CP Asymmetries in Three-Body Final States in Charged D Decays & CPT Invariance
UUND-HEP-14-BIG01
CP Asymmetries in Three-Body Final States inCharged D Decays & CPT Invariance
From Roman history about data: Caelius (correspondent of Cicero) had taken apragmatic judgment of who was likely to win the conflict and said: Pompey had thebetter cause, but Caesar the better army, and so I became a Caesarean. I. Bediaga a , I.I. Bigi b , J. Miranda a , A. Reis aa Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio deJaneiro, RJ, Brazil b Department of Physics, University of Notre Dame du LacNotre Dame, IN 46556, USAemail addresses: [email protected], [email protected], [email protected], [email protected]
Abstract
The study of regional CP asymmetry in Dalitz plots of charm (& beauty) decays givesus more information about the underlying dynamics than the ratio between total rates.In this paper we explore the consequences of the constraint from CPT symmetry withemphasis on three-body D decays. We show simulations of D ± → π ± K + K − and dis-cuss correlations with measured D ± → π ± π + π − . There are important comments aboutanalysies of recent LHCb data in CP asymmetries for B ± decays to three-body finalstates. Contents CP asymmetries & CPT constraints . . . . . . . . . . . . . . . . . . . . . 31.2 Scattering in ππ ↔ KK , ππ ↔ ππ , KK ↔ KK , πK ↔ πK . . . . . . . . 41.3 Some intriguing results: charmless three-body B ± decays . . . . . . . . . . 5 D ± → π ± K − K + and D ± → π ± π − π + D ± → π ± K + K − and D ± → π ± π + π − . . . . . . . . . 62.2 ND in D ± → π ± K + K − with D ± → π ± π + π − . . . . . . . . . . . . . . . . . 10 ‘Cause’ = symmetry, yet ‘army’ = data. a r X i v : . [ h e p - ph ] M a r Probing CP asymmetries with CPT invariance
The study of CP violation (CPV) is a portal to New Dynamics (ND). Although noobvious signal of ND has been shown in hadronic data, there are still good reasons for itsexistence: neutrinos oscillations, the existence of Dark Matter & Dark Energy, and ‘ourown existence’ are the most obvious ones.CPV is a well established phenomenon in decays of K and B mesons, but no CP asymmetry has been found in D decays – yet. The Standard Model (SM) predicts verysmall CPV in singly Cabibbo suppressed (SCS) D decays, and close to zero CPV indoubly Cabibbo suppressed (DCS) ones. The observation of CP asymmetries at O (10 − )level in charm decays would be a clear manifestation of ND. The experimental sensitivity,however, is rapidly reaching O (10 − ) with no signals of CPV. Although the observationof CPV in charm would be a great achievement in itself, one would still have the difficultproblem of disentangling ND effects from SM ones.The vast majority of experimental searches and theoretical works refer to two -bodyfinal states (FS) of the type D → P P ( P denotes light pseudoscalar mesons). From thetheory side there are large uncertainties related to the hadronic degrees of freedom thatcould easily hide the impact of ND. From the experimental side, the usual CP asymmetriesin two -body decays give ‘only’ a single number and this is not enough information tounderstand the nature of an eventual CPV signal. One needs to go beyond two -bodydecays and look at new observables. Three- and four-body decays are the natural way.First, local effects may be larger than phase space integrated ones. The asymmetryis modulated by the strong phase variation characteristic of resonant decays [1]. The CP asymmetry may change sign across the phase space, and the comparison betweenintegrated rates would dilute an eventual signal. Furthermore, the pattern of the CP asymmetry across the phase space could give insights about the underlying operators.CPV searches with charged D mesons with three-body FS is therefore a very importantprogramme.In this paper we investigate the possible patterns of CPV in three-body, SCS non-leptonic decays of D mesons. ND could produce sizable asymmetries in DCS decays –but DCS rates are small. We focus on direct CPV with ∆ C = 1 forces and explorethe correlations introduced by strong final state interactions (FSI) & CPT conservation,which is assumed to be exact.Some comments are in order: • Theoretical predictions are more complicated in D → P V with narrow vector me-son resonances V , since one deals with three-body FS and interference with otherintermediate states must be taken into account. It comes much more complicatedin D → P S with broad scalar resonances S – in particular with σ & κ . From theexperimental side, a full Dalitz plot with very large data sets is quite challenging:for example, effects such as FSI among the three hadrons must be included in thedecay model. The modeling of the S-waves is another instance of limitations of theexisting tools. 2 One alternative are model independent searches, comparing directly the D + and the D − Dalitz plots, as in [2, 3], which is a convenient first step. • Additional constraints should be used, such as
CPT invariance. It comes into ‘play’by imposing equalities of total decay rates of particle and antiparticle. Its invarianceimply also the equalities of partial rates of classes of FS where their members canre-scatter to each other. Given that three-body decays are mostly a sequence oftwo-body transitions, pocesses such as 2 π, K ¯ K ↔ π, K ¯ K and πK ↔ πK becomecrucial. In other words, CPT invariance would relate asymmetries in D → πππ with D → πK ¯ K . CPT invariance is a clear instance where the low energy hadron dynamics play animportant role: it is an unavoidable ingredient to the decay amplitude models. However,going from quarks to hadrons and understanding the dynamics of three-body FSI are realchallenges in a quantative way. Dispersion relations and chiral perturbation theory aresome of the theoretical tools needed for a more realistic description of three-body FS.This is indeed a field plenty of opportunities.
Let us consider a decay into a FS f that can proceed through two different amplitudes: T ( D + → f ) = e iφ weak1 e iδ FSI1 A + e iφ weak2 e iδ FSI2 |A | (1) T ( D − → ¯ f ) = e − iφ weak1 e iδ FSI1 A + e − iφ weak2 e iδ FSI2 A (2)In charged D mesons only direct CP violation is possible, which means Γ( D + → f ) (cid:54) =Γ( D − → ¯ f ). Computing the CP asymmetry in the partial width one hasΓ( D + → f ) − Γ( D − → ¯ f )Γ( D + → f ) + Γ( D − → ¯ f ) = − φ W )sin(∆ δ FSI ) |A / A | |A / A | + 2 |A / A | cos(∆ φ W )cos(∆ δ FSI ) | (3)with ∆ φ W = φ weak1 − φ weak2 & ∆ δ FSI = δ FSI1 − δ FSI2 . We see clearly how CP asymmetriesarise when there are differences in both weak & strong phases.However, the constraints from CPT invariance are not apparent. Suppose the decaymode f belongs to a family of n final states f n connected to each other via re-scattering.The consequences of CPT invariance (general comments on
CPT invariance are given inRef.[4, 5, 6, 7, 8, 9]) become visible, if we rewrite the decay amplitude in the form T ( D + → f j ) = e iδ fj T f j + (cid:88) f j (cid:54) = f k T f k iT resc f j f k (4) T ( D − → ¯ f j ) = e iδ fj T ∗ ¯ f j + (cid:88) f k (cid:54) = f j T ∗ ¯ f j iT resc f j f k , (5)3here amplitudes T resc f j f k describe FSI connecting f j and f k . One gets, in addition to thedirect term, a contribution to the CP asymmetry of the form:∆ γ ( a ) = 4 (cid:88) f k (cid:54) = f j T resc f j f k Im T ∗ f j T f k (6)There are subtle, but very important statements about using these equations: • Final states f n should also include modes with neutrals. In practice, decays like D + → π + π π are really hard to obtain. • CPT invariance can be satisfied in two roads:One can find that neither D ± → π ± π + π − /π ± π π /π ± K + K − /π ± K ¯ K showsevidence for CP asymmetry.At least two of them find CP violation with opposite signs.So far D ± → π ± π + π − shows no evidence about CP asymmetry [10], but the two roadsare still open. ππ ↔ K K , ππ ↔ ππ , K K ↔ K K , πK ↔ πK Early experimental results from ππ scattering presented a significant deviation from theelastic regime of the S-wave in the region between 1.0-1.5 GeV [11, 12]. The inelasticityparameter decreases, starting at 1.0 GeV get a minimum at 1.2 GeV and come back againto the unitary circle at around 1.5 GeV, going counterclockwise in the Argand circle. Asimilar inspection was performed for the P- and D-waves, but no significant deviationfrom the elastic regime was found in this energy interval.The deviation of the inelasticity in the S-wave ππ → ππ scattering is associated to acorresponding increase of the cross section of ππ → KK [13], in the same energy region.Notice that due to G-parity conservation a pair of pions can only scatter into an evennumber of pions. In other words, a initial state of two pions can produce either two pionsor two kaons.The same study performed to Kπ elastic scattering by the LASS experiment [14]showed that both S- and P-waves have an inelasticity parameter close to unity in theArgand circle, up to 1.4 GeV in the P-wave and 1.5 GeV in the S-wave. The D-wave isdominated by the resonance K (1430) that decay to Kπ with a branching fraction of about50%[15]. Therefore, re-scattering of the Kπ system is not relevant to this discussion.The energy range of the K + K − pair is 2 m K ≤ m ( KK ) ≤ m D − m π , which coincideswith the range where the inelasticity of the ππ scattering deviates from unity. CPT invariance, therefore, connects the D + → π + π + π − and the D + → π + K + K − decays,through the S-wave ππ ↔ KK scattering. A comprehensive argument should include π π and K ¯ K as well, but this will not be adressed in this paper.4 .3 Some intriguing results: charmless three-body B ± decays Recent LHCb results on charmless three-body B ± decays show sizable averaged CP asym-metries over the FS with correlations [16]: A CP ( B ± → K ± π + π − ) = +0 . ± . stat ± . syst ± . ψK ± (7) A CP ( B ± → K ± K + K − ) = − . ± . stat ± . syst ± . ψK ± . (8)It is important to note that these CP asymmetries come with opposite signs.The CP asymmetry was measured across the Dalitz plot and this is the most in-teresting result. ‘Local’ CP asymmetries come also with opposite signs, but are muchlarger: A CP ( B ± → K ± π + π − ) ‘local (cid:48) = +0 . ± . stat ± . syst ± . ψK ± (9) A CP ( B ± → K ± K + K − ) ‘local (cid:48) = − . ± . stat ± . syst ± . ψK ± . (10)‘Local’ CP asymmetries mean here: • positive asymmetry at low m π + π − just below m ρ ; • negative asymmetry both at low and high m K + K − values.There is another important aspect; asymmetries are observed in regions of the phasespace not associated to any particular resonance.A very similar effect was observed in even more CKM suppressed three-body FS,namely B + → π + π − π + and B + → π + K − K + . LHCb experiment has measured these averaged and ‘local’ CP asymmetries [17]: A CP ( B ± → π ± π + π − ) = +0 . ± . stat ± . syst ± . J/ψK ± (11) A CP ( B ± → π ± K + K − ) = − . ± . stat ± . syst ± . J/ψK ± (12) A CP ( B ± → π ± π + π − ) | ‘local (cid:48) = +0 . ± . stat ± . syst ± . ψK ± (13) A CP ( B ± → π ± K + K − ) | ‘local (cid:48) = − . ± . stat ± . syst ± . ψK ± . (14)Again it is very interesting that LHCb data in Eqs.(11,12,13,14) show CP asymmetrieswith opposite signs – as ‘natural’ by CPT invariance, no matter what forces producethem. Again, a
CPT symmetry argument has to include neutral bosons as well.In summary, the results from charmless three-body B ± decays are very intriguing.Large regional effects, diluted when phase space integration is performed, appear in re-gions not associated to resonances, and with opposite signs in FS that are related byre-scattering. Do they show impact of ND? We refer to [8, 9, 18, 19] for additionaldiscussions on this issue. 5 Simulations of D ± → π ± K − K + and D ± → π ± π − π + D ± → π ± K + K − and D ± → π ± π + π − In this section we perform simulations of the D ± → π ± K − K + Dalitz plot to illustrate there-scattering effects discussed above. The simulations are performed in the frameworkof the isobar model. It is well known that a sum of Breit-Wigners plus a nonresonatterm is not a correct representation of the S-wave[20], but the goal here is not to extractquantitative information on the resonant structure of the decay. Rather, we are interestedin the differences between D + and D − Dalitz plots that reflect CPV effects with andwithout the constraints of
CPT constraint.For the decay amplitude we use the resonant substructure is based on Dalitz plotanalysis performed by CLEO-c collaboration [21]. For simplicity, we neglect contributionsfrom amplitudes that result in decay fractions smaller than 1%. The resonant amplitudesare written as a product of form factors, relativistic Breit-Wigners and spin amplitudes.We use the following amplitudes: A φπ = A ( D + → φπ + ), A K ∗ K = A ( D + → K ∗ (892) K + ), A K ∗ K = A ( D + → K ∗ (1430) K + ), A a π = A ( D + → a (1450) π + ), and A κK = A ( D + → κ (800) K + ).The decay amplitudes are written as A = (cid:88) c j A j (15)¯ A = (cid:88) ¯ c j A j (16)with c j ≡ a j e iδ j , j = φπ, K ∗ K, K ∗ K, a π, κK . The amplitudes A j involve only CP -even,strong phases from the Breit-Wigner functions. Weak phases are included in the phase ofthe c j coefficients. CP conservation imply c j = ¯ c j for all j .The couplings c j between the j − th resonant mode and the initial state are complexfor two reasons: • Weak forces between quarks may produce phases that are opposite for anti-quarks. • the decay amplitude is affected by hadronic FSI. Strong phases due to the resonance-bachelor re-scattering are included in δ j and they are the same for hadrons andanti-hadrons.The Dalitz plot of the D ± → π ± K − K + decay is shown in Fig. 1. The prominentcontributions from the φπ and K ∗ (892) K + are clearly visible. The contribution from thebroad S-wave K − π + resonances can be seen at the edges of the s K − π + axis.The set of coefficients c j (¯ c j ) defines, thus, the decay amplitude A ( ¯ A ). In our simu-lations we assume no production asymmetries and identical detection efficiencies, so thenumber of D + and D − decays is proportional to the integral of the decay amplitudes overthe Dalitz plot, N D + ∝ (cid:90) | A | ds KK ds Kπ N D − ∝ (cid:90) | ¯ A | ds KK ds Kπ . (17)6 /c [GeV + p - K s ] / c [ G e V + K - K s Figure 1: A simulation of the Dalitz plot of the decay D → K − K + π + . The decay modelis taken from CLEO-c (see text for details) and is used as the starting point of our studies.In the case of CP conservation we have exactly the same number of D + and D − . Butif there is CPV the values of the two integrals will differ, in general. We simulate the D + and the D − Dalitz plot separately, seeding CPV in the latter. We always simulate 3 × D + → K − K + π + decays. The number of generated D − decays is defined according to theratio of the above integrals, which depends on how CPV is seeded.The averaged CP asymmetry is computed as: A CP = (cid:82) | A | ds KK ds Kπ − (cid:82) | ¯ A | ds KK ds Kπ (cid:82) | A | ds KK ds Kπ + (cid:82) | ¯ A | ds KK ds Kπ (18)The D + and D − Dalitz plots, simulated as described above, are compared using the’Miranda’ method [2, 3]. In this method the D ± Dalitz plot is divided into bins; acomparison between the D + and D − Dalitz plot is performed directly in a bin-by-binbasis, computing, for each bin, the anisotropy variable S iCP = N + i − N − i (cid:113) N + i + N − i , (19)with N + i and N − i being the i − th bin content of the D + and D − Dalitz plots, respectively.The value of S iCP is a measure of the significance of the excess of one charge especieover the other in the i − th bin. Notice that S iCP may be positive or negative. If CP isconserved, N + i and N − i will differ only by statistical fluctuations. The values of S iCP ,inthis case, are distributed according to a unit Gaussian centered at zero. As an example,we show in Fig. 2 a simulation in which CP is conserved — the same number of D + and D − decays are simulated with c j = ¯ c j . The plot on the left has the distribution of S iCP across the Dalitz plot. No region show any excess of on charge over the other, asexpected. The distribution of S iCP is shown on the plot on the left, with a unit Gaussiancentered at zero superimposed. 7 /c [GeV + p - K s ] / c [ G e V + K - K s CP S -10 -8 -6 -4 -2 0 2 4 6 8 1001020304050607080 Figure 2: A simulation of the Dalitz plot of the decays D + → K − K + π + and D − → K + K − π − . No CPV is seeded: the same set of coefficients c j is used for the simulationof the D + and the D − samples. Values of S iCP are, in this case, distributed according toa unit Gaussian centered at zero. No excess of one charge over the other is obeserved inany region of the Dalitz plot, apart from statistical fluctuations.There is a number of models for CPV beyond the SM. In this exercise we assume asimple scenario, consistent with the SM, in which CPV manifests as a difference in relativephase of the K − π + and K − K + resonances in the D + and the D − Dalitz plots. We referto this as the SM scenario. We first simulate CPV in this ’SM scenario’ (SM CPV, forshort). Then we simulate the contribution to D + → K − K + π + from the D + → π − π + π + decay via π + π − → K + K − re-scattering In this simulation we ’turn off’ the SM CPV andintroduce a small CPV effect in D + → π − π + π + . Finally the full simulation includingboth effects is performed.Our SM CPV consists in introducing a 3 ◦ difference in the relative phases of the K − K + and K − π + resonances when the D − sample is generated. This 3 ◦ difference causes a minorexcess of D − over D + resulting in an averaged asymmetry of 0.08%, beyond the currentexperimental sensitivity. The one- and two-dimensional distributions of S iCP for the SMCPV simulation are shown in Fig. 3. Large local asymmetries are observed, mostly inregions where the K − K + and K − π + amplitudes overlap. The asymmetry is modulatedby the strong phase variation of the resonances, leading to negative values of S iCP in someregions of the Dalitz plot and positive in another ones. We see how large local effects canresult in a very small averaged asymmetry. The distribution of S CP values is no longercentered at zero ( µ = − . ± . σ = 1 . ± . CPT constraint. In the D + → K − K + π + decayamplitude we now introduce the contribution from the D → π − π + π + decay throughthrough the ππ ↔ KK scattering, but keeping c j = ¯ c j . Weak phases are in generalobscured by the strong ones, but here is an instance where the existence of the strongphase favors the observation of small differences in weak phases.8 /c [GeV + p - K s ] / c [ G e V + K - K s / ndf c – – -0.3954 Sigma 0.06 – CP S -10 -8 -6 -4 -2 0 2 4 6 8 1001020304050 / ndf c – – -0.3954 Sigma 0.06 – Figure 3: A simulation of CP violation in the decay D → K − K + π + . A 3 ◦ difference inthe K ∗ K + and φπ + relative phase between D + and D − is introduced. The difference inrelative phase cause the CP asymmetry to change sign across the Dalitz plot, accordingto the phase variation of the interfering Breit-Wigners functions.The re-scattering term is ‘inspired’ in Eqs.(4, 5). For simplicity, the weak amplitudefor the D + → π − π + π + decay is represented by a complex constant, T D → π , with anunknown modulus and CP odd phase.The ππ → KK scattering amplitude is written as T ππ → KK = A ππ → KK e iδ ππ → KK . Thereal functions A ππ → KK and δ ππ → KK are taken from [13]. T ππ → KK is CP invariant.The decay amplitudes become A = c φπ A φπ + c a π A a π + c κK A κK + c K ∗ K A K ∗ K + c K ∗ K A K ∗ K + T D → π T ππ → KK ., (20) A = ¯ c φπ A φπ + ¯ c a π A a π + c κK A κK + c K ∗ K A K ∗ K + ¯ c K ∗ K A K ∗ K + ¯ T D → π T ππ → KK . (21)Before performing the full simulation, we investigate the effect of the re-scatteringterm alone , which means c j = ¯ c j . In Eqs. (20, 21) we set | ¯ T D → π | = 0 . | T D → π | and arg( ¯ T D → π ) = arg( T D → π ) + 5 ◦ . The values of | T D → π | and arg( T D → π ) are un-known. We chose arbitrary values that yield a small decay fraction of approximately2% for the re-scattering contribution. This small amount of re-scattering and the smalldifference introduced between T D → π and ¯ T D → π are sufficient to yield a CP asymmetryof approximately 0.7%, well within the current experimental sensitivity.The one- and two-dimensional distributions of S iCP for this simulation are shown in Fig.4. The effect of the global asymmetry is a displacement of the mean of the S iCP distribution(right plot). The widht of the Gaussian, σ = 1 . ± . D − over D + events towards lower valuesof m K + K − , as expected since | T ππ → KK | has a maximum near 1.2 GeV/ c .We are now ready for the full simulation. The D − sample is generated with the set of¯ c j coefficients used in the SM CPV example, whereas the re-scattering term is as describedabove. The S iCP distributions are shown in Fig. 5.9 /c [GeV + p - K s ] / c [ G e V + K - K s / ndf c – – -0.9663 Sigma 0.067 – S-10 -8 -6 -4 -2 0 2 4 6 8 1001020304050 / ndf c – – -0.9663 Sigma 0.067 – Figure 4: A simulation of the Dalitz plot of the decay D + → K − K + π + . The same set ofcoefficients c j are used for both D + and D − . A re-scattering term in the decay amplitudeis introduced (see text for details), being different for D + and D − . The distribution inthe left panel is fitted to a Gaussian with free mean and width.We do not know how large the strong re-scattering term should be, or what value theweak phase of T D → π should take. The effect of the re-scattering in the CP asymmetrydepends, naturally, on the assumed difference between T D → π and T D → π . One shouldkeep in mind that decays with neutrals must be considered in a comprehensive treatment.But with this simple simulation we show how the addition of a re-scattering contributionmay change not only the pattern of the SM-CPV asymmetry of Fig. 3, but also give riseto a global CP asymmetry. Different combinations of | T D → π | and arg( T D → π ) yieldingdecay fractions up to a few per cent were tested, always with similar results. With thisinvestigation we want to call attention to the importance of exploring the constraintsof CPT symmetry, showing how re-scattering contribution may increase both local andphase space integrated effects. D ± → π ± K + K − with D ± → π ± π + π − We discuss now one last example: how one can use the Dalitz plot to access the impactof ND. There is a number of extensions of the SM. We explore a scenario in whichND manifests as an enhancement of em CP violation effects associated with the broadscalars (like charged Higgs exchanges). These resonances populates the whole Dalitz plot,interfering with all other components. The resulting asymmetries would be spread allover the phase space.As discussed before, for the sake of simulations the use of Breit-Wigners parameteri-zation in the context of the isobar model is good enough to highlight the impact of CPV.Better tools – like refined dispersion relations [12] based on the data of low enegy strongscattering – have to be developed when it comes to analyse the large data sets from LHCb.As in our previous simulations, we use the decay amplitude from CLEO-c [21], for the10 /c [GeV + p - K s ] / c [ G e V + K - K s / ndf c – – -1.323 Sigma 0.10 – CP S -10 -8 -6 -4 -2 0 2 4 6 8 10051015202530 / ndf c – – -1.323 Sigma 0.10 – Figure 5: A simulation of CP violation in the decay D + → K − K + π + . A 3 ◦ difference inthe K ∗ K + and φπ + relative phase between D + and D − is introduced, in addition to thedifference in the re-scattering term. D ± → π ± K + K − . For the D ± → π ± π + π − we use the results from E791 [22]. CP violationis seeded as a 1% difference in the strenght of the coupling of the D + and D − mesons tothe light scalars κ and σ , plus a 1 ◦ phase difference.The distributions of the values from the Miranda procedure across the Dalitz plot withCPV seeded as described above are shown in Fig. 6. The broadness of the scalars causethe CP violation effects to be spread over a large portions of the Dalitz plots, being moreintense as one approaches the resonance nominal mass. The asymmetry pattern in thisexample is significantly different from that of the SM CPV of Fig. 3. In B transitions one has to find non -leading source of CP violation. We had emphasizedthe need to go beyond the phase space integrated CP asymmetries and probe regionaleffects on Dalitz plots of three-body B decays [2, 3, 8, 9, 18]. It is crucial to understandthe impact of ππ ↔ ππ , K ¯ K ↔ K ¯ K , Kπ ↔ Kπ and more.The landscape is very different for charm decays, where no CP violation has beenfound yet. So far theoretical and experimental efforts have focused mostly on two-bodyFS of charm mesons. This is no surprise since two-body decays are much simpler to treatthan three-body ones. However, in order to understand the possible impact of ND in aneventual observations of CP violation in charm decays, one definetely needs to go beyondthe ratio of integrated rates and study the pattern of regional CPV. This is the mainmessage of this paper.
One has to do it in steps to understand the information that thedata will give us.The SM produces only small CP asymmetries in SCS decays and very close to zero in11 /c [GeV + p - K s ] / c [ G e V + K - K s ] /c [GeV lo + p - p s ] / c [ G e V h i + p - p s Figure 6: Simulation the decays D + → K − K + π + (left) and D + → π − π + π + right. CP violation is seeded inspired in a ND scenario in which there is an asymmetry between thecoupling between the D meson and the ligth scalars .DCS one. In this respect, the mere observation of CPV in DCS decays would be a strongindication of ND. DCS rates, however, are very small and very large data sets would berequired.Singly Cabibbo suppressed decays are much more promising. Very large data setsalready exist. In this paper we have produced simulations of three-body singly Cabibbosuppressed D ± decays. We focused on the D ± → π ± π + π − /π ± K + K − and explored theconsequences of the CPT invariance. It is crucial to understand the impact of ππ ↔ ππ , ππ ↔ K ¯ K , Kπ ↔ Kπ etc.This is obviously very challenging. CPV in decays of heavy flavor involves an interplaybetween the degrees of freedom at the quark level and long distance effects of low energyhadron physics. One needs to think beyond the simple valence quark diagrams. The U -spin symmetry was invented by Lipkin [23]. Later it was applied to B decays manytimes, as one can see in these references [24]; in [25] it was suggested that one might todeal with U-spin violation of the order of 10 - 20 %.As discussed in Ref.[1] data tell us much larger violations in exclusive decays D → K + K − vs. D → π + π − and D → K + K − π + π − vs. D → π + π − π + π − – however muchless in the sum of D decays.The simulations we performed illustrates the impact of the correlations due to CPT invariance, which establishes useful connections between different FS related to each othervia strong re-scattering. FSI interactions are indeed a crucial ingredient for any accurateDalitz plot analysis with the contemporary data sets. Much more theoretical work isnecessary in order to produce better decay models.
Acknowledgments:
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