Renormalization group improved implications of semileptonic operators in SMEFT

We study implications of the four-fermion semileptonic operators at the low-energy and at electroweak (EW) scale in the framework of Standard Model Effective Field Theory (SMEFT). We show how the renormalization group (RG) running effects can play an important role in probing the generic flavour structure of such operators. It is shown that at the 1-loop level, through RG running, depending upon the flavour structure, these operators can give rise to sizable effects at low energy in the electroweak precision (EWP) observables, the leptonic, quark, as well as the Z boson flavour violating decays. To this end, we isolate the phenomenologically relevant terms in the full anomalous dimension matrices (ADMs) and discuss the impact of the QED+QCD running in the Weak effective field theory (WET) and the SMEFT running due to gauge and Yukawa interactions on the dim-4 and dim-6 operators at the low energy. Considering all the relevant processes, we derive lower bounds on new physics (NP) scale Λ for each semileptonic operator, keeping a generic flavour structure. In addition, we also report the allowed ranges for the Wilson coefficients at a fixed value of Λ = 3 TeV.


Introduction
In the absence of the discoveries of new particles at the Large Hadron Collider (LHC), the SMEFT provides an elegant framework to parameterize and quantify the effects of NP in terms of SU(3) c × SU(2) L × U(1) Y gauge invariant higher dimensional (d ≥ 5) operators [1,2]. In SMEFT, the operators are constructed using the field content of the Standard Model (SM). Excluding the flavour structures, there are in total 59 operators which conserve the baryon number [1].

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An interesting aspect of the SMEFT is its built-in gauge symmetry. This feature often leads to an intriguing pattern of correlations among low energy observables due to enforcement of the model independent relations between the Weak effective theory (WET) operators at the EW scale, on matching with SMEFT [3][4][5][6]. At the 1-loop level, through RG running, new effective operators can also be generated at the EW scale as a result of operator mixing. For instance, the four-fermion SMEFT operators can mix with the ψ 2 φ 2 D type operators which can contribute to the observables of different kinds. In this manner, the running from Λ to the EW scale can induce additional correlations [7][8][9][10][11][12]. Therefore, in an SMEFT analysis, considering such effects is very important in order to correctly predict the low energy implications of new interactions introduced at the NP scale.
By now, the complete ADMs for SMEFT as well as WET are known at the 1-loop level [13][14][15][16][17]. The recent results for ADMs in the SMEFT extended with right-handed neutrino fields can be found in refs. [18][19][20]. Based on these calculations, the tools such as wilson [21] and DsixTools [22,23] have been developed. (See also ref. [24] for a discussion on the analytic solutions to the SMEFT RGEs). Using these codes, it is possible to include the RG running effects in the theoretical predictions which can be obtained using flavio [25]. Several other packages [26][27][28] exist to facilitate the different kinds of tasks for studying the phenomenology in the SMEFT framework. As far as the present work is concerned, we have used flavio for the theoretical predictions and the RG running effects have been taken care by using the wilson package.
In this work, we focus on a subset of SMEFT operators which contain both lepton and quark fields currents: here, i,j,k,l denote the flavour indices. Such operators are known as the semileptonic operators. It is well-known that, at tree-level, these operators enter into various semileptonic decays of mesons. In this context, the operators which violate the quark flavour have been extensively studied in the literature [8,9,[29][30][31][32][33][34][35][36]. On the other hand, a generic flavour structure of these operators is not yet fully explored. In particular, the quark flavour conserving counterparts deserve more attention. A given NP model can generate both flavour violating as well as conserving operators, therefore, it is essential to know what kind of constraints apply on the later ones. One of the goals of the present work is to fill this gap by identifying all possible low energy and EW scale observables, which can be used to probe a generic flavour structure of the semileptonic operators. Concerning this matter, the ref. [37] discusses the contribution of semileptonic operators to the EWP observables, assuming flavour universality. Similarly, the refs. [7,38], pointed out the importance of EWP and lepton flavour violating (LFV) constraints on the semileptonic operators needed to resolve the B-anomalies. For more recent studies on this topic, see also refs. [30,32,39]. Due to the reasons outlined above, we will restrict ourself to the flavour structures such that there is no quark-flavour violation, to begin with, at the scale Λ. In other words, the operators in which are interested are either flavour conserving in both currents or at most they violate the lepton flavour at the NP scale. We will emphasize the importance RG running from Λ to the EW scale, through which these operators can contribute to a JHEP01(2022)107 verity of observables at lower scales. In this regard, we will first isolate the most important terms in the ADMs due to gauge couplings as well as Yukawas, i.e., the ones which are phenomenologically relevant, given the current precision of the measurements.
The remainder of the paper is structured as follows. In section 2, we will discuss our strategy, and in section 3, we discuss the SMEFT RG running of semileptonic operators. In section 4, we will identify various observables which are relevant for the different flavour structures of the operators under consideration. In section 5, we will discuss the sensitivities to the NP scales Λ for various operators. Finally, we move on to the conclusions in section 6. Additional material is collected in appendices A, B, and C.

General strategy
In SMEFT, the SM is extended with a series of higher dimensional effective operators invariant under the full gauge symmetry of the SM. In general, the SMEFT Lagrangian can be written as here, C a are known to be the Wilson coefficients. A complete list of SMEFT operators can be found in refs. [1,2]. In this work, we will focus on a subset, the four-fermion semileptonic operators:

5)
[O ed ] ijkl = (ē i γ µ e j )(d k γ µ d l ) , (2.6) [O eu ] ijkl = (ē i γ µ e j )(ū k γ µ u l ) , (2.7) [O qe ] ijkl = (q i γ µ q j )(ē k γ µ e l ) , (2.8) here, the flavour indices i, j, k, l can take values from 1 to 3 and q, u, d, , e represent the quark doublet, right-handed up-type quark, right-handed down-type quark, lepton doublet and right-handed lepton fields, respectively. The down-type quarks are chosen to be in the mass basis at the scale Λ. This corresponds to the Warsaw-down basis of the SMEFT [40]. For convenience, we divide the operators into three classes based on their flavour structure at the scale Λ: • ∆F = (0, 0): the operators which do not violate quark 1 and lepton flavours, 1 Since we work in the Warsaw-down basis, the Cabibbo-Kobayashi-Maskawa (CKM) rotations can give rise to quark flavour violating operators in the up-sector, even at the scale Λ. Note that the CKM matrix itself can be affected by NP [39,41], however such effects are beyond the scope of present analyses. In our analyses for numerics the CKM is obtained with wilson program using the input Vus = 0.2243, V ub = 3.62 × 10 −3 , V cb = 4.221 × 10 −2 , δ = 1.27 at the Z-mass scale.

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• ∆F = (1, 0): the operators which do violate lepton flavour by one unit but do not violate the quark flavour, • ∆F = (i, 1): the operators which do violate quark flavour by one unit but may or may not violate the lepton flavour, i.e, i = 0 or 1.
Since, the ∆F = (i, 1) type operators are already well studied in the literature, we will not consider them here. On integrating out the new degrees of freedom at Λ, a unique tower of SMEFT operators is generated. However, a priori it is not obvious to which observables these operators can enter into at the lower energies. This is because of three reasons. Firstly, the SU(2) L invariance can lead to correlations between different type of observables. Secondly, the pattern of mixing between different operators due to running from Λ to the EW scale is very complex in nature. Often this leads to the appearance of new operators at the EW scale, which can give rise to unpredictable correlations among low-energy observables. Finally, the choice of the flavour basis at the NP scale is not invariant with respect to the RG evolution. Therefore, it is extremely important to systematically analyze these effects for all flavour structures of the operators of our interest. This will allow us to identify all possible observables which are sensitive to these operators. With these motivations, first we will identify the most sensitive observables which can be used to probe the operators listed in (2.2)-(2.8) for the following flavour combinations: Note that none of these flavour combinations are quark flavour violating. Next, we will proceed in three steps: to begin with, we will study the operator mixing pattern due RG evolution for the two classes of the flavour structures as shown in eqs. (2.9)-(2.10). Based on this, we will then identify all possible observables which can be used to constrain a given operator directly (at tree-level) or through the operators to which it mixes into, through the RG effects (at 1-loop level). Using this information, we will finally derive the lower bounds on the scale of each semileptonic operator assuming the presence of a single operator at the scale Λ.

Renormalization group running
In this section we discuss the RG running of the SMEFT semileptonic Wilson coefficients. In general, the running is governed by the coupled differential equationṡ Here, µ is the renormalization scale andγ is the anomalous dimension matrix which is function of the SM parameters such as gauge and Yukawa couplings. In the leading-log (LL)

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approximation, the solution to these equations for running from scale Λ and µ reads In the following we analyze the RG running and operator mixing of various quark-flavour conserving semileptonic operators (shown in eqs. (2.2)-(2.8)) of our interest to various other operators. These effects can potentially relate them to new observables generated at 1-loop level and hence allow us to put additional constraints. Using the ADMs calculated in refs. [13][14][15][16] we find that, depending upon the flavour structures, the quark flavour conserving semileptonic operators can mix with two types of operators. The first category is the ψ 2 φ 2 D type operators: Here, I is the SU(2) index and D µ stands for the covariant derivative. Secondly, they also mix with the purely leptonic operators given by

12)
[O ee ] ijkl = (ē i γ µ e j )(ē k γ µ e l ). (3.13) In addition, the ∆F = (0, 0) type semileptonic operators can also mix with the fourquark operators, however we do not find any significant constraints due to such an operator mixing. In order to get the general picture, for our qualitative discussion, we will use the LL solutions to the corresponding RG equations as given by eq. (3.3). However, for the numerics, we sum the logs using full numerical solutions which have been obtained using the wilson program [21].

Evolution of ∆F = (0, 0) operators
The ADMs in the SMEFT depend on the gauge couplings as well the Yukawas. In this section, we identify the phenomenologically important terms in the ADMs of ∆F = (0, 0) semileptonic operators.

Operator mixing due to gauge interactions
First, we discuss the operator mixing of the ∆F = (0, 0) semileptonic operators due the gauge couplings. The flavour combinations for these operators are specified in eq. (2.9).

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Therefore, as an initial condition, at NP scale, only the Wilson coefficients of the operators given in (2.2)-(2.8) are assumed to be non-zero. It turns out that, these operators can mix with the ψ 2 φ 2 D-type operators, which are listed in eqs. (3.4)-(3.10), through EW interactions. Now we use the LL approximation to relate the Wilson coefficients of the semileptonic operators at the scale Λ to the Wilson coefficients of ψ 2 φ 2 D operators which get generated at the EW scale due to the operator mixing. We find Strictly speaking on the l.h.s. we should have written the difference between the Wilson coefficients at the EW scale and their corresponding values at the scale Λ, i.e., However, since at the scale Λ we assume only the semileptonic operators to be non-zero, we have C(Λ) = 0 and therefore δC(µ ew ) = C(µ ew ). Also, we have expressed the loop suppression factor and the log in terms of the quantity L, given by In eq. (3.14), the g 1 and g 2 are the EW gauge couplings at the scale Λ. It is important to note that for the first three (last four) elements on the l.h.s., only the repeated index k (i) is summed over on the r.h.s. However, on the l.h.s. indices k and i are not summed over and can take values in the range 1-3. For simplicity, we do not show the self-mixing of the operators. Interestingly, after the EW symmetry breaking, the ψ 2 φ 2 D operators are known to give corrections to the W and Z boson couplings with quarks and leptons [42][43][44][45] (see also more recent refs. [37,39,[46][47][48][49][50][51]). Therefore, the quark-flavour conserving semileptonic operators can be indirectly probed through EWP data.
In this context, one should also consider the operator C 1221 which enters into the muon decay, i.e., µ → eνν and hence affects the extraction of Fermi constant G F . At the EW scale the Wilson coefficient C 1221 can be written as here the indices on the r.h.s. are summed over ii = 11, 22 and kk = 11, 22, 33.
In addition, we find the mixing between C of mixing is driven by the gauge coupling g 2 . As before, solving the corresponding RGEs in LL approximation, one finds (3.18) here, the repeated indices k and i on the l.h.s. as well as on the r.h.s. are not summed over.
Once again, we have suppressed the self-mixing (indicated by the entries with dashes) of these operators and log term L is given by (3.16). The mixing of C can induce new charged current transitions after EW symmetry breaking.

Operator mixing due to top-Yukawa interactions
In this subsection, we isolate phenomenologically important terms in the ADMs which depend on the Yukawa interactions. In this regard, we will keep only the largest terms involving the top-Yukawa coupling. Solving the RGEs in the LL approximation, we find here, y t is the Yukawa coupling for the top-quark. In eq. (3.19), for ∆F = (0, 0) operators the flavour indices i and j can be set equal. It is worth mentioning that for simplicity, we have not included the non-standard effects due to running of the Yukawa couplings themselves, which can lead to additional contributions to the quark-flavour violating semileptonic operators at the EW scale. We will return to this point in section 4.1.2.

Evolution of ∆F = (1, 0) operators
Now we consider the ∆F = (1, 0) type semileptonic operators which violate the leptonic flavour by one unit but conserve the quark flavour (see eq. (2.10)). Apart from the selfmixing, these operators mix with the lepton flavour violating leptonic and ψ 2 φ 2 D type operators, C φ ij and C φe ij with i = j, through the EW interactions. Once again, solving the RG equations in the LL approximation, we find,

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here the repeated indices k on the r.h.s. is summed over the values 1 − 3, and i = j holds on the both sides. Also, the repeated index l on the l.h.s. can take values in the range 1 − 3 but it is not summed over. The logarithmic piece L is defined in equation (3.16). The operators on the l.h.s. contribute to the LFV W , Z boson couplings and leptonic decays. We will come to this point later with more details. Note that, below the EW scale, the semileptonic operators can mix into purely leptonic operators also through QED interaction [52,53].

Observables induced at 1-loop level
Based on our discussions in the previous section, we are now in position to identify the 1-loop induced observables to which the semileptonic operators can contribute. We will see how the SMEFT RG running can play an important role in this respect. As discussed before, because of the complicated operator mixing pattern in SMEFT, the semileptonic operators can contribute to observables of very different nature depending upon the operators with which they mix, which can be read out from the l.h.s. of eqs. (3.14), (3.17), (3.18), (3.19) and (3.20). In order to get a general picture, we will present the expressions for the relevant low energy dim-4 and dim-6 couplings in terms of semileptonic Wilson coefficients at the high scale by employing the LL approximation. Eventually, we will sum the logs with the help of numerical solutions.

Observables for ∆F = (0, 0) operators
We have identified three categories of the observables which are relevant for ∆F = (0, 0) type operators. Those are the EWP observables, flavour violating B-decays and charged current decays. In the following, we discuss how various semileptonic operators can contribute to them through operator mixing.

Electroweak precision observables
As indicated by eq. (3.14), the ∆F = (0, 0) semileptonic operators can mix with the ψ 2 φ 2 D type operators due to electroweak interactions. In addition, this mixing also depends on the top-Yukawa interactions, see e.g. eq. (3.19). Interestingly, after EW symmetry breaking the latter operators give corrections to the Z and W boson couplings with the fermions. Using the LL solutions to the RGEs, as presented before, we can express the NP contributions to these couplings directly in terms of the Wilson coefficients of semileptonic operators at the high scale Λ. In general, NP shifts in the neutral Z boson couplings with the fermions can be parameterized as here g Z = −g 2 / cos θ W and θ W represents the weak-mixing angle. NP can enter into δg ψ X with X = L, R, through three difference sources. This can be understood from the equation [49] δ(g ψ

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here, the contributions in the first two and the last term can be thought of as indirect and direct shifts to the Z boson couplings, respectively. In SMEFT, the tree-level expressions for the quantities δg Z , δ sin 2 θ W and δ(g ψ X ) ij can be found in appendix A. Also, Q ψ represents the electric charge and g ψ,SM Below we present the 1-loop contributions to all these quantities due to semileptonic operators in the LL approximation. Using the eqs. (3.14) and (A.1)-(A.7) one can obtain the NP shifts in the Z couplings with quarks: Similarly, the shifts in the Z boson couplings with leptons are found to be Here, v = 246 GeV is the vacuum expectation value and L is given by (3.16). Similarly, for δg Z and δ sin 2 θ W are found to be On the basis of the above discussions, now we point out a few important observations: • First, note that the RG induced EW corrections to δg Z and δ sin 2 θ W due to C φ 22 and C 1221 . This can be seen by inserting the LL contributions of C (3) q iikk from eqs. (3.17) and (3.14) to the latter Wilson coefficients in eqs. (A.8)-(A.9). However, this is no longer true once the contribution due to the Yukawa couplings, as shown in eq. (3.19), is included.

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• Next, in general, the top-Yukawa dependent contributions, given in eqs. (4.10)- (4.11), are larger in size as compared to the direct EW corrections to the Z-couplings shown in eqs. (4.3)-(4.9). So, for the cases in which both effects exist simultaneously, the former has a greater impact.
• Furthermore, it is evident from eq. (4.2) that the impact of the C q ii33 for ii = 11 or 22 on δ(g Y X ) ii through δg Z and δ sin 2 θ W is universal for all three families of leptons and quarks. On the other hand the shifts δ(g e L ) dir ii , δ(g e R ) dir ii and δ(g ν L ) dir ii also experience effects due to the top-Yukawa interactions, which however are lepton flavour dependent.
• Finally, the contributions of the semileptonic operators due to top-Yukawa effects do not affect the Z boson couplings to quarks directly. This is however still possible through δg Z and δ sin 2 θ W , but only for the case of C q ii33 with ii = 11, 22.
Next, we look at the impact of semileptonic operators on the W -boson couplings which can be parameterized as here the NP shifts are given as [49] δ Using the LL solutions, we find the shifts δ(ε ψ L ) dir ii in terms of the semileptonic operators to be We make the following observations for the RG induced shifts in the W couplings: • The W couplings to both quarks and leptons are universally affected for all three families by C q ii33 for ii = 11, 22 through δ sin 2 θ W .
• The top-Yukawa effects do not give direct contributions to the W couplings.
• The leptonic couplings can be directly affected by C q ii33 for ii = 11, 22, 33 through top-Yukawa interactions. This effect is however flavour dependent.
In order to quantify these effects, in appendix B we report tables for the numerical values of the RG induced shifts in the Z and W couplings at the EW scale due to semileptonic operators present at the NP scale Λ. To understand the relative importance of the top-Yukawa and gauge interactions, we present two sets of numbers with and without Yukawa RG running effects. It is evident that, whenever present, the top-Yukawa effects always dominate.

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Now, given that the Z and W boson couplings are strongly constrained by the EWP observables, this implies that the flavour conserving semileptonic ∆F = (0, 0) type operators can be indirectly constrained by the EWP measurements. The following list of operators can be constrained through this mechanism: In the next section, we will use the EWP measurements as constraints to derive the lower bounds on the NP scales of the flavour conserving semileptonic operators in given eq. (4.16).
To this end, the list of EWP observables [54][55][56][57][58][59][60] used in our analysis are taken from table 13 of ref. [39]. In addition, we have also included a recent measurement of the ratio

B meson decays
Throughout the analyses, we have used Warsaw-down basis at the high scale. In this basis, to begin with the down-type quark and lepton mass matrices are diagonal, whereas, the up-type quark matrix takes the form V † diag(y u , y c , y t ). Here, V represents the CKM matrix. However, due to the RG running of the Yukawa matrices, this choice of basis is not preserved with respect to running from Λ to the EW scale. As a result, one has to perform back-rotation to the original (Warsaw-down) basis at the EW scale. This process can generate flavour violating semileptonic operators from their flavour conserving counterparts at the scale Λ [9, 10, 21, 62]. The part of the WET Lagrangian that describes the b → s + − processes can be written as We can match them to the corresponding SMEFT Wilson coefficients as [63]: Here, ij = µµ, ee. We find that, through back-rotation, the following semileptonic operators can contribute to b → s + − observables: To constrain these operators we have used the measurements for LFUV observable such as R K ( * ) [64][65][66][67][68][69][70] and Q 4,5 = P µ 4,5 − P e 4,5 [71,72]. However, the full b → s + − data set can give rise to stronger constraints [73,74].

Charged current decays
The (V − A) × (V − A) type charged-current (c.c) interactions in low energy WET are given by effective Lagrangian q ijkl . We find Here, the CKM elements (V † ) ml are needed to rotate the up-quark states to the mass basis.
The above equation indicates that through EW corrections the singlet operator C can lead to charged current transitions at the low energy. In the following we will look at several charged current processes which can be affected through this mechanism.
K meson decays: at the quark level, the charged-current K decays involve the transition s → ueν. Therefore, these decays are driven by the WET operator (ν 1 γ µ P L e 1 )(d 2 γ µ P L u 1 ). Now, using eq. (4.23), we note that this operator can be generated at the 1-loop level from the high scale SMEFT Wilson coefficient C q 1122 , i.e., Therefore, at the low-energy, C q 1122 can lead to effects in the K L → πeν, K S → πeν, K + → πeν, and K + → ν decays. τ → Kν and τ → πν decays: the τ → Kν and τ → πν decays involve τ →ūsν and τ →ūdν transitions, respectively. At the low energy these are governed by the WET operators (ν 3 γ µ P L e 3 )(d 2 γ µ P L u 1 ) and (ν 3 γ µ P L e 3 )(d 1 γ µ P L u 1 ). However, at the EW scale these operators can be generated from high scale Wilson coefficients C respectively (see (4.23)), because π → eν and nuclear β decays: the nuclear β-decay and π → eν are controlled by the WET operator (ν 1 γ µ P L e 1 )(d 1 γ µ P L u 1 ). From (4.23), clearly at the EW scale this operator can be generated from C To summarize, we have found that the following list of high scale coefficients can contribute to various charged current decays at low energy: q 3322 , C q 3311 , C q 1111 . (4.28) Note that the Wilson coefficient C q ijkl also give rise to tree-level contributions to the charged current decays. The experimental measurements for various charged current decays are shown in table 1. In addition, the measurements for the β-decays are taken from the ref. [75].
Note that for the charged current operators, in principle the three-body τ decays such as the Belle spectrum for the process τ − → K S π − ν τ also apply [78,79]. But as shown in ref. [80] these decays give rise to similar constraints as τ − → Kν decay.

Correlations
Since a single operator can contribute to several different kinds of observables through RG running, it would be interesting to see how these are correlated to each other. For instance, the operator C (1) q 1111 can contribute to β decay as well as EWP observables. Similarly, C q 1122 contributes to EWP observables, b → s + − , as well as charged current K processes. Take another example of the operator C q 1111 , which in addition to β decay also contributes to π → eν process at tree level. In figure 1, we show correlations between constraints due to various observables in the C planes in left and right panels respectively. Clearly, in order to get a complete picture about the constrains on a given operator, it is very important to take into account all RG induced observables.

Observables for ∆F = (1, 0) operators
Next, we move on to the semileptonic operators which violate the lepton flavour by one unit and conserve the quark flavour at the scale Λ. In this case, depending upon the Dirac and flavour structures, both tree-level as well as the 1-loop generated LFV observables JHEP01(2022)107 are found to be important. The 1-loop generated processes are i → j¯ and Z →¯ i j LFV decays. These processes are found to be relevant for all ∆F = (1, 0) operators under consideration. In addition, the ∆F = (1, 0) semileptonic operators can also contribute to τ → P for P = π, φ and τ → ρ processes at the tree-level.

Z →¯ i j decays
The tree-level SMEFT contributions to the LFV Z boson couplings due to ψ 2 φ 2 D operators are given in eqs. (A.1)-(A.7). At the 1-loop level, the semileptonic ∆F = (1, 0) operators can mix with these ψ 2 φ 2 D-type operators, as shown in the eqs. (3.20) and (3.19). As a result the ∆F = (1, 0) semileptonic operators can be constrained by the Z →¯ i j LFV decays. The current experimental limits on these decays are given in table 2.

τ → 3 and µ → 3e decays
In WET, the LFV processes such as τ → 3 and µ → 3e are governed by purely leptonic operators, as given by the effective Lagrangian: Here, the superscript indicates that such operators are generated only at the 1-loop level and are set to zero at the NP scale. In SMEFT, at tree-level the corresponding four fermion leptonic operators are C ijkl , C e ijkl , and C ee ijkl which are defined in eqs. (3.11)-(3.13). In addition to this, the SMEFT can also contribute through ψ 2 φ 2 D type operators after integrating out the Z boson. Such contributions arise by combining JHEP01(2022)107 the LFV effective Z boson vertices due to SMEFT with the flavour conserving Z boson interactions in the SM [81]. The ∆F = (1, 0) semileptonic operators can give rise to both effects through operator mixing. From eq. (3.20), one can find that, at the 1-loop level through the EW interactions, the contributing four-fermion SMEFT operators can be directly generated from the semileptonic operators. In addition, the ψ 2 φ 2 D operators get contributions through both the gauge (3.20) as well as the top-Yukawa interactions (3.19). In the LL approximation, combining the four fermion and ψ 2 φ 2 D type contributions, we can express the contributing WET Wilson coefficients at the EW scale directly in terms of SMEFT Wilson coefficients of the semileptonic operators at Λ: with the factor Z = 3v 2 y 2 t L Here L is the log term defined in (3.16). Numerically, the comparison of the influence of the gauge and Yukawa interactions on the WET Wilson coefficients on the l.h.s. is presented in table 13 in appendix B.4. We find that the following set of Wilson coefficients can be constrained through this mechanism: The experimental limits used for the LFV decays of τ and µ leptons are collected in table 2.

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Here, the indices kk = 11 for τ → π , τ → ρ and 22 for τ → φ . In addition to the tree-level contributions, the semileptonic operators in SMEFT can also contribute to these operators through top-Yukawa loops. Adding the both contributions, in the LL approximation, at the EW scale we obtain:  Note that the operators involving uū, dd and ss contribute at tree-level, whereas the operators involving the third generation contribute to these processes only at the 1-loop level. Naively, the former contributions are expected to dominate, but we will give a counterexample in the next subsection. The experimental limits on the corresponding LFV decays are shown in table 2.

Correlations
Since, the semileptonic operators can contribute to the LFV processes at tree-level as well as at the 1-loop, it would be interesting to see the relative importance of the loop-level vs. tree-level LFV effects. For example, the Wilson coefficient C (1) q 1311 could in principle give a tree level contribution to τ → P e and τ → ρe processes. However, due to SU(2) L invariance, the WET Wilson coefficients with uū and dd flavours get equal contributions on matching with C

Observable
Experimental value Observable Experimental value  enter with opposite sign in the branching ratios (see e.g. eqs. (55) and (61) of ref. [39]). However, C q 1311 can still contribute at the 1-loop level through the EW corrections. On the other hand, the Wilson coefficient C q 1333 can not give tree-level contribution to these LFV processes, but it can contribute at 1-loop to τ → 3e, τ → eµ + µ − as well as τ → ρe and τ → P e processes. Interestingly, this loop induced effect on the uū and dd WET coefficients is not equal but depends on the SM couplings g d,SM L and g u,SM L and hence it is not canceled in the branching ratios of our interest (see eqs. (4.37) and (4.38)). In addition to VLL, the VLR (again with unequal sizes for the uū and dd coefficients) WET operators are also generated at the 1-loop level. In figure 2 (left), the loose constraints JHEP01(2022)107 on C (1) q 1311 as compared to C (1) q 1333 confirm these findings. On the right panel of the same figure, we show the impact of LFV constraints in the plane of C (1) q 1311 and C u 1322 Wilson coefficients. In this case, since both operators contribute at the 1-loop level, they are found to be constrained at the similar level.

Sensitivities to the NP scale Λ
On the basis of our discussion about the RG running and the various low energy and the EW scale observables identified in the previous sections, now we are in position to look at the highest possible scales for each operator that can be probed using these observables. In eq. (2.1), the Wilson coefficients are defined to be dimensionful quantities. However, these can be written in terms of the dimensionless parameters [c X ] ijkl defined by Assuming the presence of a single operator at the scale Λ, we perform combined fits for each dimensionless parameter [c X ] ijkl , using all measurements relevant for a given operator. From this we obtain the quantity Λ/ [c X ] ijkl using the central values of [c X ] ijkl from the fits. This gives us a rough estimate of the scales that can be probed for each operator. Note that in a given NP model, more than one operators can be simultaneously present which could change this simplified picture of single operator dominance. However, the goal of present work is to systematically analyze the low energy implications of each operator separately i.e, to identify the most sensitive observables and assess their potential to constrain the individual operators. In • Due to more stringent experimental limits on the branching ratio for the process µ → 3e as compared to the τ LFV modes, the ∆F = (1, 0) operators involving the 12jj flavour indices are very strongly constrained. As shown, the current lower bounds on such operators are always above the ballpark of 10 TeV.
• Among 12jj operators, the ones having jj = 33 are more strongly constrained as compared the operators with jj = 11 or 22. This can be attributed to the fact that the former operators can mix strongly with the ∆F = (1, 0) operators C  • Since the operators with the indices 13jj, 23jj contribute even at tree-level to the LFV processes such as τ → P and τ → ρ for jj = 11 or 22, as a result these operators are in general more strongly constrained as compared to operators with 1333 or 2333 flavour indices. This is due to the reason that the latter operators contribute only at the 1-loop to such processes or to purely leptonic LFV modes. However, in certain cases like C (1) q 2333 , C u 1333 and C u 2333 , the 1-loop effects due to the top-Yukawa can dominate over the tree-level effects.
• Except for C d and C ed , the ∆F = (0, 0) operators involving ii33 for ii = 11, 22 or 33 have the strongest bounds, because they contribute to the Z and W boson couplings through the top-Yukawa.
• Finally, among ∆F = (0, 0) operators, C d iijj and C ed iijj are found to be loosely constrained. In most cases, the lower bound on NP scale lies around O(1TeV) or below. This is again due to the fact that these operators do not exhibit the operator mixing due to large top-Yukawa coupling. In appendix C, we also provide the best-fit values along with 1σ errors for the dimensionless Wilson coefficients.

Conclusions and outlook
The SMEFT provides a convenient framework for parameterizing the NP effects beyond the SM. It is well known that the semileptonic operators which violate the quark flavour lead to effects in the flavour violating decays of B and K mesons at low energy, which give rise to stringent constraints. In the view of current anomalies in the B-decays, the semileptonic operators are of great interest in general. However, it is important to probe the generic flavour structure of such operators. In particular, often the quark and lepton flavour conserving operators and the ones which violate only the lepton flavour are also generated in the NP models. To probe such operators it is important to know the type of observables to which these operators contribute.
In the present paper, we address this issue. We identify the low energy and the EW scale observables which can be used to probe a generic flavour structure of the semileptonic operators. However, in order to correctly predict the low energy behaviour of these operators, it is necessary to know the operator mixing pattern due to running from NP scale to the EW JHEP01(2022)107 scale and then below. To this end, by scrutinizing the ADMs due to the electroweak gauge as well as the Yukawas interactions one can find that such operators can mix with purely leptonic and ψ 2 φ 2 D-type operators in the SMEFT. The former operators can contribute to the LFV decays of leptons, whereas the latter ones in addition also give corrections to the gauge boson couplings at the EW scale.
Therefore, first we have identified the phenomenologically relevant terms in the ADMs and then taking into account the WET and SMEFT RG running effects, we identified a list of observables which can be used to constrain the semileptonic operators having a generic flavour structure. We show that, through SMEFT RG running effects at the 1-loop level, the semileptonic operators can contribute to a variety of observables such as EWP observables, flavour violating decays of B and K-mesons − involving neutral as well as charged current transitions, the LFV decays of leptons as well as the LFV Z-boson decays. The main findings of the present study can be summarized as follows: The semileptonic operators with flavour indices iijj for ii, jj = 11, 22 or 33 contribute to the EWP observables at the 1-loop level. We have identified two different types of contributions in this context (1) the operator mixing with ψ 2 φ 2 D operators through gauge interactions affect the W and Z boson vertices at the EW scale. Then, through the top-Yukawa interactions, there are additional contributions from the operators involving third generation in the quark current, i.e., C The W and Z boson vertices also receive corrections from C (3) q 1133 and C (3) q 2233 operators via shifts in the δg Z and δ sin 2 θ W . We present the relative impact of the running due to the gauge and top-Yukawas on the W/Z couplings at the EW scale due to semileptonic operators at Λ. In addition, the quark flavour conserving operators such as, C (1) q ijkl , C ed ijkl , C d ijkl , and C qe ijkl with ijkl = 1122, 1133, 2222, 2233 can be constrained by b → s + − processes through the back-rotation effect. We have shown that the Wilson coefficients C The semileptonic operators which violate the lepton flavour, while conserving the quark flavour can contribute to the LFV decays of τ and µ leptons. Some of the semileptonic operators contribute at tree-level to τ → P and τ → ρ processes. On the other hand, purely leptonic LFV decays such as τ → 3µ and µ → 3e etc., are generated only at the 1-loop level. Again this happens through operator mixing which depends on the gauge as well as Yukawa interactions. In this regard, we find that depending upon the flavour structures the Yukawas play an important in constraining such operators. We have also studied the relative impact of RG running due to various sources on the contributing WET operators at the low scale. These results are presented in table 13 for purely leptonic WET operators and in tables 14 for semileptonic WET operators.
Finally, using the latest measurements, we derived lower bounds on the cut-off scale Λ for each semileptonic operator under consideration. We observe that depending upon the flavour structure, the RG induced constraints can lead to sensitivities to very high NP scales varying between O(1 TeV) to O(100TeV). Finally, for a fixed value of Λ = 3 TeV, we also provide allowed ranges for the semileptonic Wilson coefficients.

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Note added. While finalizing our paper we notice ref. [87] on arXiv which presented a study of the LFV decays through Z-flavour violation in the context of future colliders. The discussion on the LFV decays partly overlaps with the present work. However, the aim of the current study is not limited to the LFV modes but is to analyze both flavour conserving as well as the flavour violating processes to which semileptonic operators can contribute at tree-level or through operator mixing.

A Tree-level shifts in the dim-4 gauge boson couplings
For completeness, in this section we collect formulae for the tree-level shifts due to SMEFT in the dim-4 Z and W boson couplings at the EW scale. For this purpose we closely follow ref. [49]. The shifts in the Z boson couplings δ(g Y X ) dir ij (see eq. (4.1) for definition) get tree-level contributions due to ψ 2 φ 2 D-type operators in the SMEFT. At the EW scale for the quarks these are given by Similarly, for the leptons we have The CKM elements appearing in the expressions for δ(g u L ) dir ij are needed due to our Warsawdown basis choice for the SMEFT operators. In addition, the shifts in the parameters g Z and sin θ W are given by and
The W boson couplings can be analogously parameterized as Since, we are interested to study the effects of only the semileptonic operators on the EWP observables, we have ignored additional corrections due to C φD and C φW B operators [49]. In this regard, we have checked that these operators can not be generated from semileptonic operators via operator mixing.

B RG induced shifts in the dim-4 and dim-6 operators
Depending upon the scale and the interactions involved, there are three types of RGEs, i.e., due to the gauge and Yukawa interactions in the SMEFT and due to QCD+QED interactions in the WET. In this section, we analyze the relative impact of three different types of RG runnings on the dim-4 and dim-6 operators which contribute to the EWP observables and the LFV processes at the low energy.

B.1 Dim-4 Z boson couplings
In Coupling(m W ) Gauge Couplings Yukawa Couplings 37   Table 11. The shifts in the W boson couplings i.e., δ(ε Y L ) ii × 10 5 to fermions at µ 91 GeV due to semileptonic operator C (3) q ii33 (Λ) = 1TeV −2 at Λ = 1TeV. Here ii = 11 or 22, and jj = 11, 22, 33 = ii. In the second and third column the RG running due to gauge only and gauge + Yukawa interactions is included, respectively. Note, only the non-zero entries are shown.

B.2 Dim-4 W boson couplings
In table 11 and 12 we show impact of RG running due to Yukawas on the W couplings to fermions at the EW scale.

B.3 Leptonic dim-6 WET operators
In (QED+QCD) running, (2) WET + SMEFT running due to gauge interactions, and (3) the full WET+ SMEFT gauge + Yukawa running. As we can see, except for C d 1333 and C ed 1333 , the top-Yukawa effects always play an important role for the operators involving third generation quarks.

B.4 Semileptonic dim-6 WET operators
In  Table 13. The impact of semileptonic operators on the low energy purely leptonic ∆F = (1, 0) Wilson coefficients (in 10 −9 TeV −2 units) at µ low = 2 GeV is shown. The second column refers to the values with only WET (QED+QCD) running, the third column refers to the WET + SMEFT running due to gauge interactions, and the fourth column refers to the full WET+ SMEFT gauge + Yukawa running. Here, the indices = 11, 22, or 33 and Λ = 1 TeV. The SMEFT Wilson coefficients are set to 1 (in TeV −2 units) at Λ.

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