A novel observable for $C\!P$ violation in multi-body decays and its application potential to charm and beauty meson decays

A novel observable measuring the $C\!P$ asymmetry in multi-body decays of heavy mesons, which is called the forward-backward asymmetry induced $C\!P$ asymmetry (FBI-$C\!P$A), $A_{CP}^{FB}$, is introduced. This observable has the dual advantages that 1) it can isolate the $C\!P$ asymmetry associated with the interference of the $S$- and $P$-wave amplitude from that associated with the $S$- or $P$-wave amplitude alone; 2) it can effectively almost double the statistics comparing to the conventionally defined regional $C\!P$ asymmetry. We also suggest to perform the measurements of FBI-$C\!P$A in some three-body decay channels of charm and beauty mesons.


Introduction
CP violation is an important ingredient of the Standard Model of particle physics [1], also one of the necessary conditions for the dynamical generation of the baryonic asymmetry of the Universe [2]. It was first discovered in the neutral kaon system in the year 1964 [3], and its discoveries in B meson decay processes by the B factories confirmed the Cabibbo-Kobayashi-Maskawa mechanism of SM [4,5,6,7,8,9,10]. Recently, CP violation was also discovered in the charmed meson decay processes [11].
Intensive studies on CP violations in multi-body decays of beauty and charmed hadrons have been performed both theoretically [12,13,14,15,16,17,18,19,20] and experimentally [21,22,23,24,25,26,27,28] during the last ten years. One advantage for multi-body decays is that CP violation can be studied through the phase space distribution of the decay, namely, the regional CP asymmetries distributed in the phase space. The total decay amplitudes can be expressed as a superposition of various amplitudes, which can allow the presence of different strong phases. Because of the interference effects of these amplitudes, the regional CP asymmetries in certain places of the phase space can be very large. Up to now, the regional CP asymmetry is one of the most important and extensively studied observables associated with CP violation in multi-body decays, other than the integrated CP asymmetry. Although for four-body decay channels and baryon three-body decay channels, one can also study the CP violation associated with triple product asymmetry [29,30,31,32].
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 an observable to measure the CP violation in multi-body decays of heavy mesons, which according to our analysis below, can almost effectively double the statistics comparing to the conventionally defined regional CP asymmetries. Furthermore, this observable could potentially promote the discovery of CP violation in multi-body decays of beauty and charmed mesons.

The Forward-Backward asymmetry induced CP asymmetry
Consider a multi-body decay H → h 1 h 2 h 3 · · · h n , where H is a heavy meson, and h 1 , h 2 , · · ·, h n are light ones. We will focus on the phase space in the vicinity of a P -wave intermediate resonance X, where, the decay will be dominated by the cascade decay H → Xh 3 · · · h n , X → h 1 h 2 . The part of the phase space which we focus on satisfies (m X − σ X ) 2 < s 12 < (m X + σ X ) 2 , where s 12 is the invariant mass squared of h 1 and h 2 , 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 the relative angle between the momenta of h 1 and H in the rest frame of h 1 and h 2 system (hence, of X) as θ * 1 . Then, the part of phase space that we focus on can be further divided into two parts according to whether θ * 1 is larger or smaller than π/2. An observable, describing the forward-backward asymmetry in these two parts of the phase space, can be defined as where c θ * 1 ≡ cos θ * 1 , Γ H (c θ * 1 > 0) and Γ H (c θ * 1 < 0) are the regional decay widths of H → h 1 h 2 h 3 · · · h n in the aforementioned two part of the phase space. 1 The nonzero of A F B H→h 1 h 2 h 3 ···hn indicates that the decay amplitude of H → h 1 h 2 h 3 · · · h n in the region of phase space (m X − σ X ) 2 < s 12 < (m X + σ X ) 2 is not only just dominated by the cascade decay H → X(→ h 1 h 2 )h 3 , but other contributions, usually S-wave amplitude, are also comparable. This can be seen as follows. Suppose that the amplitude of H → h 1 h 2 h 3 · · · h n in the region of phase space (m X − σ X ) 2 < s 12 < (m X + σ X ) 2 is dominated by the cascade decay H → X(→ h 1 h 2 )h 3 · · · h n , plus an S-wave amplitude, so that it can be expressed as a coherent sum: where, the amplitude of the cascade decay H → X(→ h 1 h 2 )h 3 · · · h n can be parameterized as M H→X(→h 1 h 2 )h 3 ···hn = a P c θ * 1 , 2 while the S-wave amplitude can be parameterized as M S−wave = a S . Then, the differential decay width will take the form where a Jacobi factor corresponding to the variable transformation from s 13 to c θ * 1 is omitted. After integrating over ds 12 , dτ and c θ * 1 , one has where the angle brackets represent the phase space integration over all variables - (m X −σ X ) 2 (· · ·)ds 12 dτ , c θ * 1 is integrated from -1 to 0 or from 0 to 1, respectively. By substituting Eq. (4) into Eq. (1), the forward-backward asymmetry can be expressed as From the above equation one can clearly see that the presence of both the Pand S-wave amplitudes M H→X(→h 1 h 2 )h 3 ···hn and M S−wave results in nonzero forward-backward asymmetry A F B H→h 1 h 2 h 3 ···hn . On the other hand, if only the P -wave amplitude M H→X(→h 1 h 2 )h 3 ···hn contributes, A F B H→h 1 h 2 h 3 ···hn would simply be zero.
Up to this point, nothing is mentioned about the CP conjugate pro-cessH →h 1h2h3 · · ·h n , and hence, the CP asymmetry. It is easy to see that if CP symmetry is respected, one would simply have and A F B H→h 1h2h3 ···hn would not equal to each other. Consequently, one can introduce a new observable measuring the CP asymmetry of multi-body decay H → h 1 h 2 h 3 · · · h n , which will be called the forward-backward asymmetry induced CP asymmetry (FBI-CP A) hereafter and is defined as From the definition of FBI-CP A one can easily see that its nonzero value indeed represents the violation of CP .

Discussions on FBI-CP A
One of the motivations for the introduction of A F B CP can be explained as follows. When the S-wave amplitude are comparable with the P -wave one in the vicinity of the resonance X, the regional CP asymmetries for c θ * 1 > 0 and c θ * 1 < 0, which are conventionally defined as are correlated with each other. To see this, one just needs to reexpressed by substituting Eq. (4) into Eq. (7), where A It can be seen that there are three terms in the numerator of the above equation, corresponding to three origins of the regional CP asymmetry, A CP (c θ * 1 ≷ 0): the 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 * 1 > 0) and A CP (c θ * 1 < 0), except for the difference in the denominator, while the last one changes signs. It is easy to see that the last origin of CP asymmetry for A CP (c θ * 1 ≷ 0) is proportional to the sine of the relative strong angle between the S-and Pwave amplitudes, which, according to Watson's theorem [33], comes from the final state interaction, so that it can be large because of its nonperturbative attribute. As a consequence, the last term in the numerator of Eq. (8) can be comparable with -some times it can even dominate over-the first two terms, resulting in a substantial difference between A reg CP (c θ * 1 > 0) and A reg CP (c θ * 1 < 0). In fact, it has a good chance that the signs of A reg CP (c θ * 1 > 0) and A reg CP (c θ * 1 < 0) are opposite because of the presence of the last term in the numerator of Eq. (8). Indeed, this kind of behaviour has already been observed in B ± → π + π − K ± and B ± → π + π − π ± [24], and has been studied extensively in the literature. One interesting property of the newly defined FBI-CP A is that it is capable of isolating the CP asymmetry associated with the interference of the S-and P -waves, which can be seen by expressing It is this property which motivates the introduction of FBI-CP A. 3 3 One can see that in contrast to the conventionally defined CP asymmetry, there are extra factors 1/ |a P | 2 /3 + |a S | 2 and 1/ |ā P | 2 /3 + |ā S | 2 in the above ex-One important issue, which has to do with the integration interval of s 12 , should be pointed out here. In the above discussion, the integral of s 12 is performed symmetrically around the X resonance, i.e. (m X − σ X ) 2 < s 12 < (m X + σ X ) 2 . This may result in some cancellation when obtaining the FBI-CP A or regional CP As. This has to do with the fact that the interference term may flip its sign at s 12 ∼ m 2 X . To see this in more detail, let us first write out the Breit-Wigner in the P -wave amplitude explicitly: whereã P is introduced to isolate the Breit-Wigner factor from a P , so that the interference term in A F B , A F B CP , and A reg CP 's can be expressed as where the first term implies a sign flip at s = m 2 X . Roughly speaking, when the phases of a S andã P are about the same, the first term would dominate, and there will be large cancellation when s 12 is integrated from (m − σ X ) 2 to (s 12 + σ X ) 2 , resulting in a much smaller FBI-CP A or regional CP A for the region (m ρ − σ ρ ) 2 < s 12 < (m ρ + σ ρ ) 2 . In fact, the this kind of behaviour has already been observed by LHCb in the decay B ± → π + π − π ± , in which it is shown that the regional CP asymmetries for s 12 below and above the ρ 0 (770) mass squared, (m ρ − σ ρ ) 2 < s 12 < m 2 ρ and m 2 ρ < s 12 < (m ρ + σ ρ ) 2 , tend to take opposite signs [26,27]. The aforementioned cancellation can be circumvented by choosing different interval of s 12 . For example, for the case of B ± → π + π − π ± , one can measure the FBI-CP As defined on (m ρ − σ ρ ) 2 < s 12 < m 2 ρ and m 2 ρ < s 12 < (m ρ + σ ρ ) 2 , respectively, instead of that of the combined interval (m ρ − σ ρ ) 2 < s 12 < (m ρ + σ ρ ) 2 .
Besides the aforementioned motivation, another important advantage of FBI-CP A is that it can almost effectively double the statistics in the experiments. Consequently, as a complement to the reginal CP asymmetries, FBI-CP A can be used in searching for CP violations in some three-body decays of beauty or charmed meson, in which the CP violations are expected pression. Of course, when CP violation is small, so that the above two factors are nearly equal, A F B CP would be proportional to ℜ( a P a * S ) − ℜ( ā Pā * S ), just like the third term in the numerator of Eq. (8) does.
to be small so that higher statistics is essential. To see this, one first needs to notice that FBI-CP A can be approximated to an experimentally useful expression, which is if the CP violation is small. By comparing to the conventionally defined regional CP asymmetries A reg CP (c θ * 1 ≷ 0) in Eq. (7), it can be clearly seen that the statistics has indeed almost doubled.
It would be useful to further compare FBI-CP A with the regional CP asymmetry A reg CP (c θ * < 0 & c θ * > 0), which is defined as .
Just as the cases of B ± → π + π − π ± and B ± → π + π − K ± , although the regional CP asymmetries A reg CP (c θ * 1 > 0) and A reg CP (c θ * 1 < 0) around the vicinity of ρ 0 can be large, they tend to take opposite signs because of the interference of the S-and P -waves, hence there is cancellation when summing up the event yields to obtain the regional CP asymmetry A reg CP (c θ * 1 < 0 & c θ * 1 > 0). In fact, the CP asymmetry originated from the interference of the S-and P -waves is totally cancelled, which can be seen from the expression: Consequently, FBI-CP A may take larger values than A reg , making the former easier to observe.
From the above analysis, one can see that FBI-CP A serves as a complementary observable for CP asymmetries around the vicinity of X, along . Moreover, for some multi-body decays of B and D mesons, it has a good chance that CP violation could be first confirmed through the measurement of FBI-CP A.

Application potential to multi-body decays of charm and beauty mesons
There are a lot of channels which are suitable to perform the measurements of FBI-CP A. In the B meson sector, for channels such as B ± → π + π − K ± and B ± → π + π − π ± [24], there are very clear interference effect between S-and P -wave when the invariant mass of π + π − lies around the vicinity of the vector resonance ρ 0 (770). The regional CP asymmetries has already been measured by LHCb. We suggest to perform measurements of FBI-CP A around ρ 0 (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-CP A could potentially find evidence of CP violations in D ± → K + K − π ± [34] and D ± (s) → π + π − π ± [35]. For D ± → K + K − π ± , the resonances K * (892) and φ(1020) are clearly visible in the Dalitz plot. The forward-backward asymmetries for these two P -wave resonances are also visible. It would be interesting to check weather the CP violation shows up in FBI-CP A around these resonances. For D ± → π + π − π ± , the vector resonance ρ 0 (770) and its forward-backward asymmetry is also visible.
For an illustration, we consider the decay process D ± → K + K − π ± . From FIG. 2 of Ref. [34] one can see that the forward-backward asymmetry around the resonance K * (892) 0 is quite clear, indicating an interference effect. This interference is probably caused by the S-wave resonance K * 0 (700). If this is the case, the decay amplitudes in the phase space region around the vicinity of the resonance K * (892) 0 can then be expressed as where s X = s − m 2 X + im X Γ X (X = K * or K * 0 ), and the decay amplitudes can be parameterized as where λ q = V uq V * cq . Since λ s ≫ λ b , we can rewrite the above two amplitudes into the form and φ is the phase difference between λ s and λ b . The FBI-CP A is then approximated to be The relative strong phase between the amplitudes corresponding to these two resonances can be large because of the non-perturbative effect. As a consequence, FBI-CP A will be roughly of the order A F B CP = | λ b λs | sin φ ∼ 0.1%, which is just about the same order with the regional CP As. In order to distinct from zero for such a small value, the statistics should be large enough. In this sense, the measurement of FBI-CP A is better than that of the regional CP As, as the former can make use of the data more efficiently. Although FBI-CP A is defined through forward-backward asymmetry, one does not need to obtain FBI-CP A by means of the measurement of forward-backward asymmetry at all for this situation. Since the CP asymmetry is quite small, one just needs to measure FBI-CP A in D ± → K + K − π ± around K * (892) 0 according to Eq.
(12), from which one can see that the statistics are indeed almost doubled comparing to the regional CP asymmetry in Eq. (7). In fact, besides the above suggested decay channels, both the measurements of the forward-backward asymmetry and FBI-CP A are meaningful in other multi-body decay channels of charm and beauty meson, provided that a P -wave resonances is presented in the Dalitz plot.

Conclusion
To sum up, we introduce an observable for CP violations in multi-body decays of heavy meson, the forward-backward asymmetry induced CP asymmetry, FBI-CP A. We suggest to perform the measurements of FBI-CP A in some decay channels of charm and beauty mesons.