Effects of K-\bar K mixing on determining gamma from B^\pm ->DK^\pm

The decay $B^{\pm}\to DK^{\pm}$ followed by the subsequent decay of the $D$ meson into final states involving a neutral kaon can be used to determine the CKM angle $\gamma$. We study CP violation effects due to mixing and decay of the final state kaon. We find that ignoring these effects produce a shift in $\gamma$ of order $\epsilon_{K}/r_{B}$, an enhancement of $1/r_B$ compared to the naive expectation. We then show how to take these effects into account such that, in principle, they will not introduce any theoretical error in the extraction of $\gamma$.


I. INTRODUCTION
Interference between the b → cūs and b → ucs decay amplitudes can be used to determine the weak phase There are many hadronic final states that can be used to add information about γ [1][2][3][4][5][6][7] and, what is relevant to our case, some of them have one (or more) neutral kaon in the final state.All methods of determining γ from B ± → DK ± involve deriving a system of equations for γ in terms of the decay widths and amplitudes for the various processes involved.If all other quantities can be determined experimentally this allows for a modelindependent determination of γ.
Thus far CP violation associated with neutral kaons in the determination of γ has been neglected since the effect is expected to be much smaller than current experimental uncertainties.As more statistics become available new sources of CP violation in the overall process will become significant and will need to be taken into account.In this work we study the effects of CP violation in the kaon system on the determination of γ.Such effects were studied in other system in [8][9][10][11][12][13].As we will see, there are effects that are linear in K , and they are parametrically enhanced by 1/r B (we use standard notations that are defined below).Once CP violation in the kaon system is included it is crucial that the time dependence of the kaon mixing be taken into account.This must be done in a way which considers time dependent detector efficiencies, and thus it can only be carried out by each experiment separately as part of their analysis.

II. GLW
To understand the main issues associated with including kaon mixing and CPV effects, we start with the theoretically simplest case, which is the GLW method [1,2].We define where δ B is a strong phase and r B is a real parameter which accounts for color and CKM suppression, and is measured to be of order 10 −1 .We consider D → Kπ 0 decays, but the π 0 can be exchanged for another CP eigenstate such as ρ 0 and the same discussion would apply.We further assume The decay A(D 0 → K 0 π 0 ) and its CP conjugate are doubly Cabbibo suppressed, and there is no real complication in including them, they are unimportant for our current discussion, so for the moment we set them to zero (they are included in our final results).For the kaons we use the standard notation where in the last term we neglected direct CPV in kaon decays, that is, we set = 0. We define the time dependent asymmetry In the above t is the proper time of the kaon system, and by K S we refer to a kaon that decays into two pions (see discussion in [13]).We work to first order in K and we neglect where as usual It is useful to consider also the case where we integrate over the kaon lifetime.Following [13] we parametrize the experiment-dependent efficiency to detect the kaon by F (t) with 0 ≤ F (t) ≤ 1.We emphasize that F must be determined as part of the experimental analysis.We then define the time integrated asymmetry where the integral is from zero to infinity.To demonstrate the effect we take a simple case where F (t) = 1 and we obtain We can check few limits of the above results: 1.For the case of no CPV in kaons, that is, → 0, and to first order in r B we obtain as we should.
2. The r B → 0 limit corresponds to the case where the only source of CP violation is in the kaon system.This case was studied for τ decays in [13].Using Eq. ( 10) for r B → 0 we see that in agreement with [13].
3.Last we work to first order in r B and we get Therefore, when both effects are included, A CP (and therefore the extracted value of γ) is shifted by a term of order Re( )/r B .It is the 1/r B enhancement which makes the effect larger and somewhere at the level that is expected to be probed in the near future.

III. THE GENERAL CASE: DALITZ DECAYS A. General discussion
Our goal is to obtain a system of equations for γ in terms of various experimentally determined quantities in the most general case of a multi-body D decay such as D → K S π + π − .The quantities involved will be integrated over finite regions of the D decay phase space.Here we are using D → K S π + π − for concreteness, however our discussion will also apply to other decay modes such as D → KK + K − and also to two-body decays, such as D → K S π 0 , in which these amplitudes are not momentum dependent.We do not include other small effects like D 0 − D 0 mixing which was discussed in [4,[14][15][16].Note that, though we will not discuss it further, it is imperative to include the effects of D 0 − D 0 mixing since they will be competitive with those of CP violation in With this in mind, we begin by defining quantities associated with the three-body D decay and its CP conjugate where and ππ is either of π + π − or π 0 π 0 .Note that A and Ā are analogous to A D (s 12 , s 13 ) and A D (s 13 , s 12 ) of [4] respectively.We are assuming that we use the ππ final state to tag the kaon as a K S and we have used (5).From now on meson variables such as p, q, , x, y will be for the K unless stated otherwise.
Interference of the A and Ā amplitudes will occur through the B ± decay, where the relative phase of the interference involves γ.With the definitions ( 2) and ( 3) we can write down the amplitude for the overall process in terms of γ One can obtain the amplitude for the B + → f + by γ → −γ and A ↔ Ā. Squaring this we can write the time-dependent differential normalized widths where Eqs. (18) are our system of equations which can be used to determine γ.Note that |A| 2 and | Ā| 2 are directly measurable in D decays.In addition to these there is a phase, that of A * Ā, which is momentum dependent in the multi-body case and which must be obtained in addition to r B and δ B in order to determine γ.These equations are identical in form to those from [4], and similar to those in [2] except that we distinguish between A and Ā.The new complications are hidden in the time-dependence and CP violating parameters stored in A and Ā.
We should proceed by setting up equations for the Dalitz analysis.This is performed by integrating over bins in the D → K S ππ phase space.First, let us define the momenta of the decay products and the Mandelstam variables as in [4] In the → 0 limit one has A(s 12 , s 13 ) = Ā(s 13 , s 12 ).One way to think of this is that if the K S were a CP eigenstate then K S π + π − is a CP eigenstate but for momentum interchange.The effect due to the fact that the K S is not a CP eigenstate is that A(s 12 , s 13 ) = Ā(s 13 , s 12 ) +

O(| |).
We would like to partition the (s 12 , s 13 ) phase space into bins which are symmetric about the line s 12 = s 13 .With → 0 very simple relations exist between the various integrals above and below the s 12 = s 13 line.The CP violation in the kaon system, however, complicates this.To this end, we define integrals over bins in phase space where we have emphasized that these quantities all depend on the proper time of the kaon.
The index i labels a bin in (s 12 , s 13 ) phase space, and we will let ī denote the bin obtained by reflecting the bin i about s 12 = s 13 .We can then write a system of equations for integrals of the overall decay width in these bins.At this stage we assume that we integrate over time, and we obtain Note that the quantities T ± i are the widths for D → K S π + π − with the K S identified through its decay and thus can be determined from charm data, and we will treat these as known.
In the → 0 limit we have T − ī = T + i .In contrast, the variables C i and S i arise from interference effects and therefore cannot be directly measured in non-interfering D decays.
Next we perform the counting of the number of observables and parameters to check if we can, in principle, obtain γ.We denote by k the number of bins above the s 12 = s 13 line so that there are 2k bins in total.We consider n different B decay modes, such as B → D ( * ) K ( * ) .We see that there are 2k C i 's and 2k S i 's, n of δ f B and r f B and γ.We end up with 4k + 2n + 1 unknowns and 4kn equations.We see that for n ≥ 2 we can find k such that there are more observables than unknowns and thus γ can be determined without any approximations.
We can also determined γ in the n = 1 case using approximations.One approximation is to use a model for the Dalitz plot.Another approximation that can be made is to take advantage of the fact that the main correction to γ comes from the term that is not proportional to r B [see, Eq. ( 13)].Thus, dropping terms of order r B | | is equivalent to using C i = Cī and S i = −Sī.In this case we have only 2k + 3 unknowns with 4k equations which is solvable for k ≥ 2. Some other possibilities to reduce the number of unknowns were discussed in [4,[17][18][19][20].They are applicable here except that one must be careful to distinguish between CP conjugate quantities.

B. Including the kaon time dependence
When we outlined how methods for determining γ from B ± → DK ± can be adapted to include CP violation in the K 0 − K 0 system we glossed over the time dependence of the kaon decay and oscillations by assuming we integrate over all of time.This is not realistic since time-dependent efficiencies must be taken into account.Below we discuss the time dependence issue in more detail.
We begin by writing the time dependence explicitly where we have neglected the |A L | 2 term as it is proportional to | | 2 ∼ 10 −6 .In order to translate measurements performed in different timing windows, one must determine the various momentum dependent coefficients of the different time-dependent functions.For instance, we will need where F (t) is the time-dependent detection efficiencies defined above.To obtain such integrals we must determine both the real and imaginary parts of While in principle knowing F (t) will enable us to determine γ, care must be taken, as the whole program involves data from different experiments.For example, one uses charm decay data as an input to the B decay analysis.In principle F (t) can be very different in such experiments, so naive use of this data introduces error.In practice, we expect different experiments to have a similar F (t) so that the induced error will be small.Ultimately estimating this effect depends on the details of particular experiments, so this cannot be done in a general way.

C. Order r B | | Terms
As we have already alluded to, there are complications that arise when terms of order r B | | are included.This is because there is a new variable, roughly speaking it is the phase of A * S ĀS , which only appears in interference between D 0 and D 0 amplitudes.To understand how these terms complicate the analysis, we should introduce new definitions.First, we define and the CP conjugate amplitudes We emphasize that A D , r D , δ D and their conjugates depend on (s 12 , s 13 ) for multi-body decays.Here r D is a CKM suppression factor, but we do not know how small it is in any particular region of phase space.From here on we neglect CP violation in the D decay, and then we have example, D 0 → (π + ν) K π + π − and its CP conjugates.)From these we find The CP conjugate expressions for ĀS,L are obtained by A D → ĀD , → − , and A S → −A S .
The phase between p and q is unphysical but we adopt the convention q = x(1 − )/ √

IV. ASSUMING BREIT-WIGNER DEPENDENCE
As we alluded to previously, some unknowns can be eliminated by assuming a Breit-Wigner dependence for the D decays.The change one needs to keep in mind when generalizing the discussion of Breit-Wigner dependence in a case where kaon CP violation is neglected, is that CP conjugate amplitudes are no longer related by only an exchange of pion momenta.
We substitute A S above with a sum over Breit-Wigner functions A S (s 12 , s 13 ) = a 0 e iδ 0 + r a r e iδr A r (s 12 , s 13 ), where and the index r labels the resonance.The factor J M r depends on the spin of the resonance, where k 1 , k 2 are the spatial momenta of the particles originating from the resonance.The Breit-Wigner function is where M r is the mass of the resonance.The argument of W r depends on the particles which participate in the resonance, for example for the ρ 0 the argument of W r is s 23 .Explicit expressions for the mass dependent width Γ r ( √ s) and the other J M r 's can be found in [21][22][23][24].
In order to account for CP violation in the kaon system one cannot assume that the amplitudes are related to their CP conjugates by a momentum exchange alone, for instance The reason for this is that the K S is not an exactly even superposition of K 0 and K 0 .
Fortunately, these are related through simple momentum independent factors where expressions apply for any resonance decay of the form D 0 → K S X.
The situation is slightly different for decays in which the K S emerges from a resonance such as D 0 → K * − π + .In this case Fortunately, these are again simply related by momentum-independent factors where A * = A(K * − → K 0 π − ) = A(K * + → K 0 π + ) and we have assumed ∆(strangeness) = 2 decays to be forbidden entirely.In this way one can relate a K * to āK * .
For → 0 one only has the amplitude A S , but for finite , we also have A L , which can be decomposed just like (36).The same discussion on how to account for CP violation applies, however the relation between some of the coefficients is different.A general procedure for relating amplitude coefficients is as follows 1. Express decay amplitudes in terms of A(D → KX) for resonances which do not decay to the final state neutral kaon, where K is one of K 0 , K 0 .
2. Express decay amplitudes in terms of A(X s → KX) where X s is a resonance which decays to a neutral kaon.
3. Project out the K S,L component using the reciprocal basis to relate the various coefficients.
where the + is for the K L and the − is for the K S .

V. ESTIMATING THE ERROR IN γ
Next we estimate the error introduced on the extracted value of γ in case one use the standard analysis and neglect CP violation in the kaon system.We do so by making the following simplifying assumptions: We neglect terms of order r B | |, we assume that we integrate over all of the K S π + π − phase space, and we assume that F (t) = 1.A simple expression for γ can be obtained in terms of the difference Inverting the result for D 0∞ one finds where γ 0 is γ as determined from A CP with = 0. We find and we expect κ D /F D ∼ 1.To see this, note that in the limit where we neglect doubly Cabbibo suppressed D decays and let the D decay to a true CP eigenstate one obtains the same expression for ∆γ with κ D /F D = 1.Therefore we find ∆γ ∼ | |/r B .
There appear to be several limits in which ∆γ diverges.The first type is when any of r B , F D , or δ B vanish.None of these are problematic since they arise only when working to leading order, which is no longer justified in that case.Keeping the full expression we find sin(γ 0 + ∆γ) − sin(γ 0 ) sin(γ 0 ) = − 4κ D Re( ) which does not depend on r B , F D , or δ B .Next we see that ∆γ appears to diverge for γ 0 → π/2.This divergence reflects the fact that D 0∞ depends only very weakly on γ for γ ≈ π/2.Therefore, any source of error in D 0∞ /N would also cause a large shift in γ in this region.The uncertainty in γ using this method is then large for γ ≈ π/2, so we expect that ∆γ is also large there.This uncertainty is only intrinsic to the Dalitz method where S i is small.Where C i is small there is a similar uncertainty near γ ≈ 0. The effect is demonstrated in Fig. 1.
The best current determinations of γ from Belle and BaBar have uncertainties of roughly ±10 o [22,23].We expect |∆γ| to be of order Re( )/r B ≈ 1 o so it may be some time before the effect of CP violation in kaon mixing and decay becomes relevant.This correction is, however, very large compared to the largest irreducible theoretical error on the determination of γ from B → DK [25].

VI. CONCLUSION
The B → DK program is known to have the smallest theoretical error in any determination of weak parameters [25].As precision improves it becomes more important to look for subleading effects [26,27].We generalized the B → DK method for determining γ to account for additional sources of CP violation from mixing of final state neutral kaons.We found that γ is shifted by an amount of order /r B ∼ 10 −2 .While this effect is still below the current experimental sensitivity, it may be important in the near future.We have discussed how existing methods can be corrected to account for this effect.
29) for each bin by distinguishing between the exp(−Γ S t), exp(−Γt) cos(xΓt), and exp(−Γt) sin(xΓt) terms.When dropping order r B | | terms only the first three of these need to be determined.(Some explicit expressions of time integrals can be found in Appendix A.)

2 .
The new phase which occurs only in the r B terms which cannot be obtained through measurements of non-interfering D decays alone is that of A * D ĀD so we define θ D (s 12 , s 13 ) ≡ arg(A * D ĀD ).(35) Under an interchange of the pion momenta one has θ D → −θ D .The value of this angle at a point in phase space is unphysical, but for a particular choice of δ B it is fixed, and its momentum dependence is physical.It reduces to δ 12,13 − δ 13,12 of [4] in the → 0 limit.If one expands out the real and imaginary parts of A * Ā one will obtain various combinations of |A D | 2 , r D , δ D and trig functions of θ D , δ D , δD such as r D cos(θ − δ D ), rD cos(θ + δD ), and r D rD cos(θ + δD − δ D ).These are of course multiplied by |A D ĀD | and are ultimately integrated over bins of phase space.The full result can be found in Appendix A.
FIG. 1: ∆γ (blue line) as a function of γ, as it appears in (49) for κ D /F D = 1, r B = 10 −1 , δ B = π/2.The shaded region represents an error due to δ(D 0∞ /N ) = 0.01.For kaon mixing to be relevant, uncertainty in the CP asymmetry D 0∞ must be small enough that the blue line lies outside the shaded region.