The Impact of Dimension-8 SMEFT Contributions: A Case Study

The use of the SMEFT Lagrangian to quantify possible Beyond the Standard Model (BSM) effects is standard in LHC and future collider studies. One of the usual assumptions is to truncate the expansion with the dimension-$6$ operators. The numerical impact of the next terms in the series, the dimension-$8$ operators, is unknown in general. We consider a specific BSM model containing a charge-${2 3}$ heavy vector-like quark and compute the operators generated at dimension-$8$. The numerical effects of these operators are studied for the $t\bar{t}h$ process, where they contribute at tree level and we find effects at the ${\cal{O}}(0.5-2\%)$ level for allowed values of the parameters.


I. INTRODUCTION
One of the goals of the HL-LHC running is a precision physics program that enables a detailed comparison of theoretical and experimental predictions. Lacking the experimental discovery of any new particles, the tool of choice is the Standard Model Effective Field Theory (SMEFT) which assumes that the gauge symmetries and particles of the Standard Model provide an approximate description of weak scale physics [1]. Deviations from the Standard Model (SM) predictions are parameterized in terms of an infinite tower of higher dimension operators, where Λ is a high energy scale where some unknown UV complete model is presumed to exist. All of the new physics information resides in the coefficient functions, C i , which can be extracted from experimental data.
The SMEFT amplitude for a tree level scattering process can be written schematically as, where A SM , A It is immediately apparent that the squares of the dimension-6 contributions are formally of the same power counting in 1/Λ 4 as the interference of the dimension-8 terms with the SM result unless assumptions are made about the relative sizes of the contributions. If the process being studied is extremely well constrained (as is the case for the electroweak precision observables), it may be sufficient to include only the 1/Λ 2 contributions, as the 1/Λ 4 terms are negligible in this case [2][3][4]. Alternatively, the SMEFT could result from 1 We neglect baryon and lepton number violating operators. a strongly interacting theory at the UV scale where the A SM A (8) /Λ 4 terms are suppressed relative to the |A (6) | 2 /Λ 4 contributions [5,6]. There are, however, scenarios where the inclusion of the dimension-8 terms may be critical in order to obtain reliable results due to cancellations of the |A (6) | 2 /Λ 4 terms in specific kinematic regimes [7]. There are also scenarios where new physics effects first arise at dimension-8 such as the ZZγ coupling [8,9]. Furthermore, in weakly coupled theories, there is generically no reason to expect the dimension-8 contributions to be suppressed.
In practice, the SMEFT series is usually terminated at dimension-6 and the amplitude is that is assumed to exist at the UV scale. Such particles occur in little Higgs models [23][24][25] and in many composite Higgs models [26][27][28][29], and represent a highly motivated scenario.
Within the context of this model, the coefficients of the dimension-6 and dimension-8 operators can be calculated using the covariant derivative expansion [30,31] and matched to the SMEFT. This allows for a detailed numerical analysis of the various approximations frequently used when computing observables in the SMEFT. We consider tth associated production in the SMEFT limit of the TVLQ and are able to concretely determine the numerical relevance of the dimension-8 contributions to this process at tree level. The SM rate for tth production at the LHC is well known at NLO QCD [32][33][34][35].
In Sec. II, we review the construction of the TVLQ model and we pay particular attention to the decoupling properties of the TVLQ model. The tree-level matching to the SMEFT at dimension-8 is given in Sec. III. Phenomenological results for tth at dimension-8 in the SMEFT limit of the TVLQ are presented in Sec. IV, where we emphasize the importance of including the top decay products for SMEFT studies. We conclude with a discussion of the impact of our results in Sec. V. Appendices include a short summary of the relevant dimension-8 interactions and a brief discussion of one-loop matching in the TVLQ model.

II. THE TVLQ MODEL
We consider an extension of the Standard Model with one additional vector-like, charge-2/3 quark, denoted T 2 L , T 2 R , that can mix with the Standard Model-like top quark, T 1 L , T 1 R and call this the TVLQ model. This model has been extensively studied in the literature [36][37][38][39][40][41][42][43][44][45][46][47][48] and we briefly summarize the salient points. The SM-like third generation chiral fermions are, with the usual Higgs Yukawa couplings: Note that we will distinguish between the SM-like Yukawa couplings, and m t the physical quark masses, and the Yukawa couplings derived in the SMEFT construction of Sec. III. As The most general fermion mass terms for the charge-2/3 quarks are: Since T 2 R , T 1 R have identical quantum numbers, the m 12 term can be set to zero by a redefinition of the fields. The charge-2/3 sector is thus described by three parameters: λ t , λ T and m T .
The physical fields, t and T , with masses m t and M T , are found by diagonalizing the mass matrix with two unitary matrices, and we use the shorthand c L,R ≡ cos θ L,R , and s L,R ≡ sin θ L,R .
Useful relationships between the Lagrangian and physical parameters are, The following relationships follow from Eq. (8), with x ≡ m 2 t /M 2 T . From Eq. (9), it is clear that for fixed s L , λ T will become non-perturbative at large M T . In Fig. 1 (LHS) , we show the upper limit on s L from the requirement that λ T < ∼ 4π, along with the unitarity limit from T T → T T of s 2

L <
∼ 550 GeV/M T [49]. The M T → ∞ limit therefore requires s L → 0 for a weakly interacting theory. We also observe that the expansions in 1/M 2 T and 1/m 2 T have different counting in inverse mass dimensions for fixed s L , as is demonstrated in Fig. 1 (RHS). The ratio m T /M T quickly goes to its asymptotic limit as M T → ∞ and for s L ∼ 0.2, the ratio approaches ∼ 0.98, for example. The relations of Eq. (9) can be inverted [36]: In our phenomenological studies we will switch between Lagrangian parameters and the physical parameters to illustrate various points. We remind the reader that the physical masses are m t and M T with m t << M T and that m T is the Lagrangian parameter.
The oblique parameters place stringent limits on the parameters of the TVLQ. In Fig. 2, we update the results of Ref. [37], include the global fit results of Ref. [21] and compare with the direct search limits from TT pair production [50,51] (which are independent of s L ). We also show a comparison of current searches with projections for HL-LHC and FCC-hh and note that the HL-LHC will be sensitive to M T ∼ 1.

III. MATCHING TO SMEFT AT DIMENSION-8
In this section, we consider the M T → ∞ limit of the TVLQ model and perform the treelevel matching to the SMEFT, extending the dimension-6 results [46,47,55] to dimension-8. Since the full UV model depends on only three unknown parameters, it is particularly simple. We use the covariant derivative expansion [30,31] to integrate the heavy T out of the theory and generate the effective operators at dimension-6 and dimension-8. The resulting Lagrangian involving the SM-like top quark, t, is, where, where (2) generators τ a , and SU (3) generators T a . The dimension-6 term, L 6 , generates a non-standard normalization for the top quark kinetic energy term after electroweak symmetry breaking and the expansion of the Higgs field around its vev, so we make the gauge invariant field redefinition [56,57], where i, j are SU (2) indices. This brings the top quark kinetic energy into the canonical form.
We simplify the dimension-8 operator of Eq. (13) to extract the term contributing to the top quark Yukawa interaction, where the complete expression for δL 8b can be found in the supplemental material. The contribution to δL 8b that is proportional to the strong coupling, g s , is given in Appendix A and the momentum dependence of the dimension-8 operators is clearly seen.
The complete SMEFT Lagrangian generated from the TVLQ model to dimension-8 in-volving the top quark written in terms of the Lagrangian parameters is, We note that changing the input parameters from (λ t , λ T , m T ) to (m t , M T , s L ) using Eq. (8) re-arranges the counting in terms of inverse powers of the heavy mass [61]. Ref. [61] argues that replacing the Lagrangian mass, m T , with the physical mass, M T , improves the agreement between the SMEFT predications and those of the corresponding UV complete model in many cases. A similiar effect is found in the EFT limit of the 2HDM [62,63].
The terms contributing to the SMEFT relationship between the top mass and Higgs top Yukawa coupling are, with, It is interesting to study the behaviors of C (6) tH and C (8) quH 5 using the relationships of Eq. (9) and expanding in powers of 1/m T keeping the top quark mass fixed to its physical value. 3 Note that keeping the top quark mass fixed rearranges the counting, as does alternatively 3 Note that we are free to take a combination of three Lagrangian and/or physical parameters as inputs.
using s L and M T as inputs. To O(1/m 4 T ), The naive scalings, C 6 ∼ 1/m 2 T and C 8 ∼ 1/m 4 T are modified by terms of O(s 2 L x) when using the physical parameters.
Expanding Eq. (18) to linear order in the Higgs field, we define the top Yukawa, Y (8) t , as usual where the superscript 8 denotes the inclusion of the dimension-8 contributions. We initially fix m t (the physical top quark mass), λ T and m T , Retaining only the dimension-6 terms in the Lagrangian, In Eqs. (22) and (23), the SM is recovered in the λ 2 T /m 2 T → 0 limit, which corresponds to the s L → 0 limit. The choice to use m t as an input introduces terms due to the interdependence of the parameters.
In the small s L limit, We see that the SM limit is only recovered in the s L → 0 limit, consistent with the decoupling discussion in the previous section. Fig. 3 shows the effect of including the dimension-8 terms on the top quark Yukawa coupling and we see that it is typically less than a few percent for 500 GeV < m T < 1 TeV.

IV. PHENOMENOLOGY
We are now in a position to investigate the numerical effects of including the dimension-8 terms in the SMEFT analysis of the TVLQ and in the comparison between SMEFT and the UV complete model. As an example of the possible impact of the dimension-8 contributions, we consider tth production at the 13 TeV LHC 4 . In addition to the SM cross section, dσ SM , 4 The TVLQ model contributes to gluon fusion at one-loop, but a consistent inclusion of the dimension-8 contributions would require the double insertion of the dimension-6 contributions. The contribution to gg → h from the TVLQ is suppressed by s 2 L and is numerically small [64].
we consider various SMEFT expansions: In particular, dσ 6 and dσ 8 are of the same order in 1/Λ 4 and the difference between the two is a measure of the importance of the dimension-8 terms. In our numerical studies, we will always take Λ = m T .
The rescaling of the top Yukawa coupling at dimension-8 will give only a small difference from the dimension-6 result as demonstrated in Fig. 3. However, the dimension-8 terms introduce a momentum dependence into the tth and tthg vertices, as well as the tbW and  We next consider tth production with the tops decayed to the final state bbW + W − h.
We generate events in this final state from all tree level diagrams including intermediate top quarks to exclude pure electroweak production of W and b pairs. This includes contributions from a number of diagrams which cannot be factorized into tth production times decay.
One example of such a diagram is shown in the right-hand side of quark system, requiring it to be near the top quark mass shell: 160 GeV < m W b < 185 GeV.
We utilize the charge information of the W and b particles in performing this cut, assuming that they can be properly assigned to the correct top quark in a true experimental analysis, e.g., if they are all identified in a single large-radius top jet.
Including the full bbW + W − h final state changes the expectations from tth production without decays significantly. The diagrams where the Higgs is coupled to a W boson or b quark are not proportional to the top Yukawa, and therefore are not rescaled by the corrections to the top Yukawa as the bulk of the cross section is in the un-decayed case.
This leads to a growth in the cross section for large p T,h even at the dimension-6 level, and a change in the overall rate that is significantly different from a naive rescaling. At dimension-8, there are non-factorizable contributions with thW b vertices, which have one fewer propagator than the SM-like diagrams, and as a result, s-enhanced effects relative to the Standard Model. Finally, since the tops decay via their SU (2) L interactions, the effective operators proportional to (P R − P L ) discussed above will no longer be averaged out, and can therefore lead to additional effects at high p T,h as well. All of these effects in the amplitudes compete, and interfere with one another.
The resulting effects in Fig. 6 show that the kinematic effects apparent at dimension-6 are nearly washed out at dimension-8, and the distribution is almost flat. We emphasize that, while the overall distribution is roughly flat in p T,h , due to a combination of different effects that arise at different orders in the EFT expansion, the overall rate is different than that expected by rescaling the SM cross section by the modified top Yukawa. Note also that the size of the contributions from the dimension-8 operators are similar to the size of the dimension-6 squared terms relative to the interference contribution alone.
We also include in Fig. 6  In Fig. 7, we show the distributions for tth production including the full bbW + W − h final state in bins of |∆φ tt | and |∆η tt |, after placing a cut on the Higgs p T,h > 500 GeV. We see there are no kinematic effects in these distributions at any order in the SMEFT expansion, other than the rescaling consistent with the results in Fig. 6.
Finally, we comment on the size of the dimension-8 effects for parameters that are not experimentally excluded. We take λ T = 1.5, m T = 2 TeV, corresponding to a mixing angle The example we have considered is particularly simple, since the input parameters are not rescaled at tree level to dimension-8. It would be of interest to consider the effects of a more complicated model which generates tree level rescaling of the input parameters at dimension-8. The results of Refs. [20,66] suggest that the dimension-8 contributions may play a more significant role in such scenarios.
The UFO and FeynRules model files used to generate the TVLQ dimension-8 effects are included as supplemental material. The following terms in the tree-level dimension-8 Lagrangian, L 8 , contain non-SM gluon couplings: where indices are contracted implicitly such that terms in parentheses are SU (2) singlets.
The tth interactions that are needed for the tree level process are (with all momenta outgoing), is as given in Eq. (22).
The following are the electroweak couplings of the top quark expanded to dimension eight that occur in the tth, t → W b process: Appendix B: T Parameter in Effective Field Theory Language The oblique parameter ∆T has been calculated some time ago for the TVLQ model [64].
It is instructive to revisit this calculation using an effective field theory framework [67] and it is an example of the importance of including the one-loop matching in SMEFT calculations.
The contributions to ∆T from fermions with masses m 1 and m 2 can be expressed in terms of the function, θ + (y 1 , y 2 ) = y 1 + y 2 − 2y 1 y 2 y 1 − y 2 log y 1 y 2 − 2 y 1 log(y 1 ) + y + 2 log(y 2 ) where y i = m 2 i /M 2 Z and µ is an arbitrary renormalization scale. Neglecting the b quark mass and taking m t M Z , the t − b contribution to the SM is found from the diagrams of Fig. 8 with SM fermion-gauge boson couplings, We identify, where O HD = |H † D µ H| 2 . The coefficient function must be renormalization group evolved to the low energy scale which we take to be m t . In the TVLQ, only the top quark Yukawa coupling contributes and we have [68],