Putting SMEFT Fits to Work

The Standard Model Effective Field Theory (SMEFT) provides a consistent framework for comparing precision measurements at the LHC to the Standard Model. The observation of statistically significant non-zero SMEFT coefficients would correspond to physics beyond the Standard Model (BSM) of some sort. A more difficult question to answer is what, if any, detailed information about the nature of the underlying high scale model can be obtained from these measurements. In this work, we consider the patterns of SMEFT operators present in five example models and discuss the assumptions inherent in using global fits to make BSM conclusions. We find that including renormalization group effects has a significant impact on the interpretation of the results. As a by-product of our study, we present an up-dated global fit to SMEFT coefficients in the Warsaw basis including some next-to-leading order QCD corrections in the SMEFT theory.


I. INTRODUCTION
With the Higgs discovery in hand and the Standard Model (SM) field content complete, one of the primary goals of the LHC is to make precise measurements of SM processes, with the hope of testing its limitations. As the search for new particles has been unsuccessful as yet, much attention has shifted towards precision measurements of Higgs processes. In this direction, the SMEFT (SM effective field theory) framework becomes very useful. In the SMEFT, deviations from the SM are parameterized by a tower of SU (3)×SU (2) L ×U (1) Y invariant higher dimension operators containing only SM fields. The SMEFT is useful because it provides a consistent, gauge-invariant theoretical interpretation of the data, in which higher order corrections can be included, and connects Higgs data with electroweak precision observables at the Z and W poles, diboson data, and top quark measurements.
There have been numerous global fits to LHC and LEP data, yielding limits on the allowed values of the SMEFT coefficients [1][2][3][4][5], but thus far all of these fits are totally consistent with the Standard Model, further demonstrating the robustness of the SM.
Ultimately, however, one hopes that the SMEFT is parameterizing some high scale physics beyond the Standard Model (BSM), and it is of interest to understand how these global fits should be interpreted in this context. The goal of this work is to consider just such an interpretation in detail for a set of simple benchmark models. We consider a sample set of UV complete models where the BSM physics contains particles at mass scales above the weak scale, Λ M Z . Each model makes a prediction for the SMEFT coefficients at the high scale, C i (Λ), and typically only a small subset of dimension-6 operators are generated [6,7]. As we will see, all of our models predict particular relationships between the different coefficients generated, and we will explore the differences in fitting with these particular patterns as opposed to general values of the coefficients. We will mostly restrict ourselves to tree-level matching between the BSM physics and the SMEFT at the scale Λ, but we will also consider the renormalization group running to evolve the coefficients at leading logarithm to the weak scale [8,9], where the predictions can be compared with fits to the data. Since many of our benchmark models generate operators that are well constrained by electroweak precision observables, including these leading logarithmic effects will significantly change the interpretations of the fits. These considerations are an important first step in understanding what we can learn about possible UV complete models, and, if a deviation from the Standard Model is observed, how we can discriminate between them. This goal is sometimes referred to as the "Higgs Inverse Problem".
Understanding the sensitivity to BSM physics through the extraction of SMEFT coefficients is also useful for comparing the reach of future accelerators [10] and our calculations are part of an extended effort to understand the complementarity of the direct observation of new particles with precision measurements [11]. In addition to Higgs signal strength data, we include theory predictions with NLO QCD corrections for V V and V H (V = W, Z) production [12], and the leading logarithmic NLO QCD and electroweak corrections to Z and W pole observables computed in the full SMEFT [13]. As a by-product of our study, we obtain an update of the global fit to SMEFT coefficients in the Warsaw basis.
We begin by describing how we match our benchmark models to the SMEFT, including the effects of renormalization group evolution (RGE) of the Wilson coefficients down to the weak scale. We then summarize each of the models in turn in Section II B. The operators generated in each model are summarized in Section II C. We then perform a series of fits customized for each model in Section III, and discuss how the SMEFT fit results can differ depending on the correlations present in the underlying high scale model. Here we will further emphasize the role of the renormalization group evolution of the coefficients in the interpretation of SMEFT fit results. Finally, we conclude with our updated global fit in Section III C and a discussion of future directions in this type of study in Section IV.

II. MATCHING MODELS TO THE SMEFT
In this section, we lay out our benchmark models, and tabulate the relevant SMEFT coefficients obtained in the decoupling limit of these models. The models are chosen to be simple but representative BSM models with new particles at the ∼ few TeV scale with only a small set of unknown parameters. The full set of dimension-6 operators that we will consider, ignoring flavor, is given in Table I.

A. Matching Procedure and Renormalization Group
Here we lay out our benchmark models, and tabulate the relevant SMEFT coefficients obtained in the decoupling limits of the models. The models are chosen to be well studied BSM models with new particles at the UV scale and are quite simple models, involving a small set of unknown parameters. We assume that the new particles are at a high mass scale and integrate them out of the theory using the equations of motion at tree level, although for future studies it would be of interest to perform the matching at one-loop, since one-loop matching typically generates a much richer spectrum of SMEFT operators than does the tree level matching [7,14,15]. This procedure generates predictions for SMEFT coefficients at the mass scale Λ corresponding to the new physics, where we include the dimension-6 operators in the Warsaw basis [16,17]. The operators consist of all of the SU (3) × SU (2) L × U (1) Y operators that can be constructed from SM fields. Since we assume Λ M Z , we only consider the dimension-6 operators, O 6 i . Some of the models we consider generate effects only for third generation quarks, so we do not always assume flavor universality in the quark sector. The importance of the assumptions about flavor in the results of the global fits has been emphasized in Refs. [2,18].
We fit data from Higgs processes, diboson W W and W Z production, and electroweak precision observables (EWPOs), including the W mass and width, to the patterns of SMEFT coefficients generated in our examples. For completeness, we define the operators appearing in this work in Table I and we neglect flavor indices in this table, although we will include them in some of the models. We define H to be the SU (2) L doublet Higgs field with neutral component h+v √ 2 and, in terms of the first generation, q T L = (u L , d L ), l T L = (ν L , e L ). Our notation follows that of Ref. [16].
At tree level, the Z and W pole observables depend on, Hl , and the EWPOs are sensitive to eight combinations of these operators [13,[19][20][21], (at NLO they are sensitive to a combination of 10 operators). We also include the 2− loop contribution to M W generated by O H [22,23].
We consider tree level contributions from the following operators to Higgs data, We also include the loop contributions to Higgs production and decay from O H [24][25][26].
Finally, the diboson W W and W Z data depend on 7 effective couplings 1 , which involve the operators • Models with high scale scalar resonances: We consider a real scalar singlet model, both with and without a Z 2 symmetry, and a 2 Higgs doublet model (2HDM) in the decoupling limit.
• Models with new particles in loops: We consider two models with vector-like quarks (VLQs): One with a color triplet fermion with charge Q = 2 3 , and one with a color triplet, SU (2) L doublet of quarks with charge Q = ( 2 3 , − 1 3 ). We also briefly compare the results of the models with vector-like quarks with a model containing a heavy color triplet scalar.
There have been extensive studies in the literature computing SMEFT coefficients in these models. We summarize the models we consider below and the reader is referred to the original literature for further details.
The fits are performed in two different manners. In the first approach, we match the coefficients at the UV scale Λ to the model predictions and perform the global fit. These fits are only sensitive to the ratios C i Λ 2 and give no independent information about the UV scale. We always make the identification that Λ is the mass of the heavy particle that has been integrated out. In the second set of fits, we match the coefficients to the model predictions at Λ and then use the renormalization group to evolve the coefficients to M Z before performing the fits. The coefficients at the weak scale are then, A complete set of the relevant anomalous dimensions in the Warsaw basis is in Refs. [28][29][30]. In many cases, the RGE has a dramatic effect on the interpretation of the fits.
Hq are strongly constrained by Higgstrahlung data, while the Z → bb data constrain these operators with 3 rd generation quarks. In models where these coefficients are generated by RGE (even when they are not present at the matching scale), the constraints and the interpretations change dramatically [9].
To illustrate the importance of including the RGE when fitting to UV complete models, we consider the strongly constrained operators, Hq and O Hq [12]. O HD is not generated at tree level in any of the models we consider, while O H arises at tree level in the singlet model. Including only contributions from terms that are generated at tree level in the models we consider (see Table II), where C Ht , C Hb , C Htb , C Hτ , (C HQ ) 33 occur at tree level in the T VLQ model, and RGE generates O H at the weak scale which is strongly constrained. On the other hand, the operators generated at tree level in the 2HDM do not contribute to the RG evolution of C HD or C H and we will see that RGE has a relatively minor effect on the interpretation of this model. In our numerical results, we include the complete RGE of all the operators that contribute to our fits.

a. Singlet Scalars
One of the simplest extensions to the Standard Model is obtained by adding an additional scalar that is a singlet under the SM gauge group. The scalar potential can be constructed both with and without a Z 2 symmetry. The case without a Z 2 symmetry is particularly interesting because it can accommodate a first order electroweak phase transition for some values of the parameters [31,32]. Using the classical equations of motion [7,14,33,34], the heavy scalar can be integrated out at tree level, generating the where θ is the mixing angle between the SM-like Higgs boson, h, and the new heavy scalar and κ and m are Lagrangian parameters that are limited by the requirement that the electroweak minimum be the lowest minimum of the potential [35]. In this model, the SM-like Higgs couplings to SM particles are uniformly suppressed by a factor of cos θ and for the case with a Z 2 symmetry, there is a cancellation implying C H = 0 [14,34].
Details of the model are in Appendix A 1.

b. A Second Higgs Doublet
The 2 Higgs doublet model (2HDM) has been extensively studied in the literature, and in the limit that the new Higgs bosons are much heavier than the SM-like Higgs bosons, the Higgs couplings approach those of the SM [36]. This is the alignment limit, cos(β − α) → 0. In the exact alignment limit, SMEFT operators are not generated at tree level, but first appear at 1-loop. Away from the alignment limit, cos(β − α) 0, tree level contributions to the Higgs-Yukawa couplings are generated, along with a correction to the Higgs tri-linear coupling. To linear order in cos(β − α), the SMEFT coefficients that affect Higgs couplings to fermions, f , are [6,14,37,38], where where M is the common mass of the heavy decoupled scalars near the alignment limit.
Our results are only valid near the alignment limit, where cos(β − α) 1. Further details of the model are found in Appendix A 2.
c. Colored Extensions of the SM: an SU (3) Triplet, SU (2) L Singlet Fermion We consider a charge Q = 2 3 color triplet, SU (2) L singlet fermion, T , and call this the T VLQ (vector-like quark) model and assume that this new quark only couples to the top quark, but not to the lighter quarks. The model is parameterized by 3 parameters: m t and M T are the masses of the physical top and new heavy charge Q = 2 3 fermion respectively, and s t L is the sine of a mixing angle that defines the mixing between the left-handed charge 2 3 quarks. Integrating out the heavy fermion generates the SMEFT coefficients involving the third generation quarks only [6, 39- where O (1) , and the scale Λ is identified with M T . The corresponding coefficients for the first 2 generations are zero in this model.
Although we perform the matching at tree level, we also include O HG since it could potentially make a significant contribution to Higgs production through gluon fusion, where  Table I in terms of 3 rd generation fermions. In the decoupling limit, where s t R and s b R define the mixing between the top and bottom quarks with T and B respectively in the right-handed sector. At one-loop, O HG is also generated, where and mass, m s . At tree level, four fermion operators that do not contribute to our global fit are generated, but no dimension-6 EFT operators arise. At one loop, the colored scalar where κ is the portal coupling, (s s * ) H † H , and is defined in Appendix A 3 c. This is an example of a model where the only effect on single Higgs production is to rescale the rate and the SMEFT formalism is not necessary. The indirect consequences of the colored scalar and the corresponding SMEFT effects can be searched for in Higgs plus jet or double Higgs production [42,43].

C. Summary of Models
In Table II, we summarize the coefficients generated by our benchmark models as described in the previous sections, expressing the results in terms of the physical parameters of these models when possible. The scale Λ is consistently identified with the mass of the heavy particle in the model. More precise definitions of these parameters are given in the Appendices. We also list C HG which is generated at 1-loop in some models and give numerical values for C HG for heavy masses of 1 TeV. In all cases, we assume the decoupling limit of the models and the parameters are defined in the appendices.
Empty spaces correspond to operators not generated at tree level.

A. Methodology
We perform a series of fits to Higgs, diboson, and EWPO data with prior assumptions about the relationships between SMEFT coefficients that are motivated by our example models. We take as non-zero only those coefficients generated in a particular model and examine how that choice changes the fits and the interpretations of the fit results. The underlying goal is to see how the fits can potentially constrain the high scale models.
The EWPO fits use the data given in Table III of [13]. This occurs at 2-loops in the SM and is included using the results from Ref. [22].
The diboson (W W and W Z) and Higgstrahlung (W H and ZH) fits use the data from Table IV of [12] and we fit to linear order in the SMEFT coefficients and to NLO QCD. As shown in [12], the diboson and Higgstrahlung fits are extremely sensitive to whether the fit is performed at linear or quadratic order, with the W Z contribution being particularly sensitive to the inclusion of NLO QCD effects. Finally, the Higgs predictions use the 80 fb −1 13 TeV LHC data from ATLAS [44] and the 36 − 137 fb −1 13 TeV LHC data from CMS [45], along with the 8 TeV data given in Tables 2 and 3 of [3]. The contribution to Higgs production and decay from O H occurs at loop order and is included following the prescription of Ref. [24]. The identification of the observables with the SMEFT predictions is made using tree-level calculations (except for C HG and C H ) and compared with the results of Refs. [3,46] and reasonable agreement is found.
In the following subsections, we present results for our test models and discuss the use of the global EFT fits for extracting information about the underlying models. Hq at the weak scale, yielding shifts to the EWPOs proportional to log(Λ/M Z ). Numerically, the most important of these operators is O HD , which generates the T oblique parameter 2 . The limits from EWPOs are shown in Fig. 1 LHS as a purple dashed contour. As anticipated, the limit on C H is very weak [22,23].
The constraint from the combination of LHC Higgs and diboson data is shown as a solid blue line, and we see that measurements of the Higgs couplings provide a bound on C H of the same order as the EWPO fit. The combination of LHC data and EWPOs is shown as the solid green curve. In single Higgs data, the operators O H and O H do not generate any momentum dependence, but O H produces momentum-dependent effects in di-Higgs data which could potentially be of use for the discrimination between models [5,37,48].
While we consider first general values of C H and C H , the singlet model generates only a subset of these coefficients. In the Z 2 symmetric case, C H = 0, and C H is always less than zero (see Appendix A 1), so this class of models generates a vertical ray emanating from the origin, shown as a magenta curve in Fig. 1  With the other coefficients held fixed, the SMEFT limits can be translated into limits on the physical parameter, sin θ (defined in Appendix A 1). In the Z 2 symmetric case, it is apparent from the LHS that sin θ 0.25 for Λ = 1 TeV. More generally, we can interpret limits on the coefficients as limits on the largest allowed mixing angle as a function of the scale, Λ. For the Z 2 symmetric case, v 2 C H /Λ 2 = − 1 2 tan 2 θ, so the only scale dependence is the weak logarithmic dependence from RG evolution, and this is shown on the RHS of Fig. 1. The EWPO limits obtained by fitting the SMEFT coefficients are 2 The S parameter depends on C HW B which is not generated from RGE in the singlet model. As demonstrated in Ref. [47], in the Warsaw basis there are additional contributions from 4−fermion operators at dimension-6 that are needed to obtain a basis independent result for S. It is also interesting to note from Ref. [6], that only a very small class of models generate O HW B from dimension-6 operators and none of the models considered here fall into this class. quite similar to those extracted directly from the EWPOs in the full singlet model [49,50], demonstrating that in this case, the global fit does indeed constrain the UV complete model quite accurately, once the RGE is taken into account. While C HD also appears in the LHC Higgs and diboson data, the bounds on C H arise at tree level, and there are thus different effects in the scaling on the RHS of Fig. 1.

2HDM
The decoupling limit of the 2HDM has been extensively discussed in the literature [36,51], but here we revisit the question of what information is in the SMEFT fits in the limit where the new scalars are too heavy to be observed. In Appendix A 2, we see that the 2HDM generates O H , O uH , O dH and O eH at tree-level, assuming a small deviation from the alignment limit, | cos(β − α) | 1. In general, these operators can have arbitrary coefficients for each generation, but to avoid flavor-changing neutral currents, they are usually assumed to be proportional to the SM Yukawa matrices, so -in the limit that only the third generation Yukawas are non-zero -we can consider only the 33 components None of these operators contribute to the EWPOs at tree level, nor do they generate any of the operators in Eq. 6 at leading logarithm, so we consider only LHC constraints.
In Fig. 2, we consider the limits in the C tH -C bH plane (LHS) and the C H -C bH plane The fit to SMEFT coefficients is re-interpreted in terms of the parameters of the 2HDM in Fig. 3, for both the Type I and Type II models near the alignment limit with M = 1 TeV. These fits show good agreement with the fits in the full, UV complete 2HDM [38]. In the Type I 2HDM, the effects of the RGE reduce the value of C tH when scaling from M Z to Λ as observed in Fig. 2 and are manifest in the difference between the solid and dashed line in Fig. 3 as well.

Heavy Colored particles in Loops: T VLQ
The case of a heavy Q = 2 3 vector-like quark coupling only to the third generation leads to the operators O tH , O  We first consider the constraints on only C Hq 33 and C Hq 33 from EWPOs. The dominant effect at tree-level is in the Z → bb decay, which depends only on the combination On the RHS, we consider the limits from Higgs and diboson data. As is apparent from Table II, in the context of the T VLQ, the Wilson coefficients that are generated all come with a fixed pattern, since there is only one independent physical parameter. To illustrate the importance of this, we thus fix all the relationships between the coefficients except for C HG and plot in the C Hq 33 = −C tH /Y t vs. C HG plane. The resulting limits are highly correlated, and we see that including the effects of the RGE on the fit with only Higgs data (going from dotted black to dashed red lines) has a mild effect on the fits. Most of the limit in the vertical direction, however, comes from including the diboson data, and this dependence arises because the RGE generates O H , which is well constrained by V H and V V fits [12,56].   In Fig. 7 LHS, we directly compare the EWPO and Higgs plus diboson constraints by considering the SMEFT fits in the C Ht vs. C Hb plane. To include the effects of all the operators, we again set C tH = −Y t C Ht , and similarly set C bH = −Y b C Hb . We also include O HG with C HG = 0.65 α s /8π C Hb , as implied by Eq. 14 for M T = M B = 1 TeV. We see that, even including all of these correlations, the EWPO constraint still sets a superior bound to the Higgs plus diboson data.
In contrast with the T VLQ, the SU (2) L doublet VLQ model has two independent parameters in the decoupling limit, the two mixing angles sin θ b R ≡ s b R and sin θ t R ≡ s t R , or equivalently, s b R and the mass splitting, δM T B = M T −M B (see Appendix A 3 b for details). In the limit δM T B = 0, the mixing angles are identical, and C Hb = C Ht . This is indicated by the magenta line in Fig. 7 LHS. Allowing for a nonzero mass splitting, however, shifts this relation, as can be seen by the yellow line in Fig. 7 LHS for δM T B = 10 GeV. It is  Table II.
obvious from this shift that the EWPOs set a strong constraint on the mass splitting for fixed s b R via the RGE induced C HD . This is simply a manifestation of the strong constraint on custodial symmetry violation, which we comment more on in Appendix A 3 b. The behavior in Fig. 7 also illustrates that varying the two mixing angles sweeps out a region in the C Ht vs. C Hb plane, but that there is still a very tight relationship between all five operators generated by the model, which changes the interpretation of the global fits in this context significantly.
On the RHS of Fig. 7, we reinterpret the SMEFT bounds in the δM T B vs. s b R plane, showing both the EWPO constraint (purple) and the Higgs plus diboson constraint (blue).
We show the results for both M T = 1 TeV (solid) and M T = 5 TeV (dashed), and note that the logarithmic dependences on δM T B and s b R have opposite signs.

C. Global Fit to SMEFT Coefficients
As a by-product of our study, we present an updated global fit to the 19 SMEFT coefficients considered here in the Warsaw basis. In comparison to Ref. [3], this fit includes The bounds on operators involving fermions assume universal coefficients, except for C bH , C tH , and C τ H , which modify only the third-generation Yukawa couplings. operators. We assume universal coefficients for operators involving fermions, except for C bH , C tH , and C τ H , which modify only the third-generation Yukawa couplings.
higher integrated luminosity data from ATLAS [44] and CMS [45], as well as the NLO QCD corrections to V V and V H production with the full distributions as in Ref. [12]. The results are shown in Fig. 8 with each coefficient treated individually and in Fig. 9 when marginalizing over all the couplings. Note that, in contrast to many of our particular model fits, here we assume universal couplings to the quark operators C (1) Hq , C Hu and C Hd . This results in very different results, as the constraints now have a significant contribution from diboson and Higgstrahlung production with first-generation quarks.
Numerical values are given in Appendix B.

IV. DISCUSSION
A major goal of precision measurements at the LHC is to uncover hints of new physics through patterns of deviations from the SM. In this work, we examine how fits to SMEFT coefficients that are predicated on patterns of coefficients generated in different UV complete models give information about the high scale physics. Of particular interest to us are the assumptions made when forming inferences about the source of new physics from SMEFT fits.
Only two of our models, the Z 2 non-symmetric singlet model and the 2HDM generate a shift in the Higgs tri-linear coupling C H . In the singlet model, this shift is correlated with a non-zero C H term that can be observed in V V and V H production. In the 2HDM, the non-zero C H is directly proportional to the C f H interaction and a weak limit on C H is obtained. The 2HDM and the (T B) VLQ models generate C f H terms that can be directly measured in Higgs production at the LHC. The (T B) model also generates C Hf the fits in these cases.
In Fig. 10, we summarize our SMEFT results in terms of the physical parameters of the models and show the maximum allowed mixing angle from the global fits in each model as a function of scale. We note that these are the limits in the SMEFT where the heavy particles have been integrated out of the UV complete model. The fits are sensitive to the ratios C i /Λ 2 , modulo the logarithmic dependence from the RG running.
Our study is just the beginning of an understanding of the discrimination between UV theories from SMEFT fits [57]. Follow-up work could include information from top physics [58], a consideration of the importance of the quadratic versus linear SMEFT approximation [12], and complete 1-loop matching. It is of considerable interest to expand our study by examining further concrete example models. The most general scalar potential involving a real scalar singlet,S, and the SM SU (2) L doublet, H, is , The parameters can be redefined such that S = 0. After spontaneous symmetry break- with the physical masses, m h and M . We assume M m h . The heavy scalar can be integrated out [7,14,33,34], generating the 2 operators, O H and O H with coefficients, In terms of the physical parameters of the theory (m h , M, sin θ), where in the last lines, we take the M → ∞ limit. Ref. [37] has pointed out that in some cases, an improved agreement between the exact (singlet model) UV theory and the SMEFT can be obtained by retaining the dependence on m h in Eq. A4. The Lagrangian parameters µ and κ are limited by the requirement that the minimum of the potential be the electroweak vacuum, (see Fig. 1 of Ref. [35]), and so fixing 0 < θ < π 2 , | C H | < ∼ | 2 tan θ + 1 | C H . In the case where there is a Z 2 symmetry, the potential of Eq. A1 has A = µ = 0. In this case, there is a cancellation in Eq. A3 implying C H = 0 and the singlet vev can no longer be fixed to 0.

A Second Higgs Doublet
For the 2 Higgs doublet model, we work in the Higgs basis, where the doublets have been rotated such that only the SM-like doublet, H 1 , gets a VEV, v. In this framework, the components of the H 2 doublet can be taken heavy and we work in the decoupling limit where the 2HDM can be matched to the SMEFT coefficients [14,37,38,59]. The scalar potential is [60], The Yukawa terms are, and the parameters η f depend on the type of 2HDM and are given in Table III. In general, the Yukawa couplings are 3 × 3 matrices, but we will always take them diagonal when considering the 2HDM.
The 2HDM also generates 4-fermi interactions that do not contribute to our tree-level study. In the decoupling limit, Keeping only 3 rd generation fermion masses non-zero, Note that we need cos(β − α) M 2 v 2 to be small for decoupling [36].

Colored Extensions of the SM
Finally, we consider extending the Standard Model with new colored fields. In particular, we will consider heavy vector-like quarks, either a singlet or doublet under SU (2) L , and colored triplet scalars.

a. SU(3) Triplet SU (2) L Singlet Fermion
The T VLQ model has a charge 2 3 color triplet, SU (2) L singlet fermion. The particles in the top sector are, where ψ L , T 1 R are the SM-like left handed quark doublet and right-handed charge 2 3 quark and T 2 is the new vector-like quark. The relevant portion of the Lagrangian is, which can be expressed in terms of the physical parameters, m t , M T , sin θ t L ≡ s t L . After the mixing, the physical fermions are, and we define q 3 L = (t L , b L ) T to be the physical third generation fermion doublet. (Note that the mixing in the right-handed quark sector can be rotated away, so there is only one mixing angle in this model.) In order to obtain decoupling, the Yukawa interactions must be much smaller than the Dirac mass term, λ 2 v, λ 3 v λ 5 . In this limit [40], and Hence, decoupling requires (s t T as seen in Eq. A13.
The SMEFT coefficients that are generated at tree level are, It is clear that there is only 1 independent SMEFT coefficient in this model at tree level.
The T VLQ model generates a contribution to O HD through the running of C (1) Hq , Neglecting the b mass, matching at Λ = M T , , and evolving to m t , we find reproducing the logarithmic contribution of the UV complete T VLQ model. The complete model, however, has the s t L → 0, M T → ∞ limit [40,61], and we note that the SMEFT cannot reproduce the (numerically significant) (s t L ) 2 M 2 T /m 2 t term of the UV complete model. where and τ = 4m 2 /m 2 h , where m = m t , M T and F 1/2 → 2 3 in the m t → ∞ limit. In the high energy limit, the contribution of C HG to Higgs production is highly suppressed by the cancellation between the top loop and the T loop and there is only a very slight We next consider a model with an SU (2) L doublet and color triplet pair of vector-like fermions. We term this the (T B) VLQ model [6,40,62]. The third generation quarks in the (T, B) model are, corresponding to the scalar potential, (A term ψ L χ R can be rotated away by a redefinition of the fields.) Diagonalizing the mass matrices requires 4 angles in the left-and right-handed t − T and b − B sectors, θ t L , θ t R , θ b L , θ b R . Since there are 5 terms in the Lagrangian, there are 5 independent parameters which we take to be the physical masses and one mixing angle, For small mixing angles, The mixing in the right-handed top sector is determined from that in the right-handed bottom sector, Only C Ht and C Hb are independent and are related by Eq. A27 to the heavy masses.
We can consistently take Λ = M T or Λ = M B . From the measurement of Z → bb, the right-handed coupling to the b is small, s b R < .115 [64], corresponding to v 2 Λ 2 C Hb < .013, independent of M B . It is therefore consistent to consider the small s b R limit. Similarly to the T VLQ model, O HD is generated from the running of C Hu , C Hd and C Hud ,Ċ Neglecting the b mass and considering small s b R and | δM T B | /M T , Even in the s b R → 0 limit, the running of O HD yields a contribution to ∆T proportional to the mass splitting, giving reproducing the logarithm of the UV complete model [40,61]. Comparing with Eq. 57 of [40] we see that ∆T EF T /∆T (full) ∼ 1.4, implying that the limits obtained in the EFT will be more stringent than the actual limits in the full theory.
At one-loop, O HG is generated, where M T = M B = 1 TeV in the last equation. Note that the cancellation between the SM top quark and the T VLQ contribution that was observed in the T VLQ model is weakened due to the presence of two heavy VLQs.

c. SU(3) Triplet Scalar
Finally, we consider a model with a complex color triplet scalar, s, with charge Q = 2 3 . It is interesting to see how the predictions differ from those of the T VLQ described above.
The relevant interaction terms are, qq that do not contribute to our study, but enter at tree level in Drell-Yan and di-jet production at the LHC. At one loop, the colored scalar generates O HG , where F 0 (τ s ) = τ s 1 − τ s f (τ s ) , τ s = 4m 2 s /m 2 h , and f (τ ) is defined in Eq. A20.