A novel observable for C\!P violation in multi-body decays and its application potential to charm and beauty meson decays
aa r X i v : . [ h e p - ph ] F e b A new observable for CP violation in three-body decays and itsapplication potential to charm and beauty hadron decays Zhen-Hua Zhang ∗ School of Nuclear Science and Technology,University of South China, Hengyang, Hunan, 421001, China. (Dated: March 1, 2021)
Abstract
A new observable measuring the CP asymmetry in three-body decays, which is called the forward-backward asymmetry induced CP asymmetry (FBI-CPA), A F BCP , is introduced. This newly definedobservable has the dual advantages that 1) it can isolate the CP asymmetry associated with theinterference of the S - and P -wave amplitude from that associated with the S - or P -wave amplitudealone; 2) it can effectively double the statistics comparing to the conventionally defined regional CP asymmetry. We also suggest to perform the measurements of FBI-CPA in some three-bodydecay channels of charm and beauty hadrons. ∗ [email protected] . INTRODUCTION CP violation is an important ingredient of the Standard Model of particle physics [1], alsoone of the necessary conditions for the dynamical generation of the baryonic asymmetry ofthe unverse [2]. It was first discovered in neutral kaon system in the year 1964 [3], and itsdiscoveries in B meson decay processes by the B factories confirmed the Kobayashi-Maskawamechanism of SM [4–10]. Recently, CP violation was also discovered in the charmed mesondecay processes [11].Intensive studies on CP violations in multi-body decays of beauty and charmed hadronshave been performed both theoretically [12–20] and experimentally [21–27] during the lastten years. One advantage for multi-body decays is that CP violation can be studied throughthe phase space distribution of the decay, namely the regional CP asymmetries distributedin the phase space. The total decay amplitudes can be expressed as a superposition ofvarious amplitudes, which can allow the presence of different strong phases. Because of theinterference effects of these amplitudes, the regional CP asymmetries in certain places of thephase space can be very large. Up to now, the regional CP asymmetry is one of the mostimportant and extensively studied observables associated with CP violation in multi-bodydecays, other than the integrated CP asymmetry. Although for four-body decay channelsand baryon three-body decay channels, one can also study the CP violation associated withtriple product asymmetry [28–31].The disadvantage of the regional CP asymmetry in multi-body decays is also obvious.Once focusing on a small region of the phase space, the experimental study of regional CP asymmetries will suffer from low statistics.In this paper, other than the regional CP asymmetry, we are going to introduce a new ob-servable to measure the CP violation in three-body decays, which, according to our analysisbelow, can almost effectively double the statistics comparing to the conventionally definedregional CP asymmetries. Furthermore, this new observable could potentially promote thediscovery of CP violation in three-body decays of beauty and charmed hadrons.2 I. THE NEW OBSERVABLE
For a three-body decay H → h h h , we will focus on the phase space in the vicinityof a P -wave intermediate resonance X , where, the decay will be dominated by the cascadedecay H → Xh , X → h h . The part of the phase space which we focus on satisfies( m X − σ X ) < s < ( m X + σ X ) , where s is the invariant mass squared of h and h , m X is the mass of X , σ X is of the same order with the decay width of X , Γ X . Let us denote therelative angle between the momenta of h and h in the rest frame of h and h (hence of X ) as θ ∗ . Then, the part of phase space that we focus on can be further divided into twoparts according to whether θ ∗ is larger or smaller than π/
2. An observable, describing theforward-backward asymmetry in these two parts of the phase space, can be defined as A F BH → h h h = Γ H ( c θ ∗ > − Γ H ( c θ ∗ < H ( c θ ∗ >
0) + Γ H ( c θ ∗ < . (1)where c θ ∗ ≡ cos θ ∗ , which can be expressed by the Lorentz-invariant variables as c θ ∗ =(ˆ s − s ) / ˆ s − , with ˆ s ± = ( s , max ± s , min ) /
2, and s , max/min the maximun/minimumvalue of s kinematically allowed by the three-body decay, Γ H ( c θ ∗ >
0) and Γ H ( c θ ∗ < H → h h h in the aforementioned two part of the phasespace. The nonzero of A F BH → h h h indicates that the decay amplitude of H → h h h in theregion of phase space ( m X − σ X ) < s < ( m X + σ X ) is not only just dominated by thecascade decay H → X ( → h h ) h , but other contributions, usually S -wave amplitude, arealso comparable. This can be seen as follows. Suppose that the amplitude of H → h h h inthe region of phase space ( m X − σ X ) < s < ( m X + σ X ) is dominated by the cascade decay H → X ( → h h ) h , plus an S -wave amplitude, so that it can be expressed as a coherentsum: M H → h h h = M H → X ( → h h ) h + M S − wave , (2)where, the amplitude of the cascade decay H → X ( → h h ) h can be parameterized as M H → X ( → h h ) h = a P c θ ∗ , while the S -wave amplitude can be parameterized as M S − wave = a S . Then, the differential decay width will take the form d Γ ∝ |M H → h h h | ds dc θ ∗ Note that Γ H → h h h ( c θ ∗ ≷
0) is in fact abbreviation for Γ H → h h h (( m X − σ X ) < s < ( m X + σ X ) , c θ ∗ ≷ m X − σ X ) < s < ( m X + σ X ) has been omitted. h | a P | c θ ∗ + | a S | + 2 ℜ ( a P a ∗ S ) c θ ∗ i ds dc θ ∗ . (3)After integrating over c θ ∗ and s , one hasΓ H → h h h ( c θ ∗ ≷ ∝ |h a P i| / |h a S i| ± ℜ ( h a P a ∗ S i ) , (4)where h· · ·i = R ( m X + σ X ) ( m X − σ X ) ( · · · ) ds . By substituting Eq. (4) into Eq. (1), the forward-backward asymmetry can be expressed as A F BH → h h h = ℜ ( h a P a ∗ S i ) |h a P i| / |h a S i| . (5)From the above equation one can clearly see that the presence of both the P - and S -waveamplitudes M H → X ( → h h ) h and M S − wave results in nonzero forward-backward asymmetry A F BH → h h h . If only the P -wave amplitude M H → X ( → h h ) h contributes, A F BH → h h h wouldsimply be zero.Up to this point, nothing is mentioned about the CP conjugate process ¯ H → ¯ h ¯ h ¯ h , andhence, the CP asymmetry. It is easy to see that if CP symmetry is respected, one wouldsimply have A F BH → h h h = A F B ¯ H → ¯ h ¯ h ¯ h . On the other hand, if the CP symmetry is violated, A F BH → h h h and A F B ¯ H → ¯ h ¯ h ¯ h would not equal to each other. Consequently, one can introduce anew observable measuring the CP asymmetry of three-body decay H → h h h , which willbe called the forward-backward asymmetry induced CP asymmetry (FBI-CPA) hereafterand is defined as A F BCP = 12 ( A F BH → h h h − A F B ¯ H → ¯ h ¯ h ¯ h ) . (6)From the definition of FBI-CPA one can easily see that its nonzero indeed represents theviolation of CP . III. DISCUSSIONS ON THE NEWLY DEFINED OBSERVABLE
One of the motivations for the introduction of A F BCP can be explained as follows. Whenthe S -wave amplitude are comparable with the P -wave one in the vicinity of the resonance X , the regional CP asymmetries for c θ ∗ > c θ ∗ <
0, which are conventionally definedas A CP ( c θ ∗ ≷
0) = Γ H ( c θ ∗ ≷ − Γ ¯ H ( c θ ∗ ≷ H ( c θ ∗ ≷
0) + Γ ¯ H ( c θ ∗ ≷ , (7)are correlated with each other.To see this, one just needs to reexpressed A CP ( c θ ∗ ≷
0) as4 CP ( c θ ∗ ≷
0) = A PCP + |h a S i| + |h ¯ a S i| |h a P i| + |h ¯ a P i| A SCP ± ℜ ( h a P a ∗ S i−h ¯ a P ¯ a ∗ S i ) |h a P i| + |h ¯ a P i| + |h a S i| + |h ¯ a S i| |h a P i| + |h ¯ a P i| ± ℜ ( h a P a ∗ S i + h ¯ a P ¯ a ∗ S i ) |h a P i| + |h ¯ a P i| , (8)by substituting Eq. (4) into Eq. (7), where A S/PCP = | h a S/P i | − | h ¯ a S/P i | | h a A/P i | + | h ¯ a S/P i | . It can be seen thatthere are three terms in the numerator of the above equation, corresponding to three originsof the regional CP asymmetry, A CP ( c θ ∗ ≷ CP asymmetry associated with the S - and P -wave alone, and that associated with the interference effect between the S - and P -waves.Among these three terms, the first two are the same for A CP ( c θ ∗ >
0) and A CP ( c θ ∗ < CP asymmetry for A CP ( c θ ∗ ≷
0) is proportional to the sine of therelative strong angle between the S - and P -wave amplitudes, which, according to Watson’stheorem [32], comes from the final state interaction, so that it can be large because of itsnonperturbative attribute. As a consequence, the last term in the numerator of Eq. (8) canbe comparable with- some times it can even dominate over- the first two terms, resultingin a substantial difference between A CP ( c θ ∗ >
0) and A CP ( c θ ∗ < A CP ( c θ ∗ >
0) and A CP ( c θ ∗ <
0) are opposite because of thepresence of the last term in the numerator of Eq. (8). Indeed, this kind of behaviour hasalready been observed in B ± → π + π − K ± and B ± → π + π − π ± [24], and has been studiedextensively in the literature. One interesting property of the newly defined FBI-CPA is thatit is capable of isolating the CP asymmetry associating with the interference of the S - and P -waves, which can be seen by expressing A F BCP as A F BCP = ℜ ( h a P a ∗ S i ) |h a P i| / |h a S i| − ℜ ( h ¯ a P ¯ a ∗ S i ) |h ¯ a P i| / |h ¯ a S i| . (9)It is this property which motivates the introduction of FBI-CPA. Besides the aforementioned motivation, another important advantage of FBI-CPA is thatit can almost effectively double the statistics in the experiments. Consequently, as a comple-ment to the reginal CP asymmetries, FBI-CPA can be used in searching for CP violationsin three-body decays of beauty or charmed meson, in which the CP violations are expectedto be small so that higher statistics is essential. To see this, one first need to notice thatFBI-CPA can be approximated to an experimentally useful expression, which is One can see that in contrast to the conventionally defined CP asymmetry, there are extra factors1 / ( |h a P i| / |h a S i| ) and 1 / ( |h ¯ a P i| / |h ¯ a S i| ) in the above expression. Of course, when CP vi-olation is small, so that the above two factors are nearly equal, A F BCP would be proportional to ℜ ( h a P a ∗ S i ) − ℜ ( h ¯ a P ¯ a ∗ S i ), just like the third term in the numerator of Eq. (8) does. F BCP ≈ (cid:2) Γ H ( c θ ∗ >
0) + Γ ¯ H ( c θ ∗ < (cid:3) − (cid:2) Γ H ( c θ ∗ <
0) + Γ ¯ H ( c θ ∗ > (cid:3)(cid:2) Γ H ( c θ ∗ >
0) + Γ ¯ H ( c θ ∗ < (cid:3) + (cid:2) Γ H ( c θ ∗ <
0) + Γ ¯ H ( c θ ∗ > (cid:3) , (10)if the CP violation is small. By comparing to the conventionally defined regional CP asym-metries A CP ( c θ ∗ ≷
0) in Eq. (7), it can be clearly seen that the statistics has indeed almostdoubled.It would be useful to further compare FBI-CPA with the regional CP asymmetry A CP ( c θ ∗ < c θ ∗ > A CP ( c θ ∗ < c θ ∗ >
0) = (cid:2) Γ H ( c θ ∗ >
0) + Γ H ( c θ ∗ < (cid:3) − (cid:2) Γ ¯ H ( c θ ∗ <
0) + Γ ¯ H ( c θ ∗ > (cid:3)(cid:2) Γ H ( c θ ∗ >
0) + Γ H ( c θ ∗ < (cid:3) + (cid:2) Γ ¯ H ( c θ ∗ <
0) + Γ ¯ H ( c θ ∗ > (cid:3) . (11)Just as the cases of B ± → π + π − π ± and B ± → π + π − K ± , although the regional CP asym-metries A CP ( c θ ∗ >
0) and A CP ( c θ ∗ <
0) around the vicinity of ρ can be large, theytend to take opposite signs because of the interference of the S - and P -waves, hence thereis cancellation when summing up the event yields to obtain the regional CP asymmetry A CP ( c θ ∗ < c θ ∗ > CP asymmetry originated from the interference of the S - and P -waves is totally cancelled, which can be seen from the expression: A CP ( c θ ∗ < c θ ∗ >
0) = A PCP + 3 |h a S i| + |h ¯ a S i| |h a P i| + |h ¯ a P i| A SCP |h a S i| + |h ¯ a S i| |h a P i| + |h ¯ a P i| . (12)Consequently, FBI-CPA may take larger values than A CP ( c θ ∗ < c θ ∗ > CP asymmetries around the vicinity of X , along with A CP ( c θ ∗ > A CP ( c θ ∗ < A CP ( c θ ∗ < c θ ∗ > CP violation could be first confirmed through the measurement of FBI-CPA. IV. APPLICATION POTENTIAL TO THREE-BODY DECAYS OF CHARM ANDBEAUTY HADRONS
There are a lot of channels which are suitable to perform the measurements of FBI-CPA.In the B meson sector, for channels such as B ± → π + π − K ± and B ± → π + π − π ± [24], thereare very clear interference effect between S - and P -wave when the invariant mass of π + π − ρ (770). The regional CP asymmetries hasalready been measured by LHCb. We suggest to perform measurements of FBI-CPA around ρ (770) in these channels. For channels such as B ± → K + K − K ± or B ± → K + K − π ± [24],FBI-CPV around the P -wave resonances such as φ (1020) are also worth measuring.Similarly, measurements of FBI-CPA could potentially find evidence of CP violationsin D ± → K + K − π ± [33] and D ± ( s ) → π + π − π ± [34]. For D ± → K + K − π ± , the reso-nances K ∗ (892) and φ (1020) are clearly visible in the Dalitz plot. The forward-backwardasymmetries for these two P -wave resonances are also visible. It would be interestingto check weather the CP violation shows up in FBI-CPA around these resonances. For D ± → π + π − π ± , the vector resonance ρ (770) and its forward-backward asymmetry is alsovisible.In fact, besides the above suggested decay channels, both the measurements of theforward-backward asymmetry and FBI-CPA are meaningful in other three-body decay chan-nels of charm and beauty hadron, provided that a P -wave resonances is presented in theDalitz plot. V. CONCLUSION
To sum up, we introduce a new observable for CP violations in three-body decays of heavyhadron, the forward-backward asymmetry induced CP asymmetry, FBI-CPA. We suggestto perform the measurements of FBI-CPA in some decay channels of charm and beautyhadrons. ACKNOWLEDGMENTS
This work was supported by National Natural Science Foundation of China under Con-tracts No.11705081. [1] Makoto Kobayashi and Toshihide Maskawa, “CP Violation in the Renormalizable Theory ofWeak Interaction,” Prog. Theor. Phys. , 652–657 (1973).
2] A. D. Sakharov, “Violation of CP Invariance, C asymmetry, and baryon asymmetry of theuniverse,” Pisma Zh. Eksp. Teor. Fiz. , 32–35 (1967), [Usp. Fiz. Nauk161,no.5,61(1991)].[3] J.H. Christenson, J.W. Cronin, V.L. Fitch, and R. Turlay, “Evidence for the 2 pi Decay ofthe k(2)0 Meson,” Phys. Rev. Lett. , 138–140 (1964).[4] Bernard Aubert et al. (BABAR Collaboration), “Observation of CP violation in the B mesonsystem,” Phys. Rev. Lett. , 091801 (2001), arXiv:hep-ex/0107013 [hep-ex].[5] Kazuo Abe et al. (Belle), “Observation of large CP violation in the neutral B meson system,”Phys. Rev. Lett. , 091802 (2001), arXiv:hep-ex/0107061 [hep-ex].[6] Kazuo Abe et al. (Belle), “Observation of large CP violation and evidence for direct CP violation in B → π + π − decays,” Phys. Rev. Lett. , 021601 (2004), arXiv:hep-ex/0401029[hep-ex].[7] Bernard Aubert et al. (BaBar), “Observation of CP violation in B → K + π − and B → π + π − ,” Phys. Rev. Lett. , 021603 (2007), arXiv:hep-ex/0703016 [HEP-EX].[8] R. Aaij et al. (LHCb), “Observation of CP violation in B ± → DK ± decays,” Phys. Lett. B712 , 203–212 (2012), [Erratum: Phys. Lett.B713,351(2012)], arXiv:1203.3662 [hep-ex].[9] R Aaij et al. (LHCb), “First observation of CP violation in the decays of B s mesons,” Phys.Rev. Lett. , 221601 (2013), arXiv:1304.6173 [hep-ex].[10] R. Aaij et al. (LHCb), “Amplitude analysis of the decay B → K S π + π − and first obser-vation of the CP asymmetry in B → K ∗ (892) − π + ,” Phys. Rev. Lett. , 261801 (2018),arXiv:1712.09320 [hep-ex].[11] Roel Aaij et al. (LHCb), “Observation of CP Violation in Charm Decays,” Phys. Rev. Lett. , 211803 (2019), arXiv:1903.08726 [hep-ex].[12] Zhen-Hua Zhang, Xin-Heng Guo, and Ya-Dong Yang, “CP violation in B ± → π ± π + π − in the region with low invariant mass of one π + π − pair,” Phys.Rev. D87 , 076007 (2013),arXiv:1303.3676 [hep-ph].[13] Bhubanjyoti Bhattacharya, Michael Gronau, and Jonathan L. Rosner, “CP asymmetriesin three-body B ± decays to charged pions and kaons,” Phys. Lett. B , 337–343 (2013),arXiv:1306.2625 [hep-ph].[14] I. Bediaga, T. Frederico, and O. Louren¸co, “CP violation and CPT invariance in B ± decayswith final state interactions,” Phys. Rev. D , 094013 (2014), arXiv:1307.8164 [hep-ph].
15] Chao Wang, Zhen-Hua Zhang, Zhen-Yang Wang, and Xin-Heng Guo, “Localized direct CPviolation in B ± → ρ ( ω ) π ± → π + π − π ± ,” Eur. Phys. J. C , 536 (2015), arXiv:1506.00324[hep-ph].[16] Hai-Yang Cheng, Chun-Khiang Chua, and Zhi-Qing Zhang, “Direct CP Violation in Charm-less Three-body Decays of B Mesons,” Phys. Rev. D , 094015 (2016), arXiv:1607.08313[hep-ph].[17] Rebecca Klein, Thomas Mannel, Javier Virto, and K. Keri Vos, “CP Violation in Multibody B Decays from QCD Factorization,” JHEP , 117 (2017), arXiv:1708.02047 [hep-ph].[18] J. P. Dedonder, A. Furman, R. Kaminski, L. Lesniak, and B. Loiseau, “S-, P- and D-wavefinal state interactions and CP violation in B+- – > pi+- pi-+ pi+- decays,” Acta Phys. Polon.B , 2013 (2011), arXiv:1011.0960 [hep-ph].[19] Hai-Yang Cheng and Chun-Khiang Chua, “Branching fractions and CP violation in B − → K + K − π − and B − → π + π − π − decays,” Phys. Rev. D , 053006 (2020), arXiv:2007.02558[hep-ph].[20] Zhen-Hua Zhang, Sheng Yang, and Xin-Heng Guo, “ ρ − ω mixing contribution to the measured CP asymmetry of B ± → ωK ± ,” JHEP , 020 (2020), arXiv:2005.09157 [hep-ph].[21] Bernard Aubert et al. (BaBar), “Dalitz Plot Analysis of B+- — > pi+-pi+-pi-+ Decays,”Phys. Rev. D , 072006 (2009), arXiv:0902.2051 [hep-ex].[22] R Aaij et al. (LHCb collaboration), “Measurement of CP violation in the phase space of B ± → K ± π + π − and B ± → K ± K + K − decays,” Phys.Rev.Lett. , 101801 (2013), arXiv:1306.1246[hep-ex].[23] Roel Aaij et al. (LHCb collaboration), “Measurement of CP violation in the phase spaceof B ± → K + K − π ± and B ± → π + π − π ± decays,” Phys.Rev.Lett. , 011801 (2014),arXiv:1310.4740 [hep-ex].[24] Roel Aaij et al. (LHCb), “Measurements of CP violation in the three-body phase space ofcharmless B ± decays,” Phys. Rev. D , 112004 (2014), arXiv:1408.5373 [hep-ex].[25] Roel Aaij et al. (LHCb), “Study of the B → ρ (770) K ∗ (892) decay with an amplitudeanalysis of B → ( π + π − )( K + π − ) decays,” JHEP , 026 (2019), arXiv:1812.07008 [hep-ex].[26] Roel Aaij et al. (LHCb), “Observation of Several Sources of CP Violation in B + → π + π + π − Decays,” Phys. Rev. Lett. , 031801 (2020), arXiv:1909.05211 [hep-ex].
27] Roel Aaij et al. (LHCb), “Search for CP violation in Ξ + c → pK − π + decays using model-independent techniques,” Eur. Phys. J. C , 986 (2020), arXiv:2006.03145 [hep-ex].[28] Michael Gronau and Jonathan L. Rosner, “Triple product asymmetries in K , D ( s ) and B ( s ) decays,” Phys. Rev. D , 096013 (2011), arXiv:1107.1232 [hep-ph].[29] Michael Gronau and Jonathan L. Rosner, “Triple product asymmmetries in Λ b and Ξ b decays,”Phys. Lett. B , 104–107 (2015), arXiv:1506.01346 [hep-ph].[30] Roel Aaij et al. (LHCb), “Measurement of matter-antimatter differences in beauty baryondecays,” Nature Phys. , 391–396 (2017), arXiv:1609.05216 [hep-ex].[31] Roel Aaij et al. (LHCb), “Search for CP violation and observation of P violation in Λ b → pπ − π + π − decays,” Phys. Rev. D , 051101 (2020), arXiv:1912.10741 [hep-ex].[32] Kenneth M. Watson, “The Effect of final state interactions on reaction cross-sections,” Phys.Rev. , 1163–1171 (1952).[33] R. Aaij et al. (LHCb), “Search for CP violation in D + → K − K + π + decays,” Phys. Rev. D , 112008 (2011), arXiv:1110.3970 [hep-ex].[34] R Aaij et al. (LHCb), “Search for CP violation in the decay D + → π − π + π + ,” Phys. Lett. B , 585–595 (2014), arXiv:1310.7953 [hep-ex]., 585–595 (2014), arXiv:1310.7953 [hep-ex].