Primary Observables for Top Quark Collider Signals

At the HL-LHC and future high energy colliders, a sample of a billion top quarks will be produced, allowing precision searches for new physics in top quark decay and production. To aid in this endeavor, we characterize the independent three and four point on-shell amplitudes involving top quarks, under the assumption of $SU(3)_c\times U(1)_{em}$ invariance. The four point amplitudes factorize into primary and descendent amplitudes, where descendants are primaries multiplied by Mandelstam variables. By enumerating the allowed amplitudes, we can check for amplitude redundancies to find the number of independent terms and convert those into a Lagrangian which parameterizes these amplitudes. These results are then cross checked by utilizing the Hilbert series to count the number of independent Lagrangian operators. Interestingly, we find situations where the Hilbert series has cancellations which, if na\"{i}vely interpreted, would lead to the incorrect conclusion that that there are no primary operators at a given mass dimension. We characterize the four fermion ($ffff$) and two fermion, two gauge boson ($ffVV$) operators respectively up to dimension 12 and 13. Finally, by combining unitarity bounds on the coupling strengths and simple estimates of the branching ratio sensitivities, we highlight interesting amplitudes for top quark decay that should be studied more closely at the HL-LHC. Of those highlighted, there are both new charge current and flavor changing neutral current decays that occur at dimension 8 and 10 in SMEFT.


I. INTRODUCTION
The search for new physics beyond the Standard Model, at the LHC and beyond, has been led by the well established methods of effective field theory (EFT). To parameterize the indirect effects of new physics there are the two main paradigms of SMEFT [1,2] and HEFT [3]. These two approaches have differing assumptions about the physics at high energy scales and the relative importance of different effects.
There are however a variety of issues that can obfuscate the connection between EFTs and experimental signals. There is the large number of allowed interactions and also the complication of redundant (or incomplete) bases from equivalences due to equations of motion and integration by parts. These issues have motivated work to understand the direct connection between dimension 6 SMEFT terms and the physical observables they parameterize [4][5][6][7].
These redundancies on the Lagrangian side do not affect the predictions of physical amplitudes where external particles are on-shell. Since these amplitudes are the direct observables accessible to experiment, they provide a useful intermediary between theory and experiment. Recent work in the study of amplitudes has allowed greater insight into the independent amplitudes for a given process. In particular, the general structure of beyond the Standard Model amplitudes, given just SU (3) c × U (1) em invariance, has been analyzed, using both spinor helicity variables [8][9][10][11][12] as well as standard variables [13].
Ref. [13] was able to characterize the structure of on-shell 3 and 4 point amplitudes involving the Higgs. To complete this procedure, a set of potential on-shell amplitudes was constructed out of Lorentz invariant combinations of momenta and polarizations. By studying their Taylor expansion in the kinematic variables, a set of independent amplitudes was determined. These could then be converted into a basis of Lagrangian operators. As a cross check, the number of independent operators at each mass dimension could be determined using the Hilbert Series approach [14][15][16][17][18][19][20]. For the four point couplings, this lead to a number of primary amplitudes/operators whose multiplication by Mandelstam variables gave descendant amplitudes/operators. If these new interactions are mediated by the exchange of a massive particle, the lowest order primary amplitude would be a first approximation to the relevant phenomenology. Finally, by requiring unitarity up to an energy E max , one can place upper bounds on their coupling strength. These results, when combined with simple estimates, suggested that there are new amplitudes in Higgs decays into Zf f, Wf f, γf f, and Zγγ that could be searched for at the HL-LHC.
In this paper, we extend this study to amplitudes involving the top quark. At the HL-LHC and future TeV colliders, over a billion top quarks will be produced, allowing the study for rare decays as well as new production mechanisms. This requires understanding the general structure of four fermion operators and two fermion operators with two gauge bosons, which can result in primaries up to dimension 11. This vector space of amplitudes is spanned by these primary and descendant amplitudes, which in a model agnostic analysis can be taken to be independent [21]. Interestingly, in this classification, we find interactions (e.g. γγf f ) whose Hilbert series numerator has a complete cancellation in the coefficient for one of the terms, where a naïve inspection incorrectly concludes that there are no primary operators at a certain mass dimension. In our analysis, we have also checked that the primary and descendant structure up to at least dimension 12, going beyond the existing dimension 8 results using spinor-helicity variables [11,12]. As an initial look at the phenomenology of these operators, we give simple estimates that top quark decays for which FCNC modes (e.g. t → c(lℓ, hγ, hg, Zγ, Zg, γγ, γg)) and charged current decay modes could be interesting to search for at the HL-LHC. These simple estimates indicate that there are some decay modes that appear at dimension 8 and 10 in SMEFT that are worth studying in more detail.
The rest of this paper is organized as follows: Section II describes what amplitudes we will explore and how to determine independent amplitudes. Section III discusses the Hilbert series results for our top quark operators. In Section IV, we discuss some relevant phenomenological issues, such as unitarity bounds on coupling strengths and also rough estimates for top quark decays at the HL-LHC. Section V is the main body of results, where we list the operators for the primary amplitudes. In Section VI, we estimate which top decay amplitudes are interesting for exploration at HL-LHC. Finally in Section VII, we conclude. 3pt :qqV,qqh, 4pt :qqlℓ, qqqℓ,qqqq,qqhh,qqhV,qqV V (1) where q is a quark, ℓ is a lepton (charged or neutral), h is a Higgs boson, and V is any gauge boson. To fully characterize these 4 point interactions, we also need additional 3 point interactions for exchange diagrams, which add 3pt additional : V V V, hV V, hhh,lℓh,lℓV.
When there are identical particles involved, the form of the amplitude must respect the relevant exchange symmetry and for these, there are no amplitudes with 3 or more identical particles (note that, if we were characterizing down quark interactions, we would have to consider dddē).
In [13], a general approach for finding independent amplitudes for 3 and 4 point onshell amplitudes was presented. Here, we give a brief overview of the process and refer to that paper for further details, but will also note where changes in that approach need to be made. To characterize four point on-shell amplitudes, we form Lorentz invariants out of particle momenta, fermion wavefunctions, and gauge boson polarizations. For massless gauge bosons, we use the field strength contribution ϵ µ p ν − ϵ ν p µ , so that the amplitude is manifestly gauge invariant. Three point interactions with a covariant derivative can also give a four point contact interaction with a gauge boson; for our cases, the only one that will be relevant isqσ µν q ′ W µν , which generates aqq ′ W γ interaction. This results in a set of amplitudes M a , giving a linear parameterization of the general amplitudes M = a C a M a .
For each on-shell amplitude M a , we can associate a local Lagrangian operator, which we choose to have the lowest mass dimension possible, ca v d O −4 O a , where we've normalized its coefficient with factors of the Higgs vev to give a dimensionless coupling c a , resulting in a Lagrangian which parameterizes the on-shell amplitudes By connecting these amplitudes to Lagrangian operators, we can work in increasing mass dimension of the corresponding operators. For example,qqW W starts at dimension 5, since the lowest local operator needs two fermions and two gauge bosons, whileqqγγ will start at dimension 7. At a given mass dimension, we write out all of the amplitudes for the allowed particle helicities. In cases where there are two particles that are identical, we symmetrize and anti-symmetrize with respect to those two particles. After finding the allowed primary amplitudes for the distinguishable case, we can achieve the indistinguishable case by imposing the Bose/Fermi symmetry. We'll have more to say on that later, when we have the Hilbert series results.
For our four point amplitudes, we consider 1 + 2 → 3 + 4 scattering in the center of On-shell these have the constraints A general kinematic configuration is determined by the two continuous parameters E com and cos θ as well as the choice of helicities. However, treating p i , p f , and sin θ as independent is advantageous for finding amplitude redundancies. On-shell, one can replace even powers of these variables as sin 2 θ = (1 − cos 2 θ), . After doing this, as shown in detail in [13], the Taylor series coefficients of the amplitudes expansion in E com , p i , p f , cos θ, sin θ must all vanish if there is an amplitude redundancy. Schematically, if there are Taylor series coefficients B α , we then form the matrix ∂Bα ∂Ca , evaluate it for random numerical values for the particle masses, and numerically evaluate its singular value decomposition. The number of nonzero values in that decomposition is the number of independent amplitudes and one can find the independent ones by removing C a 's one at a time.
There are a few modifications to [13] needed to address the amplitudes of this paper. First of all, for four fermion amplitudes, we are required to have fermions in the final state. Similar to that paper, we can choose a mass configuration, either m 3 = 0, m 4 ̸ = 0 or m 3 = m 4 , to constrain the variable dependence of the kinematic variables in the fermion wavefunctions.
We have checked that this mass assumption doesn't affect the basis of independent amplitudes. Having final state fermions also results in dependence on cos θ 2 , sin θ 2 , which can be treated by replacing cos θ = 2 cos 2 θ 2 − 1 and sin θ = 2 cos θ 2 sin θ 2 and using cos θ 2 and sin θ 2 as our variables. Another complication is that the allowed SU (3) gauge invariant contractions are more diverse than before. This issue interplays with the Bose/Fermi symmetries of the amplitudes. As an example, forqqgg, interchange of the gluons must result in the same amplitude. If the gluons are contracted with an f ABC then the amplitude must also be odd under exchange of the momenta and polarizations of the gluons. On the other hand if the gluons are contracted with a d ABC then the amplitude must also be even under exchange of the momenta and polarizations of the gluons.

III. HILBERT SERIES
The Hilbert series gives a systematic way to count the number of gauge invariant independent operators, up to equation of motion and integration by part redundancies [14][15][16][17][18][19][20], which provides a useful cross check on our amplitude counting. It gives a function, whose Taylor series expansion in a parameter q gives the number of independent operators at each mass dimension [22]. In Eqn. 7, we list the Hilbert series for each of the four point operators that we will characterize. The three point and the other four point operator results can be found in [13]. , , Hqq′ qq = Hqq qq ′ = 10q 6 + 8q 7 + (10 − 2)q 8 + 8q 9 − 2q 10 (1 − q 2 )(1 − q 4 ) , 7 These fractional forms are interpretable in the following way: the numerator counts the number of primary operators and the denominator allows for the dressing of these operators with Mandelstam factors.
For example, looking at Hq qlℓ = 10q 6 +8q 7 −2q 8 , the numerator says that there are 10 dimension 6 primary operators and 8 dimension 7 primary operators. Ignore for now the −2q 8 , which we'll see denotes two constraints that appear at dimension 8. The denominator of 1/(1 − q 2 ) 2 has an expansion of (1 + q 2 + q 4 + · · · ) 2 which is just counting the number of operators from multiplying the primaries by Mandelstam factors of s, t (u is redundant to the on-shell condition). As we will see when we analyze the amplitudes of this interaction, two is an argument from counting conformal correlators that the number of primary operators is equal to the product of the spin degrees of freedom of the participating particles [18,23,24].
In our results, this is correct for all cases exceptqqqq, if one includes the negative coefficients and takes into account possible SU (3) c contractions. For example, forqqlℓ, the sum of the numerator coefficients 10 + 8 − 2 = 16 is equal to the spin counting of 2 4 . On the other hand, the case ofqqqq has further constraints from the crossing symmetry of theq and q, resulting in fewer operators.
We also note that for some denominators, the factors are (1 − q 2 )(1 − q 4 ). This results for situations where there are two identical particles in the amplitude. Assuming the two initial state particles are the identical pair, s and (t − u) 2 are the Mandelstam factors that have the correct exchange symmetry between the two particles, so we are allowed to multiply the primary by an arbitrary set of s and (t − u) 2 factors (note that the primary already has a factor of +/− when exchanging bosons/fermions).
As you'll notice in the Hilbert series list, some of the numerator coefficients are written in an unusual way, for example the (14 − 2)q 9 and (6 − 4)q 11 in H ggf f . When we evaluated the Hilbert series, these would of course have been 12q 9 and 2q 11 . However, when examining the number of independent amplitudes at dimension 9, we found 14 new primaries and 2 redundancies when 2 of the dimension 7 amplitudes were multiplied by s. In this way, the Hilbert series must be interpreted with care, as there can be hidden cancellations. In some case, there is even a complete cancellation like the (2 − 2)q 11 term for γγf f , where a naïve interpretation would have missed the new primaries at dimension 11.
The Hilbert series also allows for understanding of the constraints of Bose/Fermi symmetry. For example, for ggf f there are two symmetric contractions for the gluon SU (3) indices (δ AB , d ABC ) and one antisymmetric contraction (f ABC ), then swapping the kinematic variables of the two gluons would result respectively in a + sign for the first two and a − sign for the last one. Now, if we calculated the Hilbert series assuming photons were odd under interchange, then H asym γγf f = 2q 7 +6q 8 +(6−2)q 9 +2q 10 +2q 11 . One can then check that H ggf f = 2H γγf f + H asym γγf f as expected from the behavior under kinematic variable exchange and the allowed SU (3) contractions.
Note that unlike in [13], due to complications of enumerating all of the terms, we do not claim to have examined the full, allowed tensor structures of the amplitudes. Instead, we have checked that we agree with the Hilbert series up to dimension 13 forqqV V amplitudes and dimension 12 for four fermion amplitudes. Up to those dimensions, the numerator of these Hilbert series do not have any additional cancellations. As the Hilbert series shows, the redundancies that appear at higher dimension appear in pairs so it seems unlikely there are more, but still we cannot guarantee that others do not appear at higher dimension.

A. Unitarity
As in [13], we utilize unitarity to constrain the coupling strengths of these operators.
Since these are new couplings beyond the Standard Model, they violate unitarity at high energies. Requiring the amplitudes to satisfy perturbative unitarity up to a scale E max , gives an upper bound on the couplings. The technique follows the work [25][26][27][28], where the unitarity bounds due to high multiplicity scattering was developed (see also [29][30][31][32][33]).
To stand in for a more detailed calculation of each amplitude, we utilize a SMEFT operator realization of the amplitude to act as a proxy. As an example, consider the case of c vq qW W. This is realized by the dimension 8 SMEFT operator 1 Λ 4 (Q LH u R + h.c.)|D µ H| 2 [34]. Since we are only looking for an approximate bound, we ignore O(1) factors like √ 2, g, g ′ , sin θ W , cos θ W and only take into account factors of v. Under this approximation, The SMEFT operator has many contact interactions that violate unitarity, but we find that either the lowest and highest multiplicity give the best bound as a function of E max , so we will calculate these for all interactions and include them in our tables. For this example, the lowest multiplicity amplitude is for two quarks and two Goldstones, with a matrix element that goes as M 2→2 ≈ vE 3 max Λ 4 , where one factor of E max comes from the fermion bilinear and the other two come from the two derivatives acting on the Goldstones. This is bounded by phase space factors M 2→2 ≤ 8π [25], which translates into TeV where E TeV = E max /TeV. The highest multiplicity amplitude is for two quarks and 3 where the bound again depends on the phase space. This gives the bound c ≤ ( As this example illustrates, we generally find that the low multiplicity constraint is stronger for E max < 4πv and the high multiplicity one is stronger for energies above that.

B. Top Quark Decays
The HL-LHC will produce about 5 billion top quarks, allowing searches for rare decays as well as new production modes. Here we will consider decay modifications due to our amplitudes. The on-shell 2 and 3 body decay modes of the top quark allowed by the For this simple analysis of the phenomenology, we will approximate top decay amplitudes as a constant, assuming the top quark mass is the only relevant mass scale To estimate sensitivity, we require that the new top decays must be as large as a one sigma deviation in the Standard Model top background, which for a sample of N t top quarks gives N t δBr(t → 2) ≳ N t Br(t → 2) SM . Such a calculation gives for two and three body decays the constraints 2 Body Decays : c ≳ 5 × 10 −6 10 9 N t where we've normalized to a total sample of a billion top quarks.
For FCNC decays, such as t → c(Z, γ, g, W W, Zγ, Zg, γγ, γg, gg), the branching ratios predicted in the Standard Model (10 −12 to 10 −17 ) are too small to occur at the HL-LHC (e.g. [46][47][48][49][50]). Thus, for these decays we can ignore interference and give an estimate that works for both CP even and odd interactions. If we make an optimistic assumption that other backgrounds can be neglected, this requires that the new branching ratios Br BSM give a few events at the HL-LHC or N t Br BSM ≳ 1. Under our approximation, this gives the same bounds as Eqn. 13.
To get some sense of how well this approximation works, we've checked in a few existing FCNC searches, whether the background free assumption works at the O(1) level. As one might expect, one finds that for final states with a single gluon or photon, where hadronic backgrounds and fakes are relevant, that this is a poor assumption and gives a branching ratio bound that is too strong by two and three orders of magnitude for photon and gluon decays, respectively. Thus, estimates for these final states should be viewed as very optimistic.
However, we found that the searches with a Higgs decaying into two photons agree roughly with our bounds. Similarly, the final states with e, µ's give bounds that are correct to a factor of 2 − 3 as long as one takes into account tagging efficiencies for b (∼ 0.5), e/µ (∼ 0.8) and, when relevant, Z and W leptonic branching ratios (∼ 0.06 and 0.2). Thus, as long as one take these factors into account, these final states should be more reliable. Later, when combined with our upper bounds from perturbative unitarity, these calculations will enable us to give a simple estimate of which decay amplitudes that are worth exploring further at the HL-LHC.

V. INDEPENDENT AMPLITUDES FOR TOP QUARK PHYSICS
In the following subsections, we will list operators corresponding to the primary amplitudes for f f V V and f f f f interactions involving the top quark. We will make comparisons to the Hilbert series to show consistency with the number of independent operators, including discussions of redundancies that occur at certain mass dimensions. We will also give CP properties of the operators and unitarity bounds on the coupling constants for these interactions.
A. f f V V Amplitudes Tables I and II where the coefficients c i 's only depend on the particle masses and predict the same on-shell amplitudes as sO 26 We also list the unitarity bounds for each SMEFT operator, assuming the lowest and highest particle multiplicity. These operators can also be reworked to account forqq ′ W Z amplitudes provided we take q → q ′ and W → Z. Here, we use q ′ to denote a different quark flavor of the correct charge.
In Tables III and IV, we list the primary operators forqqZZ interactions. Reading off from the Hilbert series, we expect to see 2 operators at dimension 5, 6 operators at dimension 6, 12 operators at dimension 7, 6 operators at dimensions 8, 9, and 10, and at least 2 constraints at dimension 11. We do indeed find that there are 38 primary operators, as well as two redundancies at dimension 11, for sO 31 and sO 32 . To generate an independent set of operators, one needs to add descendants of the primaries, which involve multiplying by arbitrary powers of s and (t − u) 2 (note that (t − u) 2 respects the exchange symmetry of the Z's). However because of the redundancies at dimension 11, for O 31 and O 32 , one only We have listed all of the primary operators forqqZγ interactions in Table V. The Hilbert series tells us to expect 4 operators at dimension 6, 12 new operators at dimension 7, 8 operators at dimension 8, and 2 new operators and 2 new redundancies at dimension 9. We note that a naïve interpretation of the Hilbert series would have missed the 2 new primary operators that appear at dimension 9. We find that there are 26 primary operators, in agreement with the Hilbert series, as well as two constraints at dimension 9-sO 7 and sO 8 .
Thus for those two operators, one only needs their descendant operators t n O 7 and t n O 8 .
These operators can also be adapted to account forqq ′ W γ,qqZg, andqq ′ W g where we use a prime to denote a different quark flavor. To getqqZg operators, one replaces F µν → G µν , to getqq ′ W γ operators, one should make the replacement q → q ′ and Z → W , and to get qq ′ W g operators one needs to make the replacements q → q ′ , F µν → G µν , and Z → W . dimension 9 operators, as well as 2 operators that become redundant at dimension 9, so the analysis again finds 2 additional dimension 9 primary operators that a quick interpretation of the Hilbert series would have missed. We indeed find the 18 operators we expect from the Hilbert series analysis, as well as two operators that become redundant at dimension 9-sO 5 and sO 6 . Thus, for those two operators, we can just add their descendants t n O 5 and t n O 6 .
We list the primary operators forqqγγ interactions in Table VII. From the Hilbert series, we expect that there should be 4 operators at dimension 7, 2 operators at dimension 8, 4 operators at dimension 9, 6 operators at dimension 10, and 2 operators at dimension 11, giving 18 total primary operators in agreement with the Hilbert series. We also find that there are two new redundancies at dimension 11 for sO 7 and sO 8 . This gives rise to a complete cancellation in the Hilbert series at dimension 11 between the two new operators In Tables VIII and IX, we list all of the primary operators forqqgg interactions. The Hilbert series says that we should expect 10 operators at dimension 7, 10 operators at dimension 8, 14 operators at dimension 9, 14 operators at dimension 10, and 6 operators at dimension 11. Additionally, we find that there are 2 redundancies at dimension 9-sO 9 and sO 10 -and 4 redundancies at dimension 11-sO 21 Table VIII should Table VIII should be ready as f ABC qT A q G Bµν G C µν . Thus, for O 9,10,21,22,23,24 , we only need to add their descendants with factors of (t − u) 2 .
B. f f f f Amplitudes In Table X In Table XI,  In Table XII, we've listed the primary operators forqqq ′ q ′ interactions. Notably the Hilbert series for this has a numerator that is twice theqqlℓ Hilbert series. This factor of two is simply for the two allowed SU (3) contractions, one where the qq ′ are either in the 6 or3 representation, leading to the symmetric (S) and antisymmetric (A) operators. Again, at dimension 8, sO 9 and sO 10 are redundant to the other operators, where s = (p q + pq) 2 .
Thus one only needs to add their descendants t n O 9 and t n O 10 .
In Table XIII, we've listed the primary operators forqqqq interactions when two of the quarks are identical for the specific case of uutc. There are again two allowed SU (3) contractions, specified by whether the uu are in symmetric (S) or antisymmetric (A) combination.
In Table XIV, we've listed the primary operators forqqqq interactions when the two quarks are identical and the two anti-quarks are identical, for the specific case of uutt. There are again two allowed SU (3) contractions, specified by whether the uu are in symmetric (S) or antisymmetric (A) combination. Since we're suppressing the SU (3) indices, this makes some of the expressions look identical, with (1-3) and (4-6) being the same, as well as (13-15) and (18)(19)(20). At dimension 8, sO 2 and sO 3 become redundant and at dimension 10, sO 19 and sO 20 become redundant. Thus one only needs the descendants of O 2,3,19,20 with factors of (t − u) 2 . These four redundancies explain the two −2 terms in the Hilbert series.

VI. INTERESTING TOP DECAY AMPLITUDES FOR THE HL-LHC
Now that we have all of the results, we can compare our unitarity upper bounds on the coupling strengths with our estimate of the couplings needed for HL-LHC sensitivity to the new top quark decays in Eqn. 13, to highlight which top decay amplitudes are worth studying in more detail at the HL-LHC. In the following, we will assume we have top quark pair production, where one top quark decays into a b quark and a leptonic W , with a btagging efficiency of 0.5, a lepton tagging efficiency of 0.8, and a W leptonic branching ratio of 0.2. For the Higgs modes, we will assume it decays to photons with a branching ratio of First, let's consider two body decays of the top quark. For the charged current decays, we have t → W (b, s, d), which have left and right handed vector and tensor couplings, which can be distinguished by the lepton angular distributions [51]. In addition, the tensor operators can be constrained by top quark production [52]. For flavor changing neutral current decays, we have t → (u, c)(h, Z, γ, g), which are all actively being searched for at the LHC [35][36][37][38][39][40][41].
For all of these two body decays, there is a dimension 6 SMEFT operator that realizes the coupling, which explains why they are actively being studied. Our constraints on the coupling strengths agree that these are interesting and could potentially probe unitarity violating scales up to several tens of TeV. Now, let's consider three body decays. We do not consider all hadronic decays of the top quark since those suffer from large combinatorial backgrounds at the LHC and our estimates would be entirely too optimistic. The charged current contact interaction t → (b, s, d)(ē,μ,τ )ν has a different lepton pair invariant mass, which could be interesting to look for in terms of the quark-charged lepton invariant mass distribution. Here our estimates say that all of the dimension 6 CP even amplitudes could be interesting, even with unitarity violation occurring around 5 TeV, while the dimension 7 CP even amplitudes are interesting if unitarity violation occurs at about ∼ 3 TeV. Thus, these are worth exploring as there is room to increase the coupling for lower scales of unitarity violation. The other three body decays with a charged current interaction are t → (b, s, d)W (γ, g), which are generated at higher order in the Standard Model (we do not consider t → dW Z since this is so close to being kinematically closed and thus, our assumptions about the phase space and matrix element would be wrong.). Contact amplitudes, unlike the Standard Model processes, are not enhanced in the collinear/soft limits so these might be distinguishable. Here, we find that of the operators in Table V the  we should interpret our estimates carefully for these photon and gluon decays, the lowest dimension operators are probably the most realistic to explore.
Flavor changing decays are highly suppressed in the Standard Model, so these are very promising to search for. To start with, four fermion contact terms t → (c, u)(e, µ, τ )(ē,μ,τ ) are being searched for at the LHC in the lepton flavor violating modes to eµ [53]. Here our estimates say that dimension 6 CP even and odd amplitudes are interesting for unitarity violation above 9 TeV, while dimension 7 CP even and odd amplitudes require unitarity violation by ∼ 4 TeV. The existing CMS search probes the dimension 6 amplitudes [53], but does not look for the dimension 7 amplitudes since they appear at dimension 8 in SMEFT. We can also consider flavor changing neutral current decays involving gauge bosons, including t → (c, u)(hγ, hg, Zγ, Zg, γγ, γg, gg), but not t → (c, u)W W since it is also nearly kinematically closed. Again, our estimates are too optimistic for the decay modes that are completely hadronic, so we will focus on the other cases. For the decays with a Higgs and a photon or gluon, using the amplitudes and unitarity bounds in Table 3 of [13] and assuming the diphoton Higgs decay, we find that the dimension 6, 7, 8 operators require unitarity violation respectively by ∼ 5, 2, 1 TeV, so the dimension 6 and 7 ones are the most promising. For the decays into a Z and a photon or gluon, assuming the Z decays to ee or µµ, we find that the dimension 6, 7, 8, 9 operators in Table V There are also baryon number violating three body decays mediated by our amplitudes, t → (c,ū)(b,s,d)(ē,μ,τ ). These would have combinatorial backgrounds, but have been searched for in the past by CMS [54]. Again, theory explorations of these have focused on the dimension 6 SMEFT operators [55,56], so it would be interesting if the ones parameterized by dimension 8 SMEFT operators give distinguishable signals.
To conclude, our unitarity bounds combined with our estimates for the interesting size of couplings for top quark decays has allowed us a quick survey of which of the decay amplitudes may be worth pursuing at the HL-LHC. As the dimension of the amplitude gets larger, these two constraints become more challenging to satisfy without lowering the scale of unitarity to the TeV scale. Since the SMEFT operator realization must be at the same or higher dimension, this motivates studying in more detail top decays from many dimension 8 and a few dimension 10 SMEFT operators to determine their sensitivity at HL-LHC and future colliders.

VII. CONCLUSIONS
In this paper, we have extended an approach [13] to determine the on-shell 3 and 4 point amplitudes that are needed for modeling general top quark phenomenology at colliders. These serve as an intermediary between the observables searched for by experimental analyses and the operators in effective field theories for the Standard Model. This involved characterizing the general amplitudes for processes involving four fermions or two fermions and two gauge bosons. We were able to characterize these respectively to dimension 12 and 13, finding the structure of primary and descendant amplitudes, where descendants are primaries multiplied by Mandelstam factors. Interestingly, we find two classes of interactions whose Hilbert series numerator has a complete cancellation in the numerator. This naïvely would suggest that there are no primary operators at a certain mass dimension, but in actuality there are an equal number of new primaries and redundancies that appear at that mass dimension. This illustrates the importance of using the Hilbert series in conjunction with the amplitudes, as they complement each other in this process. We also note that our approach is a complementary check to the existing results up to dimension 8 using spinor-helicity variables [11,12] and extends the amplitude structure to higher dimension.
To provide an initial survey of the potential phenomenology, we've used perturbative unitarity to place upper bounds on the coupling strengths of these interactions. These depend on the scale where unitarity is violated E TeV = E max /TeV, with more stringent constraints as one increases E TeV . Given the expected sample of top quarks at HL-LHC, we've estimated the coupling size needed for the top quark decays to be seen over irreducible backgrounds. This allowed us to highlight the that top quark decays into both FCNC modes, like t → c(lℓ, hγ, hg, Zγ, Zg, γγ, γg), and non-FCNC modes, like t → b(W γ, W g), could be interesting to search for at the HL-LHC. Some of these highlighted modes occur at dimension 8 and 10 in SMEFT and thus would be interesting to explore how distinctive these new amplitudes are compared to existing searches. We leave such detailed phenomenology to future work.
To conclude, the high energy program at colliders is entering the phase of testing whether the Standard Model is indeed the correct description of physics at the TeV scale. To do so, we must look for new physics in the most general way, so that we can find such deviations or constrain them. On-shell amplitudes are a useful intermediary between experimental analyses and the parameterization of new physics by effective field theories. Finally, by determining the on-shell amplitude structure to high dimension and writing down a concrete basis for them, we hope this will allow the field to maximize its efforts to find what exists beyond the Standard Model.  i TeV , 0.004 TeV , 0.004 TeV 37, 38 TeV , 0.9     Thus one only needs to consider O 3,4,27,28 descendants with arbitrary factors of (t − u) 2 . (tγ µ u)(tiγ 5 Dµu) TeV , 0.9 TeV , 0.07