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The impact of flavour data on global fits of the MFV SMEFT

We investigate the information that can be gained by including flavour data in fits of the Standard Model Effective Field Theory (SMEFT) with the assumption of Minimal Flavour Violation (MFV). Starting from a theory with no tree level flavour changing neutral currents, we calculate effects in flavour changing processes at one loop, and the resulting constraints on linear combinations of SMEFT coefficients. By doing a global fit including electroweak, Higgs and low energy precision measurements among others, we show that flavour observables put strong constraints on previously unconstrained operator directions. The addition of flavour data produces four independent constraints at order TeV or above on otherwise flat directions; reducing to three when complete U(3)^5 flavour symmetry is assumed. Our findings demonstrate that flavour remains a stringent test for models of new physics, even in the most flavourless scenario.


Introduction
As the particle physics community looks forward to the upcoming LHC run, with the ultimate promise of vastly increased statistics but not significantly increased energy, attention has shifted toward developing an understanding of the subtle effects that new physics beyond the direct reach of the LHC could still have on precision measurements, using Effective Field Theory (EFT) techniques. To constrain the large number of parameters in the Standard Model Effective Field Theory (SMEFT), as many observables as possible should be used. As well as LHC measurements, we have precise data from LEP measurements, Standard Model-forbidden process experiments, high-precision measurements like parityviolating electron scattering experiments, and flavour physics experiments. Much of these have already been incorporated into EFT fits, but flavour observables have generically been applied only to explicitly flavour-symmetry violating new physics scenarios. Explicit flavour violation is in fact so well-constrained by flavour data that models which do not somehow protect themselves from generating sizeable contributions to these processes usually must be significantly heavier than the mass range at which new physics is expected to resolve the Higgs naturalness problem. Many models or simplified frameworks which are invoked to address naturalness concerns (and to be measurable at the LHC, either through direct production or indirect effects) are thus constructed to be "Minimally Flavour Violating" [1]; i.e. with new sources of flavour and CP violation only proportional to the Standard Model (SM) Yukawa matrices. This hypothesis ensures the flavour structure is similar to that in the SM, and thus significantly lowers the scale of new physics needed to be consistent with measurements in the flavour sector.
Given that the tree level contributions to flavour observables must be strongly suppressed for TeV-scale new physics, it is necessary to understand the effects at loop level. These are unavoidable even in models with no new sources of flavour breaking; loops involving W bosons will always induce flavour changing neutral currents even from flavourless interactions. In this article, we explore these loop level contributions in detail, within the framework of the MFV SMEFT. Every operator is multiplied by the minimum number of spurionic Yukawa matrices needed to make it formally invariant under the U (3) 5 flavour symmetry, which means that we begin with a theory containing no tree level flavour changing neutral currents (FCNCs). This assumption has two main motivations: one, that it allows for an approximation of the minimum, baseline effects that can be expected to be seen in flavour observables if physics beyond the Standard Model (BSM) exists; and two, that it is often used already in global SMEFT fits to electroweak and LHC data, so the value of flavour information can be analysed in this context.
The matching of flavour-singlet operators to d i → d j l + l − , d i → d j γ and down-type meson mixing processes was calculated in [2]; here we provide the matching also of operators which are necessarily Yukawa-weighted in the MFV scheme. We also provide the full oneloop matching under our flavour assumptions to d i → d jν ν processes. We note that the full one-loop matching for arbitrary flavour structures has been completed in [3], and we cross-check our results against theirs as appropriate. The additional steps provided by our calculations (including transforming to a physical mass basis and including the effects of SMEFT operators on input measurements) allow our results to be directly compared with measurements and straightforwardly incorporated into SMEFT fits.
In the next section, we lay out the flavour structure of the MFV SMEFT, and explain our assumptions. In section 3 we discuss the observables which we consider here to derive our constraints and present the linear combinations of SMEFT Wilson coefficients which contribute to those observables. We perform a simple global fit in section 4 to demonstrate the impact of flavour data, and discuss our findings in section 5. We present analytic results of the Yukawa-weighted operator matching and the relevant matching calculations for processes with final-state neutrino pairs, as well as numerical results for all matching calculations, in the Appendices.

Conventions and notation for the MFV SMEFT
We apply the MFV framework as follows. We assume that the SMEFT Lagrangian respects a U (3) 5 flavour symmetry (as well as CP invariance), broken only by the Yukawa matrices Y u , Y d and Y e . Specifically, if the flavour symmetry is written under which the SM fields are charged as then the Yukawas are assigned spurionic charges as follows such that the Yukawa terms of the SM Lagrangian are rendered formally flavour symmetric; We work in the Warsaw basis [4] of dimension 6 SMEFT operators, and define the Wilson coefficients to be dimensionful and implicitly containing a 1/Λ 2 suppression, where Λ is the scale of new physics. For the coefficient of each SMEFT operator we take only the lowest order (but non-zero) terms in the symmetry breaking parameters Y u , Y d and Y e that are needed to construct a singlet under the U (3) 5 symmetry. To illustrate this, we can take the example of the operator Q (1) Since this operator can be made into a U (3) 5 singlet by contracting the two quark doublet indices, the lowest order coefficient here requires no Yukawa insertions and is simply δ ij C (1) Hq . These assumptions ensure that the location of the CKM matrix within the quark doublet is not physical in this theory (as it isn't in the SM). Nevertheless, for concreteness of notation, we define the quark doublet as where V is the Standard Model CKM matrix. This allows us to define Yukawa matrices which are diagonal in the quark mass basis,Ŷ u andŶ d , in terms of the matrices Y u and Y d above, as follows Furthermore, we work under the approximation that the only non-zero entries of the diagonalised Yukawa matrices are the top and bottom Yukawa couplings, y t and y b . We present various quark flavour structures that occur in the SMEFT, and explore the result of our flavour assumptions on their coefficients, in table 1. We separate the Yukawas from the Wilson coefficients C a , such that all Yukawa suppressions are explicit and all Wilson coefficients may naïvely be expected to be of similar magnitude, independent of the operator classification. This also implies that while operators can be thought of as having flavour indices (which are contracted with those of Yukawa and/or CKM matrices), SMEFT Wilson coefficients in our notation do not. 1 A few operators have flavour indices which can be contracted in two different ways under the flavour symmetry (with both contractions requiring the same minimum number of Yukawa insertions). Examples include the Q ll = l p γ µ l r l s γ µ l t and Q operators. For these operators we have two independent Wilson coefficients, which we distinguish as primed or unprimed as follows; if a pair of Lorentzcontracted fields have their flavour indices contracted together (either with a Kronecker delta or a Yukawa matrix), the corresponding Wilson coefficient is unprimed, whereas if the contractions of the flavour indices and the Lorentz indices do not match up in this way, the Wilson coefficient has a prime. This is illustrated by the last two examples in Tab. 1.
In Ref. [2], we presented matching calculations for all U (3) 5 -singlet SMEFT operators to operators of the Weak Effective Theory (the EFT of the SM fields below the electroweak scale) mediating d i → d j l + l − , d i → d j γ and d idj → d jdi processes. Here, in Appendix A we present similar matching calculations for all the SMEFT operators which require quark Yukawa insertions under our flavour assumptions. These operators are listed in Tab. 2, along with the processes of interest to this analysis to which they contribute; they include dipole operators, Yukawa-like operators with extra Higgs bosons, one Higgs-fermion mixed current operator which gives rise to a W ± boson right-handed coupling, and scalar-current four-quark operator. We also calculate the matching of all MFV SMEFT operators to d i → d jν ν processes, in Appendix B.

Connecting to flavour observables
Establishing the impact of SMEFT effects on flavour observables is a multi-step process, driven primarily by the different mass scales of relevance to the problem. We define the SMEFT Wilson coefficients at the scale of new physics Λ, then find their impact on scales below the electroweak scale through matching to the Hamiltonian operators of the Weak  Transformation under   Example Operator Operator Table 2. All operators which are brought into flavour symmetric form with insertions of y t and/or y b . Tick marks indicate that the operator in that row contributes to the flavour-violating process of interest in that column.
Effective Theory (WET), 2 where the top quark and electroweak bosons have been integrated out of the theory. The WET Wilson Coefficients must then be run down to the scale of interest for any given flavour observable in order to be used straightforwardly.
On top of this, we take into account the effects of SMEFT operators which change the definition of Lagrangian parameters in terms of measured inputs. We present matching results in two common input schemes in which the electroweak input measurements are respectively {α em , m Z , G F } and {m W , m Z , G F }. For details of the procedure we refer to Section 3 of our previous paper [2] and references therein. It is necessary to include these effects in order to end up with results written in terms of measured known quantities (or equivalently, in terms of the usual SM values of the gauge, Yukawa and mass parameters). Since we work consistently to O(1/Λ 2 ), parameters which are already multiplying dimension 6 SMEFT coefficients in the results are unaffected by the input scheme choice.
Under the MFV assumptions, we may take the CKM parameters to be unshifted with respect to the SM. This can be justified by noting that, even when including NP effects, the ratio of similar processes involving different quarks will always be proportional to the same CKM ratios as in the SM. So if the values of CKM elements are fixed with an appropriate set of four or more input measurements based on ratios of observables, the extracted values will be unchanged compared to a SM fit. Recently, Ref. [5] proposed a CKM input scheme that can be applied to the general (flavour-violating) SMEFT, in which they identified four optimal inputs to fix the CKM parameters. Their particular choice of inputs was partially motivated by the fact that some processes can be complicated in the general case by the need for new unknown matrix elements and form factors, due to flavour and chirality structures in the BSM interactions which are not present in the SM. These difficulties don't arise in the MFV scenario we consider; as we shall see in more detail below, the flavour and chirality structures that result are identical to those in the SM. We therefore assume an appropriate input scheme such that the CKM matrix is unshifted in this theory (all shifts to charged-current interactions are described by corrections to the gauge coupling g 2 and/or the electroweak vev v) and FCNC processes may be used as constraints on Wilson coefficients rather than as would-be SM input parameters.
The flavour observables we consider are justified on the grounds of their (well-known) sensitivity to heavy new physics. We select observables based on processes with a down-type flavour changing neutral current, restricting attention to the theoretically well-understood (semi-)leptonic and photonic meson decays, and meson mixing. As we will see, the MFV SMEFT only enters a limited number of WET Wilson coefficients, so we use the measurements which provide the strongest constraints on these. Up-type FCNCs u i → u j will also exist in the MFV SMEFT, but their amplitude will be suppressed by O(m 2 b /m 2 t ) compared to the down-type FCNCs, due to the GIM mechanism. On top of this, theory uncertainties are generically larger for D meson processes as compared to those involving B mesons. For both reasons, up-type FCNC processes are less promising for constraining the MFV SMEFT.
The observables we will use to constrain down-type FCNCs are mostly B decay and mixing observables. This is because equivalent kaon observables are generally afflicted with large and uncertain long-distance contributions, making them less suitable for constraining heavy new physics. Exceptions are the "golden channels" K L → π 0ν ν and K + → π +ν ν, which we include in our analysis, and the observables and , which measure direct and indirect CP violation in kaon mixing and K → ππ respectively, and which we do not include since there are no new CP violating phases in our model.

Effective theory below the electroweak scale
The matching calculations presented in Appendices A and B and Ref. [2] provide the Wilson coefficients of the WET in terms of the Wilson coefficients of the SMEFT, both at the electroweak scale. Here we collect the relevant Hamiltonians that define the WET Wilson coefficients.
The WET effective Hamiltonian for d i → d j l + l − and d i → d j γ transitions to which the flavour-symmetric SMEFT matches is , where α, β are colour indices. This set of operators is identical to those present in the matching of the SM alone (note the absence of any right-handed current, primed operators) as a result of the flavour symmetry imposed. The WET effective Hamiltonian for d i → d jν ν transitions to which the flavoursymmetric SMEFT matches is Finally, the WET effective Hamiltonian governing meson mixing is 3 where α and β are colour indices, and of the semileptonic Hamiltonian (3.1). 3 Here we do not include additional terms in the operator coefficient, which are functions of m 2 c /m 2 W . Due to CKM suppression these terms are negligible for Bs and B d mixing. They should be taken account of in calculations of kaon mixing, but as mentioned, kaon mixing does not provide useful constraints on the MFV SMEFT, so we neglect it here.

Matching results
Our analytical matching results from the MFV SMEFT to the WET are given in Appendices A and B, and Ref. [2]. For convenience, we also provide the matching in numerical form as follows: where the coefficients N (1) αk are collected in Table 3, and the coefficients N (2) αk are collected in Table 4 for the {α, m Z , G F } input parameter scheme, or Table 5 for the {m W , m Z , G F } input parameter scheme. 4 These tables show at a glance which SMEFT coefficients will be important in which processes. Note that the only SMEFT coefficients whose matching is changed by the choice of input scheme are C ll , C HD , C HW B and C Hl , since these multiply the operators which enter the measured input observables in these two schemes. Throughout our calculations in the remainder of this paper, we take µ = 1 TeV.
In order to make contact with experimental observables, we must run these results from the scale m W to the appropriate scale for the FCNC observables. In the case of B (s,d) meson observables we use µ b = 4.2 GeV, and for kaon observables we run to the scale of µ K = 2 GeV, at which the relevant matrix elements have been calculated by the lattice community [6]. The running below the weak scale is calculated using Wilson [7] which incorporates the anomalous dimension matrices of Ref. [8,9].

Predictions for flavour observables
Here we detail the flavour observables we consider and how they depend on the WET Wilson coefficients. We calculate constraints from flavour on the WET Wilson coefficients, which may then be used as pseudo-observables in fits to constrain the model.
The particular structure of the MFV SMEFT means that its effects in flavour observables very closely align with those of the SM itself. This often simplifies things from a calculational point of view, although it can also limit the sensitivity of flavour observables to these NP effects. In particular, no contributions are made to operators containing right-handed flavour-changing quark currents, such as O 9 =ê 2 d j γ µ P R d i ¯ γ µ , which are negligible also in the SM due to the chiral nature of the weak interactions. This in turn ensures that the leading NP effects are linear in the SMEFT coefficients, since there is always an interference term with the SM.
We therefore expect a rather limited number of new constraints arising from the flavour observables considered. The flavour symmetry assumptions ensure that C bs 1,mix and C bd 1,mix depend on the exact same linear combination of SMEFT Wilson coefficients. The same is true for C bs L and C sd L . So, anticipating some of the calculations below, we expect to find a maximum of 6 new constrained directions corresponding to constraints on these two coefficients C bd j 1,mix and C d i d j L , as well as constraints on the three coefficients C bs 7 , C bs 9 and C bs 10 from (semi)leptonic and photonic b → s transitions, and a constraint on one linear combination of C bs 1 and C bs 2 from the width difference of B s mesons.
since the hadronic matrix elements and QCD corrections are identical for the SM and NP parts. The function S 0 (x t ) is the usual Inami-Lim function [11], given by The measured values are [12] ∆M exp These lead to constraints on the new physics parts of the WET Wilson coefficients: where Bs is the B s decay constant, B andB S are bag parameters, and C SM 1,2 and C NP 1,2 are the SM and NP values respectively of C bs 1,2 (µ b ). The SM coefficients are C SM 1 (µ b ) = −0.2451, C SM 2 (µ b ) = 1.008 at NNLO in QCD [14]. For the values of hadronic parameters we refer to [15] and references therein. The SM prediction is [16] ∆Γ SM s = (0.088 ± 0.020) ps −1 , (3.15) while the measured value is [12] ∆Γ exp From this (adding experimental and theoretical errors in quadrature), we obtain a constraint at 1σ on a linear combination of the NP WET Wilson coefficients: where [13] C bs .
( 3.18) The percentage error on the measurement of ∆Γ d , the decay rate difference of B d mesons, is much larger than that on ∆Γ s [12]. This observable is dependent on the exact same linear combination of MFV SMEFT coefficients as ∆Γ s , and would produce much weaker bounds, so we don't include it in our analysis.

K → πνν
For NP in which the neutrinos are coupled to left-handed currents only (which is the case for the MFV SMEFT), the expressions for the branching ratios are [17,18] where (3.22) is the radiative electromagnetic correction, and κ + and κ L summarise the remaining factors, including the hadronic matrix elements. The loop functions for the top and charm quark contributions, X eff and P c (X), are given in the SM by [18] P c (X) = 0.404 ± 0.024, (3.23) In the presence of the SMEFT contributions, 5 X eff becomes with µ K = 2 GeV. For the experimental limits, we take the recent upper bound from NA62 [19] B K + → π +ν ν < 1.85 × 10 −10 (90% CL), (3.26) and from the 2015 run at KOTO [20] B K L → π 0ν ν < 3.0 × 10 −9 (90% CL). (3.27) These lead to a bound on the new physics part of the WET Wilson coefficient: This coefficient barely runs, so we can take C sd Following Ref. [21], the predictions for the branching ratios in the presence of the MFV SMEFT contributions can be written (in our notation) The experimental limits on the branching ratios, measured at BaBar [22] and Belle [23] are B B 0 → K * 0ν ν < 5.5 × 10 −5 (90% CL), (3.32) which lead to bounds on the ratios, at 90% CL, of [21] B These lead to a bound on the new physics part of the WET Wilson coefficient: As above, we can take C bs L (µ b ) ≈ C bs L (m W ) since the running is very small. 5 neglecting the suppressed effect of NP in the charm loops 3.3.4 b → sγ and b → sl + l − processes Processes involving the parton level transitions b → sγ and b → sl + l − , for example branching ratios and angular observables of B → K ( * ) l + l − , and branching ratios of B → X s γ and B s → l + l − , can constrain the WET Wilson coefficients C bs 7 , C bs 9 and C bs 10 . 6 We use flavio [26] to find the vectorĈ of best fit values of the Wilson coefficients, and to numerically extract the variances and correlation matrix by expanding ∆χ 2 around the best fit point as where the covariance matrix U can be written in terms of the variances σ 2 i and the correlation matrix ρ as U ij = σ i σ j ρ ij (no sum). In this way we find constraints on the new physics Wilson coefficients of with the correlation matrix The observables that are included in this flavio fit are based on the list of rare B decay observables involving a b → s transition given in Ref. [27] (and we refer to this reference as well as Ref. [26] for the measurements and theory predictions that are included in the flavio code). Specifically, the observables used are: all relevant CP-averaged observables in semileptonic b → sµµ decays that were included in the global fit of Ref. [28], high-q 2 branching ratios and angular observables of Λ b → Λµµ, the branching ratios of B 0 → µµ and B s → µµ, the branching ratio of the inclusive decay B → X s µµ, and all observables in inclusive and exclusive radiative b → sγ decays included in the fit of Ref. [29], including B → K * ee at low q 2 . We do not fit to observables that test lepton flavour universality, or lepton flavour violation, since under our flavour assumptions these will not be altered from their SM predictions. The constrained Wilson coefficients at µ b =4.2 GeV are related to the Wilson coefficients at m W by [7]: The b → d WET coefficients C bd 7 , C bd 9 and C bd 10 depend on the same linear combinations of SMEFT Wilson coefficients under our MFV assumptions as these b → s ones. Currently, new physics in the theoretically clean b → dl + l − and b → dγ processes is not well constrained, but this should change in the future with measurements of inclusive b → d processes at Belle II [24,25].

Fit
We perform an illustrative fit, within the {α em , m Z , G F } input scheme, to demonstrate the effect that flavour observables have within a global fit assuming MFV. We assume Gaussian errors throughout and fit using the method of least-squares. We include all operators allowed by our flavour and CP assumptions.

Observables and data
Our choice of non-flavour observables is motivated by the following considerations, with the aim of getting a fair picture of the new physical information that flavour can add to a global fit: (a) we want to include observables which provide strong and up-to-date constraints on operators involving EW gauge bosons and Higgs fields, and (b) we want to include observables which depend on the same 4-fermion operators that appear at one-loop in FCNCs. With this in mind, we include the following.
• Precision electroweak observables: We use the LEPI observables and predictions from  [36], and the definitions of the observables in terms of SMEFT parameters from Tables 2 and 3 of the same reference.
• Higgs Run I: We use the combined ATLAS and CMS Run I Higgs signal strength measurements from Table 8 of Ref. [37], with the correlation matrix in Figure 27 of the same reference. For the SMEFT predictions we adopt the definitions of the observables in terms of Warsaw basis Wilson coefficients provided in the Mathematica notebook accompanying Ref. [38].
• Higgs Run II: From the CMS paper [39], we use all the signal strength measurements in Table 3, and the correlation matrix from the supplementary material. From the ATLAS paper [40] we use all the cross sections times branching ratios in Table 6 and the correlation matrix in Figure 6. For the SMEFT predictions we use the definitions of the observables in terms of Warsaw basis Wilson coefficients provided in the Mathematica notebook accompanying Ref. [38].
• W + W − production at LHC: Following Ref. [38], we use a measurement of one bin of the differential cross section of pp → W + W − → e ± νµ ± ν at ATLAS [41], using the definitions of the observable in terms of SMEFT Wilson coefficients provided in the Mathematica notebook accompanying Ref. [38].
• e + e − →qq off the Z pole: We use the data on σ had at different values of √ s from Table 6 of Ref. [42]. The original experimental results are from LEP [35] and TRI-DENT [43] and the SM predictions are taken from Ref. [44]. [42], we use observables and data on Atomic Parity Violation [45,46] and eDIS [47]. From Ref. [48], we use observables and data on neutrino-nucleon scattering (both ν e scattering data from CHARM [49] and ν µ scattering data from the PDG average [50]), deep inelastic scattering of polarized electrons [50,51], and deep inelastic scattering of muons [52].
• FCNCs: We use the constraints on WET Wilson coefficients given in Sec. 3.3.
This is a total of 186 observables. Figure 1 shows which Warsaw basis Wilson coefficients affect each set of observables, where the "4-fermion" category includes low energy precision measurements as well as e + e − →qq off the Z pole.

Fit methodology
The SMEFT corrections to all the observables are linear in the dimension 6 Wilson coefficients (under our flavour and CP assumptions and working to O(Λ −2 )), meaning that predictions for the observables can be written as a matrix equation where µ is the vector of predictions, µ SM represents the SM predictions, θ is a vector of SMEFT Wilson coefficients, and H is a matrix of functions that parameterise the SMEFT corrections. The measured central values of the observables can be represented by a vector y, with a covariance matrix V. Then the χ 2 function is The least-squares estimatorsθ for the Wilson coefficients are found by minimising χ 2 : The covariance matrix U for the least squares estimators is given by the inverse of the Fisher matrix F, defined as When the covariance matrix is diagonalised, its entries are the variances σ 2 i of its eigenvectors, which are a set of linearly independent directions in Wilson coefficient space. The eigenvalues of the Fisher matrix are therefore 1/σ 2 i . If an eigenvector direction is unconstrained by the data, its corresponding Fisher matrix eigenvalue will be zero.

Flavour in the electroweak hyperplane
If we (artificially, but for purposes of illustration) restrict attention to the set of ten Wilson coefficients that enter the Z-pole observables, Hq , C Hu , C Hd , C He , C ll }, (4.5) then it is well-known that fitting only to Z-pole observables leaves two flat directions. We define these as 7 Hq + C These flat directions must be closed by other, often less well-measured, observables. In Figure 2 we show, within the plane of the Wilson coefficients of these two flat directions, the constraints from the flavour observables (in green), which can be compared to those obtained from LEP II W W production (in orange) and LHC Higgs measurements (in blue). In all cases the Z-pole data are also included in the fit. The axes correspond to the k 1 and k 2 directions, which have been normalised to unit vectors in Wilson coefficient space. We define the norm |k| of an operator direction by assuming an arbitrary Euclidean metric δ ij in the Warsaw basis; if we write k = c · θ, where as before θ is the vector of Warsaw basis Wilson coefficients, then |k| 2 ≡ δ ij c i c j . All of the ellipses in Figure 2 are obtained by taking only this set of 10 Wilson coefficients (4.5) to be non-zero, and profiling over the 8 linearly independent directions (which are already well constrained by the Z-pole data). Under these assumptions, it can be seen that flavour is competitive with existing constraints within this highly flavourless plane.
The reason for the apparent tension between the flavour ellipse and the other ellipses in Figure 2 is due to recent measurements of b → sµµ processes, which show a pattern of deviations from SM predictions, particularly among angular observables in B → K * µµ decays [53] and measurements of exclusive branching ratios [54,55]. These processes provide some of the strongest constraints among the flavour data used, and drive the green ellipse away from the origin. These anomalous results can be explained either by lepton flavour universal new physics, or by lepton non-universal new physics which affects muons more strongly than electrons (see e.g. [56][57][58][59][60] for recent fits to WET coefficients). If the new physics explanation is lepton non-universal, it could simultaneously produce effects in ratios of decays involving muons to those involving electrons (R K and R K * ), recent measurements of which also deviate from the SM expectation [61,62] (but have no impact on our lepton flavour universal fit). For this reason most recent models invoked to explain the anomalies are lepton flavour non-universal.
Whether these anomalies will persist in new data remains to be seen, but even if they are confirmed as signs of new physics, it does not imply that there is really a tension between electroweak, Higgs and flavour data. This is because we are only studying a subset of operators in this plot, when in reality many other operators could provide the explanation for the anomalies, which would in turn shift the favoured central value within the plane of Fig. 2. Therefore under a less restrictive BSM situation, in which more operator coefficients were non-zero, it's likely that this discrepancy would be accounted for by an operator not among these ten (and which is less constrained by EW and Higgs data).
So while this limited fit gives an indication of the power of current flavour measurements when compared like-for-like with Higgs and LEP II W W measurements, it is clear that a more global picture is needed for a more realistic interpretation of the constraints, and this is the subject of the following subsections. We note, however, that our flavour symmetry assumption does not allow for lepton non-universality, and so if the R ( * ) K anomalies were confirmed with more data, it would be a sure sign that this flavour symmetry is not respected by BSM physics.

Eigensystem of the global fit
The observables we include in the fit, within our MFV flavour assumption, depend on a total of 37 Wilson coefficients in the Warsaw basis: {C H , C HW B , C HD , C HW , C HB , C HG , C W , C G , C (1) Hq , C Hu , C Hd , C He , C Hud , C uH , C dH , C uW , C dW , C uB , C uG , C ll , C    Fig. 3. The addition of flavour data changes the eigenvector directions, and therefore not all the eigenvalues can be directly compared. With this in mind, we have plotted the eigenvalues of the full fit in size order, pairing each with the eigenvalue of the fit without flavour whose corresponding eigenvector (e F ) has the largest overlap with the relevant eigenvector of the full fit (e F ), where this overlap is defined as

Higgs LEP II WW Flavour
For the well-constrained eigenvectors (towards the left of the plot), the overlap is close to 1. The five otherwise flat directions which are closed by flavour data can be seen in the plot as blue bars without an accompanying green bar. In particular, two directions which are unconstrained without flavour data are now rather well constrained (1/ |c| > 5 TeV). These directions are dependent on nearly all the Wilson coefficients in the fit, but can be written approximately as where we have dropped all terms with numerical coefficient less than 0.1, just to give an idea of their main dependence. Some or all of the remaining flat directions can probably be closed by Tevatron and LHC top observables. The study and interpretation of top data in the language of effective field theory is a very active field at the moment, see e.g. [63][64][65] for recent work in this area. Furthermore, fruitful results have been found by using flavour and top data in combination to constrain some top-containing operators (see e.g. [66][67][68]). However, to our knowledge a study of top observables within a global fit based on an exact or approximate U (3) 5 flavour symmetry has not been done, and is beyond the scope of our work.

Full flavour symmetry
We can imagine a situation in which the NP is completely flavour symmetric, and the symmetry breaking associated with the Yukawas can only enter through loops involving W bosons. In other words, the dimension 6 Lagrangian at the scale Λ only includes operators which are U (3) 5 singlets (without spurionic Yukawa insertions). In this case, the observables we include in the fit now depend on 26 Wilson coefficients in the Warsaw basis: {C H , C HW B , C HD , C HW , C HB , C HG , C W , C G , C (1) Hq , C Hu , C Hd , C He , Including all observables, we now have only one flat direction in the fit:  Fig. 4. It can be seen that one direction which is unconstrained without flavour data is now rather well constrained (1/ |c| > 5 TeV). This direction is 8 lq − 0.03 C ed + 0.01 C eu − 0.02 C lu − 0.11 C qe + 0.03 C ld − 0.07 C Hu − 0.03 C Hd + 0.02 C (1) Hq − 0.14 C (4.14)

Discussion
Our results demonstrate that flavour can add meaningful information to global fits assuming an MFV flavour structure. We have shown that flavour measurements should not be thought of as only constraining flavour-changing operators, but rather, depending on  the flavour structure of the underlying theory, they can be used to help constrain flavourconserving or bosonic operators. We find that effects in flavour can be significant even in theories where they are often neglected. The U (3) 5 flavour symmetry we have studied is the largest flavour symmetry group available for BSM physics, and it is reasonable to broadly assume that any breaking of this symmetry will only enhance effects in flavour. In this sense our findings may be taken as conservative bounds from flavour on generic new physics. However, the specific Wilson coefficient combinations that can be constrained by flavour data clearly depend rather strongly, both in their number and direction, on the flavour assumptions imposed at the scale Λ. Hence the results of our analysis cannot directly be extrapolated to other flavour scenarios which may be of interest, however we make a few comments here.
If a flavour assumption forbids tree level FCNCs (this is true, for example, of an unbroken U (2) 5 flavour symmetry among the first two generations of fermions), then the matching calculations at m W will not change considerably. The only change will be that extra terms that are independent of the top mass may now appear in the matching results involving operators that contain quarks, because GIM cancellations which occur in the MFV case will no longer happen. In the context of a global fit, the lifting of the U (3) 5 symmetry will have the effect of somewhat disconnecting the spheres of influence of the different constraints, since the flavour and Higgs observables depend strongly on top-containing operators, while the electroweak and 4-fermion observables are largely insensitive to the top.
If instead the theory at Λ contains both flavour violating and flavour conserving operators, then things become more complicated, not least because of the proliferation of Wilson coefficients. A likelihood for general flavour structures, including Higgs and electroweak observables, was recently calculated in Ref. [69]. Processes involving FCNCs will now depend on both flavour-violating operators at tree level and flavour-conserving oper-ators at loop level. The CKM must also be consistently parameterised, as discussed in Ref. [5]. Understanding the messages that we can learn from studying this general case will clearly be an important goal over the next few years.
Here we calculate the effects of each Yukawa-suppressed operator which contributes to d i → d j γ, d i → d j l + l − or down-type meson mixing, as identified in Tab. 2. Some of the Feynman diagrams for these calculations are shown in Figs. 5, 6, 7, and 8. The results of these calculations are the SMEFT contributions to the Wilson coefficients of the effective Hamiltonians defined in Sec. 3.1, at the scale m W . Since many of these contributions are quark flavour universal, if the quark flavour indices are not made explicit, C α ≡ C bs α = C bd α = C sd α is meant. In addition to individually-identified references below, these results have also been checked against [3].

A.1 Q Hud
This operator matches to C bd j 7 through diagrams in Fig. 5, as well as to C bd j 8 through diagrams similar to the second and sixth diagram in Fig. 5 (with the photon replaced by a gluon): which is in agreement with Ref. [70].

A.2 Q uW
This operator matches to C 1,mix via box diagrams involving W bosons, as well as to C 7,8,9,10 through the diagrams shown in Fig. 6 and similar: Our meson mixing result agrees with that of Ref. [71], while the other results are all in agreement with Ref. [70].

A.3 Q uB
This operator matches to C 7,9 through the third and fourth diagrams in Fig. 6: in agreement with Ref. [70].
This is in agreement with Ref. [70].

A.5 Q dW
This operator matches to C 7 , through the diagrams in Fig. 7, as well as to C 8 through a diagram similar to the second one of Fig. 7 (with the photon replaced by a gluon): which all agrees with Ref. [68].
These operators produce effects in C 7 and C 8 through the diagram in Fig. 8 (and similar with the photon replaced by a gluon): 16) in agreement with Ref. [70] (taking into account our different flavour assumptions).

B Matching to neutrino-containing operators
Here we present the matching of all operators to the coefficients C of the effective Hamiltonian in Eqn. 3.3. Since all contributions are quark flavour universal, we define C L ≡ C bs L = C bd L = C sd L . We do not show the Feynman diagrams, but they are always simply related to diagrams which match to the C 10 operator (by exchanging charged leptons for neutrinos and vice versa), so can be inferred from diagrams in Ref. [2] and in Fig. 6.
We provide results for both the {m W , m Z , G F } and {α em , m Z , G F } input parameter schemes. The full result for either input scheme is the sum of an input scheme independent piece, C S.I. L , and an input scheme dependent piece. For a fuller explanation of the different schemes, please see Ref. [2] Section 3 (and references therein).
The input scheme independent piece is The above must be added to an input scheme dependent piece. This piece in the {α em , m Z , G F } scheme is: Instead the scheme dependent piece in the {m W , m Z , G F } scheme is: are the usual Inami Lim [11] functions.

C Numerical matching results
In this appendix we present the results of our matching calculations in numerical form, as tables of the N        Table 3. Numerical values of the coefficients N αk of the matching equation Eqn. (3.5). Note that the matching to C 1,mix , C 1 and C 2 does not have any divergent diagrams so the corresponding N (1) αk coefficients are all zero (and are not listed here). All results are quark-flavour universal (C α ≡ C bs α = C bd α = C sd α ), except for results indicated by an asterisk ( * ), for which only C bdj α is meant (d j = s, d).  Table 4. Numerical values of the coefficients N αk of the matching equation Eqn. (3.5), in the {m W , m Z , G F } input scheme. All results are quark-flavour universal (C α ≡ C bs α = C bd α = C sd α ), except for results indicated by an asterisk ( * ), for which only C bdj α is meant (d j = s, d). ---6.72 ·10 −3 * -2.14 ·10 −2 * ---- Table 5. Numerical values of the coefficients N αk of the matching equation Eqn. (3.5), in the {α em , m Z , G F } input scheme. All results are quark-flavour universal (C α ≡ C bs α = C bd α = C sd α ), except for results indicated by an asterisk ( * ), for which only C bdj α is meant (d j = s, d).