980) mixingwhen the mass of the π + π − pair is in the vicinity of the f (980) resonance. Substituting this term intoEq. (13), we can get the localized CP violation of the B − → K − f → K − π + π − decay when we take thethe integration interval as [ m f − Γ f , m f + Γ f ] , which is A CP ( B − → Kf → K − π + π − ) = [0 . , . and shown in Fig. 2 (a). Substituting Eqs. (11) and (12) into Eq. (10), one can also get the totalamplitude of the B − → K − f ( a ) → K − π + π − decay with the a (980) − f (980) mixing mechanism.Then inserting it into Eq. (13), we can also get the result of the localized CP violation in the presenceof a (980) − f (980) mixing by integrating the same integration interval as above. The predicted resultis A CP ( B − → K − f ( a ) → K − π + π − ) = [0 . , . , which is plotted in Fig. 2 (b). Obviously, the CP violating asymmetry in Fig. 2 (b) is significantly larger than that in Fig. 2 (a). Thus, we conclude thatthe a (980) − f (980) mixing mechanism can induce larger localized CP violation for the B − → K − π + π − decay. However, compared with the contribution from first term in Eq. (10), that from the second termis very small and even can be ignored when calculating the branching fraction, thus we have B ( B − → K − f ( a ) → K − π + π − ) ≈ B ( B − → Kf → K − π + π − ) . Then, we calculate the branching fraction of the B − → Kf → K − π + π − decay combining the first term in Eq. (10), Eqs. (11) and (14), the theoreticalresult is B ( B − → K − f → K − π + π − ) = [6 . , . × − which is plotted in Fig. 3. This result isconsistent with the experimental result B ( B − → Kf → K − π + π − ) = (9 . +1 . − . ) × − [42] when thedivergence parameter ranges are taken as ρ ∈ [0 , and φ ∈ [0 , π ] . V. SUMMARY AND DISCUSSION
In this work, we studied the localized integrated CP violation of the B − → K − f ( a ) → K − π + π − decay considering the a (980) − f (980) mixing mechanism in the QCD factorization approach. We foundthe localized integrated CP violation is enlarged due to the a (980) − f (980) mixing effect. Without the a (980) − f (980) mixing, the localized CP violation was found to be A CP ( B − → Kf → K − π + π − ) =[0 . , . , while A CP ( B − → K − f ( a ) → K − π + π − ) = [0 . , . when this mixing effect is considered.In addition, we also calculated the branching fraction of the B − → K − f → K − π + π − decay, and obtained B ( B − → Kf → K − π + π − ) = [6 . , . × − as shown in Fig. 3, which agrees the experimental result B ( B − → Kf → K − π + π − ) = 9 . +1 . − . × − well. Since the mixing term is very small, while calculatingthe branching fraction we can take the approximation B ( B − → K − f ( a ) → K − π + π − ) ≈ B ( B − → Kf → K − π + π − ) by ignoring the a (980) − f (980) mixing effect. However, for CP violation, this mixing7 a) (b) FIG. 2: The localized CP violation of the B − → K − f → K − π + π − decay (a) without the a (980) − f (980) mixingmechanism, (b) with the a (980) − f (980) mixing mechanism.FIG. 3: The branching fraction of the B − → K − f → K − π + π − decay. effect does contribution a lot and cannot be neglected. The same situation is also expended for other B or D mesons decay channels. We thus suggest that a (980) − f (980) mixing mechanism should beconsidered when studying the heavy meson decays both theoretically and experimentally when this mixingeffect could exist. Appendix A: THEORETICAL INPUT PARAMETERS
In the numerical calculations, we should input distribution amplitudes and the CKM matrix elements inthe Wolfenstein parametrization. For the CKM matrix elements, which are determined from experiments,8e use the results in Ref. [41]: ¯ ρ = 0 . ± . , ¯ η = 0 . ± . ,λ = 0 . ± . , A = 0 . +0 . − . , (A1)where ¯ ρ = ρ (1 − λ , ¯ η = η (1 − λ . (A2)The effective Wilson coefficients used in our calculations are taken from Ref. [43]: c ′ = − . , c ′ = 1 . ,c ′ = 2 . × − + 1 . × − i, c ′ = − . × − − . × − i,c ′ = 1 . × − + 1 . × − i, c ′ = − . × − − . × − i,c ′ = − . × − − . × − i, c ′ = 3 . × − ,c ′ = − . × − − . × − i, c ′ = 1 . × − . (A3)For the masses appeared in B decays, we use the following values [41] (in units of GeV ): m u = m d = 0 . , m s = 0 . , m b = 4 . , m q = m u + m d , m π ± = 0 . ,m B − = 5 . , m K − = 0 . , m f (980) = 0 . , m a (980) = 0 . , (A4)while for the widthes we use (in units of GeV ) [41] Γ f (980) = 0 . , Γ a (980) = 0 . . (A5)The following numerical values for the decay constants are used [5, 44, 45] (in units of GeV ): f π ± = 0 . , f B − = 0 . ± . , f K − = 0 . ± . , ¯ f f (980) = 0 . ± . , ¯ f a (980) = 0 . ± . . (A6)As for the form factors, we use [5] F B → K (0) = 0 . ± . , F B → f (980)0 (0) = 0 . , F B → a (980)0 (0) = 0 . . (A7)The values of Gegenbauer moments at µ = 1GeV are taken from [5]: B ,f (980) = − . ± . , B ,f (980) = 0 . ± . ,B ,a (980) = − . ± . , B ,a (980) = 0 . ± . . (A8)9 cknowledgments This work was supported by National Natural Science Foundation of China (Projects Nos. 11775024,11705081, 11805153, 11947001 and 11605150), Natural Science Foundation of Zhejiang Province (No.LQ21A050005) and the Fundamental Research Funds for the Provincial Universities of Zhejiang Province. [1] N. Cabibbo, Phys. Rev. Lett. , 531 (1963).[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. , 652 (1973).[3] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. , 101801 (2013).[4] R. Aaij et al. [LHCb Collaboration], Phys. Rev. D , no. 1, 012006 (2020).[5] H. Y. Cheng, C. K. Chua and K. C. Yang, Phys. Rev. D , 014017 (2006).[6] J. D. Weinstein and N. Isgur, Phys. Rev. D , 588 (1983), Phys. Rev. Lett. , 659 (1982),Phys. Rev. D ,2236 (1990).[7] R. L. Jaffe, Phys. Rev. D , 267 (1977).[8] K. S. Kim and H. Kim, Eur. Phys. J. C (2017) no.7, 435.[9] J. Berlin, A. Abdel-Rehim, C. Alexandrou, M. Dalla Brida, M. Gravina and M. Wagner, PoS LATTICE ,104 (2014).[10] C. Amsler and F. E. Close, Phys. Lett. B , 385 (1995), Phys. Rev. D , 295 (1996),Phys. Lett. B , 22(2002).[11] S. G. Gorishnii, A. L. Kataev and S. A. Larin, Phys. Lett. , 457 (1984).[12] N. N. Achasov, S. A. Devyanin and G. N. Shestakov, Phys. Lett. , 367 (1979).[13] J. J. Wu, Q. Zhao and B. S. Zou, Chin. Phys. C , 848 (2010), Phys. Rev. D , 074017 (2008).[14] D. C. Colley, Phys. Lett. , 489 (1967).[15] N. N. Achasov and G. N. Shestakov, Phys. Rev. Lett. , 182001 (2004), Phys. Lett. B , 83 (2002).[16] B. Kerbikov and F. Tabakin, Phys. Rev. C , 064601 (2000).[17] O. Krehl, R. Rapp and J. Speth, Phys. Lett. B , 23 (1997).[18] F. E. Close and A. Kirk, Phys. Lett. B , 13 (2001).[19] A. E. Kudryavtsev, V. E. Tarasov, J. Haidenbauer, C. Hanhart and J. Speth, Phys. Rev. C , 015207 (2002).[20] V. Y. Grishina, L. A. Kondratyuk, M. Buescher, W. Cassing and H. Stroher, Phys. Lett. B , 217 (2001).[21] D. Black, M. Harada and J. Schechter, Phys. Rev. Lett. , 181603 (2002).[22] X. D. Cheng, R. M. Wang and Y. G. Xu, Phys. Rev. D , no. 5, 054009 (2020).[23] T. M. Aliev and S. Bilmis, Eur. Phys. J. A , no. 9, 147 (2018).[24] W. Wang, Phys. Lett. B , 501 (2016).[25] W. H. Liang, H. X. Chen, E. Oset and E. Wang, Eur. Phys. J. C , no. 5, 411 (2019).[26] M. Buescher [ANKE Collaboration], Acta Phys. Polon. B , 1055 (2004).[27] C. Amsler and N. A. Tornqvist, Phys. Rept. , 61 (2004).[28] C. Hanhart, Phys. Rept. , 155 (2004).[29] Z. G. Wang, W. M. Yang and S. L. Wan, Eur. Phys. J. C , no. 2, 223 (2004).
30] L. Roca, Phys. Rev. D , 014045 (2013).[31] F. E. Close and A. Kirk, Phys. Rev. D , no. 11, 114015 (2015).[32] V. Dorofeev et al. , Eur. Phys. J. A , 68 (2011).[33] T. Sekihara and S. Kumano, Phys. Rev. D , no. 3, 034010 (2015).[34] M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. (2018) no.2, 022001.[35] X. H. Guo and A. W. Thomas, Phys. Rev. D , 096013 (1998), Phys. Rev. D , 116009 (2000)[36] C. Wang, Z. Y. Wang, Z. H. Zhang and X. H. Guo, Phys. Rev. D , no. 11, 116008 (2016).[37] G. Lu, Y. P. Cong, X. H. Guo, Z. H. Zhang and K. W. Wei, Phys. Rev. D , no. 3, 034014 (2014).[38] M. Beneke and M. Neubert, Nucl. Phys. B , 333 (2003).[39] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B , 245 (2001).[40] N. N. Achasov and G. N. Shestakov, Phys. Rev. D , no. 1, 016027 (2017).[41] K. A. Olive et al. [Particle Data Group], Chin. Phys. C , 090001 (2014).[42] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D , no. 3, 030001 (2018).[43] C. Wang, X. H. Guo, Y. Liu and R. C. Li, Eur. Phys. J. C , no. 11, 3140 (2014).[44] H. Y. Cheng and K. C. Yang, Phys. Rev. D (2011) 034001.[45] H. Y. Cheng and C. K. Chua, Phys. Rev. D , 114014 (2013)., 114014 (2013).