Effective Theory Approach to New Physics with Flavour: General Framework and a Leptoquark Example

Marzia Bordone; Oscar Cat\`a; Thorsten Feldmann

REVIEW

Effective Theory Approach to New Physics with Flavour: General Framework and a Leptoquark Example

Marzia Bordone, Oscar Catà, Thorsten Feldmann

High Energy Physics - Phenomenology · Submitted 7 Oct 2019 · 74 references

Pith's verdict: A useful spurion catalogue and a sensible FN power-counting recipe, but the headline '11 of 24' count is an artifact of the two-parameter collapse and shouldn't be quoted as a result.

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arXiv ID1910.02641 (version 2)

Pith ID4LJ6DC36, Pith's permanent record of this paper and its review

CitationMarzia Bordone, Oscar Cat\`a, Thorsten Feldmann. "Effective Theory Approach to New Physics with Flavour: General Framework and a Leptoquark Example." arXiv:1910.02641 (2019). https://pith.science/paper/4LJ6DC36

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Editor's note

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Worth your time for the framework, not for the fit numbers. The genuinely useful part is Sec. 2: Table 2 catalogues 30 Dirac-bilinear spurions with their gauge and flavour quantum numbers, and Table 3 shows how each SMEFT four-fermion coefficient factorises into them. That's a clean extension of Feldmann-Mannel and it's the kind of table people will actually pull off the shelf when they want to write down a flavour ansatz without reinventing it. The FN power-counting as a self-consistency device (triangle inequalities guarantee the hierarchy survives RGE and operator mixing) is a good idea, cleanly explained, and the authors are honest that it's a recipe rather than a dynamical theory. The U1 matching relations (3.5-3.7) are standard and correct. So: solid method paper, competently executed, and the observation that leptoquark spurions give asymmetric Bs,d -> tau- mu+ vs tau+ mu- rates is a nice, checkable prediction that LHCb tagging could eventually test. That last point is the most falsifiable thing in the paper and deserves more prominence than it gets. Where it's soft: the '11 of 24' count. Eq. (4.2) collapses 18 complex spurion entries to two real numbers, Sec. 2.2 admits the integer charges are only defined up to +/-1, and Sec. 3.3 picks the 24 candidates with sharp integer inequalities. None of these is propagated. Loosen the c_i-alpha even modestly and the 13 rejected assignments get ~16 new directions and will likely fit just as well; the authors' own remark that all solutions fit comparably is consistent with the collapse, not the data, doing the discriminating. The reader's circularity worry is milder than stated -- Sec. 3.3 uses order-of-magnitude power counting, not the fit chi-squared, so the overlap with Sec. 4 is coarse rather than circular -- but the stress-test's core point lands. One framing nit: abstract:s2 promises a framework that 'systematically reduce[s] the number of flavour parameters', yet Table 1 shows generic FN leaves all 81 entries; the reduction is carried by the leptoquark factorisation, and Sec. 1 says so. The abstract should too. No code or data shipped, which for a MultiNest fit is a mild annoyance rather than a flaw. Who should read it: anyone building flavour ansatze for SMEFT fits, or working on leptoquark flavour structure. Send it out; ask the referees to stress-test the collapse in Eq. (4.2) and to soften the abstract's reduction claim.

Referee report

Significance

The spurion catalogue of Tables 2 and 3 is a genuinely useful reference for anyone constructing flavourful EFT interpretations of the B anomalies, and the observation that FN charges make the consistency conditions automatic via triangle inequalities is a clean and reusable argument. The paper is also commendably explicit about its assumptions: the single-coefficient collapse (Eq. 4.2), the C(1)lq = C(3)lq relation (Eq. 4.3), the fixed scale Λ = 2 TeV, and the ±1 ambiguity in the charges are all stated in the text rather than buried. The phenomenological output is falsifiable in a useful way: the asymmetric B̄s,d → τ∓μ± predictions, and their correlation with RD and B̄s → τ+τ− (Figs. 1 and 2), give Belle II and LHCb a concrete target that distinguishes these scenarios from generic leptoquark fits. The framework is a reasonable and well-motivated extension of MFV, and the paper would be a solid contribution once the quantitative claims are presented in a checkable form.

Major objections (5)

[Abstract, abstract:s2, versus Table 1 and p. 5] The abstract states that the framework 'allows us to systematically reduce the number of flavour parameters in the EFT', but Table 1 and the text on p. 5 say the opposite for the general case: 'generic FN' still carries 81 parameters, and the sentence 'there is no reduction of parameters' appears explicitly. The reduction to 18 + 2 arises only from the additional assumption that the coefficient factorises through a single leptoquark spurion, i.e. from the LQ+FN row of Table 1, not from the FN power counting itself. Since abstract:s2 is the paper's headline claim, the abstract should be rewritten to distinguish 'FN fixes the power counting but not the number of parameters' from 'spurion factorisation reduces the parameter count', exactly as Section 2.2 already does.
[Section 4 and Table 4] The one quantitative result of the paper is the fit quality, yet no χ² value is ever quoted. Section 4 says only that the global χ² 'improves the SM one by approximately a factor 3', and Table 4 reports posteriors for CQL and CDE without the corresponding χ², the number of degrees of freedom, the SM χ², or per-observable pulls. As written, the reader cannot verify the factor-of-three statement, cannot see which observables drive it, and cannot judge whether the 11 accepted scenarios are actually distinguishable. A table giving χ², Δχ² with respect to the SM, and dof for each of the 11 scenarios, plus a breakdown of the dominant pulls (the text asserts the fit is driven by the tension between W universality and RD(*), which such a breakdown would substantiate), is needed for the central claim to be checkable.
[Sections 3.3 and 4] The 24 candidate charge assignments are selected in Section 3.3 using RD(*), b → sμ+μ−, Z → νν̄, B̄s → τ∓μ±, KL → μ±e∓ and B̄d → τ−μ+, and the fit of Section 4 then uses 'all the observables discussed in Sec. 3', which includes that same subset. The '11 of 24' count is therefore not an independent validation of the framework: the discrete charges are already chosen to satisfy the data that later discriminate between them. This does not invalidate the framework, but the paper should say so explicitly and should state what the count is conditional on. A short paragraph distinguishing the power-counting selection from the fit, and noting that the two stages are not statistically independent, would be sufficient; a genuinely independent test would require holding out one of the selection observables.
[Eq. (4.2) and Section 4] Equation (4.2) collapses the 18 complex spurion entries to two real numbers, ciαQL = ±CQL and ciαDE = ±CDE. The authors acknowledge this as a simplifying assumption, but its consequence for the headline count is not examined: the 13 rejected assignments are rejected under a two-parameter ansatz, and with independent O(1) coefficients per entry they would acquire roughly sixteen additional directions and could plausibly reach the same χ². Combined with the statement in Section 2.2 that 'deviations by one unit in the exponent are not excluded', the '11 of 24' should either be softened to a statement about this restricted parametrisation, or supported by a sensitivity test. A cheap and convincing test would be to allow ciα ∈ [1/3, 3] × C with random signs for two or three of the rejected scenarios and report whether they recover the accepted χ²; if they do, the counting claim should be withdrawn and the framework claim kept.
[Table 4 and Section 3.1] Several accepted scenarios require CDE between 4.6 and 7.2 (1b, 1c, 2c, 2d, 2e) at Λ = 2 TeV. Since the matching relations (Eqs. 3.5–3.7) are tree level and the EFT expansion is in 1/Λ², coefficients of this size raise the question of whether the leading-order treatment and the perturbativity of the underlying U1 couplings are under control for those solutions. The text states that 'the values of all the Wilson coefficients are in the ballpark of what one would expect from power-counting arguments', but a coefficient of 7 is not O(1) in the usual sense. The authors should comment on the validity of the tree-level matching and of the truncation of the EFT series for the large-CDE solutions, or note explicitly that these solutions sit at the edge of the regime in which the calculation is reliable.

Minor objections (9)

[Eq. (2.7)] The second relation reads (YD)ij ∼ λ^{|biQ − bjU|}, but the down-type Yukawa should involve the right-handed down charge; it should be bjD.
[Eq. (3.25)] The scalar term contains λ^{|b3D − e3L|}, where e3L appears to be a typographical slip for b3E.
[Eq. (3.40)] The expression ends with a mismatched bracket and an ampersand: '|3 − b1L| + |2 − b2L] & 8'. This should presumably read |3 − b1L| + |2 − b2L| ≳ 8.
[Table 1, caption] The 'generic FN' row of the table contains the stray text '∼Appendices AandB' embedded in the formula, which appears to be a typesetting error and should be removed.
[Eq. (3.41) versus Table 4] Equation (3.41) states b1L = −2 (+8) and describes these as the benchmark points, but Table 4 lists b1L = −2, 10, −4 and +8. The relation between the two sets of values, and the origin of the additional −4 and 10 options, should be explained so that the reader can reconstruct the 24 candidates from the text.
[Section 3.2.1] The text states that global fits require C9 < 0 and C′9 > 0. The first is well established, but the second is not a requirement: the fits are perfectly consistent with C′9 = 0, and the subsequent sentence ('since the values for C′23229 are small, the latter condition will not be relevant') suggests the authors know this. The wording should be adjusted to say that C′9 is constrained to be small and of either sign.
[Figs. 1 and 2] The captions refer to 'all the 11 scenarios listed in Table 4' but do not give the symbol-to-scenario mapping, so the figures cannot be read against the table. A legend or an explicit key in the caption is needed.
[Section 3.2.2] The numerical values fSD(τ) = 1.5 and fSD*(τ) = 0.12 are quoted without their definition or normalisation. Since these control the relative weight of the scalar contribution to RD and RD*, which is central to the discussion in Section 4, a one-line definition or a pointer to the normalisation used in Refs. [47–49] would help.
[Footnote 1] The name is spelled 'Froggat and Nielsen' in the footnote; it should be 'Froggatt'.

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The paper

The paper claims

A Froggatt-Nielsen power-counting scheme shrinks the flavour parameter space of new-physics operators, and a vector leptoquark built on it survives low-energy flavour data.

The reading

New physics written as higher-dimensional operators has a flavour problem: the coefficient $[C_{lq}]_{ijαβ}$ alone carries 81 real numbers, so a bottom-up fit to flavour data is hopeless without extra structure. The authors propose a systematic way to supply that structure. They keep the minimal-flavour-violation idea of building operator coefficients out of spurions, but enlarge the set of spurions and assign each one a size by Froggatt-Nielsen charges, the same integer charges that reproduce the quark and lepton mass hierarchies with powers of $λ≈0.2$. Because the charges obey triangle inequalities, the resulting hierarchies are automatically stable against the products of spurions that appear in other operators. As a worked example they take the two spurions that a $U_1$ vector leptoquark would generate, $Δ_{QL}$ and $Δ_{DE}$, which collapses the 81 entries to 18 complex numbers plus an overall coefficient. Imposing the known Yukawa and CKM scalings plus experimental limits on rare and lepton-flavour-violating decays leaves 24 charge assignments, and a fit to low-energy flavour observables selects 11 that improve the Standard Model $χ^2$ by roughly a factor of three.

What carries the argument

The machinery is the spurion decomposition of the semileptonic four-fermion coefficients combined with a Froggatt-Nielsen power counting. Table 3 of the paper writes each of the ten two-quark-two-lepton operators as a sum of products of flavour spurions, for example $[C^{(1)}_{lq}]_{ijαβ} = a_0 [Δ_Q]_{ij}[Δ_L]_{αβ} + b_0 [Δ_{QL}]_{iα}[Δ^†_{QL}]_{jβ} + …$. Selecting only $Δ_{QL}$ and $Δ_{DE}$ — the spurions a $U_1$ leptoquark would generate — factorises the coefficient into $[C_{lq}]_{ijαβ} ∼ (Δ_{QL})_{iβ}(Δ^†_{QL})_{αj}$, and the triangle inequality $|b^Q_i - b^L_β| + |b^L_α - b^Q_j| ≥ |b^Q_i - b^Q_j + b^L_α - b^L_β|$ guarantees that the factorised entries never exceed the entries of an unconstrained Froggatt-Nielsen ansatz. That inequality is what makes the truncation stable under renormalisation-group running and under the appearance of the same spurions in other operators. The charges themselves are fixed step by step: the CKM matrix gives $b^Q = (3,2,0)$, the up- and down-type masses give the right-handed quark charges up to discrete ambiguities, and the charged-lepton masses give three charge differences. The remaining freedom is then pruned by requiring the correct size of $b → sμ^+μ^-$, $R_{D^{(*)}}$, $Z → ν̄ν$, $B̄_{s,d} → τμ$, and $K_L → μe$.

Extended reading notes

Core claim

The paper's central claim is that Froggatt-Nielsen charges, used not as a dynamical model but purely as a recipe for the relative size of flavour spurion entries, give a self-consistent and predictive power counting for new-physics flavour structures beyond minimal flavour violation. The scheme is defined by $Δ^{iα}_{QL} = c^{iα}_{QL} λ^{|b^Q_i - b^L_α|}$ and $Δ^{iα}_{DE} = c^{iα}_{DE} λ^{|b^D_i - b^E_α|}$, with the charges fixed as far as possible by the observed masses and mixing. Applied to the $U_1$ leptoquark, which couples through $(ȡQ γ^μ L)$ and $(d̄_R γ^μ e_R)$, this reduces the coefficient of the four-fermion operator to a product of two spurions, so that every quark-lepton transition in the model is predicted up to order-one coefficients. The authors then show that the surviving charge assignments are not unique: 11 distinct assignments fit the low-energy data about equally well, and they differ in their predictions for observables not yet measured precisely, notably the lepton-flavour-violating decays $B̄_{s,d} → τ^- μ^+$ versus $B̄_{s,d} → τ^+ μ^-$, which the leptoquark spurions make strongly asymmetric.

Load-bearing premise

The fit replaces all 18 spurion entries by two real numbers, one per spurion, taken equal in magnitude up to a sign, and additionally assumes the two leptoquark-induced Wilson coefficients are equal; if the true entries vary in size across flavour indices, the 11 surviving charge assignments and their quoted coefficient ranges are consequences of that collapse rather than of the framework.

Editorial extensions

If this is right

Every quark-lepton transition mediated by the leptoquark acquires a definite order of magnitude from the charges, so the framework makes predictions for channels that current data cannot yet constrain, including separate rates for $B̄_{s,d} → τ^- μ^+$ and $B̄_{s,d} → τ^+ μ^-$.
Because the 11 viable charge assignments give nearly identical fit quality but different predictions for $B̄_s → τ^+τ^-$ and $R_D$, a sharper measurement of either can discriminate among them.
The fit is limited by a tension between the lepton-universality tests in $W$ decays and $R_{D^{(*)}}$, a tension the authors note is generic to any effective description with tau couplings rather than specific to their scheme.
Under the assumption $C^{(1)}_{lq} = C^{(3)}_{lq}$, the framework predicts no new-physics contribution to $τ → 3μ$, $B → K^*ν̄ν$, $K → πν̄ν$, and $Z → ττ$, so any observed deviation in those channels would falsify that assumption.
The same power counting can be reused for global fits at Belle II and LHCb and for flavour-dependent collider searches at the LHC, since the spurion entries directly fix the size of the corresponding high-$p_T$ signals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The scheme is a bookkeeping device, not a theory: the order-one coefficients $c^{iα}$ are left free and the fit replaces all of them by a single number per spurion. A natural next step would be to test whether the data prefer any structure in those coefficients, which would tell us whether the charges are doing real work or only setting overall scales.
Because the Froggatt-Nielsen charges are defined in a basis fixed by a spontaneously broken $U(1)$, a UV completion that actually generates them would tie the leptoquark couplings to the same sector that generates the Yukawa hierarchies; the 11 surviving assignments are therefore a target list for model building, not just a fit outcome.
The predicted asymmetry between $B̄_s → τ^- μ^+$ and $B̄_s → τ^+ μ^-$ is a sharp signature that separates leptoquark spurions from generic scalar or vector mediators, and it is testable with the flavour tagging that both LHCb and Belle II can perform once statistics allow.
If the $R_{D^{(*)}}$ anomaly fades, the charge assignments selected here lose their main motivation and the framework reverts to a general power-counting proposal whose value would have to be judged on other observables.

Original abstract

Extending the Standard Model with higher-dimensional operators in an effective-field-theory (EFT) approach provides a systematic framework to study new-physics (NP) effects from a bottom-up perspective, as long as the NP scale is sufficiently large compared to the energies probed in the experimental observables. However, when taking into account the different quark and lepton flavours, the number of free parameters increases dramatically, which makes generic studies of the NP flavour structure infeasible. In this paper, we address this issue in view of the recently observed "flavour anomalies" in $B$-meson decays, which we take as a motivation to develop a general framework that allows us to systematically reduce the number of flavour parameters in the EFT. This framework can be easily used in global fits to flavour observables at Belle II and LHCb as well as in analyses of flavour-dependent collider signatures at the LHC. Our formalism represents an extension of the well-known minimal-flavour-violation approach, and uses Froggatt-Nielsen charges to define the flavour power-counting. As a relevant illustration of the formalism, we apply it to the flavour structures which could be induced by a $U_1$ vector leptoquark, which represents one of the possible explanations for the recent hints of flavour non-universality in semileptonic $B$-decays. We study the phenomenological viability of this specific framework performing a fit to low-energy flavour observables.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,892 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.39 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.39 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Extending the Standard Model with higher-dimensional operators in an effective-field-theory (EFT) approach provides a systematic framework to study new-physics (NP) effects from a bottom-up perspective, as long as the NP scale is sufficiently large compared to the energies probed in the experimental observables. However, when taking into account the different quark and lepton flavours, the number of free parameters increases dramatically, which makes generic studies of the NP flavour structure infeasible. In this paper, we address this issue in view of the recently observed "flavour anomalies" in $B$-meson decays, which we take as a motivation to develop a general framework that allows us to systematically reduce the number of flavour parameters in the EFT. This framework can be easily used in global fits to flavour observables at Belle II and LHCb as well as in analyses of flavour-dependent collider signatures at the LHC. Our formalism represents an extension of the well-known minimal-flavour-violation approach, and uses Froggatt-Nielsen charges to define the flavour power-counting. As a relevant illustration of the formalism, we apply it to the flavour structures which could be induced by a $U_1$ vector leptoquark, which represents one of the possible explanations for the recent hints of flavour non-universality in semileptonic $B$-decays. We study the phenomenological viability of this specific framework performing a fit to low-energy flavour observables.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Extending the Standard Model with higher-dimensional operators in an effective-field-theory (EFT) approach provides a systematic framework to study new-physics (NP) effects from a bottom-up perspective, as long as the NP scale is sufficiently large compared to the energies probed in the experimental observables.	background 1.00	0.06	0.03	asserted 0.93
s1	However, when taking into account the different quark and lepton flavours, the number of free parameters increases dramatically, which makes generic studies of the NP flavour structure infeasible.	gap 0.54	0.07	0.03	asserted 1.00
s2	In this paper, we address this issue in view of the recently observed "flavour anomalies" in $B$-meson decays, which we take as a motivation to develop a general framework that allows us to systematically reduce the number of flavour parameters in the EFT.	contribution 0.89	0.54	0.10	asserted 0.96
s3	This framework can be easily used in global fits to flavour observables at Belle II and LHCb as well as in analyses of flavour-dependent collider signatures at the LHC.	contribution 0.53	0.22	0.09	asserted 0.84
s4	Our formalism represents an extension of the well-known minimal-flavour-violation approach, and uses Froggatt-Nielsen charges to define the flavour power-counting.	method 0.90	0.76	0.08	asserted 1.00
s5	As a relevant illustration of the formalism, we apply it to the flavour structures which could be induced by a $U_1$ vector leptoquark, which represents one of the possible explanations for the recent hints of flavour non-universality in semileptonic $B$-decays.	contribution 0.88	0.74	0.07	hedged possibility 0.93
s6	We study the phenomenological viability of this specific framework performing a fit to low-energy flavour observables.	contribution 0.72	0.93	0.11	asserted 0.76

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from analysis of existing data 0.46.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Correlation between B¯ s → τ +τ − and RD for all the 11 scenarios listed in

**Figure 2.** Correlation between the B¯ d,s → τ −µ + and B¯ d,s → τ +µ − modes for each scenario. The different solutions are labeled as in

Figures reproduced from arXiv:1910.02641 by the authors.

References

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Forward citations

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How the literature uses it

33 arXiv papers cite this in 54 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 33 sentences without a confident label

Builds on it 1 paper

This construction can be extended to the effective operators of the SMEFT [ref] , resulting in hierarchical entries for the Wilson coefficients.

Small instanton-induced flavor invariants and the axion potential · Bounds from induced $θ̅$ · 0.84

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[ref] we adopted a power counting based on the well-known Froggatt and Nielsen (FN) model [ref] .

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Compares with it 2 papers

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The and models can be compared, for instance, to the $U(3)^5$ spurion analyses presented in Refs. [ref] .

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Cites it as support 1 paper

In this context, it has been shown that LQs can explain $b→sℓ^+ℓ^-$ data [ref] , $R(D^(*))$ [ref] and/or $a_μ$ [ref] , which makes them prime candidates for extending the SM with new particles.

Scalar Leptoquarks in Leptonic Processes · Introduction · 0.96

Cites it as background 11 papers

Viable flavor patterns include Minimal Flavor Violation (MFV) [ref] , $U(2)$ or $U(3)$ symmetries [ref] , flavor alignment [ref] , and Froggatt-Nielsen scenarios including leptoquark couplings [ref] .

Resolving the Flavor Structure in the MFV-SMEFT · Introduction · 0.94
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What it cites · 52 sentences without a confident label

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The statistical analysis is performed using the package MultiNest [ref] for each of the 24 FN charge assignments.

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Uses data or values from 11 papers

The magnitude of the coupling mediating the $b→sμ^+μ^-$ transition can be inferred from the global fits of [ref] .

B-decay discrepancies after Moriond 2019 · Constraints on FN charges · 1.00
The magnitude of the coupling mediating the $b→sμ^+μ^-$ transition can be inferred from the global fits of [ref] .

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The magnitude of the coupling mediating the $b→sμ^+μ^-$ transition can be inferred from the global fits of [ref] .

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Cites as background 17 papers

In order to explore physics beyond the SM it is convenient to regard the SM Lagrangian as the leading term of an effective field theory (EFT) expansion, with corrections suppressed by inverse powers of the NP scale(s) [ref] .

Dimension-Six Terms in the Standard Model Lagrangian · Introduction · 0.93
However, as soon as one includes new flavour structures, the consistency relations become a non-trivial theoretical requirement (see the discussion in [ref] ).

Minimal Flavour Violation and Beyond · Consistency conditions and Froggatt-Nielsen charges · 0.81
In the literature there exist a number of different proposals trying to achieve such a task, see e.g. [ref] .

Revisiting the vector leptoquark explanation of the B-physics anomalies · Extended MFV from $U_1$ Vector Leptoquark · 0.97

Discussion

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