Testing the Standard Model with CP-asymmetries in flavour-specific non-leptonic decays

Motivated by recent indications that the rates of colour-allowed non-leptonic channels are not in agreement with their Standard Model expectations based on QCD factorisation, we investigate the potential to study CP asymmetries with these decays. In the Standard Model, these flavour-specific decays are sensitive to CP violation in $B^0_{(s)}$--$\bar{B}^0_{(s)}$ mixing, which is predicted with low uncertainties and can be measured precisely with semileptonic decays. If there are beyond Standard Model contributions to the non-leptonic decay amplitudes, there could be significant enhancements to the CP asymmetries. Measurements of these quantities therefore have potential to identify BSM effects without relying on Standard Model predictions that might be affected by hadronic effects. We discuss the experimental prospects, and note the excellent potential for a precise determination of the CP asymmetry in $\bar{B}_s \to D_s^+ \pi^-$ decays by the LHCb experiment.


Introduction
Due to weak interactions, transitions likeB q ↔ B q are possible via box diagrams and we define the meson mass eigenstates |B q,H (H = heavy, mass M q H and decay rate Γ s H ) and |B q,L (L = light, mass M q L and decay rate Γ s L ) as linear combinations of the flavour eigenstates: with |p| 2 + |q| 2 = 1. The ratio of the magnitudes of the coefficients p and q, as well as the mass difference ∆M q = M q H − M q L and the decay rate difference ∆Γ q = Γ q L − Γ q H can be expressed in terms of the absorptive part Γ q 12 and the dispersive part M q 12 of the box diagrams, with φ q 12 = arg(−M q 12 /Γ q 12 ). To measure the a q fs parameter, which quantifies CP violation in mixing, it is necessary to study neutral mesons that mix before decaying. The general time evolution of the decay rate of neutral B q mesons, which decay with flavour opposite to that at production, is given (see e.g. [10,11]) by 2 Im( 1 λf ) 1 + |λf | −2 sin (∆M q t) . (7) Here Γ q = (Γ q L + Γ q H )/2, N f encodes a time-independent normalisation factor, including phase space effects, and the quantities λ f and λf are defined as In what follows, we will consider the flavour-specific CP asymmetry (often called semileptonic CP asymmetry), defined as 2 A q fs within the SM Within the SM we get for flavour-specific decays due to condition C1: λ f = 0 = 1/λf , the simplified time evolution with the short-hand notation X ± q (t) ≡ cosh ∆Γ q t 2 ± cos (∆M q t) .
This leads to A q fs = |A f | 2 (1 + a q fs ) − Āf 2 (1 − a q fs ) |A f | 2 (1 + a q fs ) + Āf 2 (1 − a q fs ) . (13) Note that this result for the asymmetry of time-dependent decay rates given in Eq. (9) does not depend on time. Condition C2 further givesĀf = A f and thus The SM predictions for a q fs are tiny, so that measurements of a q fs are generally considered to be null tests of the SM. Based on the calculations in Refs. [15][16][17][18][19][20][21][22][23][24][25][26], the most recent predictions [27] are Measurements of a q fs have so far been made almost exclusively with semileptonic final states (motivating the alternative notation a q sl ). The latest world averages [28], based mainly on the results of Refs. [29][30][31][32][33][34][35], are The experimental precision for these quantities is expected to increase considerably. Refs. [36,37] quote an estimated precision of ± 2 · 10 −4 for a d sl and ± 30 · 10 −5 for a s sl , achievable by the LHCb experiment with an integrated luminosity of 300 fb −1 . While for a d sl this approaches the precision necessary to test the SM prediction, this large data sample will still not be sufficient to observe a non-zero value at the SM expectation of a s fs . Nevertheless, significantly more precise results than currently available will provide stringent constraints on beyond SM contributions to Γ s 12 and M s 12 , as discussed below. The possibility to determine these asymmetries with flavour-specific non-leptonic decays has not been considered widely, as the lower yields available would result in considerably larger uncertainties compared to the semileptonic decay.
3 A q fs beyond the SM There are several possible ways that the quantities A q fs could be modified in the presence of new physics. We discuss these in turn below.

Modification of M 12
General new physics effects in the dispersive part of B mixing can be parameterised as (in the convention of [19,38]) The parameters |∆ q | are constrained to be close to unity, with around ±10% uncertainty, by the agreement of the experimental measurements [28,39,40] of the mass differences with the theoretical determinations via ∆M q = 2 |M q 12 | [24]. The new phases φ ∆ q are constrained by the measurements of the mixing phases sin 2β and sin 2β s in the golden plated modes B d → J/ψK S and B s → J/ψφ to be at most of the order of 1 or 2 degrees (except if one is willing to allow fine-tuned cancellations between new physics in B mixing and penguin diagrams contributing to the b → ccs decay). Therefore in the case of new physics only acting in M 12 , the potential sizes of a q fs could be of the order of 10 −4 . This is considerably below the current experimental accuracy, and the possible enhancement is not large enough to allow for an unambiguous observation at LHCb with 300 fb −1 .

Modification of Γ 12
The absorptive part of B mixing is in general affected by new physics as (in the convention of [38]) In this case we get constraints from the measurements of the decay rate differences ∆Γ q For ∆Γ s , experimental measurements [28,[41][42][43] agree well with theory [27] with a relative theory precision of the order of 15%. This translates into a maximal size of the new phase φ∆ s of the order of 30 • . There could also be some further, less pronounced, enhancement due to modifications in |∆ q |. Such a sizable new phase φ∆ s would lead to a strong enhancement of a s fs , close to the current experimental bound, since There is even more space for a possible enhancement of a d fs via beyond SM (BSM) effects in Γ d 12 (see also Refs. [44,45]), since there are only relatively weak experimental constraints on ∆Γ d [28,46]. This strongly motivates improved experimental measurements of a s fs and a d fs .

Modification of the B → f decay amplitude
As mentioned earlier, measurements of the rates of colour-allowed non-leptonic decays seem to deviate significantly from SM predictions [1][2][3][4]. Ref. [2] quotes for the decayB s → D + s π − a deviation of the measurement from the QCD factorisation prediction of about four standard deviations. In the case of the CKM suppressed decayB d → D + K − this deviation is even larger than five standard deviations. Commonly CKM leading, non-leptonic tree-level decays have been considered to be insensitive to new physics effects. However, general bounds on BSM effects in non-leptonic tree-level decays were systematically studied in Refs. [27,45,47], with results revealing that there is a sizable allowed parameter space for new effects, which do not violate any theoretical or experimental bound. More recently such effects have also been investigated for the case of the decayB s → D + s K − [48,49]. BSM explanations have been considered in [3,50] and challenged by collider bounds in [51].
Within the SM the decaysB s → D + s π − andB d → D + K − are flavour-specific and CP conserving. Thus, using these decay to determine the asymmetries A q fs , we expect to get the tiny values a q fs . However, if BSM effects modify the decay amplitudes, the relation between A q fs and a q fs of Eq. (14) is altered. Under the presence of general new physics contributions the decay amplitude of either The amplitudeĀf for the CP conjugate process is identical to A f up to a change in the sign of ϕ. This allows now for direct CP violation in these decays, challenging condition C2. Nonetheless, the decays are expected to remain flavour specific, since we do not see a realistic possibility to sizably violate condition C1: e.g. at the quark level the decayB s → D + s π − looks like bs → csūd, while a decay into the CP conjugate final state, triggered by an bs → scdu quark level transition would require at least dimension-nine six-quark operators. 1 Inserting 1 Condition C1 is also challenging to test experimentally, although this has been considered in [52].
into Eq. (13) leads to with the direct CP asymmetry A q dir ≈ 2r sin φ sin ϕ (formally defined in Appendix A.1, Eq. (38)). 2 To obtain the last expression in Eq. (23) we have assumed a q fs and r to be small quantities and we have expanded up to leading order in these small parameters. Allowing now for a size of r ≈ 0.1, which is indicated by the studies in [1][2][3][4], one can get -depending on the values of the phases φ and ϕ -values of up to |A q fs | = 0.2, which are several orders of magnitude larger than the SM values of a q fs . Thus, if the experimental value for A s fs (D + s π − ) or A d fs (D + K − ) differs significantly from zero, with the currently achievable experimental precision, one has an unambiguous BSM signal, independent of any theory uncertainties. Moreover, the effects of BSM contributions in M 12 and Γ 12 , which affect a s fs , can be separated from those in the decay amplitude, which affect A s fs , if we make the assumption that there is no direct CP violation in semileptonic decays which holds to excellent accuracy within the SM (since only one decay amplitude is contributing) and to some extent also beyond the SM [53,54]. In this ) has yet been experimentally measured. It is, however, likely that any large asymmetry inB s → D + s π − decays would have been spotted as this mode has been used for precise determinations of the B s oscillation frequency [40] and lifetime [55], as well as being a control channel for CP violation studies inB s → D ± s K ∓ decays [56]. In what follows we focus on thē B s → D + s π − mode as this appears to have the potential for precise measurements, but experimental studies of CP violation inB d → D + K − decays are also well motivated.

Untagged CP asymmetry
In an Appendix, we present several further possible CP asymmetries that can be determined with flavour-specific decays, that have contributions from direct CP violation and/or CP violation in mixing. In that respect we will need, in addition to Eq. (10) and (11), the decay-rate evolution for neutral B q mesons that decay with the same flavour to that at production. Assuming condition C1 is satisfied, these rates are given by [10,11] A particularly interesting observable is the untagged CP asymmetry, A q untagged , given by Inserting Eq. (10), (11), (24) and (25), we obtain = 2r sin φ sin ϕ − a q fs 1 + 2r cos φ cos ϕ + r 2 Y (t) 1 + 2r cos φ cos ϕ + r 2 − 2a q fs r sin φ sin ϕ Y (t) , with 2 Note that a q fs is defined as an asymmetry between the final states f andf , while A q dir is defined as an asymmetry betweenf and f , hence they appear with different signs in Eq. (23).
Neglecting CP violation in mixing, a q fs = 0, we find while neglecting direct CP violation would give Generally, expanding everything up to linear terms in r and a q fs , we get In contrast to Eq. (13), this asymmetry is not independent of time. It is, however, a convenient approach with which to study B 0 decays since it allows different sources of asymmetry to be disentangled. Measurements of a d fs have been made by fitting this time-dependent untagged asymmetry, using semileptonic decays in which the contribution from A d dir is expected to vanish [32,34]. For the B s case, it is experimentally convenient to measure the untagged asymmetry of timeintegrated decay rates where Expanding again up to linear terms in r and a q fs one obtains: In the case of B s decays, where the oscillation frequency is fast compared to the lifetime, the dilution factor multiplying a q fs is effectively only 0.5. Since determining A q untagged avoids the need to tag the flavour of the B s meson at production, this is therefore an experimentally attractive approach with which to measure a s fs , as quantified below. This been exploited in existing measurements with semileptonic decays where the A q dir term is assumed to be zero [33,35,57]. The same approach is also used for measurements of direct CP violation in modes where the a q fs contribution is negligible, for exampleB 0 → K − π + andB s → K + π − [14]. In this case, the use of the untagged asymmetry does not cause any dilution of the sensitivity to A q dir . Note, however, that if a d fs or a s fs were as large in magnitude as 5 × 10 −3 , at the extreme of their currently experimentally allowed ranges, this would according to Eq. (36) induce a correction of about 1 (2.5) × 10 −3 in every A dir measurement made with untagged B 0 (B s ) decays.
We now consider the experimental prospects for measurements of A s untagged inB s → D + s π − decays. The LHCb experiment appears to have by far the best prospects to determine this quantity precisely, having previously demonstrated the capability to obtain large, low-background, samples in this decay channel [40]. In addition to the existing data sample, corresponding to 9 fb −1 of pp collision data collected in Runs 1 and 2 of the Large Hadron Collider, an additional ≈ 15 fb −1 of data is anticipated to be recorded during Run 3 with an upgraded detector [58]. A new, fully softwareimplemented, trigger strategy that will be utilised during Run 3 means that LHCb will benefit from enhanced efficiency for hadronic decay modes such asB s → D + s π − . Based on the yields available in the existing data [40], and the increase anticipated to be forthcoming with Run 3, we project a sensitivity to A s untagged inB s → D + s π − decays of O(10 −3 ). If systematic uncertainties can be controlled, it will be possible to further reduce this uncertainty as a total sample of up to 300 fb −1 is collected by LHCb through operation in subsequent LHC run periods [36]. As discussed above new physics contributions to tree-level amplitudes may modify this value from its tiny SM value to O(10 −2 ) or above, and hence the experimental measurement will either discover or significantly constrain these BSM effects. Measurements of A s fs with semileptonic decays are expected to be even more precise, and will constrain the contribution to A s untagged from a s fs , assuming no direct CP violation in semileptonic decays. Indeed, the existing limits on a s fs from semileptonic measurements, which are consistent with the tiny SM expectation, are sufficient to conclude that a non-zero value of A s untagged inB s → D + s π − decays at the O(10 −2 ) level would be clear evidence for BSM effects causing direct CP violation.
Experimentally, the quantity that is directly measured is where N (X) is the total number of B 0 s → X and B 0 s → X decays observed in the data. This is related to A s untagged by The detector asymmetries, A det , will be reduced by reconstructing the D ± s meson in the D ± s → φπ ± final state. TheB s decay is then fully reconstructed in the symmetric K ± K ∓ π ± π ∓ final state with the two kaons having approximately the same momentum distribution. A small detection asymmetry will remain due to the momentum difference between the π ± originating from aB s versus a D ± s decay, but these effects can be understood using control samples [59]. (Similarly, if reconstruction and detection asymmetries of K ± mesons are well understood, the whole Dalitz plot of D ± s → K + K − π ± decays can be used to increase the available sample size.) The B 0 s -B 0 s production asymmetry in pp collisions with decays within the LHCb detector acceptance is denoted by A prod , and can be measured using the decay-time dependence of flavour-specific decays [60]. Due to fast B 0 s -B 0 s oscillations, the impact of A prod is significantly diluted by the time integral ratio in Eq. (37). The tiny residual contribution can nonetheless be calculated and corrected for. This calculation must also take into account the fact that the acceptance ǫ(t) depends on the B meson decay time, and hence enters the integrals in Eq. (37). (For completeness, the decay-time acceptance function should also be taken into account when determining Eq. (33), which impacts on the dilution factor of Eq. (36).) The asymmetries from various sources of background decays are accounted for through A i bkg which is the asymmetry of background contribution i. Each background contribution is given a weight, f i bkg , according to its relative fraction in the data. Since the background fractions are low, the sources of background are well-understood [40] and their asymmetries can be determined from control samples, this is not expected to provide a limiting systematic uncertainty.
As previously noted, by not attempting to distinguish between the mixedB s (t) → D − s π + decays and the unmixedB s (t) → D + s π − decays, there is a significant gain in the statistics available to measure A s dir . This is much greater than the factor of 2 one would naively expect from the large value of ∆M s , since one no longer requires initial state flavour tagging. LHCb has achieved a tagging efficiency for B s mesons of ǫ tag ≈ 80% and a mistag rate of w ≈ 36% [40], giving an effective tagging efficiency of ǫ tag (1 − 2w) 2 ≈ 6%. Consequently, untagged methods are highly preferable for the studies of CP asymmetries in B s mesons discussed here.
To determine A d dir for the flavour-specific B 0 → D + K − decays, on the other hand, it would be preferable to study the decay-time dependence of the untagged asymmetry as given in Eq. (32). Once experimental effects are taken into account it can be shown that fitting this distribution allows the separate measurement of the combinations A d dir + A det + a d fs /2 and A prod + a d fs /2 [34,60]. Hence it is necessary to take as an external input the value of a d fs obtained from semileptonic decays (under the assumption of no direct CP violation). The detection asymmetry A det can be determined from control samples as before, although in this case with the favoured D ± → K ∓ π ± π ± decay the final state is not symmetric so one cannot benefit from cancellations of asymmetries as in the case ofB s → D + s π − .

Conclusion
We have studied the CP asymmetries that can be investigated using flavour-specific decays, with particular attention to the non-leptonic decaysB s → D + s π − andB 0 → D + K − that have not previously been used for this purpose. Within the SM no direct CP violation occurs in these decays, and they be used to determine the flavour-specific CP asymmetry a q fs , albeit with worse precision than obtained with semileptonic decays. If new physics appears only in B mixing then semileptonic decays will still be superior in the experimental determination of the flavour-specific CP asymmetry. This changes, however, as soon as new CP violating contributions to the non-leptonic decays are allowed. In this case the tiny effects due to a q fs might be completely overshadowed by the contributions stemming from direct CP violation.
Experimentally theB s → D + s π − decay is particularly attractive, due to the large available yield, the symmetric final state, and the fact that measurements can be made without the need to determine the production flavour of the B 0 s mesons inB s → D + s π − . The untagged and time-integrated CP asymmetries depend on both a s fs and direct CP violation and are therefore sensitive to BSM effects in either. We expect a sensitivity of around one per mille for the untagged asymmetry inB s → D + s π − decays can be achieved at LHCb with Run 3 data, with improvement possible as larger data samples are collected further into the future. This will allow BSM effects causing direct CP violation in these decays to either be discovered or significantly constrained. Once the precision reaches a level that is sensitive to the Standard Model value of a s fs , one can consider the difference in A s fs values measured inB s → D + s π − and in semileptonic decays. Any significantly non-zero value of this difference would be an unambiguous signal of new physics, not relying on any theoretical estimates of non-perturbative contributions.

Acknowledgment
The work of AL and AR was supported by the BMBF project 05H21PSCLA: " Verbundprojekt 05H2021 (ErUM-FSP T04) -Run 3 von LHCb am LHC: Theoretische Methoden für LHCb und Belle II". The work of NS is supported by the ERC Starting Grant 852642 -Beauty2Charm.

A Appendix
Here for completeness we present expressions for other CP asymmetries.

A.1 Direct CP asymmetry
The direct CP asymmetry can be defined as the asymmetry of the decay-rates for neutral B q mesons that decay with the same flavour as at production (see Eqs. (24) and (25)), i.e.
2r sin φ sin ϕ 1 + 2r cos φ cos ϕ + r 2 , where the approximation is a good one for r ≪ 1. It is simply the asymmetry of the decay amplitudes squared, and hence also equal to the asymmetry of decay rates at t = 0, and to the untagged CP asymmetry in the limit of negligible a q fs .

A.2 Indirect CP asymmetry
Indirect CP asymmetry is typically defined as Since the definition of this asymmetry involves only one final state, for flavour-specific decays it does not depend on A q dir . The dominant contribution to this asymmetry is given by cos (∆M q t) / cosh (∆Γ q t/2), with a small correction proportional to a q fs . In a similar way one may also definẽ which gives up to an overall sign the same result as A q ind , when a q fs is replaced by −a q fs . Note that if we add these two asymmetries, then the leading terms cancel and the sum is proportional to a q fs : This provides a possibility to determine a q fs independently of, and with no assumption on, A q dir . In addition, we consider the time-integrated indirect CP -asymmetries Using Eqs. (10), (11), (24) and (25) we obtain where ρ q is defined in Eq. (35) and .
The time-integrated asymmetries have a leading dependence on ρ q and small corrections proportional to a q fs . Note, that this leading term cancels in the sum of the two time-integrated CP asymmetries: Since ∆Γ s ≪ Γ s ≪ ∆M s , one can expand further to get
Keeping only terms linear in a q fs and r, we arrive at with A q dir given in Eq. (38). Having no new physics in the decayB s → D + s π − , we get f (0, a s fs , φ, ϕ) = −a s fs /(2 − a s fs ) ≈ −a s fs /2, while for a sizable phases φ and ϕ and for larger values of r we can neglect a s fs and get f (r, 0, φ, ϕ) ≈ A q dir . In the approximation of keeping only the linear terms in a q fs and r the asymmetry looks as As in the case of the indirect CP asymmetires the dominant contribution to this asymmetry is given by cos (∆M q t) / cosh (∆Γ q t/2), but now the small corrections are proportional to r (in A q dir ) and a q fs . And again, one can define a similar asymmetrỹ for which we get One can get rid of the dominant contributions in A q mix andÃ q mix by considering the sum of the two, to obtain an observable that is directly proportional to 2r sin φ sin ϕ − a q fs /2: Defining the time-integrated mixed CP -asymmetries one obtains where f (r, a, φ, ϕ) is given in Eq. (51), and ρ q in Eq. (35). For the sum of the time-integrated CP asymmetries we get: