Model-independent analysis of b → d processes

We perform a model-independent analysis of | ∆ b | = | ∆ d | = 1 processes to test the standard model and probe ﬂavor patterns of new physics. Constraints on Wilson coeﬃcients are obtained from global ﬁts to B + → π + µ + µ − , B 0 s → ¯ K ∗ 0 µ + µ − , B 0 → µ + µ − , and radiative B → X d γ decays data. The ﬁts are consistent with the standard model but leave sizable room for new physics. Besides higher-statistics measurements and more data in theory-friendly bins of the dilepton mass, further complementary observables such as angular distributions of B 0 s → ¯ K ∗ 0 ‘ + ‘ − , B → ρ ‘ + ‘ − or the baryonic modes Ξ b → Σ ‘ + ‘ − , Ω − b → Ξ − ‘ + ‘ − are necessary to resolve the signiﬁcant degeneracy in the ﬁt for the semileptonic four-fermion operators. Assuming minimal quark ﬂavor violation, the b → s global ﬁt implies tight constraints on the b → d couplings, and hence allows to test this paradigm with improved data. Another beneﬁt from | ∆ b | = | ∆ d | = 1 processes is to shed light on the B -anomalies in | ∆ b | = | ∆ s | = 1 modes from a new angle. Speciﬁcally, studies of lepton ﬂavor-speciﬁc and dineutrino modes are informative on the lepton ﬂavor structure. Rare b → d ‘‘, ν ¯ ν decays can be studied at high luminosity ﬂavor facilities LHCb, Belle II, and a future Z -factory.

Flavor-changing neutral current (FCNC) transitions arise in the standard model (SM) at the quantum level, and are sensitive to new physics (NP) and its flavor structure.Rare radiative decays b → q γ and semileptonic ones b → q − + with q = s, d and = e, µ, τ are such promising probes, allowing also to test approximate symmetries of the SM such as lepton flavor universality.
Especially the b → s µ + µ − transitions have been analyzed over the past two decades with vigor, revealing an intriguing picture of discrepancies with SM predictions, referred to as the flavor anomalies, e.g.[1][2][3]: i) Branching ratios are below the SM values.ii) Angular distributions provide theoretically cleaner observables and give more than 4σ deviation from the SM in global fits, recently [4][5][6][7].Electron-muon universality violation has been evidenced by LHCb in R K , the ratio of B → Kµµ to B → Kee branching fractions [8], strengthening the trend in measurements of R K -like ratios [9] with b-decays into other strange hadrons.Interestingly, a very recent experimental update of R K and R K * [10,11] revealed consistency with the SM.Since data i) and ii) can be explained by NP in semileptonic |∆b| = |∆s| = 1 fourfermion operators with coupling to muons, the nil return of electron-muon non-universality in the new LHCb results [10,11] suggests discrepancies with the SM in b → se + e − modes, specifically similarly reduced branching ratios and distorted angular distributions.While this points to NP in flavorful processes, further scrutiny is required before firm conclusions can be drawn.This includes also cross-checks with other sectors.
In addition to benefitting from correlations among several b → d observables, model-independent EFTinterpretations also allow for probing the quark and lepton flavor structure of NP.Analyses within the SM effective theory (SMEFT) [20,21] have provided directions to shed light on the flavor anomalies and beyond by combining top, beauty, charm, and kaon, and charged dilepton and dineutrino data [22][23][24].The results of this work are providing input for the connection between third-and first-generation quark FCNCs, their link to the thirdand second-generation quark FCNCs, and tests of lepton universality in b → d ν ν decays.Specifically, we give details and updates to the analysis of [7].
The paper is organized as follows: In Sec.II, we give the effective theory framework and the |∆b| = |∆d| = 1 rare decay observables used in the analysis.Set-up and results of the global fits are presented in Sec.III.We conclude in Sec.IV.Covariance matrices are given in App. A. Uncertainty estimates of the tails of charmonia on the B 0 s → K * 0 µ + µ − branching ratios using LHCb's cuts [14] are given in App.B.

II. EFFECTIVE THEORY FRAMEWORK
In Sec.II A, we introduce the weak effective theory framework to study |∆b| = |∆d| = 1 transitions.The SM predictions for (semi-)leptonic and radiative observables, as well as its semi-analytical expressions, where NP effects are included model-independently, are presented in Sec.II B.

A. Weak effective Hamiltonian
Rare b → d µ + µ − transitions can be described by the following effective Hamiltonian with and the CKM factors λ We consider NP effects in the following dimension-six operators encode the dynamics of the heavy degrees of freedom from the SM (the top quark, the W ± and Z 0 -bosons, and the Higgs) and NP in C and can be safely neglected, in addition to (pseudo-)scalar and (pseudo-)tensor operators.In this work, we do not include the latter operators, but we work out limits on (pseudo-)scalar ones from the branching ratio of B 0 → µ + µ − decay in Sec.II B 3.

B. Rare b → d decay observables
We introduce the |∆b| = |∆d| = 1 observables taken into account in the global fits presented in Sec.III.Formulas are given in or are adapted from Refs.[18,31,32].
The differential branching ratio of B + → π + µ + µ − decays can be written as [18,33] Bπ F Bπ (q 2 ) , (8) where encodes the dependence on the Wilson coefficients and the form factors (FFs).Here, in the low q 2 region, that is, in the kinematic region where the pion is energetic in the B-meson center of mass frame we employ QCD factorization [34].We do not consider very low q 2 -bins near light resonances and those near the charmonium peaks.At high q 2 , i.e., low hadronic recoil we employ the OPEframework of [35,36].As duality is expected to work better for larger intervals, we use the largest available bin from that region.In view of the presently sizable experimental uncertainites and the selected bins we refrain from studying the impact of non-FF contributions.These include weak annihilation contributions, that are of importance at very low q 2 [37,38].In Eq. ( 9) terms induced by a finite muon mass are omitted for clarity, but are included in the numerical analysis.The functions f + (q 2 ), f T (q 2 ) (and f 0 (q 2 ) when corrections O m 2 µ are taken into account) denote the B → π transition FFs, and their determination requires the use of non-perturbative techniques.We employ the results of Ref. [39], where a fit combining lattice data at high-q 2 , and light-cone sum rules (LCSRs) data at low-q 2 has been performed.We also include the contributions from Ref. [30] in the effective coefficients.
The SM predictions of the binned branching fractions with their respective uncertainties are given in the third column of Tab.I1 .The first source of error corresponds to FFs, the second one to CKM matrix elements, and the last one reflects the fluctuations under changes of the short-distance scale µ b in the range between m b /2 and for the bin [2,4] GeV 2 , and so on.
1 [2,4] 0.80 ± 0.12 ± 0.05 ± 0.04 0.62  in units of 10 −9 GeV 2 , as defined in Eq. (10).The SM predictions are given in the third column with their three main sources of uncertainty including FFs, CKM matrix elements, and µ b , respectively.The experimental values from LHCb [12] are displayed in the last column with their statistical and systematic uncertainties, respectively.Further details on the uncertainties treatment are provided in App. A. The shaded rows indicate the bins included in the fit, see main text for details.† To allow for a direct comparison with the value provided by LHCb the last row (k = 10) corresponds to the integrated branching ratio over the full q 2region, ((m , rather than a binaverage as in Eq. (10).The measurements by LHCb [12] (fourth column in Tab.I) are in very good agreement with the SM predictions presented in the third column in Tab.I.All bins are compatible within 1 σ except for the high-q 2 bin [22, 25] GeV 2 , which agrees at 1.6 σ.We do not provide theory predictions for the low-q 2 bin [0.1, 2] GeV 2 , which suffers from hadronic uncertainties coming from the ρ, ω and φ resonances, q 2  1 GeV 2 .In addition, regions around q 2 ≈ 9.5 GeV 2 and q 2 ≈ 13.5 GeV 2 suffer from J/ψ and ψ resonances and their tails (gray bands in Fig. 1), respectively.Therefore, in the global fits presented in Sec.III we only include the theoretically clean bins k = 1, 2, 9, that are [2,4] GeV 2 , [4, 6] GeV 2 , and [15,22] GeV 2 , indicated by shaded rows in Tabs.I and II.
In Fig. 1, we compare the experimental branching fraction of B + → π + µ + µ − in bins of q 2 (black points) from LHCb [12] with the SM predictions (BGGH22, this work, yellow) from Tab. I. We compare our SM predictions to those available in the literature including Refs.[18] (APR13, blue), [40] (HKR15, red) and from lattice QCD calculations [41] (FNAL/MILC15, green).We observe very good agreement except at low q 2 with the predictions of [40], which are larger than the others.Ref. [40] includes non-FF contributions from non-local matrix elements from LCSR.On top of that, the branching ratios of [40] (red) are subject to an enhancement from the leading CKM-factor by 8% and FFs by almost 50% compared to our analysis (yellow).Taking this parametric effect into account addresses the bulk of the numerical differences between the SM theory curves for q 2 2 GeV FIG. 1. Measured branching fraction of B + → π + µ + µ − in bins of dilepton invariant mass squared q 2 (black points) from LHCb [12].SM predictions (this work, BGGH22, yellow) from Tab.I are compared to those of Refs.[18] (APR13, blue), [40] (HKR15, red) and [41] (FNAL/MILC15, green).Our prediction (yellow) for the last bin is on top of the one from FNAL/MILC15.Plot adapted from [12].See text for details.
In the presence of primed NP operators, Eq. ( 9) can be generalized by noting that due to spin parity of the external states and parity invariance of QCD the Wilson coefficients of unprimed and primed operators enter B → π + − decays only by their exact sum, C i + = C i + C i .Taking into account Eq. ( 11), the binned branching ratios Eq. ( 10) can be compactly written as with the following set of Wilson coefficients combinations The numerical values of a (Bπ) i for different bins k are given in Tab.II.Further information regarding uncertainties and correlations is provided in App. A.
The differential branching ratio of B 0 s → K * 0 µ + µ − decay can be expressed as [31] d Bs K * F Bs K * (q 2 ) , (14) where the dynamical function F Bs K * (q 2 ) reads Here, the functions A , B, ... contain the dependence on Wilson coefficients and FFs ) , where V (q 2 ), A 0,1,2 (q 2 ), T 1,2,3 (q 2 ) are the B s → K * transition vector, axial and tensor FFs, respectively.We take them from Ref. [42].As in Eq. ( 9), we do not show the muon mass corrections.They are given in Ref. [31], and are included in our analysis.Integrating Eq. ( 14) over [q 2 min , q in agreement with the first experimental evidence from LHCb [14], Equation ( 17) highlights the four main sources of uncertainty: FFs, CKM elements, the renormalization scale µ b , and the effect of the J/ψ and ψ resonances, respectively.The last uncertainty arises from the interpretation of LHCb data [14] which encompasses the resonance regions.In App.B, we provide details on the estimation of the last uncertainty, see Eq. (B7).
When chirality-flipped NP effects are turned on, thanks to Lorentz invariance and parity conservation of QCD, they enter either as a sum or difference with the unprimed coefficients, 15) can therefor be easily adapted through the following replacements in Eq. ( 16): which result in The values of a for the q 2 -region [0.1, 19] GeV 2 are given in Tab.III.Information about the uncertainty and correlations is compiled in App. A. 2 Eq. ( 21) can be expressed as well in terms of  in units of 10 −9 for the q 2 -region [0.1, 19] GeV 2 , see Eq. (20).Further details about uncertainties and correlations are provided in App. A.

scalar operators
The B 0 → µ + µ − decay is a clean observable to probe beyond standard model (BSM) physics in b → d µ + µ − transitions.In the SM, only the operator O 10 contributes which yields [43] including updated CKM values [44] in agreement with the experimental value [13] Combining Eqs. ( 22) and ( 23), we obtain The purely leptonic decay 10 , and scalar and pseudoscalar operators, Including their effects, the branching ratio can be written as [45] where Setting C P − and C S − to zero, Eq. ( 24) implies the following ranges for C 10 assuming a single NP coefficient at a time.In this work, we do not consider scalar and pseudoscalar contributions in the global fits, and the B 0 → µ + µ − branching ratio hence depends only on one combination of Wilson coefficients, C SM 10 + C 10 − .A decomposition of the branching ratio along the lines of Eq. ( 13) would result formally in three structures, 1, C 10 − , C 2  10 − , however, all three prefactors are related, so there is just one coefficient to be varied in the fit.

b → d γ
The branching ratio of the inclusive decay B → X d γ can be written as [32,47] where the numerical prefactor N = (25.7 ± 1.6) • 10 −4 is taken from Ref. [32].The factor N = (3.6±0.6)•10−3 encodes non-perturbative corrections [48], whereas P contains the perturbative SM contributions and the NP effects from O 7,8 , and P accounts for NP contributions from O 7,8 operators.We updated the SM prediction for the CP-averaged B → X d γ branching ratio from Ref. [15].The main improvement is driven by the updated CKM prefactor [44] in Eq. (32).We observe a very good agreement between the SM prediction in Eq. ( 33) and the extrapolated experimental value from BaBar [15,16], Including NP effects from C ( ) 7,8 using Ref. [32], we find the following parametrization with The central values of the factors a are compiled in Tab.IV.These factors suffer from four sources of uncertainties: renormalization scheme dependence of the ratio m c /m b (∼ 13 %), parametric uncertainties (∼ 7 %), and the perturbative scale µ b (∼ 4 %), which have been included in our analysis.
Experimental information on the exclusive radiative decay B → ρ γ is available [44] however its SM prediction has sizable uncertainties from the tensor FF T 1 (0).Here, we used T B→ρ 1 (0) = 0.24 ± 0.12 [42], see therein for earlier LCSR computations, which are consistent but have smaller uncertainties.Confronting the SM to data one obtains the upper limits Since the dependence on the coefficients is the same up to higher order contributions and non-perturbative effects in the global fit we only take the inclusive modes which gives stronger constraints into account.Once the form factor B → ρ is more accurately known the exclusive decays give stronger constraints as the measurement is more precise.

III. GLOBAL b → d FITS
We perform one-dimensional (1D), two-dimensional (2D) and four-dimensional (4D) fits to various combinations and subsets of the BSM-sensitive Wilson coefficients C The aim of performing higher dimensional fits is in particular to work out and demonstrate the impact and benefits of correlations between different observables and, for the case of the 4D fit, obtain conservative limits for global SMEFT analyses involving semileptonic four-fermion operators.

A. Fit method
Using the experimental information from b → d transitions presented in Sec.II B, we perform global fits to obtain information on the Wilson coefficients C ( ) i (collectively denoted here as θ).We work within a frequentist framework based on the approximation of Gaussian likelihood L(θ) = e −χ 2 (θ)/2 , resulting in the following χ 2 function with Here, n obs represents the number of observables included in the fit (n obs = 6), with Furthermore, ∆ (θ) is the central value of the theory prediction for the i-th observable, ∆ (exp) i is the central value of the experimental measurement of the same observable, while V (th) ij (θ) and V (exp) ij are the theoretical and experimental covariance matrices, respectively.We provide details on the computation of these matrices in App. A. To estimate the central values of the θ parameters, we use the maximum likelihood (ML) method.By construction of L(θ), the ML estimators θ correspond to the best-fit points obtained by minimizing the χ 2 (θ) function, that is ∂ χ 2 /∂ θ i | θ = 0 with i = 1, ..., n par and n par being the number of parameters (up to 8 parameters in the most general scenario).In practise, we use MIGRAD from the Python package iminuit [49] to conduct the numerical minimization.We scan for the viable parametric space of θ by initializing the MIGRAD algorithm to different initial values of θ.In some cases, we find multiple solutions with similar χ 2 min ≡ χ 2 ( θ), we always choose the one with a smaller χ 2 value.In that sense, future measurements of b → d µ + µ − modes are necessary to exclude other solutions with larger χ 2 values.
With the explicit dependence of the χ 2 function given by Eq. ( 39), we compute the errors, σ θi = (V θ ) ii , and the correlations among the fit parameters, ρ θi θj ≡ (V θ ) ij / (V θ ) ii (V θ ) jj , which are given by the matrix These matrices are provided as an output by MIGRAD when performing the minimization routine.
To determine the level of statistical significance between a given hypothesis H and the data, we also determine the p−value of the respective fit, where n dof = n obs − n par is the number of degrees of freedom.A p−value of less than 0.05 indicates strong evidence against the assumed hypothesis, as there is less than a 5% probability that H is correct.Finally, to compare the description offered by two hypotheses H and SM, we provide their relative pull (in units of standard deviations σ) where Erf is the error function, ∆χ 2 H = χ 2 SM,min −χ 2 H,min , and n H is the number of parameters in H.

B. Results for Wilson coefficient fits
In the following, we present the fit results for 1D, 2D

One-dimensional fits
We start with 1D scenarios (H 1,..., 16 ), which are compiled in Tab.V. Here, the first column displays the name of the scenario, while the second column indicates the fitted Wilson coefficient and the correlations assumed between other Wilson coefficients, as in H 9,..., 16 .The bestfit values with their 1σ and 2σ confidence intervals are given in the third and fourth columns, respectively.The last columns show the indicators of the goodness-of-fit: best-fit χ 2 value, the pull from the SM hypothesis, see Eq. ( 45), and the p−value of the fit, see Eq. (44).
Among the 1D scenarios with only one Wilson coefficient switched on at a time, H 1,...,8 , the most favored one is H 3 with NP in C 9 only, with a pull of 1.63 σ and a p−value of 95%.The second most favored scenario is H 4 with a pull of 1.50 σ and a p−value of 91%, where NP only enters in C 10 .One observes that the global fit selects the solution with the smaller NP effect from the ranges given by B 0 → µ + µ − alone, see Eq. (28).
Results of the 2D fits to Wilson coefficients C7,C8 (scenario H17, left panel) and C7,C 7 (scenario H20, right panel), see Tab.VI.We show the 1σ allowed regions of the individual observables and the combined 1, 2, and 3σ regions (red).The black star at the origin represents the SM.Results of the global b → s fit, see App.C, are superimposed (blue) and serve as a data-driven prediction assuming minimal flavor violation (MFV).Deviations between the blue area and the SM are a result of the present discrepancies in b → s processes with the SM.See text for details.
Furthermore, we perform global fits in scenarios relating two Wilson coefficients (H 9,...,14 ) in which SU (2) Linvariance is preserved in the lepton sector or those with a preference for leptonic vector-couplings.We find that the scenario H 10 with left-handed quarks and left-handed leptons, C 9 = −C 10 , is preferred by data with a pull of = −0.50 ± 0.13, see App. C. We explore additional correlations among the primed coefficients C 9,10 exploiting C 9 = ± C 10 , see scenarios H 11 and H 12 with p−values worse than the SM one.In addition, we consider C 9 = ± C 9 , which gives a pull of 1.34 σ and a p−value of 85% for C 9 = −C 9 , while a p-value of 78 % is obtained for C 9 = +C 9 .Taking into account the previous observations, we perform a fit in scenarios involving four NP coefficients, H 15 and H 16 , where the most preferred scenario is H 15 with a pull of 1.61 σ and a p−value 94%.
We observe that data shows a clear preference to include NP via C 9 , following a similar pattern as in global fits to b → s µ + µ − data, where C To summarize, while there is consistency with the SM, we also observe room for sizable, order one NP contri-butions to Wilson coefficients, significantly larger than corresponding ranges in the b → s global fit.We note also that the likelihood is quite flat, making uncertainty ranges more sensible than best-fit points.These features continue to hold in 2D and 4D fits discussed next.

Two-dimensional fits
Here we present the results for the 2D fits which are compiled in Tab.VI.We exploit scenarios composed out of two coefficients from C 7 , C 7 , C 8 , C 9 , C 9 , C 10 , C 10 in H 17,...,32 .The presently most favored scenario is H 25 with a pull of 1.22 σ and a p−value of 94%, fitting simultaneously C 9 and C 9 .We observe again a similar pattern as in the 1D fits, that, if C 9 is included in the fit, p−values are large, ∼ 90%, i.e.H 18,23,24,25,26 .In addition, we perform fits in scenarios where NP enters in a correlated way in four Wilson coefficients, H 33,...,37 .Among these, scenario H 33 , with C 9 = −C 9 and C 10 = +C 10 , features the highest pull of 1.22 σ and a large p−value of 94%.Some of these benchmark models, H 34,35,37 satisfy the relation C 10 /C 10 = C 9 /C 9 , which arises from tree-level exchange of a Z -boson [50].3. Results of the 2D fits to semileptonic four-fermion operators, see Tab.VI.Similar description as in Fig. 2.
In Fig. 3, we display contours from fits to semileptonic four-fermion operators.We observe that the complementarity between the observables is currently not as good as for dipole coefficients, leading to weaker limits on C 9 and C 10 , see the uppermost left plot of Fig. 3, displaying contours in H 23 , C 9 versus C 10 .The branching ratios of B + → π + µ + µ − , and B 0 s → K0 * µ + µ − cooperate to reduce the thickness of the annulus (red area) but do not lift the degeneracy between C 9 and C 10 .The branching ratio of B 0 → µ + µ − can in principle do so due to its dependence on C ( ) 10 only, however, the present precision is insufficient.In the future, the HL-LHC run can improve this significantly, see Eq. ( 29) for the projected sensitivity, up to a discrete ambiguity.This is illustrated by the dashed blue horizontal lines in the uppermost left plot of Fig. 3.Note also that due to the flat likelihood along the ring (red area) the best-fit point (magenta) is only shown for completeness but has little statistical preference over other points in this flat direction.The projections in Fig. 3 also make visible discrete ambiguities, for instance the two yellow bands in C 10 , C 10 (mid right-handed plot).
To remove the ambiguities additional complementary observables of b → d transitions are necessary.As well known from the early days of the b → s µµ fits, angular observables break the degeneracy in the branching ratio based fits and significantly improve the information on the NP couplings [28,51].The most simple one is the forward-backward asymmetry A FB in the lepton angle, defined as and θ is the angle between the negatively charged lepton and the decaying b-hadron in the dilepton center of mass system.In 1/2 → 1/2 + − baryon or 0 → 1 + − meson decays, where we indicate the spin of the initial and final hadron, A FB is sensitive to the product of Wilson coefficients C 9 C 10 rather than the sum of squares as the branching ratios.Therefore, in terms of constraints, the latter provide ellipses, while A FB gives hyperbolas in the C 9 -C 10 plane [52].Further observables with complementary dependence on the Wilson coefficients arise from full angular analysis.Already a rough measurement of A FB can signal new physics, and rule out large classes of BSM models and non-SM flavor structures [31].The Belle collaboration determined the sign of A FB in b → s transitions based on 113.6 ± 13 events of B → K * + − decays [53].Since LHCb evidenced Bs → K * 0 µµ with 38 ± 12 events [14], one could expect that this crucial test in the b → d sector can be performed in the not too far future.
Let us further entertain a comparison with the b → s global fits.The latter can be related to b → d processes assuming that quark flavor violation is realized minimally -induced by the known quark masses and mixing angles.Within the weak effective theory (1), this implies universal Wilson coefficients C i , allowing them to be compared between the two fits.For the primed coefficients an additional factor of down to strange quark masses is involved, . A disagreement between the resulting b → s and b → d Wilson coefficients would signal additional sources of quark flavor violation beyond the SM.In Figs. 2 and 3, we show in addition to the b → d results (red) the global fit results for the same scenarios using b → s input (blue), see App. C. The fact that the SM (black star) is not included in the b → s preferred regions is a reflection of the present anomalies in this sector.
We observe as expected from the significantly more precise data in the b → s sector, that the latter is significantly stronger constrained than the b → d one.Due to the quark mass suppression, m d /m s ∼ 0.1, the derived constraints on the primed coefficients are even stronger, and result in narrow regions in the corresponding plots.One also notices that the preferred region for b → s is inside the b → d one.Data are therefore consistent with the hypothesis of minimal quark flavor violation.To test this further requires improved data on b → d transitions.The anticipated reach in B d → µ + µ − from the HL-LHC [46] given in Eq. ( 29) probes the present MFV prediction in C 10 , see e.g., the upper left plot of Fig. 3. On the other hand, there is a sizable window for flavorful new physics in b → d modes, and new physics can be just around the corner.Observing for instance a wrong sign A FB with the next round of data would signal a breakdown of the SM, and of MFV.

Four-dimensional fits
We perform 4D fits (scenarios H 38 and H 39 ), where the numerical results are presented in Tab.VII.Results are consistent with those of the 1D and 2D scenarios, but exhibit larger uncertainties due to the larger number of parameters.This is to be expected as already good fits where obtained for 1D and 2D scenarios, so the additional degrees of freedom are not needed to improve the consistency with the data.For this reason, we do not show contour plots of the 4D fits.On the other hand, the more general situation with four independent couplings describes more general NP models.

C. Interplay with b → d ν ν and high-pT
For completeness, we compare our results with the limits extracted using other experimental information.To do so, it is convenient to introduce the following combinations of Wilson coefficients which project onto couplings to left-handed leptons.The methodology is based on [7].In Tab.VIII, we show the limits on κ bdµµ L(R) obtained in the 4D scenario H 38 (C 9 , C 9 , C 10 , C 10 ), which update those from [7] by taking into account correlations.Also given are limits using data on rare B decays into dineutrinos [7], as well as high-p T constraints [54,55].The limits from the global fit obtained in this work are a factor 30 (20) stronger than the limits extracted from dineutrino modes (Drell-Yan processes).H38).The third row displays the bound on κ bdµµ R from dineutrino modes using SU (2)L taken from Ref. [7].The last row displays upper limits extracted from high-pT data, that is, Drell-Yan processes [54,55].
Figure 4 updates the corresponding plot from Ref. [7] by taking into account the somewhat larger range of κ bdµµ R due to correlations in the global fit 3 .While the cones predicted assuming universality (purple) are consequently somewhat wider, this is a barely visible effect as the width is fully dominated by FF uncertainties, see Ref. [7] for details.

IV. CONCLUSIONS
We present the first model-independent analysis of rare radiative and semileptonic |∆b| = |∆d| = 1 processes based on the branching ratios of significantly improved.This is within the HL-LHC reach of the B d → µ + µ − branching for the Wilson coefficient C 10 (29).In the meantime, since flavorful new physics may be just around the corner, it can show up in b → d angular distributions and could rule out MFV.Note that the pattern of branching ratios suppressed with respect to the SM in b → s µµ decays continues to hold for the b → d µµ ones, although within larger uncertainties, see Fig. 1 and Eqs. ( 17) and (18).
Due to several flat directions in parameter space but also sizable uncertainties, the likelihood for the semileptonic four-fermion operators is quite flat.Consequently, improving the fit is not just a matter of higher statistics, but also of adding observables sensitive to different combinations of Wilson coefficients than the branching ratios.Such observables are the very well-known leptonic forward-backward asymmetries in semileptonic decays of b-mesons or b-baryons, see Eq. ( 46), or a full angular analysis thereof including secondary hadronic decays.We point out here opportunities from Ω − b → Ξ − (→ Λπ − ) + − , as the Ξ − → Λ π − decays selfanalyzing.The decay Λ b → n + − is also sensitive to the |∆b| = |∆d| = 1 couplings but challenging experimentally due to the neutron in the final state.We stress that already an early, first measurement of A FB , as in [53], provides key information to the global fit.Observing, for instance, a 'wrong sign'-value would rule out the SM, and point to flavorful new physics beyond MFV [31].
Rare b → d induced decays can be studied at high luminosity flavor facilities, such as LHCb [57], Belle II [58], and a possible future e + e − -collider running at the Z [59].For distributions of the dilepton mass sufficient separation from resonances is advised, and results should be provided in such bins, ideally with correlations, and without prior extrapolation over veto regions.
Improved knowledge of hadronic form factors, including B → ρ tensor ones that, for instance, would allow obtaining useful constraints from B → ρ γ data, would be desirable.With more data, also fits to complex-valued Wilson coefficients probing CP violation could be performed.
Besides testing the SM and quark flavor patterns, results of the |∆b| = |∆d| = 1 fit are key input to a novel test of lepton flavor universality using the dineutrino modes B → π ν ν, B → ρ ν ν [7], shown in Fig. 4. Further synergies from flavorful analyses in SMEFT can be anticipated.

ACKNOWLEDGMENTS
We are happy to thank Johannes Albrecht and Tom Blake for useful communication on the LHCb analysis of B 0 s → K * 0 µ + µ − .This work is supported by the Studienstiftung des Deutschen Volkes (MG) and the Bundesministerium für Bildung und Forschung -BMBF (HG) under project number 05H21PECL2.This work was per- Notes added: A recent study [60] performs a sensitivity study to NP in the B → ρ full angular distribution.
The CMS-collaboration reported recently B(B 0 → µ + µ − ) < 1.9•10 −10 [61], which is consistent with LHCb's finding (23) used in this work.We study the impact of the new CMS measurement in Eqs. ( 24), ( 28), (30), and (31) finding respectively.Figure 5 illustrates the impact of LHCb and CMS data on C 10 − in both the 1D (green) and 2D (orange) fits, with a 1σ uncertainty range.Although the B d,s → µ + µ − data from both experiments have not yet been combined, such an analysis would be highly desirable for improving the b → d, b → s fits, as evident from Fig. 5. On one hand, this analysis would eliminate the current LHCb degeneracy of C 10 − in the 1D fits, as seen in the intersection of the blue curve with the green horizontal line.On the other hand, combining the data from both experiments would greatly reduce the uncertainty of this parameter, as demonstrated by the asymptotic behavior of the ∆χ 2 functions, see Fig. 5.
added in Eq. ( 41).In addition, we do not consider correlations between different observables except for the bins of differential branching ratios of B + → π + µ + µ − decays.These assumptions result in the following block-diagonal covariance matrix where bdµ , and V additional uncertainty and can be avoided once data in theory-friendly bins are available.To make progress we model these effects using a constant-width Breit-Wigner approximation, with 'fudge factor' a R > 0 and R = J/ψ, ψ , and m R , Γ R the resonance mass and width.This simple ansatz serves to estimate the uncertainty of the interpolation but we note that a more sophisticated prescription [62] could be used for the resonance shapes.Close to q 2 ≈ m 2 R , the differential branching ratio is dominated by the resonance R, that is Here, f (q 2 ) is a function obtained from Eqs. ( 14) to (16).
To make use of the data, we exclude the veto regions and interpolate between them and give a conservative estimation of this uncertainty.In Fig. 6, we learn that the effect from resonances tends to cancel between the signal bins [0.FIG. 6. Differential branching ratio of B 0 s → K * 0 µ + µ − as a function of q 2 .The vertical gray bands around the J/Ψ and the ψ correspond to the regions vetoed by LHCb from the branching ratio measurement [14].The red dashed line depicts the central value of the non-resonant SM contribution.The orange region illustrates the SM behavior with both the non-resonant and the resonant contributions, obtained by scanning independently over both relative phases δ J/ψ,ψ ∈ [0, 2π].Details on the blue and red areas in the veto regions are given in the text.
and then perform an interpolation between them (blue and red solid lines).Allowing for different strong phases on both sides of the veto intervals (blue region in Fig. 6) gives a more conservative uncertainty estimation than assuming the same strong phases (red region in Fig. 6), that is, the red area is inside the blue one.In the fits we employ the more conservative approach and assume that the strong phases are not correlated and take maximum/minimum values on both sides, corresponding to the blue area.The resulting symmetrized uncertainty of ε model is obtained as ε model = (δε J/ψ ) 2 + (δε ψ ) 2 = 3.6 • 10 −9 , (B7) corresponding to the last uncertainty in Eq. ( 17), with For comparison, the model uncertainty on the interpolation for the red region is smaller, ε model = 2.2 • 10 −9 .We also note that the fraction of the full (after cuts and interpolation) to the signal branching ratio is 1.33 ± 0.24, consistent but with larger uncertainty that the one employed by LHCb from the B 0 → K 0 * µµ analysis, 1.532 ± 0.001 (stat) ± 0.010 (syst) [63].

Appendix C: Global b → s fits
In this appendix, we provide the results of the global fit to b → sµ + µ − , γ data including the update on B(B 0 s → µ + µ − ) from the CMS-collaboration [61].We follow the same approach as Ref. [7], where the python package flavio [64] was employed, and refer there for details.We do not include the R K and R K * measurements from LHCb [10,11] since these observables also include electron modes, and would require lepton-specific fits.We recall that dielectron modes are presently much less probed experimentally than dimuon ones and have therefore also not been included in the b → d fits presented in this work.to b → sµ + µ − , γ data for some exemplary 1D and 2D scenarios at the scale µ b .Best-fit values with its 1σ uncertainties are displayed in the third column.We also provide the χ 2 /dof value and respective pull from the SM hypothesis.
, and 4D scenarios floating different combinations of NP Wilson coefficients.The main results of the fits are given in Tabs.V to VII.Constraints on NP Wilson coefficients C i , C i are given at the b-quark mass scale.The SM hypothesis gives a good fit to b → d µ + µ − and b → d γ data (n obs = 6), resulting in χ 2 min = 3.77 and a p−value of 71 %.

9 =
−0.85 ± 0.21 gives a good fit, see App. C. Therefore, future data is very welcome to confirm or refute this pattern in other flavor sectors, such as b → d transitions.

10 B
B → Xd γ H23 best fit SM best fit b → s

10 B
fit b → s best fit b → s B → Xd γ
and B → X d γ decays.The detailed numerical outcome of the global fits to Wilson coefficients of dipole C in Tabs.V to VII.We find that data are consistent with the SM, but also leave sizable room for new physics.Contributions to Wilson coefficients can be order one in b → d processes, significantly larger than corresponding ranges in the b → s transitions, as shown in Figs.2 and3.The preferred regions of the b → d Wilson coefficients (red areas), collapse to the smaller, included ones from b → s fits (blue areas) in models with minimal quark flavor violation, indicating consistency with MFV.To test this paradigm at the level of the current b → s prediction the b → d fit needs to be 3 The ranges from [7] are κ bdµµ L = −3 ± 5, κ bdµµ R = 0 ± 4. For the correlation shown in Fig. 4 only κ bdµµ R matters.

TABLE I .
SM non-resonant binned branching fractions B

TABLE II .
Central values of a 2.

TABLE IV .
Central values of a ( BX d ) i

TABLE V
(44)t results for Wilson coefficients in the 1D scenarios (H1,...,16) at the scale µ b .Best-fit values and 1σ (2σ) uncertainties are displayed in the third and fourth (fifth) columns.The indicators of the goodness-of-fit are provided in the last columns: χ 2 function evaluated at the best-fit point; the PullH i in units of standard deviation, as in Eq. (45); and the p−value pH , as in Eq.(44).The SM has χ 2 SM = 3.77, with a p−value of 71%.

TABLE VII .
Fit results for the 4D scenarios (H38,39).Further notation as in Tabs.V and VI.