Model-independent bounds on new physics effects in non-leptonic tree-level decays of B-mesons

We present a considerably improved analysis of model-independent bounds on new physics effects in non-leptonic tree-level decays of B-mesons. Our main finding is that contributions of about $\pm 0.1 $ to the Wilson coefficient of the colour-singlet operator $Q_2$ of the effective weak Hamiltonian and contributions in the range of $\pm 0.5$ (both for real and imaginary part) to $Q_1$ can currently not be excluded at the $90\%$ C.L.. Effects of such a size can modify the direct experimental extraction of the CKM angle $\gamma$ by up to $10^{\circ}$ and they could lead to an enhancement of the decay rate difference $\Delta \Gamma_d$ of up to a factor of 5 over its SM value - a size that could explain the D0 dimuon asymmetry. Future more precise measurements of the semi-leptonic asymmetries $a_{sl}^q$ and the lifetime ratio $\tau (B_s) / \tau (B_d)$ will allow to shrink the bounds on tree-level new physics effects considerably. Due to significant improvements in the precision of the non-perturbative input we update all SM predictions for the mixing obervables in the course of this analysis, obtaining: $\Delta M_s = (18.77 \pm 0.86 ) \, \mbox{ps}^{-1}$, $\Delta M_d = (0.543 \pm 0.029) \, \mbox{ps}^{-1}$, $\Delta \Gamma_s = (9.1 \pm 1.3 ) \cdot 10^{-2} \, \mbox{ps}^{-1}$, $\Delta \Gamma_d = (2.6 \pm 0.4 ) \cdot 10^{-3} \, \mbox{ps}^{-1}$, $a_{sl}^s = (2.06 \pm 0.18) \cdot 10^{-5}$ and $a_{sl}^d = (-4.73 \pm 0.42) \cdot 10^{-4}$.


Introduction
Motivations for flavour physics are manifold. Standard model parameters, like the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] or quark masses are determined very accurately in this field. Moreover the quark-sector is the only sector, where CP violating effects have been detected so far -since 1964 in the Kaon sector [3] and since 2001 also in the B-sector [4,5]. Very recently CP violation has been measured for the first time in the charm sector [6], which might actually be an indication for physics beyond the standard model (BSM) [7,8]. Considering that CP violation is a necessary ingredient for creating a baryon asymmetry in the universe [9], flavour physics might shed some light on this unsolved problem. In addition flavour physics is perfectly suited for indirect new physics (NP) searches, because there are many processes strongly suppressed in the standard model (SM) but not necessarily in hypothetical NP models. And, last but not least, a comparison between experiment and theory predictions can provide a deeper insight into the dynamics of QCD. In recent years experimental flavour physics entered a new precision era, which was initiated by the B-factories at KEK and SLAC (see e.g. [10]) and the Tevatron at Fermilab [11,12]. Currently this field is dominated by the results of the LHCb collaboration [13,14], but also complemented by competing results from the general purpose detectors ATLAS and CMS, see e.g. [15,16]. The corresponding dramatic increase in experimental precision, demands complementary improvements in theory. Besides calculating higher orders in perturbative QCD or more precise lattice evaluations, this also means revisiting some common approximations by investigating questions like: How large are penguin contributions? How well does QCD-factorization [17,18,19,20] work? How large can duality violation in the Heavy Quark Expansion (HQE) (see e.g. [21,22,23,24,25,26,27,28] for pioneering papers and [29] for a recent review) be? How sizeable NP effects in tree-level decays can be? Some of these questions have been studied in detail for quite some time. There is e.g. a huge literature on penguin contributions, see e.g. [30,31] for reviews. Others gained interest recently, for instance duality violations [32]. In principle all these questions are interwoven, but as a starting point it is reasonable to consider them separately. The assumption of no NP effects at tree-level in non-leptonic b-decays was already challenged after the measurement of the dimuon asymmetry by the D0-collaboration [33,34,35,36], see e.g. [37].
And after the measurements of B → D ( * ) τ ν by BaBar, Belle and LHCb [38,39,40,41] for the case of semi-leptonic b-decays. Compared to numerous systematic studies of NP effects in the Wilson coefficients of the penguin operators Q 7 , Q 9 and Q 10 , see e.g. [42,43,44,45,46], we are not aware of systematic studies for NP effects in the Wilson coefficients for non-leptonic tree-level decays, except the ones in [47,48,49,50,51]. The aim of the current paper is to considerably extend the studies in [47,49] by incorporating two main improvements: 1. A full χ 2 -fit is performed instead of a simple parameter scan. To implement this step we use the package MyFitter [52] and allow the different nuisance parameters to run independently. This will allow us to account properly for the corresponding statistical correlations.
2. Instead of simplified theoretical equations we include full expressions for the observables under investigation.
The recent work in [50,51] concentrates exclusively on the transition b → ccs, while we consider in this paper all different hadronic decays, that occur in the SM on tree-level. Moreover in this work we consider only BSM effects to the tree-level operators Q 1 and Q 2 , while [50,51] investigates also effects of four-quark operators that do not exist in the SM. Whenever there is some direct overlap between the work in [50,51] we directly compare the results. Any realistic BSM model that gives rise to new tree-level effects will also give new effects at the loop-level, which are not considered in the current model independent approach. In that respect this work can be considered as an important building block of future model dependent studies. The paper is organised as follows: In Section 2 we describe briefly the theoretical tools to be used: we start with the effective Hamiltonian in Section 2.1, then in Section 2.2 we introduce the Heavy Quark Expansion and in Section 2.3 we review basic concepts in QCD factorization relevant to this project. Next in Section 3, we outline our strategy for performing the χ 2 -fit. We discuss all our different constraints on NP effects in non-leptonic tree-level decays in Section 4. The bounds on individual decay channels are organized as follows: b → cūd in Section 4.1, b → uūd in Section 4.2, b → ccs in Section 4.3, b → ccd in Section 4.4. Additionally, in Section 4.5 we present observables constraining more decay channels. Our main results are presented in Section 5: fits for the allowed size of BSM effects in the tree-level Wilson coefficients based on individual decay channels will be discussed in Sections 5.1 -5. 3. In particular we focus on the channels which can enhance the decay rate difference of neutral B 0 d -mesons ∆Γ d and we calculate these enhancements. Flavour-universal bounds on the tree level Wilson coefficients will be presented in Section 5.5, with an emphasis on the consequences of tree-level NP effects on the precision in the direct extraction of the CKM angle γ. In Section 6 we study observables that seem to be most promising in shrinking the space for new effects in C 1 and C 2 . Finally we conclude in Section 7 and give additional information in the appendices. Since there has been tremendous progress (see e.g. [53,54]) in the theoretical precision of the mixing observables, we will present in this work numerical updates of all mixing observables: ∆Γ q in Section 4.3.2, ∆M q in Section 4.4.1 and the semi-leptonic CP asymmetries a q sl and mixing phases φ q in Section 4.5.

Basic formalism
In this section we provide an overview of the basic theoretical tools required for the description of our different flavour observables, this includes: the effective Hamiltonian, the Heavy Quark Expansion for inclusive decays and mixing observables. A quick review of QCD factorization for exclusive, nonleptonic decays is also provided. In addition we fix the notation to be used during this work.

Effective Hamiltonian
We start by introducing the effective Hamiltonian describing a b-quark decay into a pp q final state via electroweak interactions, with p, p = u, c and q = s, d: The Fermi constant is denoted by G F , additionally we have introduced the following CKM combinations Moreover C i denote the Wilson coefficients of the following dimension six operators: Here α and β are colour indices, e k is the electric charge of the quark k (in the penguin operators the quark flavours are summed over k = u, d, s, c, b), e is the U (1) Y coupling and g s the SU (3) C one, m b is the mass of the b-quark and F µν and G µν are the electro-magnetic and chromo-magnetic field strength tensors respectively. In this work we consider NP effects that will affect the tree-level operatorsQ q, pp 1 andQ q, pp 2 by modifying their corresponding Wilson coefficients. In our notationQ q, pp 1 is colour non diagonal andQ q, pp 2 is the colour singlet, the QCD penguin operators correspond toQ q 3−6 and the electro-weak penguin interactions are described byQ q 7−10 . Different bases compared to the one in Eq. (3) are used in the literature. Our notation agrees with the one used in [55] and [56], here C 8g is negative because we are considering −igγ µ T a as the Feynman rule for the quark-gluon vertex. In [19] a different basis is taken into account, whereQ 1 andQ 2 are interchanged and Q 7γ andQ 8g have a different sign (this is equivalent to the sign convention iD µ = i∂ µ + g s A µ a T a for the gauge-covariant derivative). A nice introduction on effective Hamiltonians can be found in [57], and a concise review up to NLO-QCD in [55].
The Wilson coefficients C i with i = 1, 2, ..., 10, 7γ, 8g in Eq. (1) are obtained by matching the calculations of the effective theory and the full SM at the scale µ = M W and then evolving down to the scale µ ∼ m b using the renormalisation group equations according to where the NLO evolution matrix is given by [19] U(µ, M W , α) = U(µ, µ W ) + α 4π R(µ, µ W ).
The matrix U(µ, µ W ) accounts for pure QCD evolution, on the other hand R(µ, µ W ) introduces QED effects as well. We write at NLO [19] U(µ, M W , α) where α s (µ) denotes the strong coupling at the scale µ calculated up to NLO-QCD precision and α is the electro-magnetic coupling. At LO the evolution matrix U(µ, M W , α) reduces to The NLO-QCD corrections are then introduced through J, the explicit expressions for U 0 and J are given in Eqns. (3.94)-(3.98) of [55]. The anomalous dimension matrices γ (0) s and γ (1) s required for these evaluations can be found in Eqn. (6.25) and Tables XIV and XV of [55]. To introduce QED corrections we calculate R 0 and R 1 using Eqns. (7.24)-(7.28) of [55], the anomalous dimension matrices used are γ  Tables XVI and  XVII of [55]. The initial conditions for the Wilson coefficients have the following expansion at NLO as pointed out in [19] the electroweak contributions C and it is therefore treated as a NLO effect, it contains the NLO scheme dependency. This approach differs from the one followed by [55], where the contribution of C  [55] and Section 3.1 of [19], the results presented for C (1) e in [19] are based in the calculations of [58]. It should be further stressed that when applying Eq. (4) we consistently dropped products between NLO contributions from U(µ, M W , α) and NLO effects from C(M W ) but we have taken into account products between NLO contributions from U(µ, M W , α) and LO contributions from C(M W ) and vice versa.

Heavy Quark Expansion
The effective Hamiltonian can be used to calculate inclusive decays of a heavy hadron B q into an inclusive final state X via With the help of the optical theorem the total decay rate in Eq. (9) can be rewritten as with the transition operator consisting of a non-local double insertion of the effective Hamiltonian. Expanding this bi-local object in local operators gives the Heavy Quark Expansion (see e.g. [21,22,23,24,25,26,27,28] for pioneering papers and [29] for a recent review). The total decay rate Γ of a b-hadron can then be expressed as products of perturbatively calculable coefficients Γ i times non-perturbative matrix elements O D of ∆B = 0-operators of dimension D = i + 3: with O D = B q |O D |B q /(2M Bq ). The leading term Γ 0 describes the decay of a free b-quark and is free of non-perturbative uncertainties, . At order 1/m 2 b small corrections due to the kinetic and chromomagnetic operator are arising, at order 1/m 3 b we get e.g. the Darwin term inΓ 3 , but also phase space enhanced terms Γ 3 , stemming from weak exchange, weak annihilation and Pauli interference. The numerical values of the matrix elements are expected to be of the order the hadronic scale Λ QCD , thus the HQE is an expansion in the small parameter Λ QCD /m b . Each of the terms Γ i with i = 0, 2, 3, ... can be expanded as In our investigation of the lifetimes we will use Γ (0) 0 and Γ (1) 0 from [59], which is based on [60,61,62,63,64,65], Γ (0) 3 from [50] based on [66,67] and Γ (1) 3 from [68,69]. The matrix elements of the dimension six operators were recently determined in [70]. The HQE can also be used to describe the off-diagonal element Γ 12 of the meson mixing matrix.
The matrix element Γ q 12 can be used together with M q 12 to predict physical observables like mass differences, decay rate differences or semileptonic CP-asymmetries, see e.g. [31].

QCD Factorization
In our analysis we included different observables based on non-leptonic B meson decays such as: B → Dπ, B → ππ, B → πρ and B → ρρ. To calculate the corresponding amplitudes we used the expressions available in the literature obtained within the QCD Factorization (QCDF) framework [17,18,19,20]. In this section we briefly summarise the QCDF results relevant for the evaluation of some of our flavour constraints. Consider the process B → M 1 M 2 , in which a B meson decays into the final states M 1 and M 2 , where either M 1 and M 2 are two "light" mesons or M 1 is "heavy" and M 2 is "light" 1 . If both M 1 and M 2 are light, then the matrix element M 1 M 2 |Q i |B of the dimension six effective operators in Eq. (3) can be written as In the right hand side of Eq. (18) F B→M 1,2 j (m 2 2,1 ) represents the relevant form factor to account for the transition B → M 1 (and correspondingly for B → M 2 ) and Φ M (u) is the non-perturbative Light-Cone Distribution Amplitude (LCDA) for the meson M , see Fig. 1. Notice that Eq. (18) is written in such a way that it can be applied to situations where the spectator quark can end in any of the two final state light mesons. If the spectator can go into only one of the final mesons, this one will be labelled as M 1 and just the first and the third terms on the right hand side of Eq. (18) should be included. The functions T I,II are called hard-scattering kernels and can be calculated perturbatively. The kernel T I contains, at higher order in α s , nonfactorizable contributions from hard gluon exchange or penguin topologies. On the other hand, nonfactorizable hard interactions involving the spectator quark are part of T II . When in the final state the mesons M 1 is "heavy" and M 2 is "light", then the corresponding QCDF formula for the matrix element M 1 M 2 |Q i |B becomes where the meaning of the different terms in Eq. (19) are analogous to those given for Eq. (18).  , which for the purposes of our discussion will be termed "Topological Amplitudes" (TA). The TA α p i (M 1 M 2 ), for p = u, c, have the following generic structure at NLO in α s [20]  where C i are the Wilson coefficients calculated at the scale µ ∼ m b , and the subindex in the coefficient C i±1 is assigned following the rule The Wilson coefficients inside the squared bracket in Eq. (20) will be modified to allow for NP contributions as discussed below, see Section 3, and N c denotes the number of colours under consideration and will be taken as N c = 3. The global factor N i (M 2 ) multiplying the square bracket corresponds to the normalisation of the light cone distribution for the meson M 2 , and is evaluated according to the following rule function H i (M 1 M 2 ) can be written in terms of the leading twist LCDAs of M 1 and M 2 , Φ M 1 and Φ M 2 respectively, and the twist-3 LCDA of M 1 , Φ m 1 , as [20]: 4,9,10) , , The analogous expressions for H i (M 1 M 2 ) when M 1 and M 2 are two longitudinally polarised light vector mesons can be found in [85]. We provide the functions H i (M 1 M 2 ) for the processes relevant to this project in Appendix B. The global coefficients A M 1 M 2 and B M 1 M 2 presented in Eqs. (21) depend on form factors and decay constants and are given in Eq. (B1) also in Appendix B.
We want to highlight two sources of uncertainty arising in Eq. (21). The first one stems from the contribution of the twist-3 LCDA Φ m 1 (y). Since this function does not vanish at y = 1, the integral 1 0 dyΦ m 1 (y)/ȳ is divergent. To isolate the divergence we follow the prescription given in [20] and write The divergent piece of Eq. (22) is contained in X H . The remaining integral 1 0 dy/[ȳ] + Φ m 1 (y) is finite (for instance for a pseudo scalar meson Φ m 1 (y) = 1 and trivially 1 0 dy/[ȳ] + Φ m 1 (y) = 0 ). Physically X H represents a soft gluon interaction with the spectator quark. It is expected that X H ≈ ln(m b /Λ QCD ) because the divergence appearing is regulated by a physical scale of the order Λ QCD . A complex coefficient cannot be excluded since multiple soft scattering can introduce a strong interaction phase. Here we use the standard parameterisation for X H introduced by Beneke-Buchalla-Neubert-Sachrajda (BBNS) [18] where Λ h ≈ O(Λ QCD ) and ρ H ≈ O(1).
The second source of theoretical uncertainty in Eqs. (21) that deserves special attention is the inverse moment of the LCDA Φ B corresponding to the B meson. Following [17] we write where λ B is expected to be of O(Λ QCD ). We provide more details about the values for X H and λ B used in this work at the end of this subsection. Next we address the contributions from weak annihilation topologies, see . The numerical subscript k describes the Dirac structure under consideration: k = 1 for (V − A) ⊗ (V − A), k = 2 for (V − A) ⊗ (V + A) and k = 3 for (−2)(S − P ) ⊗ (S + P ). The annihilation coefficients are expressed in terms of a set of basic "building blocks" denoted by A i,f k . Where the subindex k also denotes the Dirac structure being considered as previously explained, and the superindices i and f denote the emission of a gluon by an initial or a final state quark as shown in Fig. 5. The coefficients A i,f k relevant for this work can be found in Appendix B. The final expressions for annihilation are the result of the convolution of twist-2 and twist-3 LCDA with the corresponding hard scattering kernels; as in the case of hard spectator scattering, there are also endpoint singularities that are treated in a model dependent fashion. To parameterize these divergences, we follow once more the approach of BBNS. Thus, in analogy with hard spectator scattering we introduce [18] To finalize this subsection we discuss the numerical inputs used in our evaluations of λ B , X H and X A . As indicated in Eq. (24), the inverse moment of the LCDA of the B meson introduces the parameter λ B . The description of non-leptonic B decays based on QCDF requires λ B ∼ 200 MeV [20,86]. In contrast, QCD sum rules calculations give a higher value. For instance, in [87] the result λ B = (460 ± 110) MeV was found. In [88] the usage of the channel B → γ ν was proposed in order to extract λ B experimentally. This study was updated by [89], where subleading power corrections in 1/E γ and 1/m b were also included. Based on this idea, the Belle collaboration found [90] λ B at the 90% C.L. and it is expected that the Belle II experiment improves this result [89]. Interestingly the experimental bound in Eq. (26) is compatible with the QCD sum rules value quoted above and other theoretical approaches, including the one in [91] where the value λ B = (476.19 ± 113.38) MeV was obtained. For the purposes of our analysis, we consider the following result calculated in [92] with QCD sum rules: As discussed above, the calculation of hard spectator interactions and the evaluation of annihilation topologies, leads to extra sources of uncertainty associated with endpoint singularities that are power suppressed. As indicated in Eqs. (23) and (25) they can be parameterized through the functions X H (ρ H , φ H ) and X A (ρ A , φ A ) respectively. Using these models, we account for the hard spectator scattering power suppressed singularities through the parameters ρ H and φ H . Correspondingly, we introduce ρ A and φ A to address the analogous effects from annihilation topologies. Based on phenomenological considerations we will take into account the intervals [48,93] 0 < ρ H,A < 2, 0 < φ H,A < 2π, which correspond to a 200% uncertainty on |X H | and |X A |.
To evaluate the central values of our observables we take ρ H,A = 0, or equivalently X H = X A = ln m B /Λ h . Finally, we calculate the percentual error from X A and X H , by estimating the difference between the maximum and the minimum values reached by the hadronic observables when considering the intervals in Eq. (28), and then we normalize by two times the corresponding central values.

Strategy
Consider the effective Hamiltonian in Eq. (1) written in terms of the basis in Eq. (3). We introduce "new physics" in the Wilson coefficients {C 1 , C 2 } of the operatorsQ 1 andQ 2 following the prescription where in the SM In this paper we present possible bounds on ∆C 1 and ∆C 2 at the matching scale µ = M W and work under the assumption of "single operator dominance" by considering changes to each Wilson coefficient independently, e.g. to establish constraints on ∆C 1 (M W ) we fix ∆C 2 (M W ) = 0 and vice versa. This is the most conservative approach, if we allow both parameters to change simultaneously this can result into partial cancellations leading to potentially bigger NP allowed regions for {∆C 1 (M W ), ∆C 2 (M W )}. Since the theoretical formulae for our observables are calculated at the scale µ = m b , we evolve down the modified Wilson coefficients C 1 (M W ) and C 2 (M W ) up to this scale using the renormalisation group formalism described in Section 2.1. We consider NP to be leading order only, therefore we treat the SM contribution . Notice that, even though at the scale µ = M W the only modified Wilson coefficients are C 1 (M W ) and C 2 (M W ), the non diagonal nature of the evolution matrices propagates these effects to all the other Wilson coefficients undergoing mixing at µ = m b . Hence, when writing expressions for the different physical observables, it makes sense to consider NP effects in C i (m b ) even for i = 1, 2.

Statistical analysis
The values of ∆C 1 (M W ) and ∆C 2 (M W ) compatible with experimental data are evaluated using the program MyFitter [52]. The full statistical procedure is based on a likelihood ratio test. The basic ingredient is the χ 2 function whereÕ i,exp andÕ i,theo are the experimental and theoretical values of the i−th observable respectively and σ i,exp is the corresponding experimental uncertainty. The vector ω contains all the inputs necessary for the evaluation ofÕ i,theo and will be written as In Eq. (32) we are making a distinction between {∆C 1 (M W ), ∆C 2 (M W )} and the rest of the theoretical inputs, which have been included in the subvector λ.
Examples of the entries inside λ are masses, decay constants, form-factors, etc. Notice that our main target is the determination of ∆C 1 (M W ) and ∆C 2 (M W ), however, the components entering λ are crucial in defining the uncertainty of our observables and hence in establishing the potential values of ∆C 1 (M W ) and ∆C 2 (M W ). In this respect, we will say that the elements inside λ are our nuisance parameters, and that the determination of the possible NP values compatible with data are obtained by profiling the likelihood with respect to {∆C 1 (M W ), ∆C 2 (M W )}. During our analysis the elements of {∆C 1 (M W ), ∆C 2 (M W )} are assumed to be complex and, as indicated in the argument, the initial evaluation is done at the scale µ = M W . The statistical theory behind the χ 2 -fit software used, e.g. MyFitter [52], can be found in the documentation of the computer program. Here we only summarize the key steps involved in our analysis: 1. We first define the Confidence Level CL for the χ 2 -fit. Following the criteria established in [47,49] for our study we take which is equivalent to 1.64 standard deviations approximately. 2. Then, we establish a sampling region on the plane defined by the real and the imaginary components of {∆C 1 (M W ), ∆C 2 (M W )}. The sampling region is observable dependent. In our case we opt for rectangular grids around the origin of the complex plane defined by ∆C 1 (M W ) and ∆C 2 (M W ). Notice that the origin of our complex plane corresponds to the SM value. The number of points in our test grid depends on three factors: the numerical stability of our algorithms, on the time required to compute a particular combination of observables and the size of the NP regions determined by them. 3. Each one of the points inside the sampling grid described in the previous step corresponds to a null-hypothesis for the components of ∆C 1 (M W ) and ∆C 2 (M W ). We test our null-hypothesis values using a likelihood ratio test considering the confidence level established in the first step. For a combination of multiple observables several nuisance parameters are involved and the full statistical procedure becomes time and resource consuming. Hence, the parallelization of our calculations using a computer cluster became necessary. We did our first numerical evaluations partially at the Institute for Particle Physics and Phenomenology (IPPP, Durham University). The results presented in this work were obtained in full using the computing facilities available at the Dutch National Institute for Subatomic Physics (Nikhef).

Individual Constraints
In this section we present the different observables considered during the analysis. From Sections 4.1 to 4.3 we focus exclusively on observables that constrain individual b decay channels, in our case: b → cūd, b → uūd, b → ccs and b → ccd. In Section 4.5 we will study observables that affect multiple b decay channels. In what follows and unless stated otherwise, the SM predictions as well as the experimental determinations are given at 1 σ, i.e. 68% C.L.. However the allowed NP regions for C 1 and C 2 are presented at 1.64 σ, i.e. 90% C.L..

4.1.1.B 0
d → D * + π − Our bounds will be established using the ratio between the decay width for the non-leptonic decayB 0 d → D * + π − and the differential rate for the semi- This observable was proposed by Bjorken to test the factorization hypothesis [94], it is free from the uncertainties associated with the required form factor to describe the transition B → D * and offers the possibility of comparing directly the coefficient α D * π 2 calculated using QCDF against experimental observations. At NLO the TA α D * π 2 [18] is given by where the termB inside the square bracket cancels the renormalisation scheme dependence of the Wilson coefficients C d, cu , which in naive dimensional regularisation requiresB = 11. The kernel F (u, x c ) includes QCD vertex corrections arising in the decay b → cūd and has to be evaluated at x c =m c (m b )/m b before being convoluted with the light-cone distribution Φ π associated with the π − meson in the final state. For the explicit evaluation of Eq. (34) we use the updated determination of the TA α D * π 2 at NNLO calculated in [95] The annihilation topologies contributions are taken into account through where and with the parameters X A are given in Eq. (25) and the factors r π χ and r D * χ quoted in Eq. (B1). Using the numerical inputs given in A we find corresponding to z = 0.225, the partial contributions to the total error are shown in Table 1. The SM result is dominated by the contribution of C 2 , thus we will get from R D * π strong constraints on C 2 and relatively weak ones on C 1 . To compute the experimental result we use [95] dΓ Parameter Relative error together with [96] Br to obtain Our χ 2 -fit provides the 90 % confidence level regions allowed by ∆C d,cu

Observables constraining b → uūd transitions
We proceed to describe the constraints to the NP contributions ∆C d,uu 1,2 (M W ) entering in the CKM suppressed quark level transition b → uūd. Our bounds are obtained taking into account both the branching ratios, but also the CP asymmetries of the decays B → ππ, ρπ, ρρ and using again QCDF for the theoretical description. The combination of CP-conserving and CP-violating observables significantly shrinks the allowed region for ∆C d,uu 2 (M W ).  (M W ) allowed by the observable R D * π at 90% C.L.. The black point corresponds to the SM value. Since R D * π is dominated by C 2 , we get strong constraints on C 2 and relatively weak ones on C 1 .

R ππ
Our first observable is the theoretical clean ratio [94] where − = µ − , e − and α ππ 1 , α ππ 2 are the TA associated with the decays B → ππ which were introduced in a generic way in Eq. (20). The dependence of R ππ is now symmetric in C 1 and C 2 , so both Wilson coefficients will be constrained in an almost identical way. Notice that the denominator in Eq. (44) refers to the differential distribution dΓ(B 0 d → π + −ν )/dq 2 evaluated at q 2 = 0, where q 2 is the four momentum transferred to the system composed by the − andν . In Eq. (44), our sensitivity to NP enters through the decay B + → π + π 0 which is to a good degree of precision a pure tree level channel. We neglect hypothetical BSM effects inB 0 d → π + −ν for = e, µ, see e.g. [97] for a recent investigation of such a possibility. The observable R ππ is theoretically clean since it does not depend on the CKM matrix element |V ub |, which cancels in the ratio. Moreover, at leading order in α s it is independent of the form factors F B→π (0) which account for the hadronic transition B → π. However, these parameters enter in the coefficients α ππ 1,2 once the spectator interaction contributions H ππ are taken into account. More precisely, they appear in the ratio B ππ /A ππ inside H ππ , see Eqs. (B1) and (B14). Currently, the coefficients α ππ 1,2 in Eq. (44) are available up to NNLO in QCDF [86,98,99,100]. In order to optimize the computation time of our χ 2 -fit, we have accounted for the NNLO effects using the following formula Where in Eq. (45): (µ 0 ) corresponds to the fully programmed NLO expression for the amplitude α ππ 1,2 . For this term, the renormalization scale is kept fixed to the value µ 0 = m b whereas the rest of the input parameters are allowed to float. are the NNLO version of the amplitude α ππ 1,2 . We are interested in the NNLO results because of the reduction in the renormalisation scale dependency with respect to the NLO determination. Therefore during the χ 2 -fit we have treated the coefficients α NNLO,ππ 1,2 as nuisance parameters given by [92] α NNLO,ππ where the error indicated arises only from the renormalization scale uncertainty. Alternatively, we also tested the numerical values provided in [86] which give consistent results once the uncertainties arising by varying µ and µ h 2 are taken into account.
We predict the SM value of R ππ to be with the partial contributions to the total error shown in Table 2. To calcu-Parameter Relative Error Table 2: Error budget for the observable R ππ . Here X H accounts for the endpoint singularities from hard scattering spectator interactions. F B→π + (0) is the relevant form factor for the transitions B → π. The parameter λ B is the inverse moment of the LCDA of the B meson and a π 2 is the second Gegenbauer moment for the π meson.  late the experimental result, we consider the following updated value for the branching fraction for the process B + → π + π 0 [101]

R ππ
together with the product [102] |V ub F B→π which was extracted via a fit to data including experimental results from BaBar, Belle and CLEO [103,104,105,106,107] under the assumption of the SM, neglecting the mass of the light leptons and keeping the mass of the B * meson fixed. Using the inputs indicated in Eqs. (48) and (49) we obtain the following result for the experimental value of R ππ This determination is in agreement with the result given in [86], however, the uncertainty is reduced by nearly 50% due to the update on the product |V ub F B→π

S ππ
Since our NP contributions are allowed to be complex, we are exploring the possibility of having new CP violating phases. We can constrain these effects through the time-dependent asymmetries where we have neglected the effects of the observable ∆Γ q entering in the denominator -this is only justified for the case of B d -mesons. The symbol f in Eq. (51) denotes a final state to which both, the B 0 q and theB 0 q meson can decay, for q = d, s. The mixing induced (S f ) and direct CP asymmetries with the parameter λ q f given by In Eq. (53) the amplitude for the process B 0 q → f has been denoted as A q f and the one forB 0 where M d 12 is the contribution from virtual internal particles to the B 0 q −B 0 q mixing diagrams. For instance in the case of B d mesons we get Notice that the observable S f , in Eq. (52), is particularly sensitive to the imaginary components of ∆C 1 (M W ) and ∆C 2 (M W ).
HereĀ π + π − and A π + π − denote the transition amplitudes for the processes B 0 d → π + π − and B 0 d → π + π − respectively. They have been calculated in [20] using the QCDF formalism briefly described in Section 2.3. The explicit expression forĀ π + π − is To determine the remaining amplitude A π + π − , the CP conjugate of the expression in Eq. (57) has to be obtained. The parameters λ u,c in Eq. (57) correspond to products of CKM matrix elements as defined in Eq. (2). Notice that our sensitivity towards NP in tree level enters mainly through α ππ 2 , which according to Eq. (20) has a leading dependency on ∆C d, uu 2 (M W ). Therefore, the observable S ππ yields to strong constraints on ∆C d, uu 2 (M W ), while giving weak ones in ∆C d, uu 1 (M W ). Besides the TA α ππ 2 , which is introduced in our analysis at NNLO following the prescription shown in Eq. (45), there are now also contributions from QCD and electroweak penguins given byα 4 p,ππ and α 4 p,ππ EW respectively. Finally β p,ππ 4 accounts for QCD penguin annihilation and β p,ππ 4,EW for electroweak penguin annihilation. All the TA can be calculated using Eq. (20) together with the information presented in Appendix B. At leading order in α s , the normalization factor A ππ introduced in Eq. (B1), which depends on the form factor F B→π + (0) and the decay constant f π , cancels in the ratio given in Eq. (56). However it appears again once interactions with the spectator are taken into account. This leads to small effects in the error budget of O(1 %) and O(0.1 %) from F B→π + (0) and f π respectively, see Table 3. Our theoretical prediction for the SM value of the asymmetry S ππ is For the corresponding experimental value we have [96] S Exp ππ = −0.63 ± 0.04, showing consistency with the SM estimation in Eq. (58). The relevant constraints on ∆C d, uu Parameter Relative Error

S ρπ
We also included the mixing induced CP asymmetry associated with the decays B d ,B d → ρπ. Our evaluation is based in the following definition with the partial contributions given bỹ with The individual amplitudesĀ π + ρ − andĀ ρ + π − for the processesB 0 which is compatible with the current experimental average [96] S Exp πρ = 0.06 ± 0.07.
The relative errors from each one of the inputs for S πρ are presented in Tables  4 and 5, it can be seen that this observable is highly sensitive to the CKM input γ leading to a relative uncertainty of O(100%). This is related to the fact that in the ratio λ ρπ given in Eq. (62) we have: and which lead to a very strong cancellation on the resulting imaginary component. The allowed NP regions for ∆C d,uu

R ρρ
To obtain extra constraints on NP contributions to the tree level Wilson coefficients for the transition b → uūd we include the ratio where A ρ − ρ 0 and A ρ + ρ− are the amplitudes for the processes In terms of TAs they can be written as [85,108] Parameter Relative Error is the form factor for the transition B → ρ, a ρ 2 is the Gegenbauer moment for the leading twist LCDA for the ρ meson.
Parameter Relative Error Total 194.57% Here we expect a stronger dependence on C 1 compared to C 2 . As indicated in Eq. (69), in addition to the tree level contributions α ρρ 1,2 , we can also identify QCD α ρρ 4 and electroweak penguins α ρρ 7,9,10 . Moreover QCD penguin annihilation topologies enter through β p,ρρ 3,4 . On the other hand electroweak penguin annihilation is given by β p,ρρ 3,4,EW . The expressions for the topological amplitudes obey the structure indicated in Eq. (20) and can be calculated explicitly using the information provided in Appendix B. Currently α ρρ 1,2 are available up to NNLO, we introduce these effects following the same procedure used for the determination of α ππ 1,2 . Thus, we apply Eq. (45) under the replacements For the corresponding NNLO components we use [92] The uncertainty shown in Eq. (70) has its origin in higher order perturbative corrections, we have taken this as the corresponding renormalization scale uncertainty when treating α NNLO,ρ L ρ L 1,2 as nuisance parameters. Our SM determination for R ρρ is The experimental result for R ρρ is obtained by calculating the ratio of The partial contributions to the error budget are presented in Table 6 and the constraints derived for ∆C d,uu Fig. 10. We do not show the associated regions for ∆C d,uu 2 (M W ) because, for R ρρ , the results are weaker than those derived from other observables in our study.

Observables constraining b → ccs transitions
In this section we study bounds for ∆C s,cc 1,2 (M W ) stemming from Br(B → X s γ), the mixing observable ∆Γ s , the CKM angle sin(2β s ) and the lifetime ratio τ Bs /τ B d . These observables give very constrained regions for ∆C s,cc 1,2 (M W ).
The processB → X s γ is of mayor interest for BSM studies for several reasons.
To begin with, within the SM it is generated mainly at the loop level (its Parameter Relative Error On the theoretical side there has been a huge effort on the determination of this observable; the most precise results available are obtained at NNLO. Here we consider [110] Br where the energy of the photon satisfies the cut The calculation of the branching ratio for the processB → X s γ can be written as [111] Br(B → X s γ) In Eq. (76), P (E 0 ) and N (E 0 ) denote the perturbative and the non-perturbative contributions to the decay probability respectively. They depend on the lower cut for the energy of the photon in the Bremsstrahlung correction E 0 shown in Eq. (75). Using the parameterisation given in Ref. [112] we write E 0 = m 1S b /2 1 − δ and choose δ such that the lower bound in Eq. (75) is saturated. The perturbative contribution P (E 0 ) is given by [111] P with K ij = δ i7 δ j7 +O(α s ).
In order to account for the NNLO result in Eq. (74) we write Where • Br NLO (B → X s γ) is the branching ratio for the processB → X s γ calculated at NLO including NP effects from ∆C s,cc 2 (M W ). All inputs are allowed to float except the renormalisation scale, which is fixed at µ 0 = m b . Our calculations are determined using the anomalous dimension matrices provided in [112]. NP contributions are introduced according to Eq. (29). They propagate to the rest of the Wilson coefficients C i after applying the renormalisation group equations, described in Section 2 of Ref. [112].

• Br
SM, NLO 0 (B → X s γ) is the SM branching ratio for the processB → X s γ calculated at NLO and evaluated at the central values of all the input parameters and then kept constant during the χ 2 -fit.
• Br SM, NNLO (B → X s γ) is the SM branching ratio for the process B → X s γ calculated at NNLO and allowed to float within the uncertainty associated with the renormalisation scale. In the case of the theoretical result given in Eq.(74) this corresponds to 3% of the central value [110].
The partial contributions to the final error are described in Table 7. The allowed regions for ∆C s,cc 1 (M W ) and ∆C s,cc 2 (M W ) are shown in Fig. 11, where it can be seen how this observable imposes strong constraints on ∆C s,cc 2 (M W ). The bounds in Fig. 11 are consistent with those reported in [51] once a 68% C.L. is taken into account.

∆Γ s : Bounds and SM update
The decay rate differences ∆Γ q and the semileptonic asymmetries a q sl arising from neutral B q meson mixing are sensitive to the tree-level transitions b → uūq, b → ucq , b → cūq and b → ccq for q = s, d. We will, however, show below that for the decay rate difference of B s -mesons our BSM study is completely dominated by the b → ccs transition, yielding therefore strong constraints to ∆C s,cc 1 (M W ) and ∆C s,cc 2 (M W ). The definitions of the observables ∆Γ q and a q sl in terms of Γ q 12 /M q 12 were introduced in Eqs. (16) and (17). Since, as explained in Section 2.2, the elements Γ q 12 are determined from the double insertion of H |∆B|=1 ef f Hamiltonians, there are leading order contributions originating from the insertion of two current-current operatorsQ ab,q j for ab = uu, uc, cc and j = 1, 2, see Eq. (3). Additionally, there are also double insertions from a single current-current Q ab 1,2 and a penguin operatorQ 3,4,5,6 . In this section, we will only include NP effects to Γ q 12 , while we neglect tree level NP contributions to M q 12 (these contribution are discussed in Section 4.4.1 and they yield considerably weaker bounds for the observables ∆Γ q and a q sl ). To show the dominance of the b → ccs contribution for B s -mixing, we decompose Γ q 12 into partial contributions Γ q,ab 12 , where the indices ab = uu, uc, cc indicate which"up" type quarks are included inside the corresponding effective fermionic loops. Thus, the expression for Γ q 12 /M q 12 becomes We have used here the unitarity of the CKM matrix: λ For the ratio of CKM elements we obtain Within the SM we find a very strong hierarchy of the three contributions in Eq. (80). The by far largest term is given by c q and it is real. The second term proportional to a q is GIM and CKM suppressed -slightly for the case of B d mesons and more pronounced for B s . Since λ is complex, this contribution gives rise to an imaginary part of Γ q 12 /M q 12 . Finally b q is even further GIM suppressed and again slightly/strongly CKM suppressed for B d /B s mesons -this contribution has also both a real and an imaginary part. According to Eqs. (16) the decay rate difference ∆Γ q , given by the real part of Γ q 12 /M q 12 , is dominated by the coefficient c q -stemming from b → ccq transitions -and the coefficients a q and b q yield corrections of the order of 2 per mille. The semi-leptonic asymmetries are given by the imaginary part of Γ q 12 /M q 12 (c.f. Eq. (17)), which in turn is dominated by the coefficient a q , with b q giving sub-per mille corrections and no contributions from c q . Allowing new, complex contributions to C 1 and C 2 for individual quark level contributions we get the following effects: 1. The numerically leading coefficient c q can now also obtain an imaginary part. 2. The GIM cancellations in the coefficients a q and b q can be broken, if b → ccq, b → cūq , b → ucq and b → uūq are differently affected by NP. If there is a universal BSM contribution then the GIM cancellation will stay. 3. The CKM suppression will not be affected by our BSM modifications.
For the real part of Γ s 12 /M s 12 , we expect at most a correction of 2 per cent due to a s and b s , even if the corresponding GIM suppression is completely lifted -thus ∆Γ s is even in our BSM approach, completely dominated by c s and gives therefore only bounds on b → ccs. In the case of B d mesons, the corrections due to a d and b d could be as large as 40 per cent -here all possible decay channels have to be taken into account -except we are considering universal BSM contributions to all decay channels. Since ∆Γ d is not yet measured, we will revert our strategy and use the obtained bounds on the Wilson coefficients C 1 and C 2 to obtain potential enhancements or reductions of ∆Γ d due to BSM effects in non-leptonic tree-level decays. Considering the imaginary part of Γ s 12 /M s 12 , we can get dramatically enhanced values for the semi-leptonic CP asymmetries, if C 1 or C 2 are complex, which will result in an imaginary part of the GIM-unsuppressed coefficient c q . On the other hand new contributions to e.g. only b → cūq or b → ucq would have no effect on c q , but they could lift the GIM suppression of the coefficient a q and thus lead to also large effects. Therefore the semileptonic CP asymmetries are not completely dominated by the b → ccq transitions. Next we explain in detail how to implement BSM contributions to C 1 and C 2 in the theoretical description of Γ q 12 . Each one of the functions Γ q,ab 12 in Eq. (80) are given by [75] Γ q,ab The coefficients α 1 and α 2 in Eq. (84) include NLO corrections and are written in the MS scheme as Furthermore, the expressions for G q,ab and G q,ab S in Eq. (84) are decomposed as with F q,ab and F q,ab S encoding the perturbative contributions resulting from the double insertion of current-current operators. Finally, P q,ab and P q,ab S contain the perturbative effects from the combined insertion of a currentcurrent and a penguin operators. In terms of the tree-level Wilson coefficients C q,ab 1 and C q,ab 2 , the equations for F q,ab and F q,ab S have the following generic structure F q,ab = F q,ab 11 C q,ab 1 (µ) 2 + F q,ab 12 C q,ab 1 (µ)C q,ab 2 (µ) + F q,ab where the individual factors F q,ab 11,12,22 are available in the literature up to NLO Where the following definition for the ratio of the masses of the bottom and charm quarks, evaluated in the MS scheme [75], has been used The functions K cc 1,2,3 inside Eq. (89) are given by K cc 1 (µ) = 2 3C s,cc 1 (µ)C s,cc 3 (µ) + C s,cc 1 (µ)C s,cc 4 (µ) + C s,cc 2 (µ)C s,cc 3 (µ) , K cc 2 (µ) = 2C s,cc 2 (µ)C s,cc 4 (µ), K cc 3 (µ) = 2 3C s,cc 1 (µ)C s,cc 5 (µ) + C s,cc 1 (µ)C 6 (µ) + C s,cc 2 (µ)C s,cc 5 (µ) + C s,cc 2 (µ)C s,cc 6 (µ) , and the expression for the NLO correction function F cc p (z) is The Wilson coefficients inside Eqs. (91) should be calculated by introducing NP deviations at the scale µ = M W and then running down their corresponding values to the scale µ ∼ m b through the renormalization group equations, for details see the discussion in Sec. 2. In Appendix A, we provide details on the numerical inputs used. Since there was tremendous progress [53,54] in the theoretical precision of the mixing observables we will present in this work numerical updates of all mixing observables: ∆Γ q below, ∆M q in Section 4.4.1 and the semi-leptonic CP asymmetries a q sl and φ q in Section 4.5. For our numerical analysis we use results for Γ q,(0) 12,3 ,Γ q,(1) 12,3 and Γ q,(0) 12,4 , from [56,68,71,72,73,74,75] and for the hadronic matrix elements the averages presented in [54] based on [78,70,79] and [80,81,82,83], as well as the dimension seven matrix elements from [84]. The new SM determinations for ∆Γ s and ∆Γ d are The error budgets of the mixing observables ∆Γ s and ∆Γ d are presented in Tabs. 8 and 9 respectively. Compared to the SM estimates for ∆Γ s stemming from 2006 [75], 2011 [114] and 2015 [31] we find a huge improvement in the SM precision. In addition, the current SM predictions are based for the first time on a non-perturbative determination [84] of the leading uncertainty due to dimension seven operators. All previous predictions had to rely on vacuum insertion approximation for the corresponding matrix elements. To further reduce the theory uncertainties, improvements in the lattice determination would be very welcome or a corresponding sum rule calculation. The next important uncertainty stems from the renormalisation scale dependence, to reduce this a NNLO calculation is necessary. First steps in that direction have been done in [115]. In the ratio ∆Γ q /∆M q uncertainties due to the matrix elements of dimension six are cancelling -so for a long time this ratio    was considerably better known than the individual value of ∆Γ s . Due to the huge progress in determining precise values for these non-perturbative parameter, this advantage is now considerably less pronounced, see Table  10.  The resulting regions for ∆C s,cc 1 (M W ) and ∆C s,cc 2 (M W ) allowed by ∆Γ s are presented in Fig.12.

S J/ψφ
The mixing induced CP asymmetry for the decayB s → J/ψφ, given as   decaysB 0 s → J/ψφ and B 0 s → J/ψφ respectively. The required theoretical expressions have been calculated explicitly within the QCDF formalism in [116]. The equation for the decay amplitude obeys the structure where the the proportionality constant has been omitted since it cancels in the ratio λ s J/ψφ . The amplitudes α J/ψφ i appearing in Eq. (98) obey the structure given in Eq. (20). The required expressions for the vertices and hard-scattering functions can be found in the appendix. The index h = 0, ± indicated in Eq. (98) makes reference to helicity of the particles in the final state. During our analysis we average over the different helicity contributions. Therefore we take where each one of the asymmetries S h J/ψφ , are determined individually considering the corresponding amplitude A h J/ψφ for h = 0, ±. Neglecting penguin contributions our theoretical evaluation leads to sin(2β SM s ) = 0.037 ± 0.001, which numerically coincides with 2β SM s within the precision under consideration. The error budget is shown in Table 11. On the experimental side we use the average [96] 2β Exp s = 0.021 ± 0.031.
The effect of S J/ψφ on the allowed values for ∆C s,cc 1 (M W ) is not as strong as the results derived from other observables. However we included it in our analysis for completeness. For this reason we do not show the individual constraints from S J/ψφ and present only its effect in the global χ 2 -fit described in Section 5.4.  effects in both b → ccs and b → cūd the individually large effects will hugely cancel. We also neglect the currently unknown contribution of the Darwin term. Using the results presented in [50] we write
The experimental result for the ratio is [96] τ Bs To estimate the NP contribution (τ Bs /τ B d ) NP we consider the following function [50] where x c = m c /m b and C denotes the following combination of tree-level Wilson coefficients The non-perturbative matrix elements of the arising four-quark ∆B = 0 operators are parameterised in terms of the decay constant f Bs and the bag parameter B 1 , B 2 , 1 and 2 , which we take from the recent evaluation in [70].
The numerical values used are listed in Appendix A. The NP contribution to the lifetime ratio can be written as where in the second term in Eq. (107) we have dropped the NP contributions ∆C s,cc 1 (µ) and ∆C s,cc 2 (µ). Our bounds for ∆C s,cc 1 (M W ) are shown in Fig. 13, the corresponding results for ∆C s,cc 2 (M W ) turn out to be weak and therefore we do not display them. We would like to highlight the consistency between our regions and those presented in [51] which were calculated at the 68% C. L..

Observables constraining b → ccd transitions
We devote this section to the derivation of bounds on ∆C d,cc 1 (M W ) and ∆C d,cc 2 (M W ) from sin(2β d ) and B → X d γ. In our final analysis we also included contributions from a d sl which will be described in more detail in Section 4.5.

sin(2β d ) and SM update of ∆M q
In our BSM framework mixing induced CP asymmetries can be modified by changes in the tree-level decay or by changes to the neutral B-meson mixing. The first effect was studied in Section for the case of B s → J/Ψφ and found to give very weak bounds. Thus we will not consider them here. The second effect is also expected to give relatively weak bounds, but since the lack of strong bounds on new contributions to b → ccd we will consider it here -in the b → ccs we neglected it, because of much stronger constraints from other observables. We can constrain ∆C d,cc which can be evaluated by applying the generic definition of the CP asymmetry shown in Eq. (52) and using Where in Eq. (109), A J/ψK S andĀ J/ψK S correspond to the amplitudes for the processes B 0 → J/ψK S andB 0 → J/ψK S respectively. We study here modifications of q/p| B d , while we neglect the change of the amplitudes A J/ψK S andĀ J/ψK S -since an exploratory study found much weaker bounds. The definition of q/p| B d in terms of the B d matrix element M d 12 is given in Eq. (54). In the SM we have The dimension six effective |∆B| = 2 operator Q d 1 in Eq. (111) is given bŷ and the Wilson coefficient C |∆B|=2 (m t , M W , µ) corresponds to   where the factorη accounts for the renormalization group evolution from the scale m t down to the renormalization scale µ ∼ m b [77] and S 0 (x t ) is the Inami-Lim function [76] S 0 ( Using the new averages presented in [54] for the hadronic matrix elements (based on the non-perturbative calculations in [78,70,79] and [80,81,82,83]) we get the new updated SM results where we observe a huge reduction of the theoretical uncertainty, see Tables  12 and 13 Following [117] we evaluate the full combination at the scale µ c = m c , where the extra contribution to the SM |∆B| = 2 Hamiltonian in Eq. (111) is given byĤ with The set of HQET operators required in Eq. (120) arê Thus, our full determination of M d 12 is given by where the |∆B| = 2 operatorQ d 1 is matched at the scale µ c = m c intoP 0 [117]. The required matrix elements for the numerical evaluations are [70] with the values for the Bag parameters as indicated in Appendix A. Our theoretical result -neglecting contributions from penguins -is Parameter Relative error the full error budget in the SM can be found in Table 14. Notice that, the contributions from double insertions of the |∆B| = 1 effective Hamiltonian are relevant only when ∆C d,cc 2 (M W ) = 0, hence they do not appear in Table 14. On the experimental side we use the average from direct measurements [96] sin(2β Exp d ) = 0.699 ± 0.017, (126) our results for the allowed regions on ∆C d,cc 2 (M W ) are shown in Fig. 14.

4.4.2.B → X d γ
The branching ratio of the processB → X d γ allows us to impose further constraints on the NP contribution ∆C d,cc 2 (M W ). For the theoretical determination, we used the NNLO branching ratio for the transitionB → X d γ given in [118] B NNLO On the experimental side we consider [119,120,121] B Exp The NP regions on ∆C cc,d Fig. (15). Our treatment forB → X d γ is analogous for the one followed forB → X d γ, therefore our discussion here is rather short and we refer the reader to the details provided in Section 4.3.1.

Observables constraining multiple channels
Several observables like ∆Γ q , τ (B s )/τ (B d ) and the semi-leptonic CP asymmetries are affected by different decay channels. We have shown that ∆Γ s is by far dominated by the b → ccs transition, ∆Γ d has not yet been been measured. In τ (B s )/τ (B d ) a new effect in the b → ccs transition roughly cancels a similar size effect in a b → cūd transition, thus we have assumed for this observable only BSM effects in the b → ccs transition. Below we will study constraints stemming from a q sl , which is affected by the decay channels b → ccq, b → cūq, b → ucq and b → uūq.
Nevertheless, these observables yield already, with the current experimental precision, strong bounds on C 1 and C 2 due to the pronounced sensitivity of Im(Γ q 12 /M q 12 ) on the imaginary components of the ∆B = 1 Wilson coefficients. The regions for ∆C 1 (M W ) and ∆C 2 (M W ) allowed by the observables a s sl and a d sl are presented in Figs.16 and 17 respectively where for simplicity we have assumed the universal behaviour for j = 1, 2. As discussed in Section 4.3.2 different BSM effects in individual decay channels could lift the severe GIM suppression and lead to large effects, while the scenario given in Eq.(132) is dominated by b → ccq transitions. However, in Secs. 5.1, 5.2 and 4.4 we will also study the effects of a d sl on the different b-quark decay channels b → uūd, b → cūd, and b → ccd independently.

Global χ 2 -fit results
So far, we have limited our discussion to constraints derived from individual observables. In this section, we present, as the main result of this work, the resulting regions for ∆C 1 (M W ) and ∆C 2 (M W ) obtained after combining observables for the different exclusive b quark transitions. We will investigate three consequences of BSM effects in non-leptonic tree-level decays.
1. The allowed size of BSM contributions to the Wilson coefficients C 1 and C 2 , governing the leading tree-level decays. 2. The impact of these new effects on the possible size of the observable ∆Γ d , which has not been measured yet. Notice that, if one sigma deviations are considered, the current experimental uncertainty associated with ∆Γ d , see Eq. (95), allows enhancement factors within the interval On the other hand, if the confidence interval is increased up to 1.65 sigmas, i.e. 90% C.L., then the potential effects in ∆Γ d become The measured value of the dimuon asymmetry by the D0-collaboration [33,34,35,36] seems to be in conflict with the current experimental bounds on a d sl and a s sl , see e.g. the discussion in [122]. An enhanced value of ∆Γ d could solve this experimental discrepancy [123], at the expense of introducing new physics in ∆Γ d and potentially also in a s sl and a d sl . If all BSM effects in the dimuon asymmetry are due to ∆Γ d , then an enhancement factor of 6 with respect to its SM value is required. On the other hand, if there are also be BSM contributions in a s sl and a d sl , then the BSM enhancement factor in ∆Γ d can be smaller. 3. The impact of these new effects on the determination of the CKM angle γ. Within the SM, this quantity can be extracted with negligible uncertainties from B → DK tree-level decays [124,125,126,127,128,129]. This quantity is currently extensively tested by experiments, see e.g. [130,131] and future measurements will dramatically improve its precision to the one degree level [132]. This observable is particular interesting since direct measurements, e.g. LHCb [130], seem to be larger than bounds from B-mixing [53] 3 .
Therefore, in Sections 5.1 to 5.3 we combine our bounds from the b → uūd, b → cūd and b → ccd transitions, and evaluate the corresponding potential enhancement in ∆Γ d . We do not present the allowed regions for the NP contributions related to the channel b → ucd, since the bounds are expected to be rather weak considering that our only bound will arise from a d sl . In Section 5.4 we report the maximal bounds on ∆C 1 (M W ) and ∆C 2 (M W ), assuming universal BSM contributions to all different quark level decays. Hence, we combine all our possible bounds regardless of the quark level transition and asses the implications on the measurement of the CKM angle γ. The target of this part of analysis, is to update the investigations reported in [49] in the light of a far more detailed study of BSM effects in non-leptonic treelevel decays. In particular we account here for uncertainties neglected in the former study and we also make a very careful choice of reliable observables.
We have also included the contour lines showing the potential enhancement of the observable ∆Γ d . Accounting for the uncertainties in theory and experiment we find the following 90% C.L. intervals for ∆Γ d due to NP at tree level: for ∆C d,uu Thus only moderate enhancements of ∆Γ d seem to be possible, while a reduction to up to −39% of its SM values is still possible. This scenario could thus not be a solution for the dimuon asymmetry. find the possibility of huge enhancements/reductions of ∆Γ d : Based on the bounds shown in Eq. (140), we find that this scenario could solve the dimuon asymmetry. Since the experimental bounds for ∆Γ d are saturated in the case of ∆C d,cu We find again that this scenario could solve the tension between theory and experiment found in the measurement of the dimuon asymmetry. Considering the results shown in Fig. ( As can be seen on the l.h.s. of Fig. 20 C 1 is only weakly constrained by the semi-leptonic CP asymmetries, here additional information stemming from ∆Γ d will be important to shrink the allowed regions. for a = u, d and b = u, d. This procedure allows us to obtain the maximal constraints for our NP contributions. Making a combined χ 2 -fit is time and resource consuming, consequently we select the set of observables that give the strongest possible bounds. For ∆C 1 (M W ) this includes: R D * π , S ρπ , ∆Γ s , Br(B → X s γ) and a d sl and for ∆C 2 (M W ) we use: R D * π , R ππ , ∆Γ s , S J/ψφ and τ Bs /τ B d . We show in Fig. 21 our resulting regions from which we extract We can see from Eqs. (148) and (149) Fig. (22). We find that at 90% C.L. only O(20%) deviations on ∆Γ d with respect to its SM value can be induced, which is in a similar ballpark as the SM uncertainties of ∆Γ d and can clearly not explain the D0 measurement of the dimuon asymmetry.

NP in non-leptonic tree and its interplay with the CKM angle γ
As is well known [124,125,126,127,128,129] the CKM phase γ can be determined from the interference of the transition amplitudes associated with the quark tree level decays b → cūs and b → ucs with negligible theory uncertainty within the SM [135] 4 . At the exclusive level, this can be done with the decay channels B − → D 0 K − and B − →D 0 K − . The ratio of the two corresponding decay amplitudes can be written as where the r B stands for the ratio of the modulus of the relevant amplitudes.
The resulting phase has a strong component, denoted as δ B , and a weak one, which is precisely CKM γ. New effects in C 1 and C 2 can lead to huge shifts in γ: the left side of Eq. (150) will be modified according to [49] r where The ratios of matrix elements in Eq. (152) have not been determined from first principles, to provide an estimation we use naive factorization arguments and colour counting to obtain [49], [136] Eq.(151) gives a particularly strong dependence of the shift in γ on the imaginary part of C 1 ; approximately we get [49] We are now ready update the study presented in [49] on the effects of NP in C 1 and C 2 on the precision for the determination of the CKM angle γ, our results are presented graphically in Fig. (23). The current uncertainties in our knowledge of the value of C 1 seem to indicate an uncertainty in the extraction of the CKM angle γ of considerably more than 10 • , thus much higher than the current experimental uncertainty of around five degrees [130,131]. Interestingly direct measurements give typically larger values than the ones obtained by CKM fits [137,138] or extracted from B-mixing [53]. Even more interestingly future measurements will dramatically improve the precision of γ to the one degree level [132] and our BSM approach would offer a possibility of explaining large deviations in the extraction of the CKM angle γ. We would like, however, to add some words of cautions: for a quantitative reliable relation between the deviations of C 1 and the shifts in the CKM angle γ, the non-perturbative parameter r A and r A have to be known more precisely ansatz. We can explore the effects of modifying these values on CKM-γ. For instance, consider an alternative scenario where r A is twice the value presented in Eq. (153), while r A remains fixed. This is equivalent to assigning an uncertainty of 100% to r A and taking the upper limit. The results for this new scenario are presented in Fig. 24, where the shifts δγ CKM have been halved with respect to those found in Fig. 23, however the absolute numerical values of about ±5 • , still represent huge effects on the CKM angle γ itself.
Here clearly more theoretical work leading to a more precise understanding of r A and r A is highly desirable.

Future prospects
In this section we will present projections for observables, that are particularly promising to further shrink the allowed regions of NP contributions to non-leptonic tree-level decays. We have already studied the impact of BSM effects in non-leptonic tree-level decays on the observables ∆Γ d and the CKM angle γ in detail. More precise experimental data on ∆Γ d will immediately lead to stronger bounds on the ∆B = 1 Wilson coefficients, it could also exclude the possibility of solving the D0 dimuon asymmetry with an enlarged value of ∆Γ d . Alternatively, if the measured values of ∆Γ d will not be SMlike, we could get an intriguing hint for BSM physics. In order to make use of the extreme sensitivity of the CKM angle γ on an imaginary part of C 1 more theory work is required to make this relation quantitatively reliable. If this is available, then already the current experimental uncertainty on γ will exclude a large part of the allowed region on ∆C 1 -or it will indicate the existence of NP effects. Below we will show projections for improved experimental values on the lifetime ratio τ Bs /τ B d and the semi-leptonic CP asymmetries, as well as commenting on consequences of our BSM approach to the recently observed flavour anomalies.
6.1. τ Bs /τ B d As already explained, the lifetime ratio τ Bs /τ B d can pose very strong constraints on the Wilson coefficients C 1 and C 2 , if we e.g. assume that BSM effects are only acting in the b → ccs channel. In Fig. 25 we show future projections, assuming the errors will go down to 2 per mille or even one per mille. On the l.h.s. of Fig. 25 we assume that the current experiment value will stay -in this case a tension between the SM value and the experimental measurement will emerge. On the r.h.s. of Fig. 25 we assume that the future experimental value perfectly agrees with the SM prediction. In this case, the imaginary part of Im∆C 1 will be considerably constrained, this is a very interesting possibility since according to Eq. (154) Im∆C 1 is precisely Im ∆C s,cc Im ∆C s,cc Current status τB s /τB d =1.0006 ±0.0020 τB s /τB d =1.0006 ±0.0010 τ B s /τ B d Figure 25: Future scenarios concerning the behaviour of τ Bs /τ B d . In the left panel the central experimental value of the lifetime ratio is assumed to remain unchanged in the future whereas the uncertainties will be reduced. In the right panel, the theoretical and experimental values for the lifetime ratio are supposed to become equal. the driving force for large deviations in the CKM angle γ.

Semi-leptonic CP asymmetries
The experimental precision for the semi-leptonic CP asymmetries is still much larger than the tiny SM values for these quantities. Nevertheless already at this stage a q sl provide important bounds on possible BSM effects in the Wilson coefficients. The experimental precision in the semi-leptonic CP-asymmetries will rise considerable in the near future, see e.g. Table 1 of [139] from where we take: δ (a s sl ) = 1 · 10 −3 LHCb 2025 (155) δ (a s sl ) = 3 · 10 −4 Upgrade II (156) We show the dramatic impact of these future projections on the BSM bounds on the Wilson coefficients in Fig. 26.

Rare decays
As discussed in [50,51] NP effects in the b → ccs transitions can induce shifts in the Wilson coefficient of the operator This result offers an interesting link with the anomalous deviations in observables associated with the decay B → K ( * ) µ + µ − , where model independent explanations with physics only in C 9 require ∆C eff 9 µ=m b = −O(1). In order to account for NP phases we use the results presented in [140] where ∆C 9 is allowed to take complex values leading to the constraints shown in Fig. 27. Here both C 1 and C 2 get a shift towards negative values. BSM in effects in non-leptonic tree-level can in principle explain the deviations seen in lepton-flavour universal observables, like the branching ratios or P 5 ; they can, however, not explain the anomalous values of flavour universality violating observables like R K . Future measurements will show, whether the bounds, obtained in Fig. 27 should be included in our full fit.

Conclusions and outlook
In this work we have questioned the well accepted assumption of having no NP in tree level decays, in particular we explored for possible deviations with respect to the SM values in the dimension six current-current operatorsQ 1 (colour suppressed) andQ 2 (colour allowed) associated with the quark level transitions b → qq s and b → qq d (q, q = u, c). We evaluated the size of the NP effects by modifying the corresponding Wilson coefficients according to C 1 → C 1 + ∆C 1 , C 2 → C 2 + ∆C 2 , for ∆C 1,2 ∈ C; we found that sizeable deviations in ∆C 1,2 are not ruled out by the recent experimental data. Our analysis was based on a χ 2 -fit where we included different B-physics observables involving the decay processes: We also considered neutral B mixing observables: the semi-leptonic asymmetries a s sl and a d sl as well as the decay width difference ∆Γ s of B 0 s oscillations and the lifetime ratio of B s and B d mesons. Finally we also studied the CKM angles β, β s and γ. For the amplitudes of the hadronic transitionsB 0 d → D * π,B 0 d → ππ,B 0 d → πρ andB 0 d → ρρ andB s → J/ψφ we used the formulas calculated within the QCD factorization framework. We have identified a high sensitivity on ∆C 1,2 with respect to the power corrections arising in the annihilation topologies and in some cases in those for the hard-spectator scattering as well. It is also important to mention that the uncertainty in the parameter λ B used to describe the inverse moment of the light cone distribution for the neutral B mesons is of special importance in defining the size of ∆C 1 and ∆C 2 . For the mixing observables and the lifetime ratios we have benefited from the enormous progress achieved in the precision of the hadronic input parameters, thus we have also updated the corresponding SM predictions: We have made a channel by channel study by combining different constraints for the decay chains b → uūd, b → cūd, b → ccs and b → ccd; we also performed a universal χ 2 -fit where we have included observables mediated by b → qq s decays as well. The universal χ 2 -fit provides the strongest bounds on the NP deviations, we found that whereas for the independent channel analyses the corresponding deviations can much larger. We have analysed the implications of having NP in tree level b quark transitions on the decay width difference of neutral B 0 d mixing ∆Γ d -note, that the most recent experimental average is still consistent with zero. We found that enhancements in ∆Γ d with respect to its SM value of up to a factor of five are consistent with current the experimental data. Such a huge enhancement could solve the tension between experiment and theory in the D0 measurement for the dimuon asymmetry. Thus we strongly encourage further experimental efforts to measure ∆Γ d , see also [141]. Next we evaluated the impact of our allowed NP regions for ∆C 1 and ∆C 2 on the determination of the CKM phase γ, where the absence of penguins leads in principle to an exceptional theoretical cleanness. We found that γ is highly sensitive to the imaginary components of ∆C 1 and ∆C 2 and our BSM effects could lead to deviations in this quantity by up to 10 • . It has to be stressed, however, that for quantitative statements about the size of the shift δγ the ratios of the matrix elements D 0 K − |Qū cs 1 |B − / D 0 K − |Qū cs 2 |B − and D 0 K − |Qc us 1 |B − / D 0 K − |Qc us 2 |B − have to be determined in future with more reliable methods. So far only naive estimates are available for these ratios. Finally we studied future projections for observables that will shrink the allowed region for NP effects -or identify a BSM region -in non-leptonic tree-level decays. Here τ (B s )/τ (B d ) and the semi-leptonic CP asymmetries seem to be very promising.
While PDG gives Λ The PDG value for m c (m c ) correspond to m c (m b ) = 0.947514, which will be used 5 for the analysis of the mixing quantities ∆Γ q and a q sl . For the top quark pole mass we use the result obtained from cross-section measurements given in [101] m We use the averages of the B mixing bag parameters obtained in [54] based on the HQET sum rule calculations in [79,70,78,158] and the corresponding lattice studies in [159,160,161,162,163]:  corresponds to the primed bag parameter of [84]. For the remaining two operators we are using equations of motion [71] B qR For the determination of the uncertainties of the ratios of Bag parameter, we first symmetrized the errors of the individual bag parameter. Based on the updated value for the bag parameter B q 1 given above and the lattice average (N f = 2 + 1 + 1) for f Bq presented in [149] -based on [80,81,82,83] Additionally, for the determination of the contributions of the double insertion of the ∆B = 1 effective Hamiltonians to M d 12 we require the following Bag parameters at the scale µ c = 1.5 GeV (see [70]) where we have used Finally we take the lifetime bag parameter from the recent HQET sum rule evaluation in [70] -here no corresponding up to date lattice evaluation exists Using CKMfitter-Live [137] online, we perform a fit to the CKM elements |V us |, |V ub |, |V cb | and the CKM angle γ excluding in all the cases the direct determination of the CKM angle γ itself. Our inputs coincide mostly with the CKMfitter-Summer 2018 analysis, however in order to be consistent with our main study we modify the following entriesm t (m t ),m c (m c ),B s 1 and the ratiosB from which we obtain |V ub | |V cb | = 0.08833 ± 0.00218.
The full set of CKM matrix elements is then calculated under the assumption of the unitarity of the 3 × 3 CKM matrix.

B1.3. Penguin functions
To simplify the following equations we have denoted M = π, ρ when the corresponding expressions apply to both π and ρ mesons. In addition we have used for p = u, c, although in practice we consider s u = 0.