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Determination of the angle gamma using multibody D decays in B^+- ->D K^+-

We describe a method for determining gamma using B^+- ->D K^+- decays followed by a multibody D decay. In the talk we focus on K_S pi^- pi^+ final state, but other modes such as D ->K_S K^- K^+ and D ->K_S pi^- pi^+ pi^0 can also be used. The main advantages of the method are that it uses only Cabibbo allowed D decays, and that large strong phases are expected due to the presence of resonances. Since no knowledge about the resonance structure is needed, gamma can be extracted without any hadronic uncertainty.


Basic idea
The theoretically cleanest way of determining the angle is to utilize the interference between the b → cūs and b → ucs decay amplitudes [1][2][3][4][5][6][7], see Fig. 1. The salient feature of these transitions is that they involve only distinct quark flavors and therefore do not receive any penguin contributions. In the original idea by Gronau and Wyler (GW) [1] the B ± → D CP K ± decay modes are used, where D CP represents a D meson which decays into a CP eigenstate. The dependence on γ arises from the interference between the B ± → D 0 K ± and B ± → D 0 K ± decay amplitudes. The main advantage of the GW method is that, in principle, the hadronic parameters can be cleanly extracted from data, by measuring the B ± → D 0 K ± and B ± → D 0 K ± decay rates. In practice, however, measuring γ in this way is not an easy task. Due to the values of the CKM coefficients and color suppression, the ratio between the two interfering amplitudes, r B [see Eq. (4)], is expected to be small, of order 10% − 20%. This reduces the sensitivity to γ, which is roughly proportional to the magnitude of the smaller amplitude. It also leads to experimental difficulties in measuring the color suppressed B − → D 0 K − mode (and its charge conjugate) preventing a straightforward application of the * Talk given by J. Zupan, based on [8].
GW method [2]. There exist a number of extensions of the original GW proposal which avoid the problem by not relying on the measurement of B − → D 0 K − amplitude [2,3,6]. Instead several different decay modes of D mesons such as quasi two-body D decays with one particle a resonance (e.g. D 0 → K * + π − [2,3]) are used. Since these are really three body decays (for instance in the example mentioned, K * + decays strongly to K 0 π + or K + π 0 ) one can pose the following questions : • Can one use the complete phase space of such threebody D decays for γ extraction?
• Is it possible to avoid fits to Breit-Wigner forms in doing the Dalitz plot analysis?
As we show in the following, the answer to both of these questions is positive. The first question was raised already in [3], however, most of the results and applications we present are new. For the sake of concreteness, we concentrate on the D → K S π − π + decay mode, while an extension to a larger set of the decay modes can be found in [8]. The advantage of using the chosen decay chain is threefold. First, one expects large strong phases due to the presence of resonances. Second, only Cabibbo allowed D decay modes are needed. Third, the final state involves only charged particles, which have a higher reconstruction efficiency and lower background than neutrals. The price one has to pay is that a Dalitz plot analysis of the data is needed. We describe how to do the Dalitz plot analysis in a model-independent way, and explore the advantages gained by introducing verifiable model-dependence. The final balance between the advantages and disadvantages depends on yet-to-be-determined hadronic parameters and experimental considerations. Finally, we mention that an equivalent formalism to the one we present below has been independently developed by Atwood and Soni in [9].
and define the amplitudes The same definitions apply to the amplitudes for the CP conjugate cascade B + → DK + → (K S π + π − ) D K + , with the change of weak phase sign γ → −γ in (4). We have set the strong phase of A B to zero by convention, so that δ B is the difference of strong phases between the two amplitudes.
For the CKM elements, the usual convention of the weak phases has been used. The value of |A B | is known from the measurement of the B − → D 0 K − decay width using flavor specific decays of D 0 . The amplitude color suppressed and cannot be determined from experiment in this way [2]. The color suppression together with the experimental values of the ratio of the relevant CKM elements leads to the theoretical expectation r B ∼ 0.1 − 0.2 (see recent discussion in [7]).
For the three-body D meson decay we define where s i j = (p i + p j ) 2 , and p 1 , p 2 , p 3 are the momenta of the K S , π − , π + respectively. We also set the magnitude A 12,13 ≥ 0, such that δ 12,13 can vary between 0 and 2π. In the last equality the CP symmetry of the strong interaction together with the fact that the final state is a spin zero state has been used. With the above definitions, the amplitude for the cascade decay is where P D is the D meson propagator. Next, we write down the expression for the reduced partial decay width where d p denotes the phase space variables, and we used the extremely accurate narrow width approximation for the D meson propagator. The moduli of the D decay amplitude A 12,13 can be measured from the Dalitz plot of the D 0 → K S π − π + decay. To perform this measurement the flavor of the decaying neutral D meson has to be tagged. This can be best achieved by using the charge of the soft pion in the decay D * + → D 0 π + . However, the phase δ 12,13 of the D meson decay amplitude is not measurable in B-factories without further model dependent assumptions. If the three-body decay D 0 → K S π − π + is assumed to be resonance dominated, the Dalitz plot can be fit to a sum of Breit-Wigner functions, determining also the relative phases of the resonant amplitudes. This is further discussed in section 4. The other option is to use data from charm factory, where weighted averages of the sine and the cosine of the relevant phase difference may be measured (see section 3). Here we assume that no charm factory data is available and develop the formalism without any model dependent assumptions.
Obviously, to compare with the data, an integration over at least some part of the Dalitz plot has to be performed. We therefore partition the Dalitz plot into n bins and define where the integrals are done over the phase space of the i-th bin. The variables c i and s i contain differences of strong phases and are therefore unknowns in the analysis. The variables T i , on the other hand, can be measured from the flavor tagged D decays as discussed above, and are assumed to be known inputs into the analysis.
Due to the symmetry of the interference term, it is convenient to use pairs of bins that are placed symmetrically about the 12 ↔ 13 line, as shown in Fig. 2. Consider an even, n = 2k, number of bins. The k bins lying below the symmetry axis are denoted by index i, while the remaining bins are indexed withī. Theī-th bin is obtained by mirroring the i-th bin over the axis of symmetry. The variables c i , s i of the i-th bin are related to the variables of theī-th bin by while there is no relation between T i and T¯i. Note that had one used 12 ↔ 13 symmetric bins centered on the symmetry axis, one would have had s i = 0.
Together with the information available from the B + decay, we arrive at a set of 4k equationŝ These equations are related to each other through 12 ↔ 13 and/or γ ↔ −γ exchanges. All in all, there are 2k + 3 unknowns in (10), so that the 4k relations (10) are solvable for k ≥ 2. In other words, a partition of the D meson Dalitz plot to four or more bins allows for the determination of γ without hadronic uncertainties. This is our main result.
When c i = 0 or s i = 0 for all i, some equations become degenerate and γ cannot be extracted. However, due to resonances, we do not expect this to be the case. Degeneracy also occurs if δ B = 0. In this case, γ can still be extracted if some of the c i and/or s i are independently measured, as discussed in the following sections.

Improved Measurement of c i and s i
So far, we have used the B decay sample to obtain all the unknowns, including c i and s i , which are parameters of the charm system. We now show that the c i and s i can be independently measured at a charm factory [10][11][12]. This is done by running the machine at the ψ(3770) resonance, which decays into a DD pair. If one D meson is detected in a CP eigenstate decay mode, it tags the other D as an eigenstate of the opposite CP eigenvalue. The difference Figure 3. The interference between the decays of D mesons originating from ψ(3770) allow for a measurement of c i and s i at charm factories. Shown is a decay allowing for determination of c i . between the two decay widths gives [8] where we have defined D 0 ± ≡ (D 0 ± D 0 )/ √ 2. As stated above, obtaining these independent measurements reduces the error in the measurement of γ by removing k of the 2k + 3 unknowns.
In addition, if one of the D mesons decays into a non-CP eigenstate, we are sensitive to the s i variables as well. Consider for instance a ψ(3770) decaying into a D,D pair, of which one decays into K S π + π − and the other decays into some general state g. The partial decay width corresponding to the i−th bin of the K S π + π − Dalitz plot and the j−th bin of the g final state's phase space is where T g j , c g j , s g j are defined as in (8). In particular, if one chooses g = K S π + π − and j = i (or j =ī) one measures s 2 i . If, on the other hand, g is a CP even (odd) eigenstate, s g j = 0, T g j = T ḡ j = ±c g j and equation (13) reduces to (12).

Assuming Breit-Wigner dependence
If the functional dependence of both the moduli and the phases of the D 0 meson decay amplitudes A D (s 12 , s 13 ) were known, then the analysis would be simplified. There would be only three variables, r B , δ B , and γ, that need to be fit to the reduced partial decay widths in Eq. (7). A plausible assumption about their forms is that a significant part of the three-body D 0 → K S π − π + decay proceeds via resonances. These include decay transitions of the form D 0 → K S ρ 0 → K S π − π + or D 0 → K * − (892)π + → K S π − π + , as well as decays through higher resonances, e.g., f 0 (980), f 2 (1270), f 0 (1370) or K * 0 (1430). An important feature is that there exists an overlap region of Cabibbo allowed D 0 → K * − (892)π + , D 0 → K S ρ 0 decays, where the variation of strong phase will be large, allowing for the extraction of γ.
The decay amplitude can then be fit to a sum of Breit-Wigner functions and a constant term. Following the notations of Ref. [13] we write where the first term corresponds to the non-resonant term and the second to the resonant contributions with r denoting a specific resonance. The functions A r are products of Breit-Wigner functions and appropriate Legendre polynomials that account for the fact that D meson is a spin 0 particle. Explicit expressions can be found in Ref. [13].
One of the strong phases δ i in the ansatz (14) can be put to zero, while others are fit to the experimental data together with the amplitudes a i . The obtained functional form of A D (s 12 , s 13 ) can then be fed to Eq. (7), which is then fit to the Dalitz plot of the B ± → (K S π − π + ) D K ± decay with r B , δ B and γ left as free parameters.

Discussions
The observablesΓ ± i defined in (10) can be used to experimentally look for direct CP violation. Explicitly, Nonzero a CP requires non-vanishing strong and weak phases. Due to the resonances, we expect the strong phase to be large. Therefore, it may be that direct CP violation can be established in this mode even before the full analysis to measure γ is conducted. With more data, γ can be extracted assuming Breit-Wigner resonances (cf. section 4). Eventually, a model independent extraction of γ can be done (cf. section 2 and 3).
The above proposed method for the model independent measurement of γ involves a four-fold ambiguity in the extracted value. The set of equations (10) is invariant under each of the two discrete transformations The discrete transformation P π is a symmetry of the amplitude (6) and is thus an irreducible uncertainty of the method. It can be lifted if the sign of either cos δ B or sin δ B is known. The ambiguity due to P − can be resolved if the sign of sin δ B is known or if the sign of s i can be determined in at least some part of the Dalitz plot for instance by fitting a part of the Dalitz plot to Breit-Wigner functions.
The r B suppression present in the scheme outlined above can be somewhat lifted if the cascade decay B − → DX − s → (K S π − π + ) D X − s is used [5,7]. Here X − s is a multibody hadronic state with an odd number of kaons (for instance K − π − π + , K − π 0 or K S π − π 0 ). The same formalism as outlined above applies also to this case with trivial changes [8]. In addition to using different B modes, statistics may be increased by employing various D decay modes as well. An interesting possibility are Cabibbo allowed, D → K S π − π + π 0 , K − K + K S , and Cabibbo suppressed, D → K − K + π 0 , π − π + π 0 , K S K + π − decay modes, to which our formalism applies with very minor changes [8].
In conclusion, we have shown that the angle γ can be determined from the cascade decays B ± → K ± (K S π − π + ) D . The reason for the applicability of the proposed method lies in the presence of resonances in the three-body D meson decays that provide a necessary variation of both the phase and the magnitude of the decay amplitude across the phase space. The fact that no Cabibbo suppressed D decay amplitudes are used in the analysis is another advantage of the method and leads to a sensitivity on γ at order O(r B ). However, it does involve a Dalitz plot analysis with possibly only parts of the Dalitz plot being practically useful for the extraction of γ. In reality, many methods have to be combined in order to achieve the required statistics for a precise determination of γ [4].