Analytic Resolution of Puzzle in $B\to K \pi$ decays

We present a systematic method to extract each standard model (SM)-{\it like} hadronic parameter as well as new physics parameters in analytic way for $B\to K\pi$ decays. Using the analytic method to the currently available experimental data, we find two possible solutions analytically equivalent: one showing the large SM-{\it like} color-suppressed tree contribution and the other showing the large SM-{\it like} EWP contribution. The magnitude of the NP amplitude and its weak phase are quite large. For instance, we find $|P^{NP}/P| = 0.39\pm0.13$, $\phi^{NP}=91^\circ\pm15^\circ$ and $\delta^{NP}=8^\circ\pm27^\circ$, which are the ratio of the NP-to-SM contribution, the weak and the relative strong phase of the NP amplitude, respectively.


I. INTRODUCTION
Brilliant progress of the B factory experiments sheds light on the study of rare B decays, which are crucial for testing the standard model (SM) and detecting any hints beyond the SM. Especially, B → Kπ decays are of great importance not only for investigating new physics (NP) due to the property of penguin dominance but also examining one of angles of Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle [1,2,3]. Many elaborate theoretical calculations based on QCDF [4], PQCD [5,6] and SCET [7] have been done for physical observables within the SM. But, some experimental data have shown considerable discrepancy from the theoretical estimation, inspiring searching NP in B → Kπ decays.
The ratios are expected to satisfy R c ≈ R n within the SM [3]. Before ICHEP-2006, those experimental values had shown a significant discrepancy, but as time passes they were getting closer to each other [8]. Current data updated by March 2007 in HFAG [9] show R c = 1.12 ± 0.07 and R n = 0.98 ± 0.08, which are consistent with the SM expectation. On the other hand, the CP asymmetry measurements still show a disagreement with the SM prediction. The SM naively expects A CP (B 0 → K + π − ) ≈ A CP (B + → K + π 0 ) for the direct CP asymmetry and (sin 2β) K S π 0 ≈ (sin 2β) ccs = 0.68 for the mixing-induced CP asymmetry. But the current experimental data show (sin 2β) K S π 0 − (sin 2β) ccs = −0.30 ± 0.19.
The recent PQCD result for the difference of the above direct CP asymmetries is 0.08 ±0.09, which is actually consistent with the data. However, the PQCD prediction A CP (B + → K + π 0 ) P QCD = −0.01 +0.03 −0.05 still has 1.5σ difference from the current experimental data A CP (B + → K + π 0 ) EXP = 0.050 ± 0.025. Moreover, the difference of the mixing-induced CP asymmetry from the PQCD prediction is 0.065 ± 0.04, which shows about 2σ off the data.
Searching for NP via the electroweak penguin (EWP) processes in the B → Kπ decays has drawn lots of attention for a long time, especially based on various specific NP scenarios such as SUSY models [10], flavor-changing Z ′ models [11], four generation models [12], and so on. On the other hand, numerous model-independent attempts have been also made in search of NP within the quark diagram approach [13,14,15,16,17,18]. According to re-parametrization invariance (RI) which was recently proposed in Ref. [19,20], any NP contribution can be absorbed into the SM amplitudes always in pair: for example, both the color-suppressed tree and the EWP amplitude. Thus, we would like to point out that the large enhancement of the color-suppressed tree amplitude and the EWP amplitude can be simultaneously understood by the single NP contribution with a non-zero NP weak phase within the model-independent analysis.
Our main goal in this work is to propose a systematic method for extracting each hadronic parameters in the presence of the single NP contribution under the consideration of RI. It will be shown that the parametrization with the additional NP contribution can be modified into the same form of the parametrization of the SM. The complete analytic solution for each hadronic parameters in this SM-like parametrization will be given in terms of the experimental data, and also their numerical values. Therefore, once the experimental data are given, one can pinpoint the hadronic parameters and will be able to directly compare to the theoretical estimations. For the extraction of NP parameters, the additional theoretical inputs are needed. To this end, we adopt two different schemes, one is flavor SU(3) symmetry and the other is PQCD prediction. It is discussed that how this NP contribution depends on the weak phase γ.

II. PARAMETRIZATION AND REPARAMETRIZATION INVARIANCE
In the quark diagram approach [21,22], the decay amplitudes of four B → Kπ modes are described as under the redefinition of Each topological parameter represents strong penguin (P), electro-weak penguin (P EW ), exchange penguin (EP), tree (T ), color-suppressed tree (C) and annihilation (A) topologies, respectively. The superscript C on the penguin parameters denotes a color-suppressed process. It is understood that each parameter includes both the weak phase and the strong phase in it. Each penguin parameters are involved in three terms associated with the internal quark exchanges. They can be manipulated by using unitarity of the CKM matrix. Note that the CKM factors relevant to each parameter are V * tb V ts for the P tc , P EW , P C EW and V * ub V us for the T , C, A, P uc . The relative sizes among these parameters are roughly estimated within the SM [22] as where λ ∼ 0.2 from the Wolfenstein parametrization [23]. For the relative size of |P uc |, one can roughly estimate that Note thatP u andP c are smaller thanP t [24], and more precisely it can be estimated that 0.2 < |P uc /P tc | < 0.4 within the perturbative calculation [25]. Therefore, we assume |P uc /P tc | ∼ O(λ 3 ) for our analysis. It has been generally argued that the NP effects, if present, are the size of the EWP amplitude or smaller in B → Kπ decays. Thus we neglect all the minor contributions smaller than |C|, such as A and P uc , for simplicity. (We also neglect P C EW , since the |P C EW | is expected to be smaller than |C| [14,16].) Therefore, in our analysis the limit of NP sensitivity would be the order of |C| (∼ λ 2 P tc ) at most.
Explicitly showing the weak phase γ and the strong phases δ, the decay amplitudes can be rewritten as where P ≡ |P tc |, r T ≡ |T /P tc |, r C ≡ |C/P tc |, r EW ≡ |P EW /P tc |, which are defined to be positive. We set the strong phase of the penguin contribution P to be zero so that all the other strong phases are relative to it. It is also used that V * tb V ts = −|V * tb V ts |. We assume that the weak phase γ can be measured from elsewhere. Then the number of unknown parameters in the above decay amplitudes is 7 (P, r T , r C , r EW , δ T , δ C , δ EW ) within the SM.
We again emphasize that this approximated parametrization is the most efficient way to probe new physics up to the order of |C|. Now we introduce a single NP contribution coming through the EWP (or the color suppressed tree) contribution such as where P N is defined to be positive, and φ N and δ N are weak and strong phase of the NP term, respectively. Then the two decay amplitudes in Eqs. (14) and (15) are modified by simply adding the NP term in the EWP contribution: where r N ≡ P N /P . It has been introduced that any single decay amplitude can be separated into two decay amplitudes which have arbitrary weak phases θ and η, respectively, unless θ and η are equal or modulo π [19]. Since any physical results should not be changed, it is called reparametrization invariance (RI). More explicitly, any phase term e iφ can be separated as where the phases θ and η are arbitrarily chosen, satisfying θ − η = 0 (mod π). This is a simple algebraic identity. Due to this identity, the NP amplitude can be re-expressed as Here, the weak phases γ and 0 are chosen in order to match with the weak phases of the color-suppressed tree and EWP amplitudes. Then those two terms can be absorbed into the parameters of the color-suppressed tree and EWP leading to the following parametrization which have the same form of the SM parametrization with the following modified parameters The NP amplitude is now absorbed into the SM parameters of the color-suppressed tree and EWP. Therefore, color-suppressed tree amplitude in these SM-like parametrization can be affected by the NP contribution of EWP unless φ N = 0 as shown in Eq. (23).

III. ANALYTIC SOLUTIONS FOR THE SM-LIKE PARAMETERS
In this section, we present the analytic solutions for the SM-like parameters [26]. For the first step, we rewrite Eqs. (12), (13), (21), and (22) as where A ij denote magnitudes of the decay amplitudes of B → K i π j and α ij represent their complex phases (ij = {0+, +−, +0, 00}). We put a bar on top of the amplitude parameters in case of the CP conjugate modes. It should be noted that these SM-like parametrization is including the NP contribution, namely the one coming into EWP sector, via RI. Table I shows current experimental data for the B → Kπ decays [27,28]. We use the notation for the branching ratios and CP asymmetries compatible with HFAG [9]:  9.9 ± 0.6 A CP (K 0 π + ) −0.029 ± 0.039 ± 0.010 0.03 ± 0.03 ± 0.01 0.18 ± 0.24 ± 0.02 0.009 ± 0.025 0.030 ± 0.039 ± 0.010 0.07 ± 0.03 ± 0.01 −0.29 ± 0.23 ± 0.02 0.050 ± 0.025 The number of parameters is 7 (P, r T , r M C , r M EW , δ T , δ M C , δ M EW ), while 9 observables are available in B → Kπ decays. Since A 0+ CP automatically vanishes in our parametrization, we discard the data. Setting aside the mixing induced CP asymmetry data S K S π 0 , we use the remaining 7 experimental data in order to determine the 7 parameters. ¿From Eq. (25) we easily get the solution for P in terms of the observable by taking into account the phase space factor: Combining Eqs. (25) and (26), one finds [16] that where R is given [2] by The analytic solutions for δ T and r T are obtained in terms of the observables from the above equations are Using the experimental data given in Table I, we obtain numerical values of r T and δ T .
Since the second value of r T is unreasonably larger than the SM expectation, which is around 0.15, we safely choose the first one as our solution.
It is instructive to represent the phases α 00 andᾱ 00 geometrically as in Fig. 2. The figure shows the famous isospin quadrangle in a complex plane depicting the isospin relation among the decay amplitudes for B → Kπ: The notation A(B → K i π j ) ≡ A ij e iα ij is used in the figure and A ij (1,2) corresponds to the case of α 00 (1,2) . The isospin quadrangle can be geometrically constructed as follows. The two complex values of A(B + → K 0 π + ) and A(B 0 → K + π − ) in the complex plane are fixed from the solutions shown above. Subsequently, the value xe iζ is determined, where FIG. 2: The isospin quadrangles in a complex plane displaying the isospin relation among the decay amplitudes for B → Kπ. A ij (1,2) corresponds to the case of α 00 (1,2) . (43). Since the magnitudes A +0 and A 00 are directly determined from the measurements, we find two distinct solutions for A(B + → K + π 0 ) and A(B 0 → K 0 π 0 ) which are expressed as A +0 (1,2) and A 00 (1,2) in Fig. 2. Two sides of the quadrangles denoted by the diamond marks (and the circle marks) are equal in length to each other. The quadrangles in the figure have been constructed by using the present experimental data. We recall that the weak phase γ has been used as an input in Eq. (32).
Next, we find analytic solutions for r M C , r M EW , δ M C , δ M EW in terms of the observables, using α 00 andᾱ 00 determined in Eqs. (45) and (46). To this end, we use Eq. (28) and its CP conjugate version. They can be rewritten as where It is straightforward to find the solutions for r M C , r M EW , δ M C , δ M EW as a function of y,ȳ and η,η from Eqs. (48) and (49) : We note that there occurs no ambiguity in the above equations. Therefore, we have found the analytic solutions for the 7 parameters: (P, r T , δ T ) without ambiguity, and (r M C , r M EW , δ M C , δ M EW ) with a four-fold discrete ambiguity which stems from α 00 andᾱ 00 given in (45) and (46).  Table II shows the result for each case. The prediction for S K S π 0 is also given for each case. Provided precise measurement of S K S π 0 , one can choose consistent solutions with the data of S K S π 0 among these 4 cases. Then, as mentioned before, one can analyze each hadronic parameters of the solutions, comparing to given theoretical estimation such as PQCD and QCDF. The Case 3 solution is discarded because its prediction for S K S π 0 is quite different from the current data. The solutions for Cases 2 and 4 are our favorites because their predictions for S K S π 0 are consistent with the data within 1σ error. The Case 2 solution shows large color-suppressed tree than the typical SM estimation, while the Case 4 solution presents large EWP, where both cases suggest considerable NP contribution.
Please note that many authors uncovered that the anomalous behaviors of the experimental data could be accommodated with the enhancement of the EWP amplitude [13,14] as well as an additional weak phase in the electroweak sector [15,16,17], and a few authors have also found that the color-suppressed tree amplitude would be the main source of NP in the B → Kπ modes [17,18]. Due to our analytic approach, we can find two solutions

IV. EXTRACTING NEW PHYSICS PARAMETERS AND DISCUSSION
Finally, we would like to solve Eqs. (23) and (24) for the NP parameters r N , δ N and φ N . The left-hand side of Eqs. (23) and (24) has 4 parameters which can be obtained from the analytic solution shown above. But the number of unknown parameters on the right-hand side is 7 (r C , δ C , r EW , δ EW , r N , δ N , φ N ) Thus there is no model independent way to extract NP parameters without additional theoretical inputs. We need at least 3 additional inputs in the color-suppressed tree and the EWP sector in order to determine NP parameters. Here, we adopt two different schemes for the additional theoretical inputs: one is flavor SU (3) symmetry, and the other is recent PQCD calculation.
Using the flavor SU(3) symmetry, we estimate the color-suppressed tree amplitude from the B → ππ decays amplitudes, following the Ref. [16], where the C ππ is color-suppressed tree amplitude of B → ππ decays. The EWP amplitude is also associated with the tree and color-suppressed amplitudes under flavor SU (3) symmetry [30] as The parameters r T and δ T can be obtained within the B → Kπ modes as in Eq. (39).
And the parameters r C and δ C are given by Eqs. (56) and (57) combined with Eq. (33).
Subsequently r EW and δ EW can be obtained from the above equation. We summarize the result: r EW e iδ EW = (0.14 ± 0.04) e i(9±10) • .

SU(3)
On the other hand, The Recent PQCD calculation for the B → Kπ decays gives [6] r C e iδ C = (0.039) e −i61 • , r EW e iδ EW = (0.12) e i22 • . Nevertheless, we adopt above 4 additional inputs in order to get rid of discrete ambiguity.

PQCD (60)
We define the following quantities: The parameters of ∆r C , ∆δ C , ∆r EW , and ∆δ EW can be extracted using above additional theoretical inputs. Then, we can easily see from Eqs. (23) and (24) that the following relation should be satisfied: And, we find the solutions of NP parameters as For the δ N , two different solutions are possible as shown in Eq. (64). Since the strong phase of NP contribution is expected to be small, we choose the one with close to the δ EW . The numerical values for the solution with the two different schemes of theoretical inputs are shown in Table III. We can see that the result is consistent each other for both schemes of theoretical input. Note that in both cases, for both schemes of theoretical input, the magnitude of the NP amplitude is quite large and its weak phase is also sizable. Since the experimental value of γ still has large uncertainties, we investigate how our NP solutions depend on these experimental results. We perform a minimum χ 2 analysis to get the NP solution in order to simply see the dependence. After employing four additional inputs of r C , δ C , r EW , δ EW from flavor SU(3) symmetry, the number of unknown parameters is 6 (P, r T , δ T , r N , φ N , δ N ) while we can use 8 available experimental data excluding A 0+ CP . The fitting result as a function of γ is shown in Fig. 3. As we can see, the NP contribution is not much sensitive to γ.

V. CONCLUSIONS
In this work, we present complete analytic method for analyzing the hadronic parameters with the single NP contribution under consideration of reparametrization invariance. It is shown that any single NP contribution in the color-suppressed tree sector or EWP sector can affect both the SM parameters of color-suppressed tree and EWP. We show the analytic solution for every parameters of SM-like parametrization, and also for the NP parameters.
Therefore one can pinpoint each hadronic parameters and compare them to the theoretical estimations once the precise experimental data are given. There were 4 possible solutions for the SM-like parameters which can be chosen rightfully by considering mixing induced CP asymmetry data. Consequently, it could be understood simultaneously that the two different intriguing solutions occur: one is large color-suppressed tree and the other is large EWP. We obtain the solution for the NP parameters after adopting additional theoretical input. The solution shows quite large NP contribution and sizable weak phase of it.

ACKNOWLEDGMENTS
The work of C.S.K. was supported in part by CHEP-SRC and in part by the KRF Grant