Leading directions in the SMEFT: Renormalization Effects

The stability of the electroweak scale, challenged by the absence of deviations in flavor physics, prompts the consideration of SMEFT scenarios governed by approximate SM flavor symmetries. This study examines microscopic theories that match onto a set of $U(3)^5$-symmetric dimension-6 operators. Renormalization group mixing from the ultraviolet to the electroweak scale yields significant phenomenological constraints, particularly pronounced for UV-motivated directions. To demonstrate this, we explore a complete suite of tree-level models featuring new spin-0, 1/2, and 1 fields, categorized by their irreducible representations under the flavor group. We find that for the leading directions, corresponding to a single-mediator dominance, RG mixing effects occasionally serve as the primary indirect probe.


I. INTRODUCTION
Given a perceptible gap between the new physics (NP) scale and the electroweak (EW) scale, the Standard Model Effective Field Theory (SMEFT) [1][2][3][4][5][6] emerges as a robust theoretical framework for describing deviations from the Standard Model (SM).The SMEFT Lagrangian is an infinite series of higher-dimensional local operators built from the SM fields obeying gauge and Poincaré symmetries.The respective Wilson coefficients (WCs) encapsulate the short-distance effects of a broad spectrum of models beyond the SM (BSM).In the absence of a clear direction toward a specific BSM scenario, such a framework provides a convincing path forward, informing phenomenological studies and data interpretation.
The minimal number of independent WCs (an operator basis) is rendered finite at each order in the inverse powers of the cutoff scale controlled by the canonical dimensions.Yet, the size of this space rapidly increases with the growing canonical dimension, but also with the number of families [7].Specifically, for leading-order baryonnumber conserving operators at dimension six, the parameter count rises from 59 for a single active generation to a striking 2499 for three generations [8].This escalation underscores the complexity introduced by the flavor degrees of freedom.
On the other hand, the fermion kinetic terms enjoy a large U (3) 5 global symmetry owning to the three copies of five different gauge representations.The Yukawa interactions induce a rather peculiar explicit breaking, giving rise to exact and approximate flavor symmetries in the SM.The absence of violation of the implied selection rules in precision flavor experiments, such as ∆F = 2 transitions, charged lepton flavor violation, and electric dipole moments, already imposes stringent constraints on NP which does not lie far above the EW scale [9].Indeed, a viable TeV-scale physics, anticipated by the Higgs hier-archy problem and driving direct searches at the energy frontier, should not excessively violate the approximate flavor symmetries.This reasoning motivates the introduction of flavor power counting in the SMEFT, allowing for more focused analyses.Indeed, flavor symmetries prove to be very beneficial in charting the space of the SMEFT [10,11].
In this work, we consider a class of microscopic theories that integrate out to U (3) 5 -symmetric dimension-6 basis made up of only 47 operators.These are the leading operators in the minimal flavor violation (MFV) [12] power counting and represent the most minimal complete operator basis of interest for global fits of top, Higgs, and electroweak data [13][14][15][16].As such, it constitutes an important initial playground towards more complicated global analyses, such as those based on the U (2) 5 flavor symmetries [17][18][19][20]. 1  Restricting ourselves to U (3) 5 -symmetric operators at the ultraviolet (UV) matching scale, the main focus of this investigation is on the renormalization group (RG) effects between the UV and the EW scales.These effects are governed by the SMEFT anomalous dimension matrix computed in [8,22,23] (see also [24]) and implemented in numerical tools [25] such as wilson [26], DsixTools [27,28] and RGESolver [29].While it has become a common practice to automatically include these effects, for example, in smelli [30], our work aims to pinpoint the most constrained linear combinations of operators at the UV matching scale resulting solely from RG mixing.To deepen the understanding of these effects, we provide simplified analytical expressions supported by the full numerical results.Our key interest lies in identifying RG-induced contributions to precision observables at low energies, which offer stronger or comparable bounds to those from tree-level processes.Upon examination, noteworthy cases involve four-and two-quark operators, where the RG bounds rival those from top quark production [31], echoing recent findings for topspecific operators [32].
To demonstrate the significance of RG effects, we examine a full set of tree-level mediator models matching onto the U (3) 5 -symmetric operator basis at the UV scale.A complete spectrum of mediator fields with spin 0, 1/2, and 1, along with their SM and U (3) 5 flavor representations, has been comprehensively identified and matched to the universal basis in Ref. [33], building upon [34].This matching process has defined a finite set of leading directions -UV-motivated linear combinations of the WCs, warranting thorough examination.This paper performs a complete RG analysis of all these leading directions, going beyond the tree-level phenomenology presented in [33].Our central findings are showcased in Tables II, III and IV, comparing the RG bounds on a comprehensive set of four-and two-quark leading directions with the tree-level bounds.
This paper is structured as follows: in Section II we identify crucial RG equations, Section III gives an overview of the most sensitive low-energy observables, expressing them in terms of the WCs at the UV matching scale in the leading-log approximation.In Section IV, we derive a comprehensive set of bounds on the leading directions, which are then compared against exclusions from direct searches for selected benchmark models.The summary and the future outlook are presented in Section V.

II. RG EFFECTS FROM FLAVOR-BLIND UV
The full U (3) 5 -symmetric operator basis is defined in Appendix B of Ref. [33].Focusing on the phenomenologically important RG effects, we consider a subset of operators at the UV matching scale involving quarks. 2  Our starting point is the Lagrangian where the sum goes over all four-quark and two-quarktwo-ϕ operators defined in Table I above the double line.
Here, q and ℓ denote the left-handed quark and lepton doublets, while u and d denote the right-handed up and down quark fields.ϕ denotes the Higgs doublet.Flavor indices are i, j = 1, 2, 3, and the summation over repeated indices is assumed.In the rest of the paper, the labels assigned to the WCs, which we treat as dimensionful parameters, correspond directly to the labels of the operators from Table I.Starting from Eq. ( 1) at the UV matching scale, the RG equations [8,22,23] determine the non-zero dimension-6 WCs at the EW scale, where the subsequent matching to the low-energy effective field theory 2 For all leading directions involving leptons, we have verified that tree-level effects consistently dominate (see Section IV).The same is true for purely bosonic operators with the exception of O ϕ already discussed in [33]. A.

C
(3) qq q q q q ℓ ℓ W FIG. 1. Representative RG diagrams.See Section II for details.
(LEFT) [35] is performed at the tree level. 3There are three categories of important RG effects depicted in Figure 1: • A. Four-quark operator mixing into an EW boson vertex.
• B. Four-quark operator mixing with two insertions of Yukawa interactions.
• C. Four-quark operator mixing into a semileptonic operator.
While gauge interactions are flavor-diagonal (FD), the Yukawa-dependent part of the anomalous dimension matrix leads to FD and flavor-violating (FV) effects.For example, when matched to the LEFT, the resulting operators from category A produce both FD and FV Z couplings.The latter ones play a role in ∆F = 1 processes such as b → sℓℓ decays. 4abel Operator

O
(1)D qq 5 -symmetric dimension-6 SMEFT operators appearing at the UV matching scale in this work are listed above the double line.Other operators enter through the RG mixing.Our notation closely follows Ref. [33].All operators considered are Hermitian, ensuring that their corresponding WCs are real.

A. Vertex corrections
As depicted by the top diagram in Figure 1, operators O ,3) ϕq by closing the loop with an up-type quark and emitting two Higgs fields.See Ref. [22] for the flavor-generic RG expressions.These contributions are proportional to Y u Y † u , leading to y 2 t -enhanced effects. 5All such effects, including the running of the operators from the ψ 2 H 2 D class, can be described by the following 5 Throughout this work, we will not discuss small contributions d , although they are included in our numerical studies.
set of RG equations Ċ(1) ϕq,FD δ pr , Ċ ϕq,FD δ pr , where the abbreviation of the form Ċ ≡ 16π 2 µ d dµ C is used, and we introduce the linear combinations of WCs in the flavor-symmetric basis6

C
(1) ϕq , C When the operators O (1,3) ϕq , O ϕu and O ϕd , generated via Eq.( 2), are matched to the LEFT, they induce modified Z and W couplings to quarks [35], contributing to important FD, as well as FV observables, see Section III.

B. Four-quark operators
The four-quark operators given in Table I mix among themselves under RG equations, leading to interesting ∆F = 1 FV effects as illustrated by the middle diagram in Figure 1.The system of equations, simplified to in-clude only Y u -dependent terms, is as follows: Ċ( 1) where we introduce the linear combinations qd,FV ≡ C (1) qu,FV ≡ C (8)  qu − 4C E uu , C ud .
(5) As in previous cases, these are formulated in terms of the WCs of operators from the flavor-symmetric basis.The RG equations presented here are not exhaustive but capture only the phenomenologically relevant terms.

C. Semileptonic operators
For the upcoming analysis of the low-energy observables, as it unfolds, keeping only the RG terms proportional to g 2 2 , while neglecting the ones proportional to g 2 1 , there is only one relevant RG-generated semileptonic operator, which appears as a result of the O ℓq [8].The RG equation for O (3) where we introduce To reemphasize, our numerical analysis in Section IV does not employ such approximations.Nonetheless, the above equations effectively approximate the most sensitive RG effects.

III. RG-INDUCED LOW-ENERGY PROBES
Having detailed the relevant RG equations, this section focuses on their impact on key low-energy probes.We address this by solving these equations using a leadinglog approximation, which allows us to express low-energy (pseudo)-observables in terms of the WCs at the UV matching scale.

A. b → sℓℓ
Flavor-violating Z couplings to quarks, represented by the O (1,3) ϕq matching to LEFT, are effectively constrained from rare meson decays to charged leptons or neutrinos. 7 Presently, available data from charged leptons is more constraining, while final states with neutrinos can provide a complementary test in the future.Furthermore, correlated effects are predicted in all down-quark FV neutral currents, including b → s, b → d, and s → d transitions.In fact, the measurements of b → sℓℓ decays provide the most sensitive probe of semileptonic interactions with MFV structure, see Ref. [37].
Rare b decays are described by the weak Hamiltonian where G F is the Fermi constant and α is the finestructure constant, while the local operators are defined as and C 9,10 denote the short-distance contributions.NP contributions through modified Z couplings to quarks predict lepton flavor universality.In addition, C NP 9 is suppressed due to the small Z couplings to the leptonic vector current.Conversely, C NP 10 receives significant NP contributions.
In our framework, contributions to this observable are generated by the RG mixing of the four-quark operators (Section II B), along with the four-quark operators mixing into the gauge boson vertex corrections (Section II A). 7 Down-quark FV is absent when ϕq at the EW scale. 8Note, that finite one-loop matching contributions are subleading when compared to large log-enhanced RG effects [13,36].
Solving the RG equations ( 2) and ( 4) in the leading-log approximation, then matching the SMEFT onto the JMS LEFT basis [35], and using the master formula provided in [44,45], we obtain ϕq,FV + C (3) ϕq,FV ϕq,FV + C (3) ϕq,FV where N ∆S=1 = (1 TeV) 2 .We only keep the contributions from the largest hadronic matrix elements, P (1,8) dd which capture all of the relevant effects in this study.Note that, in accordance with the aforementioned references, µ f = 160 GeV is used.As a final comment, although we start with strictly real WCs at the UV scale (see e.g.Ref. [46] for a discussion regarding imaginary WCs), through the process of RG mixing and matching, we end up with Im [V * ts V td ], which then enters the CPviolating observable ε ′ /ε. 9

C. W mass
We choose (G F , m Z , α) as an input parameter set and can therefore predict the value of m W both in the SM, including up to two-loop corrections [47], and in U (3) 5 symmetric SMEFT [48][49][50].We express the combined prediction as where θ is the Weinberg angle, s x ≡ sin x, and c x ≡ cos x. 10  The operator class ψ 2 H 2 D 2 from the U (3) 5 -symmetric basis (where ψ denotes quark fields) contributes to m W already at the leading log, by mixing into O ϕD and O (1) (3) 9 This effect is independent of the flavor basis used.In the up basis, two CKM matrix element insertions arise when rotating d quarks to mass eigenstates, while in the down basis, they emerge from rotating the Yu matrix in the RG equations. 10It should be noted that Eq. ( 12) generally includes an additional term proportional to C ϕW B [48][49][50].However, operators from Table I do not mix into this operator at the one-loop level.
Solving these equations, we find Interestingly, the four-quark operators considered in this work only contribute to δm 2 W beyond the leading-log approximation.Specifically, all four-quark operators in Table I except color octets contribute at this order. 11For example, O(Y 4 u ) contribution proceeds through C D,E uu → C ϕu → C ϕD and similarly at O(g 4  2 ), we have ℓℓ .Here, we rely on a fully numerical solution of the RG equations using wilson [26].
As for the measured value, we consider the latest PDG combination of m exp.

D. Z pole observables
As discussed in Section II A, the operators considered in this work can lead to modifications of the FD Z-boson couplings with quarks.Such effects can be constrained from electroweak precision tests (EWPT) with on-shell Z bosons [53,54].These constitute various partial decay width ratios, forward-backward asymmetries, and leftright asymmetries [53,55].We utilize an extensive list of observables with correlated experimental errors, employing smelli [30] to build a custom EWPT likelihood focused solely on Z-pole observables.We exclude m W (analyzed separately in Section III C) and W -pole observables sensitive to modified W couplings to SM fermions, as they are not phenomenologically competitive.The constructed likelihood relies crucially on flavio [51] due to its database of experimental measurements [41,[55][56][57] and the implemented theoretical predictions of the considered observables at the scale µ = m Z , including SM and BSM contributions [3,58].In the subsequent numerical analysis, we use wilson [26] to run and match the WCs, capturing also beyond leading-log effects.

E. β-decays
Following the discussion in Section II, the only RG contributions to charged-current processes in our framework go through either modified W couplings with left-handed quarks due to O ℓq contact interaction, see Eq. ( 6).Both of these match onto 11 Color octets are absent because loops with color octet operator insertions in the relevant mixing cascades are always proportional to Tr(T A ), which is identically zero.
the same low-energy V − A operator.The low-energy Hamiltonian is The NP contributions to the left-handed currents have been absorbed into Ṽux as Ṽux = V ux (1 + ϵ x L ), where V ux are elements of the unitary rotation matrix [59,60].The effects of nonzero ϵ x L can be probed through the violation of the (first row) CKM unitarity Solving the respective RG equations in the leading-log approximation and matching onto Eq. ( 15), we obtain (3) for all x = d, s, b.The CKM unitarity test then reduces to ∆ CKM ≈ 2ϵ x L at linear order in WCs, with no summation over x.

F. Atomic parity violation
Atomic parity violation (APV) is sensitive to parityviolating couplings of electrons to quarks.Experiments report the weak charge, defined as where Z (N ) is the number of protons (neutrons).The general expressions for g eu AV and g ed AV are given in [59].Upon solving the RG equations ( 2) and ( 6) in the leadinglog approximation, we get ϕq,FD , (3) The most precise measurements are done with the cesium ( 133 Cs) atom, for which up to small radiative corrections [67][68][69].The prediction for the cesium weak charge obtained in the SM is Q Cs,SM W = −73.23(1)[68][69][70] which includes radiative corrections, while the current experimental value as reported by PDG is Q Cs,exp.W = −72.82(42) [68,71,72].

IV. LEADING DIRECTIONS
The natural next step in a systematic bottom-up approach is to construct UV completions of the U (3) 5symmetric dimension-6 basis.An exhaustive leadingorder classification yields a finite set of possibilities within the scope of perturbative short-distance NP.The UV/IR dictionary of Ref. [34] is a collection of all possible SM gauge representations of new scalar, fermion, and vector fields that match onto dimensions-6 operators in the SMEFT at the tree level.Expanding on this, Ref. [33] further imposed U (3) 5 flavor symmetry to the UV Lagrangian, categorizing new mediator fields into irreducible representations (irreps) and defining the flavor coupling tensors.The exact symmetry limit is highly predictable -each nontrivial flavor multiplet leads to mass degenerate states, which once integrated out, match to a single Hermitian operator in the U (3) 5 -symmetric basis with a well-defined sign for the obtained WC. 12 Each case predicts a direction in the WC parameter space, denoted as a leading direction.A general tree-level matching result is a linear combination of these directions.
Needless to say, a finite number of scenarios featuring a single mediator dominance is of particular phenomenological importance.To this purpose, Ref. [33] conducted a thorough tree-level analysis for each leading direction, reporting a compendium of bounds based on the available data.This study extends the previous analysis by incorporating the RG effects.Upon a detailed case-by-case examination, we find that a substantial number of scenarios have RG-induced bounds competitive with tree-level bounds.These cases, along with their tree-level matching formulas, are presented in Table II for scalars, Table III for vectors, and Table IV for fermions.
These tables explain our initial choice of operators in Table I.The tree-level bounds on the UV mediators which match onto the four-quark operators (scalars and vectors) are from the top quark production [31], while those generating two-quark-two-ϕ operators (fermions) were constrained from a combined low-energy fit of (semi)leptonic operators [59].In Section IV A, we compare the tree-level with the RG-improved constraints for all mediators where the latter are numerically important. 13In Section IV B, we finally compare these indirect  In the first column, we collect the labels for each mediator along with their irreps under the SM gauge group.In the second column, we list the flavor irreps, while in the third and fourth columns, we provide the normalization and the linear combination of the generated operators from the U (3) 5 -symmetric basis (see Table I).In the remaining columns we collect the 95% CL constraints discussed in Section IV A.
constraints with direct searches for two benchmark cases, revealing an interesting interplay.

A. Improved EFT limits
In this section, we derive a set of RG-improved bounds on all leading directions in the operator space of Table I.In Section III, the observables are calculated using the leading-log approximation to elucidate RG-induced effects.However, the results presented here rely on solving the RG equations numerically.This approach, as compared to leading-log expressions, offers several improvements.It includes effects beyond the leading log, such as δm 2 W from four-quark operators discussed in Section III C, ensures resummation of large logs, and accounts for terms proportional to g 1 and other smaller parameters, which were omitted in the leading-log expressions but are included in numerical calculations.
We use the open-source python package wilson [26] to numerically solve the RG equations. 14The UV matching scale is set at µ i = 3 TeV, and we run down to renormalization scale µ i , although this dependence is only logarithmic.In contrast, their dependence on the mass M X and coupling y X is quadratic, scaling as ∝ y 2 X /M 2 X . 14We checked that our leading-log formulae nicely agree with wilson where appropriate.
low energy scales, relevant for the observables discussed in Section III. 15 We separately construct χ 2 functions for each observable and a combined χ 2 , considering both theoretical and experimental uncertainties.These functions are then used to derive 95% CL constraints on the effective mass of each mediator, represented as a massto-coupling ratio.If a single number is presented, it is to be understood as a lower limit, whereas a reported interval corresponds to a 2σ preferred range.
The results for all scalar mediators are given in Table II.We observe that b → sℓℓ plays an important role in constraining the linear combinations of O (1)D,E qq and O (3)D,E qq operators.This is most apparent in the cases of ω 1 ∼ 6q and Υ ∼ 6 q , where the bounds obtained using b → sℓℓ reach ≳ 3 TeV.For the ζ ∼ 3 q and Ω 1 ∼ 3q , we obtain bounds which are comparable to the tree-level results, still ≳ 2 TeV.On the other hand, ε ′ /ε bounds turn out to be less stringent, however, for φ ∼ ( 3q , 3 u ), ω 4 ∼ 3 u and Ω 4 ∼ 6 u , the bounds are still around 1 TeV.Z-pole observables impose significant bounds on the same leading directions constrained by b → sℓℓ, extending to multi-TeV ranges, and in other cases, the bounds are comparable to those derived from top data.Regarding D qq 0.7 1.6 0.4 0.5 1.2 0.5 0.2 1.9 qd + 3O qu + 3O

+ 9O
(3)D qq − 3O (3)E qq 0.9 0.6 -0.7 [0.7, 3.5] 0.7 0.2 0.9 ud + 3O   (3)D,E qq operators.In the case of ζ ∼ 3 q and Υ ∼ 6 q , β decays provide a highly competitive lower bound on the effective mass, while in the remaining two cases (ω 1 ∼ 6q and Ω 1 ∼ 3q ), we obtain a preferred range for the effective mass.The APV observable Q Cs W yields in many cases constraints exceeding 1 TeV, for example ω 4 ∼ 3 u , ζ ∼ 3 q , and Υ ∼ 6 q .The modification of m W , an effect beyond the leading log, is highly relevant for the ω 1 ∼ 6q , Ω 4 ∼ 6 u , Υ ∼ 6 q and Φ ∼ ( 3q , 3 u ) irreps, where the obtained bounds are ≳ 1 TeV.In summary, the combined fit notably improves the bound for ω 1 ∼ 6q and Υ ∼ 6 q compared to using only top data, while in other cases, it yields a modest enhancement of the bounds, although the RG-induced observables are competitive.Numerical results for the vector mediators are pre-sented in Table III.Highly important bounds due to b → sℓℓ are set for B ∼ 8 q , W ∼ 8 q , and H ∼ 1. ε ′ /ε gives a bound above 1 TeV for certain irreps, such as B 1 ∼ ( 3d , 3 u ) and Y 1 ∼ ( 3d , 3q ), which are comparable to the bounds from top data.The Z-pole observables, β decays, Q Cs W and m W seldom reach the 1 TeV level.However, they are often still comparable to the constraints from top data alone.In some instances, Z-pole observables and β decays provide a preferred range rather than a lower limit for the effective mass.
Lastly, for completeness, in Table IV, we collect the constraints on fermion mediators.Tree-level bounds in Ref. [33] were derived using a combined fit of low-energy observables from Refs.[59,73], exceeding 3 TeV in most cases as shown in Table 9 of Ref. [33].Our analysis focuses on the most relevant observables from these likelihoods, presented individually.While tree-level effects primarily dominate, RG-induced constraints remain significant.As detailed in Section III C, for UV fermion mediators, the modification of m W at leading-log order proves to be a stringent constraint in most cases, with bounds exceeding 3 TeV for D ∼ 3 q , Q 7 ∼ 3 u , and T 1 ∼ 3 q .For U ∼ 3 q and D ∼ 3 q , it is notable that b → sℓℓ, an RG-induced constraint, sets a lower limit around 2 TeV.Other constraints, largely at the tree level, are similar to those reported in Ref. [33] with minor differences.

B. Direct searches
This subsection examines the direct search sensitivity for two heavy mediators (ω 1 ∼ 6q and Q 7 ∼ 3 u ) primarily constrained at the EFT level by the RG effects discussed in this paper.We review the relevant LHC collider constraints from single and pair production for both cases.
Consider the scalar diquark ω 1 ∼ 6q .Table II shows that for this mediator, the primary EFT constraint arises from RG mixing of O qq into O ϕq , affecting quark FV Zboson couplings and impacting b → sℓℓ processes.The obtained constraint in the coupling versus mass plane (µ i = M ω1 ) is depicted by the green shaded region in Figure 2, demonstrating the breaking of the pure power-law dependence on M/y from the previous subsection where µ i = 3 TeV (dashed green line).For M ω1 = 3 TeV, the two approaches align, but for significantly lower or higher mass values, the discrepancies increase as expected.
The considered diquark couples to gluons and can be pair-produced in proton collisions, with a cross section set by its gauge representation and mass.Considering the ATLAS [74] and CMS [75] searches for pair-produced colored resonances decaying to jets, we obtain a lower limit on the diquark mass of M ω1 > 700 GeV following [76]. 16his constraint is represented in the magenta color on Figure 2.Moreover, the diquark couples to quark pairs, most notably there is a component of the flavor multiplet that couples to valence quarks. 17This leads to important constraints from direct searches of dijet resonances from ATLAS [79,80] and CMS [81,82], which have been recast for a generic mediator in Ref. [76].We use the results from the latter reference to obtain the constraints on Figure 2 presented in orange.There are two notable abrupt cuts in the shown contour: firstly, the mass interval of the searches stops at 5 TeV at most, explaining the vertical cut in the contour, and secondly, we horizontally cut the contour at the point where the partial decay 500 1000 2000 5000 width of the resonance Γ ω1 = M ω1 |y qq ω1 | 2 /(2π) is equal to 10% of its mass.Above this line, we deem the resonance to be too broad to respect the narrow-width approximation as assumed in obtaining the constraint, and only the EFT constraint applies.Ultimately, we do not consider the parameter space in which the ratio Γ ω1 /M ω1 > 50%.
The limited parameter space permitted by b → sℓℓ and not excluded by direct searches, visible in the upper part of Figure 2, could be explored through broad (and heavy) resonance searches.This coincides with the region of interest for contact interaction searches in dijet production, which utilize angular distributions [83,84].Regrettably, the constraints on the WCs obtained in these studies are not directly transferable to our work because the dijet invariant mass underlying these bounds lies in the multi-TeV range, where a mediator-based description is more suitable.A further challenge is that ATLAS and CMS analyze a narrow range of operators, unsuitable even within the restrictive U (3) 3 flavor structure.We recommend that experimental collaborations adjust future dijet data interpretations accordingly.
We analyze the vector-like quark Q 7 ∼ 3 u for our second example.As Table IV indicates, the most significant EFT constraint is the RG-induced shift in the W -boson mass, imposing a lower limit of over 4.5 TeV on the massto-coupling ratio.Concurrently, LHC searches for pairproduced vector-like quarks, particularly ATLAS, have established a lower bound of 1.3 TeV for top quark partners [85].This limit applies to the mass of Q 7 since part of its multiplet interacts with the top quark.It is worth noting that single production searches for thirdgeneration vector-like quarks [86] offer similar, though slightly less stringent probe, due to the parton den-sity suppression.A dedicated search for first-generation vector-like quarks would be beneficial.

V. CONCLUSION
This paper explores the intricate phenomenology emerging from renormalization group equations in the SMEFT.We rigorously assess the most significant RG mixing patterns by focusing on microscopic theories whose dominant effects are captured with a U (3) 5symmetric dimension-6 operator basis at the UV scale.
A key finding is that RG-induced effects on low-energy precision observables often lead to constraints on fourquark (and two-quark) operators that rival or surpass those derived from tree-level processes, notably in topquark physics.We thoroughly investigate a wide array of single mediator models that match onto the U (3) 5 symmetric basis at the tree-level, identified as leading directions.We provide a comprehensive RG analysis of these directions, extending beyond the earlier tree-level study [33].The key outcomes of this analysis are detailed in three tables: Table II for scalar mediators, Table III for vectors, and Table IV for fermions.Moving forward, our numerical analysis indicates that the observables we have examined are probing tree-level physics at the TeV scale, characterized by order one couplings, where direct searches can offer complementary probes, as depicted in Figure 2. A pivotal aspect of future work will be the thorough examination of dijet data within the SMEFT framework and across explicit mediator models, improving the direct search strategies for leading directions, as well as quantifying the validity of the EFT interpretation.Another promising direction is the exploration of U (2) 5 flavor symmetry in a similar context.This lower symmetry increases the set of tree-level mediators, adding more complexity and perhaps offering a better benchmark for the physics lying beyond the SM.
corresponding RG equations are given as ĊϕD = 24y 2 t C

FIG. 2 .
FIG.2.The leading EFT and direct searches constraints at the 95% CL in the mass-coupling plane of the scalar diquark ω1 ∼ 6q.See Section IV B for details.

TABLE III .
Vector mediators.For the description see the caption of TableII.

TABLE IV .
Fermion mediators.For the description see the caption of TableII.