The pollution to the $K\pi$-puzzle from the isospin-breaking $\pi^0-\eta-\eta'$ mixing effect

The influence of the isospin-breaking $\pi^0\!\!-\!\eta\!-\!\eta'$ mixing effect on the $CP$-asymmetries of $B\to K\pi$ processes is examined for the first time. It is found that this mixing effect brings a large uncertainty both to the $CP$-asymmetry sum rule of $B\to K\pi$ processes and the $CP$-asymmetry difference of the $B^+\to K^+\pi^0$ and $B^0\to K^+\pi^-$, obscuring the significance of the $K\pi$-puzzle. This uncertainty can be so large that it is even possible to explain the $K\pi$-puzzle by the $\pi^0\!\!-\!\eta\!-\!\eta'$ mixing effect {\it alone}.


I. BRIEF INTRODUCTION TO THE Kπ-PUZZLE
It is believed that the decay processes B → Kπ are good probes for new physics (NP) beyond the Standard Model (SM), as the tree-level amplitudes are suppressed, making them more sensitive to the potentially NP contributions.Based on the isospin consideration, the amplitudes of the aforementioned weak decays B → Kπ are related through [1][2][3] from which a sum rule between CP asymmetries of B → Kπ processes can be deduced [4]: This CP -asymmetry sum rule can be further simplified into a more crude relation between the CP asymmetries of B + → K + π 0 and B 0 → K + π − [5]: which is clearly in contradiction to the latest world average of the CP -asymmetry difference between these two aforementioned processes [6][7][8][9][10][11], ∆A CP (Kπ) This is basically a short version of the long-standing "Kπ-puzzle".

II. THE POLLUTION OF THE π
The basic idea is very simple.Since there are π 0 s in the final states of these B → Kπ decay processes, the isospin-breaking π 0 −η−η ′ mixing effect [26][27][28][29][30] takes place.Although this effect seems to be negligible at first sight as it is small [31] -perhaps that is why the π 0 −η −η ′ mixing effect has never been put on the table dealing with the Kπ-puzzle-this kind of isospin-breaking effects could potentially affect CP asymmetries more badly than expected [32].

IB
represents the correction of the π 0 − η − η ′ mixing effect to the CP asymmetry of B + → K + π 0 , which, after some algebra, can be expressed as Note that two strong phases between the amplitudes of B ± → K ± η (′) and B ± → K ± π 0 , θ and θ ′ , are presented explicitly.From Eq. ( 5) and ( 6) one can roughly see that the π 0 −η−η ′ mixing correction term can not be simply neglected, provided that the CP asymmetry parameter A K + π 0 CP is about the same order with the mixing parameters ǫ and/or ǫ ′ .Similarly, there is also a correction term to the CP asymmetry of B 0 → K 0 π 0 , which reads Since Eqs. ( 2) and (3) are obtained under the ignorance of the the π 0 −η−η ′ mixing effect, the π 0 s in these two equations are in fact the isospin eigenstates π 3 s.Consequently, Eqs.
(2) and (3) should be rewritten as and respectively.It should be pointed out that all the branching ratios and CP asymmetries containing π 3 in the final state are strictly speaking not physical observables.This however, can be fixed with the aid of Eq. ( 5) by rewriting Eqs. ( 8) and ( 9) in terms of the physical branching ratios and CP asymmetries with π 0 s containing in the final states.The CPasymmetry sum rule and the CP -asymmetry difference now read and respectively, where ∆ IB accommodates the π 0 −η −η ′ mixing correction and takes the form One interesting behaviour of this modification is that although Eq. ( 3) relates only the CP asymmetries of B 0 → K + π − and B + → K + π 0 , the isospin-breaking correction term ∆ IB in Eq. ( 11), however, not only contains the contribution of the process B + → K + π 0 , but also contains that of the process B 0 → K 0 π 0 .The latter turns out to be numerically even more important in ∆ IB .
We are now in a position to estimate the impact of the the isospin-breaking correction term ∆ IB to the Kπ-puzzle.Although the amplitudes in Eqs. ( 6) and ( 7) can be calculated theoretically or extracted from data, the four relative strong phases between these amplitudes, θ, θ ′ , θ, and θ′ , are non-perturbative, preventing us from an accurate prediction of ∆ IB .What we can do is to give an rough estimation of the possible range of ∆ IB by treating the four strong phases as free parameters varying from 0 to 2π independently.Based on this strategy, with the amplitudes borrowed from Ref. [23], ∆ IB is estimated to be In TABLE I, with different values of ǫ and ǫ ′ from different references, the corresponding allowed range of ∆ IB is calculated via Eq.( 13).Note that the mixing parameters ǫ and ǫ ′ take quite different values throughout the literature, resulting in quite different ranges of the ∆ IB , as is presented in the last column of this table.From TABLE I one can see that the isospin-breaking-correction term ∆ IB can be as large as a few percent, which indicates that the influence of the π 0 −η−η ′ mixing effect to the CP -asymmetry sum rule of B → Kπ and the CP -asymmetry difference of the B + → K + π 0 and B 0 → K + π − can not be simply ignored.It is even possible that the Kπ-puzzle can be explained by the π 0 −η −η ′ mixing effect alone.Take the CP -asymmetry difference between B + → K + π 0 and B 0 → K + π − as example.If we treat the ∆ IB -term as an uncertainty, with the mixing parameters from Ref.
[29], the CP -asymmetry difference now should read as from which one can clearly see that the uncertainty caused by the π 0 −η−η ′ mixing effect can be even larger than the other experimental uncertainties combined.Hence the significance of the nonzeroness of this CP difference is considerably reduced to less that 3 standard deviations.

III. CONCLUSION
In conclusion, the contribution of the isospin-breaking π 0 − η − η ′ mixing effect to the CP asymmetries of B → Kπ is investigated for the first time in this paper.It is found that the π 0 −η −η ′ mixing effect pollutes the Kπ-puzzle more dramatically than the naive expectation.This pollution is imbedded in the parameter ∆ IB , which brings quite a large uncertainty to the CP -asymmetry sum rule of B → Kπ and the CP -asymmetry difference of the B + → K + π 0 and B 0 → K + π − .The analysis of this paper shows that it is even possible to explain the Kπ-puzzle by the π 0 −η −η ′ mixing effect alone.Consequently, the implications of the Kπ-puzzle should be reconsidered.

TABLE I .
The π 0 −η−η ′ mixing parameters quoted from different references and the corresponding range of ∆ IB .Only the central values of ǫ and ǫ ′ are used when obtaining the range of ∆ IB .