Evidence of A CP ( D 0 → π + π − ) implies observable CP asymmetry in the D 0 → π 0 π 0 decay

Inspired by the recent measurement of CP asymmetry in the individual mode on LHCb, we study CP asymmetry of the D → ππ system in the isospin and topological analysis. The ratio between penguin and tree amplitudes P/ ( T + C ) in the D → ππ system is found to be greater than two in most values of the relative strong phase. And D 0 → π 0 π 0 is a potential mode to reveal the CP asymmetry of the order of 10 − 3 , which would be observed by Belle II in the future. The large CP asymmetry in the D → ππ system might be understood in the t -channel ﬁnal-state interaction.


I. INTRODUCTION
CP asymmetry in D meson decay, which is defined as provides a window to test the Standard Model (SM) and search for new physics (NP) in the up-type quark weak decay in hadrons.The LHCb collaboration observed CP asymmetry of charm decay in 2019 [1], After that, many experimental efforts are devoted to the measurement of CP asymmetry and mixing parameters in charm system [2][3][4][5][6][7][8].In theoretical aspect, there are two controversial viewpoints for the observed CP asymmetry difference in literature, regarding it as signal of new physics [9][10][11][12], or the non-perturbative QCD enhancements to penguin [13][14][15][16][17][18][19][20][21].It attributes to the large ambiguities in evaluating penguin topologies and the absence of more information given by experiments.
CP asymmetry in the individual mode is more significant compared to the difference between two decay modes because it allows us to extract more knowledge of non-perturbative QCD.The newest data indicates a very large U -spin breaking in the D 0 → K + K − and D 0 → π + π − modes, which is beyond the naive expectations of ε ∼ m s /Λ QCD ∼ 30% [23].
In this work, we analyze the implications of the new measurement of CP asymmetry in the By applying the isospin and topological analysis, we show the ratio between penguin and tree amplitudes in the D → ππ system is greater than 2 in most values of strong phase.And CP asymmetry in the D 0 → π 0 π 0 decay could reach to be O(10 −3 ), which is available on Belle II in the future.The large CP asymmetry in the D → ππ system might be understood in the t-channel final-state interaction (FSI) [24][25][26][27][28][29][30][31][32][33][34][35][36][37].
This paper is organized as follows.In Sec.II, we analyze CP asymmetry in the D → ππ system in the isospin and topological analysis.In Sec.III, we try to explain the large CP asymmetry in charm in the final state interaction.And Sec.IV is a short summary.
Isospin decompositions of the D 0 → π + π − , D 0 → π 0 π 0 and D + → π + π 0 modes are in which A 3/2 and A 1/2 are the amplitudes with ∆I = 3/2 and ∆I = 1/2, respectively.Topological decompositions of the D → ππ modes can be expressed as where The contributions from the penguin operators O 3−6 and the chromomagnetic penguin operator O 8g are neglected in Eqs.(8)∼ (10).T , C and E denote the tree amplitudes and P and P b denote the penguin amplitudes.P b is the difference between P d and P s , P b = P d − P s , and P = P s .P d and P s are the topologies with d and s in the quark loop respectively.In literature such as [38], P is written as penguin plus penguin annihilation diagrams, and P break is defined as P break = P s −P d = −P b .In order to math the isospin amplitudes, the quark compositions of D and π mesons are defined as 8)∼ (10).By comparing Eqs. ( 8)∼ (10) with Eqs. ( 5)∼ (7), the relations between isospin amplitudes and topological amplitudes are found to be Eqs. ( 11) and ( 12) can also be derived from the effective Hamiltonian of charm decay by analyzing the isospin structure of tree and penguin operators, see literature such as Ref. [39] for details.
In the SM, λ b is much smaller than λ d , λ b /λ d ∼ O(10 −4 ) [40].The last term in Eq. ( 12) can be neglected safely in the branching fractions.We define an approximate ∆I = 1/2 amplitude without the λ b P term as A 1/2 can be written as with magnitude A s 1/2 and relative strong phase The strong phase of A 3/2 is usually set to be zero.The isospin amplitudes A 3/2 and A 1/2 can be extracted from the branching fractions of three D → ππ modes which are given by [40] The partial decay width Γ is parameterized to be There are two solutions for the strong phase δ I .Superscripts n and p are used to distinguish the negative and positive solutions.Eq. ( 17) is consistent with the result given by Ref. [41].
To analyze CP asymmetries in the D → ππ modes, we parameterize the penguin amplitude P as in which P and δ p are the magnitude and strong phase (with respect to A 3/2 ) of penguin amplitude respectively.γ is phase parameter of the CKM matrix, known as φ 3 in the unitarity triangle.In the SM, γ is fitted to be 1.144 ± 0.027 [40].The weak phase of λ d is negligible compared to γ.With the isospin and topological amplitudes, CP asymmetries of the D 0 → π + π − and D 0 → π 0 π 0 decays are derived to be In Eqs.(19) and (20), the first term in numerator is the CP asymmetry in A 1/2 and the second term is the interference between A 1/2 and A 3/2 .
There are two non-determined parameters in Eqs. ( 19) and ( 20), P and δ p .If the scenario of no new physics effect is assumed, we can solve P as a function of δ p according to the experiment result of A CP (D 0 → π + π − ).The ratio between penguin and tree amplitudes P/(T + C) dependent on δ p is plotted in Fig. 1.One can find P/(T + C) is greater than 2 in most values of δ P in both negative and positive δ I .It suggests the penguin topology is enhanced by non-perturbative QCD in the D → ππ system.More generally, it is possible that the large penguin topologies exist in other singly Cabibbo-suppressed charmed hadron decay modes, leading to observable CP asymmetries.
Topology P b is comparable to the tree amplitudes, contributing to a large SU (3) F breaking effect in the singly Cabibbo-suppressed charm decays and affecting the branching fractions.Thus P b cannot be neglected in the global fit of the D meson or baryon non-leptonic weak decays.
With the function of P (δ p ), we get the function of A CP (D 0 → π 0 π 0 ) dependent on δ p , which is plotted in Fig. 2. A CP (D 0 → π 0 π 0 ) is expected to be O(10 −3 ) at most values of δ p , which is available on Belle II at 50ab −1 data set.At some particular values of δ p , A CP (D 0 → π 0 π 0 ) could FIG.2: CP asymmetry in the D 0 → π 0 π 0 decay dependent on δ p in the cases of negative (left) and positive (right) δ I The horizontal pink shadow is 1σ experimental limitation to date [40,42].The blue shadow is the expected statistical uncertainties on Belle II at 50ab −1 data set [43].In experiments, the ratio R is used to test new physics in the ∆I = 3/2 amplitude, which is defined by [45,46] With the isospin and topological amplitudes of D → ππ modes, ratio R is derived to be FIG. 4: Final-state interaction effects: s-channel (resonance) and t-channel (rescattering) contributions to one particle exchange.
in the SM.According to Eq. ( 22), ratio R is determined by CP asymmetry in the ∆I = 1/2 amplitude.The dependence of ratio R on δ p is plotted in Fig. 3.It is found that R ∼ O(10 −3 ) in the most values of δ p .

III. ESTIMATION IN THE FINAL-STATE INTERACTION
In the D → ππ system, the penguin contribution is λ d P d + λ s P s .Considering all the tree amplitudes are proportional to λ d in the D → ππ modes, we write penguin amplitudes as 8)∼ (10).The CP asymmetry is induced by the interference between λ b P s with other decay amplitudes.In this section, we discuss how large the penguin amplitude P s could be in the final-state interaction.
For the two-body heavy meson weak decay, the FSI effect can be modeled as exchange of one particle between two particles generated from the short-distance tree emitted process.There are s-channel and t-channel contributions in the final state interaction, which are depicted in Fig. 4.
In the s-channel contribution, the resonance state in the D → ππ decay has the quantum number J P C = 0 ++ derived from the final states.Ref. [18] suggests that f 0 (1710) playing an important role in enhancing the penguin amplitude.However, the CP asymmetry ratio between D 0 → π + π − and Compared to Ref. [23], the K * K * → ππ scattering is included in the final state interaction.Since the weak vertex of triangle diagram is a short-distance T diagram, the factorization approach is used to estimate the weak decay amplitude.It is parameterized as the decay constant of the emitted meson and the transition form factor of another meson.For the amplitude of total triangle diagram, there are several calculational methods, in which the treatment of hadronic loop integration is different [24][25][26][27][28][29][30].In this work, the intermediate states are treated to be on their mass shell and then the optical theorem and Cutkosky cutting rule are used to calculate the absorptive part.The calculation details can be found in Ref. [27].
Considering the exchanged meson being generally off-shell, a form factor is introduced as to compensate the off-shell effect [47], where t and m t are the momentum square and mass of the exchanged meson respectively.F (t) is normalized to unity at the on-shell situation

IV. SUMMARY
We studied CP asymmetry in the D → ππ system based on the isospin and topological analysis.
According to the new measurement of CP asymmetry in the D 0 → π + π − decay, we concluded that CP asymmetry in the D 0 → π 0 π 0 decay could reach to be O(10 −3 ), which is available on Belle II in the future.Besides, the large CP asymmetry in the D → ππ system might be explained by the t-channel final-state interaction.

FIG. 1 :
FIG. 1: Ratio of penguin and tree amplitudes P/(T + C) dependent on the strong phase δ p in the cases of negative (left) and positive (right) strong phase δ I .The gray bands represent the uncertainties, and the blue solid and dashed lines are P/(T + C) = 2 and −2 respectively.

8 R (× 10 - 3 )FIG. 3 :
FIG.3: Ratio R dependent on δ p in the cases of negative (left) and positive (right) δ I , in which the horizontal pink shadow is 1σ experimental limitation to date taken from HFLAV[44].