Primary Observables for Indirect Searches at Colliders

We consider the complete set of observables for collider searches for indirect effects of new heavy physics. They consist of $SU(3)_{\rm C}\times U(1)_{\rm EM}$ invariant interaction terms/operators that parameterize deviations from the Standard Model. We show that, under very general assumptions, the leading deviations from the Standard Model are given by a finite number of `primary' operators, with the remaining operators given by `Mandelstam descendants' whose effects are suppressed by powers of Mandelstam variables divided by the mass scale $M$ of the heavy physics. We explicitly determine all 3 and 4-point primary operators relevant for Higgs signals at colliders by using the correspondence between on-shell amplitudes and independent operators. We give a detailed discussion of the methods used to obtain this result, including a new analytical method for determining the independent operators. The results are checked using the Hilbert series that counts independent operators. We also give a rough sketch of the phenomenology, including unitarity bounds on the interaction strengths and rough estimates of their importance for Higgs decays at the HL-LHC. These results motivate further exploration of Higgs decays to $Z\bar{f}f$, $W\bar{f}f'$, $\gamma \bar{f}f$, and $Z\gamma\gamma$.


Introduction
Searches for physics beyond the standard model (BSM) fall roughly into two classes: direct searches for signals from new particle production, and searches for indirect effects of new particles that are too heavy to be produced.These search strategies are complementary, and both are important.In this paper we will focus on experiments looking for indirect effects of new heavy physics, with an emphasis on channels that include the Higgs boson.Even if the LHC does not discover new physics, the constraints from these searches will guide further exploration of the energy frontier, similar to the precision measurements performed at LEP.
The main points of this paper are the following: • We propose a theoretical framework for precision measurements that is meant to be intermediate between experimental searches and theoretical interpretations (such as SMEFT, HEFT, or theoretical models).
• All local vertices that parameterize deviations from the Standard Model (SM) can be written as a linear combination of 'primaries' and 'descendants.'There are finitely many primaries, and descendants are (roughly) higher-derivative corrections to the primaries.
• If these vertices arise from integrating out heavy physics at the scale M , the effect of the descendants is expected to be suppressed by powers of E 2 /M 2 relative to the primary operators, where E is the energy scale of the experiment.If the primary vertices are unsuppressed, the descendants can be neglected, and the leading BSM effects are parameterized by the primaries.This leads to a larger (finite) basis of 'leading' deviations from the SM than either HEFT or SMEFT, with a clear physical motivation.
• The classification of the primary operators was initiated in Refs.[1,2] (where they are called 'stripped contact terms').This paper extends these results to operators of arbitrarily high dimension, as well as operators with identical particles, and operators with both massive and massless particles.We give the complete determination of all primary operators relevant for Higgs phenomenology.This is in itself a nontrivial theoretical problem, and we have developed new tools to do this.
• For each primary operator, we estimate the scale at which the operator gives rise to violation of tree-level unitarity at high scales.This is used to determine the theoretically motivated range for the coupling constants of the primary vertices.
• Although this paper is primarily focused on the theoretical problem of determining the primary operators, we give rough estimates for the importance of primary operators for Higgs decays.We identify a number of vertices that occur at higher orders in the SMEFT expansion that can potentially give observable effects.
In the remainder of this introduction, we expand on the points above.
We are considering models where the leading deviations from the SM arise from particles with mass of order M that are too heavy to be directly produced in experiments.In this case, the effects of the heavy particles on experimental observables with energy scale E M can be parameterized by adding additional local operators to the effective Lagrangian.In this paper, we argue that very general physical considerations lead us to expect that the leading observed deviation from the SM comes from so-called 'primary' vertices.
The coefficient of a local interaction generated by heavy particles depends on their mass M , as well as the couplings of the heavy particles to SM fields.A single exchange of a heavy particle will generate an infinite series in derivatives from expanding the propagators of the heavy particles, for example where s ∼ E 2 is the energy scale of the experiment.In this example, it is clear that the subleading terms in the expansion will be subdominant, and it is sufficient to keep only the first term in the expansion as long as we are searching for the leading deviation from the SM.This is the idea that we are attempting to make precise for general local vertices from a 'bottom-up' perspective.
To do this, we must take into account the fact that local operators are a redundant description of new physics effects, due to the freedom to perform field redefinitions and integration by parts.Identifying a complete and independent basis of operators is very nontrivial.In this paper, we use the fact that there is a one-to-one correspondence between local operators and local on-shell amplitudes [1][2][3][4][5][6][7].The correspondence is simply that the onshell amplitude is the Feynman rule for the corresponding operator.In this correspondence, the integration by parts ambiguity for operators corresponds to momentum conservation in the amplitude, and the freedom to perform field redefinitions in the operator basis is fixed by the on-shell conditions of the amplitudes.The parameterization in terms of amplitudes is not only technically convenient; it has physical meaning because the corresponding amplitudes give an estimate of the contribution of the operator to physical processes, even if the particles involved are not on shell.
Using the correspondence with amplitudes, we now define what we mean by 'primary' and 'descendant' operators/amplitudes.A general local on-shell amplitude can be written as a linear combination of 'spin structures' consisting of Lorentz invariant products of momenta and particle wavefunctions, times a series expansion in the Mandelstam invariants p i • p j , where p i are the momenta of the particles.For 3-point functions, all the kinematic invariants can be written in terms of the masses of the particles, while for 4-point functions these can be written in terms of the Mandelstam variables s and t and the masses of the SM particles.An example of a BSM correction to an on-shell 4-point amplitude is where the numbering scheme matches the convention in the later table .Here, the first term in each line defines the 'primary' vertex, while the infinite series in Mandelstam variables are the 'descendants.'Although the choice of primary operators is not unique, any choice defines a basis for the same linear space of on-shell amplitudes.
We now consider the relative importance of the various terms above for low-energy experiments.The couplings c 1 and c 9 depend on the couplings of the heavy particles to the SM fields, as well as the mass scale M of the new physics.However, very generally, we expect that the coefficients α 1,9 and β 1,9 will be of order unity.The reason is that the higher order terms in the expansion in Mandelstam variables probes the kinematic corrections to the amplitudes coming from the momentum expansion of heavy propagators, as in Eq. (1.1).This tells us that we expect that we can neglect the effects proportional to α 1 and β 1 compared to the term proportianal to c 1 , but this argument gives us no information about whether c 1 is expected to give larger effects than c 9 .The second operator has a higher mass dimension than the first, but the coefficient c 1 may be suppressed compared to c 9 due to the structure of the couplings of the new physics to the SM fields.
The primary effective operators that correspond to Eq. (1.2) are given by Note that the operators are written in terms of physical mass eigenstate fields, as in HEFT.
Although there is no a priori limit on the dimensionality of the primary operators, there are finitely many such operators that are relevant for collider searches, because such searches are only sensitive to 3-and 4-point functions.Therefore, we only need to consider functions with up to 4 fields, and the expansion in derivatives is truncated if we drop the descendants. 1he result is that there is a finite basis of primary operators relevant for collider searches.
The argument that primary operators dominate over descendants can fail if the fundamental physics has special features that suppress the primary operators, but not their descendants.This occurs in models where the Higgs boson is a pseudo Nambu-Goldstone boson (PNGB).This important case can be treated in the strongly interacting light Higgs (SILH) EFT framework [8].In those models, the dominant interactions of the Higgs field are invariant under a shift symmetry h → h + λ + • • • , and therefore operators involving derivatives of h can be more important than operators with fewer derivatives.For example, Ref. [9] showed that in such models, the coupling ∂ ρ h∂ ρ hG µν G µν is phenomenologically relevant, and is more important than h2 G µν G µν , even though the former is a descendant of the latter.Another exception comes from the scenario discussed in [10], where multipole interactions of composite gauge bosons can be enhanced relative to monopole interactions by a factor of a strong coupling.These examples show that we cannot claim that the primary operators always dominate.Even for 'generic' new physics models with no special structure, it is possible that the primary operators could be suppressed by accidental cancelations.Although the primary operators do not give the leading observables for all possible UV models, they do give a large number of physically motivated observables that we believe are worth investigating further.
We emphasize that the framework that we are proposing is an intermediate step to connect experimental observations with theoretical models or EFTs such as SMEFT [11][12][13][14][15][16][17] or HEFT [18][19][20][21][22]. Assuming that the dominant BSM effects come from tree-level diagrams involving the primary vertices, they give well-defined theoretical predictions for experimental searches that can be used to optimize searches and report results. 2 At the same time, these vertices can be computed in any theoretical model or EFT, so the experimental results can be used to constrain these models.In this approach, the primary operators are treated as independent observables in experimental searches.Any theoretical model will have many correlations between these observables that depend on the details of the physics at the scale M .However, given the large number of possible models of new physics, we believe that it is sensible to take a 'bottom-up' point of view and treat the operators in L BSM as independent observables for purposes of performing individual collider searches.Correlations between different observables in a given model can be taken into account in global fits, while experimentalists can focus on making the best measurement of as many observables as possible.
We now comment on the relation this paper to previous work.Ref. [23] identified the leading contact interactions arising from dimension-6 SMEFT operators, calling them 'BSM primaries' (see also the 'Higgs basis' [24] and 'Pseudo-Observables' frameworks [25,26]).Our work can be viewed as generalizing this to all orders in the SMEFT expansion.There are a number of motivations to go beyond leading order in the SMEFT expansion.Dimension-6 SMEFT parameterizes the leading effects of decoupling physics, namely heavy particles with mass M that decouple in the limit M → ∞ with dimensionless couplings held fixed.However, dimension-6 SMEFT does not correctly parameterize the leading effects from non-decoupling physics, for example the models and scenarios discussed in Refs.[27][28][29][30].For non-decoupling physics, M cannot be much larger than 1 TeV [31,32], but this is the scale being probed in many measurements at the high-luminosity LHC.For example, if there is a deviation in the hZZ coupling of the size of the current 1σ constraints, the SM violates unitarity at a scale of order 5 TeV [33].For new physics at such low M , higher-dimension SMEFT operators could well be important.For example, we believe it is important to independently measure the h tt and hh tt couplings, even though these are related by dimension-6 SMEFT.
Returning to the discussion of previous work, Ref. [34] analyzes observables to all orders in the SMEFT expansion, but only gives results for 3-point functions.Refs.[1,2] analyze both 3-point and 4-point couplings using the massive spinor-helicity formalism. 3They gave results only for the case that all particles are massive and distinguishable, and did not identify primary operators to arbitrarily high dimension for all 4-point functions.We have tackled the additional subtleties required to treat massless and/or indistinguishable particles, and give explicit results that we believe are complete.Where our results overlap, we agree with [2], after correcting an error in their work that was pointed out by us.This paper is organized as follows.In Sec. 2 we give an overview of the three and fourpoint interactions that are needed to parametrize Higgs signals at colliders.In Sec. 3, we give details about how we enumerate amplitudes, determine the independent primaries, and check the counting using Hilbert series.In Sec. 4, we discuss the unitarity and precision electroweak constraints on the coefficients of these operators and give estimates for the corrections to Higgs decay rates.In Sec. 5, we list the primary operators for three and fourpoint amplitudes and based on the unitarity bounds and rough phenomenology estimates, determine which operators for Higgs decay are interesting at the HL-LHC.In Sec. 6 we give our conclusions and discuss future directions.

Scope of Paper
The aim of this paper is to classify the primary operators that are relevant for Higgs signals at hadron and lepton colliders.Specifically, we focus on all 3-and 4-point couplings that are relevant for Higgs decays, di-Higgs production, and Higgs associated production.In this section, we specify the couplings that we will study in the remainder of the paper.We will also define our notation and normalization conventions, and comment on ambiguities in the operator basis associated with off-shell 3-point couplings.
< l a t e x i t s h a 1 _ b a s e 6 4 = " I z Fig. 1: Three and 4-point couplings relevant for Higgs decays and Higgs production.Dashed lines denote the Higgs particle, solid lines denote fermions, and wavy lines denote any of the SM gauge bosons γ, g, W , or Z.The crossed-out diagrams are not relevant because they vanish on shell (see text).
< l a t e x i t s h a 1 _ b a s e 6 4 = " I z 5 i R 3 7 Fig. 2: Exchange contributions to 4-point amplitudes involving BSM 3-point couplings that do not contain the Higgs.The notation is the same as Fig. 1.We do not show diagrams involving Higgs exchange that involve 3-point functions already shown in Fig. 1.We also do not show diagrams involving two Higgses and a neutral gauge bosoon, since these vanish on shell (see text).

Topologies and Couplings
The 3-point and 4-point couplings that involve at least one Higgs boson are shown in Fig. 1 (2.1b)Some of these three-point couplings vanish on-shell, and we have crossed these out above. 4n addition, there are 3-point couplings that do not involve the Higgs which contribute to some of the 4-point processes that get contribution from the couplings in Eq. (2.1b).These are shown in Fig. 2.These couplings are given by where we have again crossed out couplings that are not allowed on-shell.
These interactions parameterize the BSM contributions to general 2-body and 3-body decays of the the Higgs boson.They also parameterize the BSM contributions to the production of a single Higgs, a pair of Higgs, and Higgs associated production via the processes ( f f, gg, W + W − , ZZ) → (h, hh, hZ, hγ, hg) Note that the hhhZ, hhhγ amplitudes can be used to calculate exchange diagrams for hhh production, e.g.f f → (Z * , γ * ) → hhh, but fully characterizing the 5-point amplitude would require us to classify the 5-point couplings hhh f f .Because of the large number of couplings that we are considering, we will use a uniform notation for their couplings.The operators contributing to a 3-or 4-point coupling X = ABC or ABCD will be denoted by O X i , where i runs from 1 to the number of primary operators of type X.For a primary operator O with mass dimension d(O), we write the coupling as where v = 246 GeV is the Higgs VEV, and c O is a dimensionless coefficient.Note that if c O ∼ 1 we expect the effects of the inserting such an interaction into an electroweak process to be roughly of order the SM contribution, since in that case all couplings are order unity, and all mass scales are of order 100 GeV.For operators that are present in the SM Lagrangian, the coupling c O is related to the associated 'κ parameter' by where c is the coefficient of O in the SM Lagrangian.

Off-Shell Ambiguities
The correspondence between local on-shell amplitudes and EFT couplings completely removes any basis ambiguity as long as the EFT couplings are used at tree-level and on shell.
However, some of our processes of interest involve 3-point couplings where particles are exchanged and thus potentially off-shell.In this case, there are residual ambiguities in the basis.These are straightforward to remove, but we discuss them here for completeness.
To explain the point, it will be sufficient to consider a simple example, the coupling h f f .If all particles are on shell, then this interaction is equivalent to the higher-derivative couplings For off-shell kinematics, these operators parameterize 'form factor' corrections to the minimal on-shell coupling h f f .They parameterize the ambiguity in continuing the coupling h f f offshell.We can use field redefinitions to reduce any linear combination of such couplings to the minimal three-point coupling h f f [35][36][37][38].However, making such a field redefinition also shifts the values of some of the 4-point couplings so that amplitudes that involve both the 3-point and the 4-point couplings remain invariant.The conclusion is that the choice of basis for 3-point functions is part of the definition of the basis for the 4-point couplings.Said in another way, if we allow for the most general local and on-shell 3 and 4-point interactions, then using them in Feynman diagrams generate the most general 3 and 4-point on-shell amplitudes in an expansion in Mandelstam invariants.

Independence of Operators/Amplitudes
In this section, we explain the methods we used to determine a basis for the independent primary operators.This is done in 3 steps: • Enumerating an over-complete basis of amplitudes • Determining the independent primary amplitudes • Checking the result against the Hilbert series counting We will give a short summary of each of these steps before going into the details in the subsections below.
The first step is to find an over-complete basis of local amplitudes for a given process.These basis elements are scalar monomials in the momenta and wavefunctions of the particles involved.They are Lorentz invariant, so the indices are contracted using the metric and the Levi-Civita tensor.When there are no indistinguishable particles, we can omit monomials where the momenta are contracted with other momenta, since these can be written in terms of Mandelstam invariants and masses.Operators with indistinguishable particles can be treated by appropriately symmetrizing these amplitudes, as we will discuss below.In this way, we obtain a finite number of amplitudes such that any local amplitude is a linear combination of these amplitudes and their Mandelstam descendants.This step is done by hand, and in some cases we used Mathematica [39] to enumerate the index contractions.
The second step is to find the independent primary interactions.The fact that these are parameterized by on-shell amplitudes turns this into a problem about linearly independent functions.We proceed order by order in the number of powers of momenta.Note that the number of momenta determines the mass dimension of the amplitude (and the corresponding EFT operator), so we are also working order by order in the operator dimension.We first determine the linearly independent amplitudes of lowest dimension that do not contain inner products of momenta.We look for linear relations of the form where the basis amplitudes are denoted by M a (p, s, m), where p denotes the momenta, s the spins, and m the masses of the particles.The notation reminds us that the coefficients in the linear relations can depend on the masses, but not the momenta and spins of the external particles.These amplitudes have no Mandelstam factors are thus are guaranteed to be primary amplitudes, since there is no operator that they can be descendants of.Then we consider operators of higher dimension, including Mandelstam descendants of primary operators found in earlier steps that have the same dimension.Eventually, we reach a dimension where all of the amplitudes at that dimension are linear combinations of the Mandelstam descendants of operators we already have. 5At that point, we know that we have found all of the primary amplitudes.We used several methods to find the linearly independent amplitudes, including a new analytic method, and these are described below.
Finally, we compare the results to the Hilbert series counting of operators of different dimension [40][41][42][43][44][45][46][47].The Hilbert series gives a direct counting of the primary operators up to certain redundancies, which we review below.
We now turn to a detailed description of each of these steps.

Enumerating the Local Amplitudes
The first step is to enumerate all possible local amplitudes of a given topology and symmetry that do not involve any Mandelstam invariants.We will explain the procedure using the example of the hZ f f coupling.The most general form of the corresponding amplitude is The choice of the channel is arbitrary, and does not affect the results. 6We do not use massive spinor-helicity variables because momentum conservation is a quadratic constraint in terms of them, while if we work with 4-momenta we can simply write all possible functions of the 3 independent momenta.Also, the Mandelstam variables are manifest when the amplitude is written in terms of the 4-momenta.
For the amplitude Eq. (3.2), the problem reduces to enumerating all possible Γ µ .This is obtained by forming all possible 4-vectors formed from p µ 1,2,3 and γ µ with indices contracted with the spacetime metric and up to one power of the Levi-Civita tensor (since products of Levi-Civita tensor can be written in terms of Kronecker deltas).We omit terms where the momenta are contracted with other momenta, since these are Mandelstam descendants of other amplitudes.This gives a finite list of operators that includes all primary operators.In this way, we find Note that terms containing γ 5 and γ µ γ 5 implicitly contain one power of the Levi-Civita tensor since γ 5 ∝ µνρσ γ µ γ ν γ ρ γ σ .We have omitted terms that can obviously be simplified by equations of motion, for example / p 1 u 1 = m 1 u 1 and p 3 • * 3 = 0.There are several complications that are not illustrated in the present example.The first involves amplitudes containing massless gauge bosons, which for us means photons and gluons.In operator language, there are local interactions involving massless gauge bosons that arise from expanding covariant derivatives.However, these do not give rise to gauge invariant local amplitudes because they are always accompanied by exchange diagrams involving the same interaction.For example, the W W Z BSM coupling µνρσ (W + µ ↔ D ν W − ν )Z σ contributes to the amplitude W W Zγ both through a a direct 4-point coupling and an exchange diagram with a SM W W γ vertex.In the amplitude approach, we find the W W Z local amplitude when characterizing the 3-point amplitudes, and the gauge invariant operator is parameterized by the usual replacement ∂ µ → D µ acting on charged fields.
The gauge invariant local on-shell amplitudes involving massless gauge bosons must satisfy the Ward identity, and are therefore proportional to the combination p µ ν (p) − p ν µ (p).In the operator language, these correspond to gauge invariant operators involving the field strength tensor.
Another complication that is not illustrated in our example above occurs when we have identical particles.For 3-point functions, this is a simple matter of symmetrizing the amplitudes, but it is nontrivial for 4-point functions because they can depend on Mandelstam invariants.In this case, some of the primary amplitudes may contain powers of the Mandelstam invariants because the operators do not satisfy the appropriate Bose/Fermi symmetries without them.For the operators we consider, we only have identical bosons, and we discuss the relevant cases below.
Two identical bosons: We want to find a basis for the primary amplitudes M(1234) where 1 and 2 are identical bosons.We find these starting with the amplitudes where 1 and 2 are distinguishable and then symmetrizing 1 ↔ 2. To do this, we first write a basis for the distinguishable amplitudes M(1234) that do not contain any Mandelstam invariants.We then define the symmetric and antisymmetric combinations We then construct all Mandelstam descendants of these operators that are symmetric under 1 ↔ 2. This exchange acts on the Mandelstam invariants as t ↔ u, so the most general such amplitude symmetric under 1 ↔ 2 can be written as where F and G are polynomial functions of their arguments.We see that the amplitudes of the form M+ (12; 34) and (t − u) M− (12; 34) are an over-complete basis for the primary operators in this case, and the higher order terms in F and G give the descendants.
Three identical bosons: Now we want to find a basis for the primary amplitudes M(1234) where 1, 2, and 3 are identical bosons.In this case, we proceed by first symmetrizing with respect to 1 ↔ 2 as above, and then symmetrize the results with respect to the remaining symmetries.This implies that the most general symmetric amplitude has the form The amplitudes generated in this way are not guaranteed to be Mandelstam descendants of primary operators of 3 identical particles.Such descendants have the form descendants : M(123; 4) = J(stu, s 2 + t 2 + u 2 ) M(123; 4), (3.9)where M is a primary amplitude and J is a polynomial.(Note that s + t + u = 3m 2 1 + m 2 4 .)Because of this issue, we cannot claim that we have rigorously enumerated all primaries to arbitrarily high mass dimension.The Hilbert series determines the maximum dimension of the primaries if we assume that there are are no relations among operators at lower dimension (see discussion below).The results we obtain are compatible with the Hilbert series, so this would require a cancelation in the Hilbert series between the new primary operators and a constraint that appears at the same mass dimension.This appears to be unlikely, but we cannot rigorously rule it out.We emphasize that our methods correctly classify all the operators up to the highest dimension that we checked.For example, we have determined all operators of the form hγγγ and hggg up to dimension 15, and we will see that this is more than sufficient for the phenomenology of Higgs decays at the HL-LHC.

Independence of Amplitudes: Numerical Methods
We now describe the methods used to determine which of the amplitudes are independent.This means that we have to find all linear redundancies of the form Eq. (3.1).In this section we describe 'brute force' numerical methods similar to those used in previous works [48].
We start with a basis of amplitudes M a with a = 1, . . ., n.The first approach is to construct an n × N matrix X whose rows consist of the values of M a for N n values of p and s and at fixed values for the masses.This matrix can be written as where the index (p, s) runs over N kinematic configurations p, including all possible choices of the helicities s for each configuration.For each linear redundancy Eq. (3.1), this matrix satisfies C • X = 0, so the redundancies are associated with the singular values of X.
Equivalently, we can consider a rectangular matrix Y whose columns are given by derivatives of the amplitudes with respect to the independent kinematic variables, evaluated at a canonical kinematic point p 0 : Here the notation ∂ n /∂p n is schematic: it means that we consider a large number of mixed partial derivatives with respect to the independent kinematic variables (see below).We again include all possible choices of the spin variables s for each ∂ n /∂p n .We expect that this will work for any choice of kinematic point p 0 , but we chose to expand the amplitudes around threshold in several channels.
We find that both of these methods work well for moderately large matrices, typically less than around 1000 columns.However, for sufficiently large matrices, the numerical methods will find more 'nonzero' singular values because of the effects of round-off errors in the numerical calculation.This can be addressed using a smaller numerical tolerance, and checking for robustness of the results by looking at different kinematic configurations.

Independence of Amplitudes: Analytical Method
The shortcomings of the numerical approaches described above motivated us to develop an analytical approach, which we now describe.To explain it, we will need to be specific about the kinematic variables involved.In the center of mass frame for a 12 → 34 process, we can write the momenta as where ) There are 2 independent kinematic variables, which can be taken to be p i and θ, for example.
For vector bosons, the polarization vectors can be taken to have the form Here e x,y are the coefficients of the transverse polarizations (linear combinations of helicity ±1), while e z is the coefficient for the longitudinal polarizations (helicity 0).For massless vectors, only the transverse polarizations are present.
Let us first consider a 4-point amplitude involving only vector and scalar particles (no fermions).From Eqs. (3.12) and (3.14), we see that these are polynomials in the variables p i , p f , E 1,2,3,4 , sin θ, cos θ. (3.15) If these variables were independent of each other, then finding the linear redundancies Eq. (3.1) would be a simple matter of requiring that the coefficient of each monomial vanishes.However, there are in fact only 2 independent variables.Nonetheless, we show that there is a sense in which we can in fact treat the amplitude as a polynomial in a set of independent variables.
To illustrate the idea, suppose that the amplitudes were polynomials in cos θ and sin θ only.These are not independent because of the relation cos 2 θ + sin 2 θ = 1.We consider the polyomial to be a function of the two complex variable c = cos θ, and s = sin θ.We can use the relation to eliminate all powers of s larger than one, so that we can write the redundancy condition as where P (c) and Q(c) are polynomials in c and since we are working with an upper bound on the operator dimension, they are also finite polynomials.Even though s and c are not independent, we claim that the constraint that the function vanishes implies that the polynomials P and Q vanish identically, just as if s and c were independent variables.To see this, note that we can view the right-hand side of Eq. (3.16) as a function of c alone, with s = √ 1 − c 2 .For general coefficients C a , there are singularities in the complex c plane that are branch cuts starting at c = ±1.In order for this function of c to vanish identically, the coefficient of this singularity must vanish, which implies that the polyomial Q vanishes identically: (3.17) Once this condition imposed, Eq. (3.16) implies We can extend this method to include the full set of kinematic variables in Eq. (3.15).We consider the remaining variables to be a function of E cm , which we think of as a complex variable.Then p i,f are given by These have branch point singularities at 4 points, E cm = ±(m 1 ± m 2 ), ±(m 3 ± m 4 ).The energies E k can be written in terms of E cm using We can use Eqs.(3.19) and (3.20) to eliminate the dependence on E k and even powers of p i,f .The resulting function of E cm has 1/E n cm singularities, which we eliminate by multiplying by E N cm for some sufficiently large N .The result has the form of a polynomial in E cm , s, c, p i , p f with at most linear powers of s, p i , and p f : where P, . . ., W are polynomials in E cm and x.Because s, p i , and p f all have different singularity structure when written as functions of E cm and c, we can treat all of the variables in Eq. (3.21) as independent when solving the constraints, which again requires that all of the polynomials separately vanish.
Extending these ideas to amplitudes involving fermions is nontrivial because the spinor wavefunctions contain factors of E ± p i,f .We were able to extend the method to amplitudes involving 2 fermions, for special choices of the fermion masses.Taking the fermions to be the incoming particles, the spinor wavefunctions are functions of E 1,2 ± p i , for example where s = 1, 2 is the spin label and ξ 1,2 are a basis for 2-component spinors.The analytic method can be extended for the following special cases: The amplitude is proportional to the product of spinor wavefunctions for particles 1 and 2, which contain the following square root structures: The constraints therefore have the same form as Eq.(3.21).
• m 2 = 0: In this case we have so p i no longer has a branch cut singularity as a function of E cm , but p f does.The spinor wavefunctions contain the following square root structures: The amplitudes are proportional to one factor from Eq. (3.26a) and one factor from Eq. (3.26b), so the nonzero amplitudes are all proportional to √ E cm √ p i .By multiplying by √ E cm √ p i E N cm for some N , the constraints can therefore be written as a polynomial in E cm , s, c, p f that is linear in s and p f : where P, . . ., S are again polynomials in E cm and c.The same argument above therefore shows that we can treat all of the variables in Eq. (3.27) as independent when solving the constraints.
We find that both methods find the same sets of independent amplitudes with 2 fermions, and that these methods also agree with the numerical methods for generic masses.This is reassuring, since we do not expect the independent amplitudes to be different for special choices of the fermion masses.
To summarize, given that the redundancies require the polynomials to individually vanish, we can analyze the number of independent amplitudes by choosing the kinematic variables E cm , p i , p f , c, s where we treat them independently, as long as we've replaced factors of s 2 , p 2 i , p 2 f in terms of c and E cm .It would be interesting to generalize this analytic argument to general amplitudes, for example involving 4 fermions.The method relies on the fact that the singularities of the amplitudes are simple square root branch cuts.In comparison to the spinor helicity formalism, the local amplitudes are polynomials in spinor-helicity variables.These variables are also not independent, but the constraints they satisfy are quadratic polynomial equations.It is natural to speculate that this underlying structure allows us to generalize the results above beyond special kinematic points.

Hilbert Series
An important check of our results is the Hilbert series that counts the number of independent EFT operators, described in Refs.[40][41][42][43]46].The Hilbert series counts the number of operators at a given mass dimension, taking into account symmetry constraints as well redundancies due to integration by parts are field redefinitions.7The Hilbert series for our trilinear interactions are the following: Here q is a parameter that counts the mass dimension of the operators.The power of q in each term is the mass dimension of the operator, and the coefficient gives the number of operators at that dimension.So for example, H h f f = 2q 4 implies that there are 2 operators with dimension 4.
The Hilbert series for our four-point interactions are the following: ) , H hggg = 2q 7 + 2q 9 + 4q 11 + 6q 13 + 2q 15  (1 − q 4 )(1 − q 6 ) , , H hγγγ = 2q 11 + 4q 13 + 2q 15  (1 − q 4 )(1 − q 6 ) , ) , q 5 + 6q 7 + 8q 9 + 7q 11 + 5q 13  (1 − q 4 )(1 − q 6 ) , ) . (3.29) The denominators represent the infinite series of Mandelstam descendants.For the couplings where all particles are distinguishable, this factor is given by which counts the series of products of the two independent Mandelstam variables (s and t say).For couplings containing indistinguishable particles, the denominator factor is modified because the series of Mandelstam variables is constrained by symmetry.For example, for hh f f in the channel hh → f f , the independent symmetric Mandelstam invariants are s and (t − u) 2 .The denominator factor is given by which counts the series of products of s and (t − u) 2 .For hZZZ, the independent symmetric Mandelstam invariants are s 2 + t 2 + u 2 and stu, which is matched by the denominator factor This suggests that the the numerator factors simply count the number of primary operators at each dimension.While this is the simplest interpretation, it is not necessarily correct.The reason is that there can be relations between Mandelstam descendants of independent primary operators.For example, two lower dimensional primaries may become redundant at higher mass dimension when one includes enough Mandelstam factors.If there are n such relations that arise at dimension d, this is parameterized in the Hilbert series by an infinite series which subtracts off the redundant terms in the Hilbert series.(The remaining positive terms in the Hilbert series must of course ensure that the coefficient of each power of q is positive.)In fact, negative terms in the numerator of the Hilbert series appear for 4-fermion couplings, which are not considered in this work.Although all of the coefficients in the numerators of the Hilbert series above are positive, it is possible that there are relations at the same mass dimension that we have new primaries.In other words, the coefficient of q d in the numerator is equal to the number of independent primaries minus the number of relations between Mandeltam descendants that appear at dimension d.For all operators other than those that contain 3 identical particles, our methods determine all primary operators up to arbitrary mass dimension independently of the Hilbert series.In these cases, the Hilbert series is used only as a check, and we find that the coefficients in the numerators do in fact count the number of primary operators in all cases.
For the case of 3 indistinguishable particles, as discussed in §3.1, our methods do not guarantee that there are no additional primary operators at dimensions higher than we have explicitly checked.In these cases, the primary operators we find are equal to the coefficients of the numerators of the Hilbert series above, so we again have agreement with the Hilbert series.However, we cannot exclude the possibility that at higher dimensions there are additional primary operators with an equal number of additional constraints at that dimension.Even if this is the case, we have determined all primary operators up to dimension 13, and any additional operators are unlikely to be phenomenologically relevant.

Phenomenology from the Bottom Up
In this section, we discuss some of the basic phenomenology of the operators that we have found.We first show that unitarity bounds can give us an upper bound on the couplings of the SM deviations.As emphasized in [31], any new interaction that is not included in the SM implies that tree-level unitarity is violated at some energy scale, and this scale can be estimated without a complete EFT framework.Assuming an energy scale where unitarity is valid to, enables us to to give an upper bound on couplings of the interactions.In this section, we will describe the assumptions and methods that we use to obtain these bounds.We also give rough estimates of the size of physical effects of the new interactions for Higgs decays.Comparing these to the unitarity bounds gives an idea of which operators may be plausibly large enough to be observed in upcoming Higgs searches.

Perturbative Unitarity Bounds
We now describe how we place bounds on the coefficients of the primary operators from unitarity considerations.It is a classic result that the SM is the unique theory with the observed particle content that does not violate tree-level unitarity at high energies [49] (see [50] for a purely on-shell derivation).Therefore, any deviation from the SM will lead to a violation of tree-level unitarity at some scale, which can be used to bound the scale of new physics.We now turn this around to determine the allowed coefficients of the primary interactions such that the scale of unitarity violation is larger than some value, for example 1 TeV.This gives a theoretical upper bound on the deviations from the SM that can be used to decide which searches are sufficiently motivated to carry out.
As emphasized in [31,33,51,52], the unitarity bounds can be obtained from a purely bottom-up perspective (without assuming any EFT power counting), but the unitarity bounds do depend on what assumptions we make about other couplings.To illustrate this, we consider the coupling hh tt.We want to know whether this coupling could possibly be the first observed sign of new physics.In order for this to be the case, we must assume that the BSM contribution to the h tt coupling is suppressed, due to the greater sensitivity of experiments to this coupling.This assumption affects the unitarity bounds on the hh tt couplings, as we will now explain.
The strongest constraint on the hh tt coupling from unitarity violation at the highest energies comes not from the 4-particle amplitudes such as hh → tt, but from higher-point amplitudes involving longitudinally polarized W and Z bosons.This arises because the hh tt coupling ruins cancelations that otherwise ensure tree-level unitarity of these higherpoint amplitudes.As shown in Refs.[31][32][33], these can be understood at the level of the Lagrangian using the Goldstone boson equivalence theorem [49,53,54].The point is that gauge invariance implies that couplings like hh tt have associated dependence on the triplet of eaten Nambu-Goldstone fields G, and the amplitudes for the Nambu-Goldstone bosons are the same as the longitudinal W and Z bosons in the high-energy limit, which can be determined by replacing tt → 1 For simplicity, we only consider amplitudes of the form ttG m from h n tt couplings, so we can then expand the expressions above to give Note that c t,2 gives rise to amplitudes of the form ttG n for n ≥ 4, but these can be canceled by other couplings.Because we are assuming that c t,1 is small, its contribution cannot cancel the contribution to the ttG 4 and ttG 5 couplings, but the higher couplings can be canceled by the unconstrained couplings c t,n for n ≥ 3. We can therefore use the ttG 4 and ttG 5 couplings to obtain a unitarity bound on c t,2 .We see that with the assumptions that we are making, the hh tt coupling effectively behaves like a dimension 8 operator at high energies.This can also be understood from the perspective of SMEFT, as we will discuss below.
In general, we compute the unitarity bounds for 4-point couplings under the assumption that the 3-point couplings are sufficiently small that their contribution to the unitarity bound can be neglected.If a deviation from the SM is observed in any channel, one would obviously want to perform a complete analysis including all experimental constraints, but we believe that the bound we are presenting is appropriate for the purpose at hand.
To calculate the unitarity bounds from higher-point processes such as tt → G 5 , we use the results of Refs.[31,33].We will use a simplified version of these estimates that neglects some numerical factors of order 1.A coupling of n distinguishable scalars can be written and the associated scattering amplitudes are The unitarity bound on this amplitude is [33] M(φ where is the total massless phase space for k distinguishable massless particles with total energy E, where we have neglected a combinatoric factor 1/(k − 1)!(k − 2)! .By ignoring those combinatorial factors, the combination Φ k Φ n−k that appears in Eq. (4.6) is independent of k, and we do not have to optimize the number of incoming and outgoing particles.If we require that unitarity is satisfied up to some maximum energy E max , we obtain the unitarity bound For a fermion coupling we have and we obtain the bound In this way, we obtain the approximate unitarity bounds where E TeV is E max measured in TeV.Even though these estimates were obtained by ignoring combinatoric factors in the phase space and matrix elements, they agree well with the results of [33], where all such factors are included.
Which of the unitarity bounds in Eq. (4.12) is the strongest depends on the scale E max .For asymptotically large values of E max , the process with the most particles gives the strongest bound, but for low values of E max the process with the smallest number of particles dominates.If we neglect combinatoric factors, these bounds cross at the NDA scale E max ∼ 4πv ∼ 3 TeV.In the tables, we will give the unitarity bounds in terms of E TeV , since 1 TeV is roughly the scale that has been probed by measurements at the LHC.
Although every Higgs interaction can be understood from the bottom-up approach described above, we find it convenient to use SMEFT operators as a proxy for calculating the unitarity bounds in our tables.Specifically, for 3-point functions, we use the lowest-dimension SMEFT operator as a proxy, while for 4-point functions, we use a combination of SMEFT operators of lowest dimension that does not modify the 3-point functions.This is motivated by the fact that 3-point functions are generally more constrained by experiments.In the example of hh tt, we use a combination of the H † H QL Ht R and (H † H) 2  QL Ht R SMEFT operators, and assume that the deviation in h tt is suppressed by a cancelation between them.This could be viewed as an accidental cancelation, or it may be that the SMEFT power counting simply does not hold for new physics at low scales.The SMEFT approach predicts that interaction behaves as a dimension-8 operator with at most 7-particle interactions, just as we found from the bottom-up point of view.When we estimate the unitarity bounds for couplings such as hZ tt and hW tb, we will assume that they come in combinations that preserve custodial symmetry, since this gives weaker unitarity constraints.The fact that custodial symmetry is straightforward to incorporate in SMEFT is another reason we make use of it.
Let us illustrate the use of SMEFT operators to obtain the unitarity bounds with the example of the coupling hZ µ tL γ µ t R .We assume that the 3-point coupling Z µ tL γ µ t R is not modified, so this requires a cancelation between the SMEFT operators (H where the additional factors of m Z and v on the right-hand side come from expanding the Higgs doublets and covariant derivatives. 8(We are ignoring order-1 numeric factors, since we are performing a rough calculation.)We see that at high energies, the unitarity growth is that of a dimension-8 operator, and that we can consider amplitudes with a maximum of 7 particles.The fastest energy growth at high energies can be read off from the amplitude Processes with 7 particles such as tt → ZG 4 trade one derivative (power of energy) with an additional Z boson and give a slightly weaker bound at high energies.At lower energies, the bound comes from the processes such as As mentioned above, with these approximations all of the unitarity bounds become degenerate at E max ∼ 4πv ∼ 3 TeV, so it is sufficient to compute the one with the bounds for the processes with the largest and smallest number of particles.

Precision Electroweak Constraints
Precision electroweak measurements also give stringent constraints on corrections to the SM.In our approach, primary operators that are not directly constrained by precision electroweak measurements are simply treated as independent.For example, µ decays constrain one linear combination of the W ¯ ν couplings, but allow large deviations in individual couplings if there is a cancelation in the combination that controls the µ decay rate.From a bottom-up perspective, precision electroweak constraints are similar to naturalness constraints, since they can be satisfied by fine-tuning different contributions to the same process.
However, the degree of cancelation required to obtain an observable signal is an important factor in deciding which observables are sufficiently well-motivated to merit further investigation.We therefore performed estimates of loop-induced precision electroweak corrections, even though we are not working in a complete EFT framework.That is, we treat the primary operators as interaction terms in an SU (3) C × U (1) EM invariant EFT, and estimate the size of loop corrections with a UV cutoff Λ that we identify with the scale of new physics.We have not analyzed all of the primary operators, but we generally find that requiring the absence of cancelations in precision electroweak observables gives weaker constraints than the unitarity constraints as long as we assume that the new physics satisfies custodial symmetry.
As an example of a strong constraint in the absence of custodial symmetry, we consider the operator hhZ µ Z µ .Closing the Higgs loop gives a quadratically divergent contribution to the Z mass.If this is not canceled by a custodial preserving contribution to the W mass, we obtain the constraint on the coefficient where Λ TeV is the cutoff in TeV units and we are using the operator numbering in Table 12.If we identify Λ with the unitarity violating scale E max , the precision electroweak constraint is stronger than the unitarity constraint for Λ < ∼ 40 TeV (see Table 12).Approximate custodial symmetry can significantly weaken this constraint, but its implementation in EFT is subtle (see [55]).Therefore, we will not attempt to estimate corrections to precision electroweak observables that are sensitive to custodial symmetry violation.
We now some examples of the precision electroweak constraints for some of the operators that are the most promising for Higgs decay phenomenology (see §5.3 below).For example, the CP-even operators hZ µ Z µ and hZ µν Z µν give a 1-loop contribution to the Z kinetic term, generating a correction equivalent to the S parameter.This gives the constraints where we have used the operator numbering in Table 1.These are weaker than the corresponding unitarity constraints.
Next, we consider the CP-even hZf f couplings in Table 3.At one loop these induce a correction to Zf f couplings, which are highly constrained by LEP.Operators 1 and 2 induce a correction to the vector and axial-vector Z couplings, and give which are comparable to the unitarity bounds for Λ ∼ TeV, but are otherwise weaker.
Operator 5 corrects the coupling iZ µ ψ↔ ∂ µ ψ, which flips the fermion helicity.This has a weaker constraint at LEP because it does not interfere constructively with the SM Z coupling.Using the results of [56], we find the weak constraint Operator 7 corrects the coupling ∂ µ Z µ ψψ, which vanishes on shell.To get a nonzero correction, we must go to higher loop, and this will give weak constraints.Operators 9 and 11 correct the coupling i Zµν ψγ µ ↔ ∂ ν ψ, which gives the constraint which is weaker than the unitarity bound.
The general pattern that we find is that the unitarity bounds are more sensitive to the UV scale than the precision electroweak observables, at least if we neglect the corrections to the W and Z masses that violate custodial symmetry.It would be interesting to give a more complete analysis, including constraints on CP-odd operators, but we leave this for further work.

Estimates for Higgs Decays
We now perform some crude estimates determine what ranges of BSM couplings can be probed in Higgs decays at the LHC.Specifically, we will estimate the corrections to the branching ratios of Higgs decays to determine which operators can give an observable number of Higgs decays.These couplings can then be compared to the unitarity bounds discussed above to determine whether it is motivated to search for a particular coupling.
We will focus on operators that are not present in the SM.In the case where the BSM operator O modifies a Higgs coupling, the phenomenology can be studied in the so-called 'κ framework' [57].The κ parameter associated to O is given by Projections for the sensitivity of the HL-LHC to various κ parameters can be found in Ref. [58].We will therefore focus on couplings that are not present in the SM.
We are interested in the sensitivity to Higgs decays at the HL-LHC, where we expect about N h ∼ 10 8 Higgses to be produced with 3 ab −1 .Estimating the SM Higgs branching ratios to the decays we consider, we find that they all have branching ratios larger than 10 −8 so that all of these searches have a SM background.Thus, looking at total decay rates, we should compare the new contribution to the fluctuations in the SM Higgs background.
If this is satisfied, there is at least the possibility to distinguish the new contribution from the SM Higgs background.
We begin by considering the case where the interference between the BSM and the SM contribution is negligible.This may occur because the SM contribution is so small that the BSM contribution dominates.Another interesting case is where the BSM contribution is CP odd.If the measurement performed is sufficiently inclusive that it weights CP conjugate final states equally, the interference term between CP-even and CP-odd amplitudes cancels.This occurs for example in the total rate summed over final state spins.Measurements of differential distributions may be sensitive to interference terms, but these are beyond the simple estimates performed here and should be studied on a case-by-case basis.
We will estimate the size of the BSM contribution assuming that the matrix element of the decay is constant, and that the decay is not phase space suppressed.The matrix elements for 2-and 3-body decays due to the insertion of a BSM operator O are then approximated by where d O is the dimension of the operator O.The corresponding decay rates are approximated by To be of interest, we need to compare this deviation to the fluctuations in the SM Higgs background Eq. (4.22), which is conservative since many of these will have additional backgrounds.This gives the bounds 2-body, no interference: where BR SM is the branching ratio of the decay in the SM.The estimates for higher-dimension operators are more uncertain due to the high powers of ratios of scales involved.Now we consider the case where there is significant interference with the SM.In this case, we obtain a rough estimate by also approximating the SM amplitude as a constant.For example, for 2-body decays this gives The correction to the decay rate due to the BSM operator O is then To be observable, the difference in the number of Higgs decays compared with the SM must be larger than the fluctuations in the SM background, as in Eq. (4.22).In this case, we find that the dependence on Γ SM cancels out in the bound, and we obtain the bounds: 3-body, interference: Note that comparing to the no interference case, we see that when there is interference it allows better coupling sensitivity since we've estimated that BR SM 10 −8 .
These approximations made above are very crude, and are intended only as a rough guide.It will be interesting to compare them with detailed phenomenological studies, but we leave that for future work.In §5, we will combine these estimates with the unitarity bounds to identify some BSM operators that are worthy of further study.

Results
In this section, we present our results for the independent primary operators for the 3point and 4-point amplitudes.We do not consider flavor-violating operators and leave such generalizations for future work.Equivalently, our results are presented for a single generation of quarks and leptons.This section consists mainly of the tables of operators, with some brief comments in the main text.We then use the results to discuss the most promising primary observables for Higgs decays.

3-Point Couplings
We begin with the 3-point couplings.These are equivalent to on-shell 3-point amplitudes (for complex momenta), which have no Mandelstam invariants.Therefore, all 3-point functions correspond to primary observables in our terminology.This problem has been previously studied by many authors, see for example Refs.[1,4,34].Our main focus is the enumeration of the 4-particle observables, but we have taken a fresh look at the 3-point functions to check our approach.
The 3-point functions involving the Higgs boson are shown in Table 1, and the additional 3-point functions needed for Higgs processes that do not involve the Higgs boson are shown in Table 2.The table gives the CP of the operator, a SMEFT operator that contains the interaction, and the unitarity bound for the coefficient of the operator, where the normalization for the couplings is defined by Eq. (2.4). 9 We have attempted to identify a SMEFT operator of lowest dimension, but this can be nontrivial and our method for determining it isn't systematic.Finding lower dimension operators would weaken the bounds for unitarity constraints for large E max .To connect to the 'Higgs basis' [24], we note that the interactions that appear at dimension 6 in SMEFT can be read off from our tables.In some cases, we find that some four point functions in [24] are redundant due to field redefinitions.
For the triple gauge boson couplings, we note that our approach differs from the classic work [59] in that we are performing a systematic low-energy expansion of the kinematic dependence.As explained in §2.2 above, this necessarily involves an interplay between 3point and higher-point couplings.We have put the effects of possible 'form factors' of our 3-point couplings into higher-point couplings.Ref. [59] instead defines this in terms of form factors whose momentum dependence must be specified to define a model for experimental searches.In particular, they include form factors for couplings of the form O W W Z 4,5 with Z µ replaced by A µ even though these couplings are not U (1) EM gauge invariant.(They restore gauge invariance by using a specific non-local form factor for these couplings that contains massless poles.)We believe that our approach is more physically transparent and can be systematically matched to EFT frameworks such as SMEFT.
ETeV , 20 TeV Table 1: Three point functions that involve the Higgs boson.We write , while F µν and G µν are the field strength tensors for the photon and gluon, respectively.We have omitted the color indices of the gluon fields.The last column gives the maximum allowed value for the coupling c defined in Eq. (2.4) allowed by tree-level unitarity, where E TeV is the unitarity violating scale in units of TeV.

4-Point Couplings
Our results for 4-point operators are summarized in Tables 3-13.The notation is hopefully self-explanatory; to save space, we have used There are several cases for which we do not provide separate tables, because the operators can be read off from other tables by simple substitutions: • hg f f can be obtained from hγ f f in the bottom part of Table 3 with the substitution The operators are SU (3) C gauge invariant only if the fermions are quarks.
• hZgg can be obtained from hZγγ in Table 4 with the substitution • hhZZ can be obtained from hhW W in Table 12 by replacing W µ → Z µ .When this is done, the operators numbered 5, 7 and 8 vanish by symmetry, so there are only 6 nonzero operators in this case.
• For hhhh the only primary operator is h 4 , and we have not made a table for that.
There are other cases where the results are closely related, but additional corrections must be made.For example, we can take operators involving Z and convert them to operators with a photon, by taking Z µ → A µ and forming the field strength for the photon by using derivatives and anti-symmetrizing.This allows hZ f f, hZgg to be respectively converted to hγ f f and hγgg.
The tables list the primary operators.In the on-shell amplitude language, the remaining amplitudes are obtained by multiplying each operator by a power series in the Mandelstam variables.In the operator language, these correspond to operators with additional derivatives with the Lorentz indices contracted between them.For operators where all particles are distinguishable, this is simply a series in the Mandelstam variables s, t, and u (with s + t + u fixed).For operators with identical particles, these additional terms must be appropriately symmetrized.For hZγγ, hγgg (Table 4), hZZγ (Table 8), hh f f (Table 11), hhW W , hhZγ and hhγγ (Table 12), we can add arbitrary powers of s and (t − u) 2 .For hγγγ (Table 5), hggg (Table 6), hZZZ (Table 10), hhhZ and hhhγ (Table 13), we can add arbitrary powers of s 2 + t 2 + u 2 and stu.As an example, adding a factor of s 2 + t 2 + u 2 to h∂ µ Z ν Z µ Z ν can be done by adding four derivatives, i.e.
The tables give unitarity bounds on the coefficients of the operators (see §4.1).As one might expect, the unitarity bounds become more stringent for operators of higher dimension.These bounds should be used only as a very rough guide, especially for the operators with high mass dimension.
Our final results are in full agreement with the Hilbert series counting in all cases (see §3.4).We also agree with the results of Ref. [2] in all cases where they overlap.We found a discrepancy in the results for hW W Z (see Table 9) in an earlier version of their paper, but our results agree after they identified and corrected a mistake.Our results also include massless particles, the effects of symmetrization for identical particles, and we have found all primary operators to arbitrarily high dimension, at least in the cases where there are two or fewer identical particles (see §3.1).

Primary Observables for Higgs Decay
We now use the results in the tables to identify promising primary observables to search for new physics in Higgs decays.We limit ourselves to CP even operators, so that it is clear that there is interference with SM processes.(Also, CP-odd new physics effects may be suppressed by approximate CP symmetry.)In this case Eq. (4.28) gives an estimate for the minimal value of the coefficients in order for the new contribution to the decay to be observable at the HL-LHC.We compare this to the bound on the coefficient arising from the unitarity bounds in the tables.
In this way, we find that the following operators are potentially observable at the LHC assuming a unitarity violating scale above 10 TeV: where V = W, Z.The next class of operators are those that are potentially observable with a unitarity violating scale between 1 and 10 TeV: . (5.3) These are presumably already constrained, and not as theoretically motivated as the others.
We see that there are a large number of observables that worthy of further investigation.This motivates searches for BSM effects in Higgs 2-body decays, as well as 3-body decays to Z f f , W f f , γ f f , g f f , Zγγ, γgg, and ggg.The decays to strongly-interacting particles are likely very challenging due to QCD backgrounds that we have neglected.We note that some detailed phenomenological studies on the effects of higher-dimension operators on 3-body decays have already been performed.For example, [60] considers effects equivalent to some of the operators above in the decay h → e − e + µ − µ + , but not all of them.We leave further detailed study of these effects for future work.

Conclusions
This paper has analyzed the most general observables that parameterize the indirect effect of new heavy physics at colliders.An important conceptual point is that the space of these observables is finite, with a finite basis that can be enumerated.This can be most easily seen in the language of on-shell amplitudes: any local amplitude can be written as a linear combination of a finitely many 'primary' amplitudes, each of which is multiplied by an infinite series in Mandelstam invariants.Under very general physical assumptions, the additional Mandelstam invariants are suppressed by powers of a heavy mass scale M , and the leading approximation is given by the first nonzero term in this expansion.Each primary amplitude can be associated with a local operator, up to the usual ambiguities from integration by parts and integration by parts.However, these ambiguities do not change the on-shell amplitude, so we can make the simplest choice when defining the operator basis.
The major results of this paper are a systematic method for determining all primary operators, and an explicit determination of the 3-point and 4-point primary operators relevant for Higgs signals at colliders.The 3-point on-shell amplitudes have no Mandelstam invariants, so there is a finite list of 3-point operators, which has previously been found in the literature Refs.[1,4,34].Partial results for primary 4-point functions have been given in [1,2], and our results agree where they overlap.
The correspondence between local on-shell amplitudes and EFT operators has been invaluable in this work.For example, we found that if the on-shell amplitudes are expressed in a specific set of kinematic variables, the amplitudes can be treated as polynomials in the kinematic variables for purposes of determining the linearly independent amplitudes.This allows us to efficiently and reliably determine the independent amplitudes.The Hilbert series that counts independent operators is also an invaluable check on these methods.The primary operators are a natural set of observables for searches for new physics at colliders, and they can be matched onto theoretical models or EFT frameworks (such as SMEFT or HEFT).We have considered the unitarity and precision electroweak constraints on these observables, and made a first pass at determining which may be promising for searches for new physics in Higgs decays.In particular, the three-body decays into Z f f , W f f , γ f f , and Zγγ are estimated to be of interest at the HL-LHC.Investigating the phenomenology of these observables is an obvious direction for future work.
It is our hope that this framework will prove useful for the LHC program of constraining (or discovering!) the indirect effects of new particles too heavy to be produced.Under the general assumptions made here, the primary observables are independent of each other, and experiments can measure them without worrying about correlations with other observables.These results can then be compared with predictive theoretical frameworks.In subsequent work, we plan to study experimental strategies for carrying out such searches and reporting the results in a way that can be compared with searches in other channels, or in future colliders.
Note: As this paper was being finalized, Ref. [61] appeared on the arXiv, which overlaps with some of our results.This paper presents a general method for obtaining independent HEFT operators, but does not distinguish between primaries and descendants.They give results for operators only up to dimension 8, and these operators are in agreement with our results.
R 5 / K Y b 7 a T C l C a o w r L W b 0 5 E o R d L s 7 L y / j t s w v D G 8 e 6 c 7 q B r + / c r P T D g H K 6 g A x 7 0 w I c n 6 E M A D B A + 4 B O + 4 I d c k g 6 5 X r 2 2 y H p z B o 2 Q 2 1 9 q D W I L < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " E d 7 n B 4 d d 4 d U j w M T + i

Table 2 :
Additional three point functions needed to calculate 4-point amplitudes involving the Higgs.The notation is the same as in Table1.Here σ µν = i 4 [γ µ , γ ν ], σ a are Pauli matrices, and

Table 3 :
Primary operators for couplings of the form hZ f f and hγf f .As noted in the text, the hW f f operators can be obtained from the hZ f f operators by the replacement Z f f → W f f , and the hg f f operators can be obtained from the hγ f f operators by the replacement F µν → G µν .Dν D ρ HB µν Bµρ + h.c.F νρ Fµσ∂ρ∂ σ Zν − iH † DρD σ Dν H∂ µ B νρ Bµσ + h.c.

Table 4 :
Primary operators for hZγγ.As noted in the text, the hZgg operators can be obtained from these by the replacement F µν → G µν .