Constrain the UT angle $\gamma$ by CP violation parameters in $B^0 \to \pi^+ \pi^-$

We calculate the tree and penguin amplitudes in the $B^0\to\pi^+\pi^-$ decay channel employing the perturbative QCD factorization approach. Using the amplitudes as input with the theoretical uncertainties sufficiently considered, we constrain the UT angle $\gamma$ to $53^\circ\leq\gamma\leq70^\circ$, from the measurements of the CP violation parameters $C_{\pi^+\pi^-}$ and $S_{\pi^+\pi^-}$ in $B^0\to\pi^+\pi^-$. The U-spin breaking effect between $B^0\to\pi^+\pi^-$ and $B_s^0\to K^+K^-$ is estimated to be around 30\%.


Introduction
In the Standard Model, the quark mixing is described by the Cabbibo-Kobayashi-Maskawa (CKM) matrix [1], in which the nonzero phase angle induces the Charge conjugation Parity (CP) violation in weak interaction. For recent developments on the CKM matrix, one can refer to the review [2]. It is important to examine the unitarity of the CKM matrix, since any deviation would indicate new physics beyond the Standard Model. The three angles of the well-known unitarity triangle (UT), which are defined by α ≡ arg[−(V td V * tb )/(V ud V * ub )], β ≡ arg[−(V cd V * cb )/(V td V * tb )] and γ ≡ arg[−(V ud V * ub )/(V cd V * cb )], have been measured by experiments and the present averages are [3] α = (85.4 +3. 9 −3.8 ) • , sin 2β = 0.682 ± 0.019, γ = (68.0 +8.0 The angle γ is the least known one among the three angles. Methods were proposed to extract γ from the tree-dominated modes B → DK, known as the GLW method [4], the ADS method [5], and the Dalitz-plot method [6], with different final states of D decays. Combining the B → DK measurements performed by Belle, BaBar, CDF and LHCb [7], the CKMfitter group [8] obtained the above average for γ. Recently, the LHCb collaboration made two new measurements [9]. Alternatively, γ can also be determined by the U-spin analysis on the two-body charmless B decays, B 0 → π + π − and B 0 s → K + K − [10]. A combination with the channels B 0 → π 0 π 0 and B + → π + π 0 makes the analysis more sophisticated [11]. Recently, following the method proposed in Ref. [11], the LHCb collaboration performed the U-spin and isospin analysis and obtained [12] which has a smaller central value than the world average in Eq. (1).
In this letter, we constrain the UT angle γ from B 0 → π + π − , with the help of factorization approach to calculate the tree and penguin amplitudes. Similar ideas have been used to constrain α from B 0 → π + π − [13], and to constrain γ from B 0 s → D ± s K ∓ [14]. However, neither of them got strong constraint on the corresponding UT angle for lack of precisely measured experimental results at their time. Recently, the CP violation parameters in B 0 → π + π − have been precisely measured [15], and the weighted averages of the results are given by [12], with the statistical correlation ρ(C π + π − , S π + π − ) = −0.007. The high precision of the parameters indicates the possibility that our constraint on γ is comparable to the world average in Eq. (1) and the results given in Ref. [12]. The method can also be applied to B 0 s → K + K − . The rest of the paper is organized as follows. In Sec. 2, the relevant formulas for the CP violation parameters in the channels B 0 → π + π − and B 0 s → K + K − are listed. In Sec. 3, we introduce our strategy for the numerical analysis and obtain the constraints on γ from the two channels, between which the U-spin breaking effect is also estimated. In Sec. 4, we conclude.

Theoretical formalism
For B 0 → π + π − , the relevant effective Hamiltonian is given by where O 1,2(3−10) are the tree (penguin) 4-quark operators, and C 1−10 are the corresponding Wilson coefficients. After we apply some factorization approach to calculate the hadronic matrix elements π + π − |O i |B 0 , the amplitude of B 0 → π + π − can be expressed as where T and P are the tree and penguin amplitudes, respectively. Defining with d and θ real-valued, we obtain the expression for the CP violation parameters For C π + π − and S π + π − , we have accepted the convention in the letter [12], where q and p are the coefficients in the mass eigenstates p|B 0 ± q|B 0 . Similarly for B 0 s → K + K − , one has where the real-valued parameters are defined by with T (P ) representing the tree (penguin) amplitude in B 0 gives the mixing phase in the B 0 s −B 0 s mixing system.

Numerical Analysis
The present average of the UT angle β is given in Eq. (1), which has a two-fold ambiguity 2β → π − 2β. A series of measurements [16] prefer that cos 2β is positive, so we accept Choosing the sample values for d and θ, de iθ = 0.23e i2.4 , we can then obtain the γ dependence of C π + π − and S π + π − , as shown in Fig. 1. The experimental 1 σ allowed regions are also displayed. Fig. 1(b) shows that S π + π − is very sensitive to the change of the angle γ, and at meanwhile precise measurements for S π + π − have been performed. This indicates that γ is potentially to be strongly constrained in our method, though there are considerable theoretical uncertainties in any factorization approach.  In the perturbative QCD (PQCD) approach base on the transverse momentum factorization [17], hadronic matrix elements are factorized into convolutions of the calculable hard kernels and the non-perturbative meson wave functions which are however universal. The PQCD approach has been applied in analysis on hadronic B meson decays, successfully making predictions for both branching ratios and CP violation [18,19]. Especially for B 0 → π + π − , the PQCD prediction of the branching ratio is (5.8 +3.0+0.5+0.4 −2.1−0.4−0.3 ) × 10 −6 [19], which is consistent with the experimental result (5.12 ± 0.19) × 10 −6 [3]. Therefore, we employ the PQCD approach to calculate the tree and penguin amplitudes here. The formulas for calculating the leading-order decay amplitudes are given by Eqs. (50 -61) in Ref. [19]. We also include the next-to-leading-order corrections to the B → π transition form factors, of which the twist-2 and -3 contributions have been studied in Ref. [20] and [21], respectively.
To perform a reliable analysis, we need to sufficiently take into account the uncertainties introduced by the calculation of the hadronic matrix elements. In the calculation, we adopt the updated non-asymptotic distribution amplitudes [22], where C α n (2x − 1) are the well known Gegenbauer polynomials with x the longitudinal momentum fraction of the quark in pion. The values of the Gegenbauer moments, a π 2 and a π 4 , have been determined in the global fit to the data of the pion electromagnetic form factor [23], which yields a π 2 = 0.17 ± 0.08, a π 4 = 0.06 ± 0.10.
To keep it safe, we double the error bars in the numerical analysis. In Ref. [24] where the joint resummation was performed for the pion transition form factor in the transverse-momentum factorization formalism, the authors found that their prediction for the form factor with a π 2 = 0.05 agrees well with the experimental data. Our choice for the range of a π 2 covers this value. As for the other non-perturbative parameters η π 3 and ω π 3 , we accept the values taken in Ref. [22], also with doubled error bars. The shape parameter in the distribution amplitude of the B 0 meson [25] is taken value in the range We also consider the uncertainties caused by the unknown next-to-leadingorder corrections characterized by the choice that Λ QCD ∈ [0.20, 0.30] and a 20% variation of the factorization scale. Taking values for the theoretical parameters randomly in the ranges covering their uncertainties, we perform the PQCD calculation and obtain 99 points of (d, θ), which are shown in Fig. 2. At each point of (d, θ), we perform the global fit of β and γ to the experimental results of the CP violation parameters in Eq. (3) and that of β in Eq. (11). Then, we combine the 1 σ allowed regions of all fits at the 99 points, and regard it as our constraint on γ and β. As shown in Fig. 3, the constraint on γ is 53 We can also perform the similar analysis to B 0 s → K + K − using Eqs. (9) and (10), though the experimental results for the CP violation parameters are much less precise, which are given by [12] C K + K − = 0.14 ± 0.11, S K + K − = 0.30 ± 0.13, with the statistical correlation ρ(C K + K − , S K + K − ) = 0.02. To improve the precision on the determination of γ, β s is expressed in terms of β and γ. However, the B 0 s → K + K − constraint 20 • ≤ γ ≤ 150 • is still too loose. As a byproduct, we also estimate the U-spin breaking effect in the two channels B 0 → π + π − and B 0 s → K + K − , which is parameterized by The PQCD result is In the letter [12], the U-spin breaking effect is parameterized by two relative magnitudes r D and r G with the corresponding phase shifts θ r D and θ r G , d e iθ = de iθ 1 + r G e iθr G 1 + r D e iθr D .
Assuming the parameters range within the region the authors obtained γ = (63.5 +7.2 −6.7 ) • . This region can fully cover the PQCD result (including the uncertainties), so we conclude that the assumption about the U-spin breaking in Ref. [12] is reasonable.

Conclusion
We extract the UT angle γ from the precise experimental results of C π + π − and S π + π − given in the letter [12], with the tree and penguin amplitudes in B 0 → π + π − calculated in the PQCD approach. Including the theoretical uncertainties, we constrain 53 • ≤ γ ≤ 70 • at 68% probability. Through the similar method, the angle γ is also constrained in the range 20 • − 150 • by the measurements of C K + K − and S K + K − . The U-spin breaking effect between the two channels is found to be smaller than 50%, which indicates that the results in the letter [12] are reliable.