Constructing massive on-shell contact terms

The purely on-shell approach to effective field theories requires the construction of independent contact terms. Employing the little-group-covariant massive-spinor formalism, we present the first systematic derivation of independent four-point contact terms involving massive scalars, spin-1/2 fermions, and vectors. Independent three-point amplitudes are also listed for massive particles up to spin-3. We make extensive use of the simple relations between massless and massive amplitudes in this formalism. Our general results are specialized to the (broken-phase) particle content of the electroweak sector of the standard model. The (anti)symmetrization among identical particles is then accounted for. This work opens the way for the on-shell computation of massive four-point amplitudes.


Introduction
On-shell methods find a natural application in the framework of effective field theories (EFTs), which provide a model-independent description of particle interactions, positing just the low-energy particle content and symmetries. In the context of the standard-model effective field theory, they have for instance been used to elucidate the patterns observed in the renormalization of dimension-six operators [1] and in their interference with the standard model [2], which are somewhat obscured in the Lagrangian approach, and to derive anomalous dimensions beyond leading order in the loop and EFT expansions [3][4][5][6]. Many of these applications only involve massless amplitudes, for which on-shell techniques are especially powerful. EFTs describing massive particles are of particular importance however. At energies probed experimentally, the masses of standard-model particles are often relevant and the interactions of the heaviest ones may hold the key to a better understanding of electroweak symmetry breaking. New particles postulated beyond the standard model are moreover typically massive. Whether these masses are sub-eV or above TeV, their effects are often important. The little-group-covariant massive-spinor formalism of ref. [7] (see also ref. [8]) facilitates the study of these theories from an on-shell perspective.
Beyond the applications described above, the on-shell program allows for an alternative description of general EFTs, substituting the classification and enumeration of independent operators by that of independent on-shell amplitudes [9][10][11][12][13][14][15][16][17][18]. A general n-point tree-level amplitude can in principle be sequentially derived from three-point amplitudes, as well as four-and higher-point contact terms. Both renormalizable and non-renomalizable three-point amplitudes must first be identified. These can be glued together to form the factorizable parts of tree-level four-point amplitudes whose analytic structure is characterized by single poles due to internal propagators [7,19]. For recursion relations involving massive particles, see e.g. refs. [20][21][22]. The remainder of the four-point amplitude consists of pole-free non-factorizable contact terms which map to higher-dimensional operators in a Lagrangian formulation. Their classification therefore replaces that of EFT operators contributing to the four-point amplitude of interest. Repeating this process to form higherpoint amplitudes and isolating their local contact terms, one can characterize general EFTs and construct their amplitudes up to a given dimension. For massive amplitudes, a general procedure for the construction of contact terms is lacking. In this paper, we take a first step in this direction, deriving bases for four-point non-factorizable contact terms in the massive-spinor formalism of ref. [7].
We cover general three-point amplitudes for spins up to 3, and four-point amplitudes involving scalars, spin-1/2 fermions, and vectors. Our results extend and complete several previous analyses. Four-point contact terms featuring a massive scalar or vector plus gluons were classified up to dimension-13 in ref. [11]. The massive three-point amplitudes generated by bosonic dimension-six standard-model operators were presented in ref. [14]. Several four-point contact terms required to restore unitarity were also identified in ref. [14] (see also ref. [23]). Renormalizable [24] and non-renormalizable [15] contributions to general three-point amplitudes featuring particle of the standard-model electroweak sector were classified. Reference [15] also derived the general four-point fermion-fermion-vector-scalar contact terms. Massless contact terms were classified in ref. [16] for spins 0, 1/2, 1, 2 and up to dimension-eight. An alternative classification of independent massless amplitudes in terms of momentum twistors was presented in ref. [18].
We will discuss three types of bases: a. the spinor-structure bases comprising the basic independent building blocks which span generic amplitudes. These replace the standard independent Lorentz invariants constructed out of external polarizations; b. the analogous stripped-contact-term (SCT) bases, comprising the basic building blocks required for spanning the pole-free non-factorizable amplitudes; and c. the contact-term bases which span the full non-factorizable amplitudes. The latter are relevant for EFT classification and on-shell computations. Throughout, we exploit the massless limits of massive spinor structures. In the formalism of ref. [7], the spinors associated with massive particles are bolded, to imply symmetrization over little-group indices, while those of massless particles remain unbolded. The massless limit can then be obtained, in most cases, by unbolding the massive structures [7]. In fact, the reverse process, which we refer to as bolding, yields an elegant derivation of spinor-structure bases. These are obtained by starting with a judicious choice of independent massless amplitudes and covariantizing them with respect to the massive little group.
We apply our results to derive the general four-point amplitudes of the standard-model electroweak sector, for one generation, without imposing SU(2) L ×U(1) Y . We list all contributions of dimension ≤ 6, as well as several other contributions of interest, which only arise at higher dimensions. The symmetrization over identical electroweak bosons greatly A schematic description of our analysis. A set of spinor structures can be reduced to a spinor-structure basis, whose dimension is determined by counting 3d conformal field theory (CFT 3 ) correlators. The P -matrix of inner-products is useful in this process, and all redundancies are removed. Alternatively, the set of spinor structures can be reduced to a stripped-contact-term (SCT) basis, by eliminating local linear combinations, namely those with prefactors containing only positive powers of the Lorentz invariants and masses. This is appropriate for the construction of manifestlylocal contact terms. Finally, the contact-term basis, which corresponds to an operator basis, is obtained by appending positive powers of the independent Lorentz invariants to the elements of the SCT basis, and dropping newly redundant combinations. constrains the minimal dimension at which certain structures can first appear. Since the standard-model gauge symmetry is not imposed, the contact-term bases we derive can be used to parametrize general EFT extensions of the standard model, including both stronglycoupled theories, with the EFT scale near the electroweak scale, and weakly coupled extensions, featuring hierarchically-separated scales, with the Higgs VEV v ≪ Λ. The results furthermore automatically include the full v/Λ expansion, thus encoding the geometric interpretation of generic standard-model EFTs in the Lagrangian formulation [25][26][27][28].
The basics of our approach are outlined in section 2. We then turn to a classification of the independent spinor structures contributing to three-and four-point amplitudes, without imposing any symmetry requirements apart from Lorentz, and treating all particles as distinguishable. Section 3 addresses three-point amplitudes. In section 4, we study four-point amplitudes, starting with a quick review of the massless case, and proceeding to amplitudes involving massive scalars, spin-1/2 fermions and vectors. These general results are then specialized to the particle content of the electroweak sector of the standard model in section 5. At this stage, the (anti)symmetrization required by spin statistics is in particular accounted for, while colour and flavour structures are left implicit. A roadmap of the different basis constructions and their inter-relations is sketched in fig. 1.

Basics
We use the little-group-covariant massive-spinor formalism [7]. A massive amplitude carries 2s i symmetrized little-group indices for each external particle of spin s i . This symmetrization is implicit in the boldface notation of ref. [7]. We use i, j, . . . to label external particles, and I, J, . . . = 1, 2, to denote massive little-group indices. Apart from section 5, we treat external particles as distinguishable.
The non-factorizable part of the amplitude can be written as a sum, where {I} collectively denotes the little-group indices of all external particles, while L n is a polynomial of the Lorentz invariantss ij = 2p i · p j and, for five or more external legs, of the antisymmetric contractions of four momenta ǫ(p i , p j , p k , p l ) ≡ ǫ µνρσ p µ i p ν j p ρ k p σ l . Each spinor structure S {I} n is the minimal product of angle and square spinor brackets, possibly containing momentum insertions, obtained by stripping the contact term off all Lorentz invariants. For clarity, we always consider manifestly local contact terms which involve no negative power of Lorentz invariants (or spinor products, in the massless case). Our main objective is the construction of contact-term bases, spanning eq. (1). As an intermediate step, we identify the sets of independent spinor structures {S {I} } from which contact-term bases can be generated by multiplying each element S {I} by Lorentz invariants. We refer to these sets as bases of stripped contact terms (SCTs). For the purpose of constructing an SCT basis, a spinor structure is redundant if it can be expressed as a linear combination of the other structures, with prefactors involving only non-negative powers of Lorentz invariants and masses (in the massive case). Locality can therefore be kept manifest at all stages. From an EFT point of view, forbidding negative powers of dimensionful quantities also avoids removing lower-dimensional operators in favour of higher-dimensional combinations. Inverse powers of the masses are only required in the i [i/m i combinations appearing in the polarization vectors of particles of spin-1 and higher.
Another basis of interest is the spinor-structure basis, which can be used to span a generic amplitude. Here a different notion of redundancy is appropriate, with no restriction imposed on the prefactors. In particular, the prefactors generically take the form of rational functions of the invariants in this case. Bases of SCTs are therefore typically larger than spinor-structure bases. Note however that an SCT basis cannot be obtained simply by adding some elements to a spinor structure basis (see discussion in section 4).
As we will see, the construction of the spinor-structure bases can directly rely on the massless limit of spinor structures. Indeed, the transparent relations between massless and massive amplitudes in the spinor formalism [7] will be heavily exploited throughout. Massless amplitudes are particularly simple. For each helicity amplitude, the spinor-structure basis collapses to a single element. All spinor products of identical little-group weight are equal, up to a rational function of Lorentz invariants. Using the massless relations between spinor products, including Schouten identities, it is straightforward to identify SCT bases for massless amplitudes [16].

Three-point amplitudes
Formally written in terms of complex momenta, massless three-point amplitudes are unique. One spinor structure at most is allowed for each helicity configuration: (2) The spinor structures trivially map to contact terms as all Mandelstam invariants vanish in massless three-point amplitudes. Non-analyticity (i.e. non locality) is allowed in these unphysical objects, and disappears when they are consistently glued into physical four-point amplitudes.
A formula for the construction of massive three-point amplitudes was provided in ref. [7]. However, the spinor structures it yields are neither mutually independent, nor of minimal mass dimension. The example of the three-vector W W Z amplitude detailed in ref. [15] is illustrative in this respect. As in the massless case, independent spinor structures and contact terms are identical since Mandelstam invariants are functions of the masses. A counting derives from angular momentum conservation. The number of independent massive three-point spinor structures corresponds to that of irreducible representations in the addition of the three spins [15,29]. For s 1 ≤ s 2 ≤ s 3 spins, eq. (4.42) of ref. [29] gives: The number of parity-even and parity-odd structures are respectively found to be (n 3-pt + 1)/2 and (n 3-pt − 1)/2. Let us proceed to the systematic construction of those independent spinor structures. The resulting amplitudes, for spins up to 3, are listed in Table 2. Similarly to the massless case, one can start from an ansatz involving spinor bilinears. The little-group constraints require that each i], i spinor appears 2s i times. No distinction is made between square and angle spinors at this point and we denote them jointly as i) ≡ i] or i . Schematically, our starting point is thus This expression is only possibly sensible when all powers are positive. This occurs for i s i − 2 max j (s j ) ≥ 0, i.e. when the highest spin is smaller than the sum of the others. In our schematic notation, any positive power can be distributed on either angle or square bracket bilinears: (ij) n represents any [ij] k ij n−k product for n + 1 values of k = 0, ..., n. The ansatz above thus contains terms in total. When the highest spin is equal to the sum of the two others, i.e. i s i − 2 max j (s j ) = 0, this number actually matches that of independent massive three-point spinor structures given in eq. (3). For examples of massive three-point amplitudes already worked out explicitly in the literature (e.g. in ref. [15]), one can verify that eq. (4) then generates the correct independent spinor structures. The ansatz of eq. (4) notably fails when the highest spin is larger than the sum of the two others: i s i −2 max j (s j ) < 0. The largest spin is in this case irrelevant to the counting of independent structures in eq. (3); each irreducible representation in the addition of the two smallest spins contributes to the counting by as many units as its dimension; the sum of these dimensions is however equal to the product of the dimensions of the two lowest spins. So n 3-pt = i (2s i + 1) (2 max j (s j ) + 1) in this case. The i s i − 2 max j (s j ) < 0 condition is equivalent to the one introduced in ref. [16] indicating that momentum insertions are required to form massless non-factorizable contact terms. While no momentum insertion was possible in the massless case, one single structure of the [j(k − l)j form is available in the massive case. Its use allows to reduce the number of spinors left to be contracted for max j (s j ) by an even number. The remaining contractions are that of a spinor structure with maximal spin max(s i ) reduced by any integer. Reducing it until it equals the sum of the other two spins, i.e. i s i − 2 max(s i ) = 0, the ansatz of eq. (4) is again applicable.
For a somewhat more subtle reason, the ansatz of eq. (4) also fails when the highest spin is smaller than the sum of the two others: i s i − 2 max j (s j ) > 0. This only occurs when min(s i ) ≥ 1. For example, in the case of three spin-1 massive vectors, one would expect 8 different possibilities: It was however shown in ref. [ equality then applies. This means that the three-vector amplitude has 8−1 = 7 independent structures. Note eq. (7) can be rewritten as a parity-odd contraction of polarization tensors and momenta ǫ(ε 1 , ε 2 , ε 3 , p 1 + p 2 + p 3 ) which vanishes trivially because of momentum conservation. 1 It can be multiplied by additional spinors. One relation is then induced for each possible contraction of these. They are given by the ansatz of eq. (4) for all spins reduced by one unit. Using eq. (5), one counts j ( i s i − 2s j ) possible contractions. Eventually, our constructive procedure yields the number of independent massive threepoint spinor structures expected from eq. (3) and the number of irreducible representations in the spin sum: 1 Dividing both sides of eq. (7) by m1m2m3 and taking the massless limit, one obtains the so-called Dual Ward Identity, or U(1)-decoupling identity [30] for the colour-ordered amplitudes, The assignment of operator dimensions for massive three-point amplitudes is nontrivial and requires a study of their high-energy limit, as performed in ref. [15] for examples relevant to the electroweak sector of the standard model. Specific combinations of spinor structures are singled out in this process.

Four-point amplitudes
Before addressing the construction of massive amplitudes, let us first recall some properties of massless contact terms.

Massless case
In the massless case, spinor structures factorize into products of angle-and square-brackets, since momentum insertions can be decomposed as [ijk = [ij] jk . Different spinor structures contributing to the same helicity amplitude are related by rational functions of the invariants. Thus, the spinor-structure basis for each helicity amplitude is one-dimensional. The SCT basis from which manifestly local contact term can be built is typically larger. Thus for example, to span the four-fermion (++++) amplitude arising at dimension-six, both [13][24] and [14][23] are needed, even though these structures are related via, At dimension eight however, the sum ofs 14 [13][24] ands 13 [14][23] is redundant. The SCT basis in this case therefore has two elements, which can be chosen as [13][24] and [14] [23].
Since there are two independent Lorentz invariants in four-point amplitudes, the full set of contact terms can be obtained from these, multiplied by powers of, for instance,s 13 and s 14 , and dropping the combination in eq. (9) starting at dimension eight. A systematic construction of massless n-point SCTs generated by operators of dimension eight at most was performed in ref. [16]. We focus here on massless four-point amplitudes, but include all dimensions. The Schouten identities, are used to eliminate occurrences of both [14][23] and 14 23 . We employ the following procedure to account for momentum conservation. Products of the type The resulting SCT bases for particles of spin-2 at most are displayed in Table 3, 4, 5, together with the lowest dimension at which each term is generated. To our knowledge, this is the first time bases involving spin-3/2 particles are presented. For spins higher than one, we use the convention, where n is the number of external legs. The full set of contact terms, including higherdimensional contributions, are obtained by dressing these spinor structures with positive powers of Lorentz invariants, and using relations like eq. (9) to remove redundant terms.

Massive case
We now turn to the main part of the paper, namely the construction of massive contact terms. We focus primarily on four-point amplitudes involving scalars, spin-1/2 fermions and vectors, generically denoted s, f , v, respectively.

Spinor-structure bases
We begin with an analysis of spinor-structure bases, and discuss two methods for their construction. One is purely based on massive spinor structures, and the second relies on their massless limits. The tools we develop are then applied to the derivation of SCT bases, which is somewhat more involved.

Dimension
The dimension of the spinor-structure basis in four-and higher-point amplitudes can be derived by counting conformal correlators in three dimensions [10,31,32] and is given by It is interesting to note that, in the four-point case, this result matches the counting of independent three-point amplitude gluings (see also sec. 5.5 of ref. [7] for an approach similar in spirit). To see this, introduce a fictitious mediator of spin S ≥ max{s 1 + s 2 , s 3 +s 4 } and consider the factorization of the four-point A 4 (s 1 , s 2 , s 3 , s 4 ) amplitude on the two A 3 (s 1 , s 2 , S) and A 3 (s 3 , s 4 , S) three-point ones. As discussed in the previous section, there are (2s 1 + 1)(2s 2 + 1) independent A 3 (s 1 , s 2 , S) amplitudes, and (2s 3 + 1)(2s 4 + 1) independent A 3 (s 3 , s 4 , S) ones. So we recover the stated result for the dimension of the spinor-structure basis. It would be interesting to examine whether such a procedure can be generalized to higher-point amplitudes.
As it also applies to higher-point amplitudes, the counting above implies that no new spinor structure is required in amplitudes featuring additional scalars. Thus for example, the f f ss and vsss spinor structures also form complete sets for f f sss and vssss amplitudes.
Purely massive construction Spinor structures of a definite particle content form a vector space on the tensor-product space of their little-group indices. An inner product can therefore be defined as ( This can be viewed as the spin-summed interference between two structures, and is a scalar function of the masses, Mandelstam invariants and, for n > 4, fully antisymmetric ǫ(p i , p j , p k , p l ) contractions. Given a set of spinor structures {S m }, we can construct the matrix of inner products, or P -matrix [33][34][35], P mn = (S m , S n ). All its elements are independent if det P = 0. A set of i (2s i + 1) spinor structures with a non-vanishing P -matrix determinant therefore forms a basis. A spinor-structure basis can then be constructed by iteratively drawing structures from an initial over-complete set. Such an initial set can easily be formed by multiplying spinor contractions with zero, one, or two momentum insertions (namely, [ij], ij , ijk], and ijki , [ijki]) while ensuring that 2s i spinors appear for each particle of spin s i . Note that, in four-point amplitudes, structures like ijkl can be reduced using momentum conservation.
P -matrix eigenvectors of vanishing eigenvalues yield relations between spinor structures. When just one structure is redundant, the relation S m = j =m x j S j can be obtained by solving the j =m P ij x j = P im linear system for x j . This method notably allows for an algorithmic derivation of relations like those of eqs. (7), (17), (18), and (19). It also applies to higher-spin and higher-point amplitudes. Large systems, where analytical solutions becomes intractable, can be studied numerically. Thus for example, relations between spinor structures can be extracted by sampling over masses and momenta, to fit a sufficiently general ansatz.
Bolding: a massless-based construction Massive spinor structures can be classified according to the helicity configuration of their unbolded massless counterparts. 2 The vvf f spinor structure 12 [21][34] for instance unbolds tos 12 [34], and can therefore be labelled as (00 1   2   1 2 ), or just (00++) when there is no ambiguity about the spin of each particle. We will refer to these labels as helicity categories.
A given massive amplitude has exactly i (2s i + 1) helicity categories. Since massless spinor structures of different helicities do not interfere, the P -matrix associated to any set becomes block-diagonal in the massless limit, with one block per helicity category. It is therefore straightforward to construct a spinor-structure basis by selecting one representative spinor structure in each helicity category. The resulting set contains i (2s i + 1) elements which are guaranteed to be independent, since their P -matrix is diagonal for zero masses. If none of the initial structure vanishes in the massless limit, all diagonal elements are also non-vanishing. This method applies to higher spins and higher-point amplitudes as well.
Note that the representative spinor structures in each helicity category can moreover be derived from massless amplitudes. To do so, one simply bolds all spinors. To form massive vector amplitudes, massless scalar amplitudes featuring momentum insertions are required, with the replacement p i → i [i. In this sense, a massive spinor-structure basis can be obtained simply by bolding massless amplitudes. What is at work behind this magic is of course the little group. The massless amplitudes constitute certain components of the massive ones. The bolding amounts to covariantizing them with respect to the full massive little group.

Stripped-contact-term (SCT) bases
To construct SCT bases, one can proceed in analogy to the purely massive construction of the spinor-structure bases described above, but eliminate spinor structures only if they are redundant in the SCT sense. Here too, the reduction can be greatly simplified by using the massless limit. As in the massless case, the reduction of an over-complete set of spinor structures to an SCT basis is conceptually performed by eliminating, among all possible structures, only those which can be expressed as linear combinations of others with prefactors involving no negative powers of Lorentz invariants. Therefore, unlike in the construction of spinor-structure bases, one needs to explicitly examine the relations between spinor structures. A first step of reduction can conveniently be performed by relying mostly on the massless limit, treating one helicity category at a time. A second step of reduction is possible in some cases, and requires knowledge of the mass-suppressed terms in spinor relations. In particular, as we will see, inverse powers of the masses can appear when eliminating redundant structures at this stage. Our results for the SCT bases are summarized in Table 1, which lists the structures obtained following the first and second reductions.

Massless and massive spinor relations
Remarkably, all massive relations between four-point spinor structures have non-trivial massless limits. 3 A relation in which each term contains explicit positive powers of the masses could be employed to eliminate the representative of a helicity category in the spinor-basis construction. The counting of eq. (12) would then be invalidated. Thus, any relation between massive spinors can be obtained by starting with its massless counterpart, and finding the O(m) corrections. We can characterize the relation according to the helicity category of its massless limit. The masssuppressed terms involve spinor structures of different helicity categories. We sometimes refer to the process of finding the O(m) corrections to a relation as mass-completing.  [34]. One of them can thus dropped in the SCT basis construction. This first reduction can be systematized by considering the massless limit of the Pmatrix. The relations associated with the eigenvectors of vanishing eigenvalues become relatively simple in this limit, so one can easily identify redundant spinor structures. Here, these are given by linear combinations of other structures with prefactors containing only non-negative powers of the invariants. They obviously involve only spinor structures of the same helicity category. Thus, structures that are redundant in the massless limit can also be eliminated in the massive case, up to mass corrections involving structures of lower dimensions. The set of independent spinor structures after this first reduction therefore corresponds almost exactly to the massless SCT. Note that only the massless limits are used at this stage. The mass corrections to these relations are not needed at all.

First reduction
One caveat to the above discussion involves massive spinor structures which vanish when unbolded. These must be examined with special care. We have found only one example in which the discussion above does not hold, in the (0000) helicity category of the vvvv amplitude. It involves the fully antisymmetric contraction of four polarization vectors ǫ (ε 1 , ε 2 , ε 3 , ε 4 ) that unbolds to ǫ(p 1 , p 2 , p 3 , p 4 ) which vanishes in a four-point amplitude.
Here, one obtains (using eq. (17) in the second step): (ε 1 , ε 2 , ε 3 , ε 4 ) (15) holds in both the massive and massless cases, four of the five terms it involves are independent in the massive case but only three remain in the massless limit.
Second reduction The first reduction described above is adequate for the treatment of spinor structures of maximal helicity categories. Further reductions are possible however for particles of spin ≥ 1, which feature helicity categories smaller than their spins, or nonmaximal (|h i | < s i ). For the amplitudes we consider here, this only occurs for vectors within the zero-helicity category. To obtain a sensible normalization, the relevant spinor structures are assigned factors of inverse masses, such as for an external vector i. This combination indeed appears in the massive polarization vectors, ε i ≡ √ 2 i [i /m i . Including this 1/m i factor is therefore needed to correctly map the spinor-structure dimension to that of the operator generating it. We can also motivate this normalization directly at the level of on-shell spinor amplitudes. Longitudinal-polarization amplitudes coincide with scalar amplitudes in the high energy (or massless) limit of Higgsed theories. These are the Goldstone amplitudes, which scale as p i /m i and therefore feature a momentum insertion i [i/m i . Covariantizing this with respect to the massive little group yields i [i/m i . Amplitudes involving such combinations may be badly behaved at high energies. Indeed, they only arise in isolation in EFTs, like low-energy QCD, with cutoffs not parametrically separated from vector masses. On the other hand, full amplitudes arising from Higgsed theories like the standard-model EFT can only have energy-growing behaviours matching the highest 1/Λ power they involve.
In the construction of EFT bases, redundant structures (or operators in the Lagrangian formulation) can be eliminated in favour of lower-dimension ones. Thus, the inverse mass normalization of eq. (16) has important implications from the point of view of constructing the SCT basis, since it affects, and specifically lowers, the dimension associated with spinor structures containing i [i combinations. As a result, additional structures can be eliminated compared to the naive first reduction which relies solely on the massless limit.
After the first step of reduction, a limited number of spinor structures remain in each helicity category. The second step of reduction generically requires the full massive relations between spinor structures of different helicity categories. One starts from a massless relation between structures in a given helicity category and mass-completes it. Though tedious, the latter is a well defined problem. Fortunately, only a limited number of relations are relevant in this second reduction. Notably, relations of maximal helicity cannot be used to eliminate any structures. 4 They do not admit negative powers of the masses and therefore only become relevant at higher dimensions, in the construction of contact terms, where linear combinations involving Lorentz invariants appear. Only relations involving non-maximal helicity categories, like vectors of zero-helicity categories, thus need to be considered. Still, they cannot possibly allow for any further reduction if they arise at a dimension higher than that of all of the structures remaining after the first step of reduction. Eventually, only a limited subset of massless relations therefore needs to be mass-completed.
Let us illustrate this discussion with a few examples. The first step of reduction left two spinor structures in the (++++) helicity category of the f f f f amplitude: [12] [34] and [13][24] (see Table 1). The massless relation between their unbolded counterparts, s 24 [12][34] +s 12 [13][24] = 0, is analogous to eq. (9). Following the prescription of eq. (11), it arises from operators of dimension eight. No inverse masses are required since the helicity categories of all particles are maximal: |h i | = s i . As f f f f spinor structures remaining after the first step of reduction have dimension seven at most (see last column of  [24] combination is indeed seen to be redundant. In practice, the second step of reduction towards a SCT basis does not allow to eliminate any spinor structure in amplitudes involving only fermions and scalars. In these cases, massive SCT bases can directly be obtained by bolding massless ones (collecting for instance the relevant helicity amplitudes in Table 3). As a second example, let us therefore consider the f f vs amplitude involving a vector. Two [12] which allows to eliminate the [12] 3123 spinor structure [15]. The parity-flipped analogue of the above equality also allows to eliminate the (−−+0) structure.

Case-by-case discussion
In rest of this section, we detail the reduction towards SCT bases for each spin content in turn. Our results are collected in Table 1. Arrows keep track of the two-step reduction process. For brevity, we do not explicitly list structures obtained by flipping square and angle brackets (parity), or by permutations of external particles. These are collectively counted by n perm . The number of independent SCTs (i.e. the dimension of SCT bases) is provided under n SCT . Where it matches the dimension of the spinor-structure basis n s , the SCT basis can actually be used as spinor-structure basis. Note however that a non-trivial reduction may still be needed in these case. We leave implicit the factors of 1/m i required in the presence of i [i spinor combinations to determine the dimension d op of the operator generating each structure.
ssss One single spinor structure -a constant-appears in the ssss massive SCT basis.
vsss The parity-even combination of (±000) spinor structures is reducible through the mass-completion of thes 13 One can therefore retain only one of the two or, more symmetrically, employ the parity-odd combination: There thus remain 3 irreducible spinor structures in the SCT basis.
f f ss The massive SCT basis is simply obtained by bolding the massless one and contains 4 elements.
vvss The parity even combination of (±∓00) spinor structures can be reduced thanks to the mass-completion of the [12] The SCT basis therefore eventually contains 9 massive spinor structures.
f f vs A spinor-structure basis for the f f vs amplitude was obtained in ref. [15] and also forms a SCT basis. As discussed above, the (±±∓0) structures are reducible thanks to the mass-completion of the [12][131 s 23 = [12][323 s 13 equality in eq. (18). One therefore remains with a basis of 12 spinor structures.
f f f f As discussed already, no further reduction is possible among the 18 spinor structures relevant for the f f f f amplitude. Once dressing SCTs with Lorentz invariants to form contact terms, eq. (17) and its parity conjugate make thes 24 [12][34] +s 12 [13] [24] combination and its parity conjugate redundant at dimension eight.  ε 2 , ε 3 , p 4 ). Unlike in the similar (0000) category of the vvvv amplitude, the number of irreducible massive spinor structures eventually matches that of massless structures. Multiplying eq. (18) by [12], one can eliminate [12] 2 3123 in the (++−0) category. Two combinations of (+−00) structures can moreover be removed 5 thanks to the equality One can therefore keep their difference only.
Ultimately, 38 spinor structures remain in the SCT basis of the vvf f amplitude. There are thus only two additional elements compared to the spinor-structure basis.
vvvv The SCT reduction of the vvvv amplitude is the most involved. Masscompleting the (+++0) category relation [23]  (26) A similar procedure also applies to structures related by particle permutations and parity flips.
The mass-completion of the  Four combinations of (+000) spinor structures can be expressed in terms of those of the (0000) category, as can be seen by multiplying eq. (17) and its parity conjugate by 12 34 , 13 24 and [12] [34], [13] [24], respectively. One could for instance remove the four (000±) structures.
Eventually, one therefore counts a total of 85 structures in the SCT basis of the vvvv amplitude, compared to the 81 of the spinor-structure basis. Once Mandelstam invariants are appended to SCTs to form the contact terms, four redundancies arise at dimension ten in the (++++) category. They are for instance obtained by multiplying eq. (17) and its parity conjugate by [12][34], [13][24] and 12 34 , 13 24 , respectively.

Contact-term bases
Finally, contact-term bases are obtained by appending positive powers of the Lorentz invariants to the elements of the SCT bases. Relations that could not be used to remove spinor structures from the SCT bases, such as eq. (17), can now be used to eliminate certain combinations at higher dimensions. They are determined from the massless limit, and no additional mass-completion is required at this stage. As only two Mandelstam invariants are independent in four-point amplitudes, each of the structures in Table 1 can for instance be multiplied by increasing powers of s 12 and s 13 : So far, we have only considered distinguishable external states. The (anti)symmetrizations required by spin statistics in the presence of identical particles can however become involved. These are only performed, in the next section, when specializing our results to the particle content of the electroweak sector of the standard model. Ultimately, flavour and gauge structures should be combined with the Lorentz structures discussed in this paper.

Electroweak particle content
Our general results can be specialized to the electroweak sector of the standard model which features a single massive scalar h, massive fermions ψ, ψ ′ and vectors W ± , Z. We work in the broken electroweak phase, not imposing the SU(2) L × U(1) Y symmetry. Electric charge conservation is assumed. The two species of fermions are taken such that the total electric charge of the ψ c ψ ′ W + vanishes. Thus, they represent u, d or ν, e pairs of any generation. Baryon-and lepton-number conservations are assumed too. We focus primarily on Lorentz structures generated by operators of dimension six at most. The minimal dimension at which contributions arise in each helicity category is indicated under min{d op } in Table 1. Colour and flavour structures are left implicit and could only be non-trivial for four-fermion operators (massless gluons are not considered). Spin statistics requires amplitudes involving identical particles to be (anti)symmetric under their exchange. In the case of fermions, when more than a single generation is considered, the flavour structure can always be chosen to achieve an overall antisymmetrization. Parity-conjugate structures are not explicitly listed. Permutations among the helicity category assignments of different particles of identical spins are also understood.
ssss The only scalar being the Higgs particle h, the only four-point amplitude pertaining to the ssss category has hhhh particle content. At dimension four, a constant amplitude is generated. At dimension six, the Bose symmetrization reduces thes ij + perm. combination to a sum of square masses. At dimension eight, a non-trivial amplitude of the forms 12s34 +s 13s24 +s 14s23 is for instance manifestly symmetric. vsss The only electroweak particle content possible for massive vsss amplitudes is Zhhh. In the (0000) helicity category, the Bose symmetrization for the three scalars does not allow for a [121 /m 1 + perm. structure that would have corresponded to an operator of dimension five. The first amplitudes therefore arise at dimension seven. An example is the manifestly symmetrics 34 [121 /m 1 +s 24 [131 /m 1 +s 23 [141 /m 1 combination. Note that in the (+000) category, the [1231] − 1231 structure also vanishes upon symmetrization given its antisymmetry under the exchange of all three scalars. This can easily be seen when expressing it as ǫ(ε 1 , p 2 , p 3 , p 4 ) by using momentum conservation to eliminate p 1 in eq. (20). An antisymmetric combination of Mandelstam invariants like (s 12 −s 13 )(s 12 −s 14 )(s 13 −s 14 ) could be used to form a contact term of dimension thirteen.
f f ss The ψ c ψhh particle content is the only one possible with f f ss spin assignment. At dimension five, the [12] contribution is found in the (++00) category, without additional factors of Mandelstam invariants. At dimension six, the Bose symmetrization does not allow for the [132 + (3 ↔ 4) structure that would otherwise arise in the (+−00) helicity category. This structure only starts contributing at dimension eight, in combinations like [1(3 − 4)2 (s 13 −s 14 ).
vvss Both ZZhh and W + W − hh massive particle contents match the vvss spin assignment. At dimension four, the [12] 12 /m 1 m 2 structure arises in the (0000) helicity category. The standard-model contact term, which unitarizes this amplitude, is of this form. At dimension six, thes 34 [12] 12 /m 1 m 2 contact term is symmetric under the exchange of the two scalars. It is incidentally also symmetric under the exchange of the two vectors, so that it is present in both ZZhh and W + W − hh cases. Still at dimension six, one finds [12]

Conclusions
We addressed the construction of contact-term bases required to form massive on-shell three-and four-point amplitudes. This is a crucial step in the application of on-shell methods to the classification and computation of EFT amplitudes. We derived general expressions for four-point amplitudes involving massive scalars, spin-1/2 fermions and vectors, as well as for all three-point amplitudes involving particles of spins ≤ 3. Some of the techniques we described also apply to higher-spin and higher-point contact terms. The massless limit was extensively exploited. Its most powerful application is the derivation of the independent building blocks of generic amplitudes: spinor-structure bases. These can be obtained from appropriate sets of massless amplitudes written in terms of massless spinors, by bolding the spinors into the massive spinors of ref. [7]. A similar procedure applies to the derivation of contact terms involving scalars and fermions only. Contact terms featuring vectors and higher spins are more challenging, and require non-trivial massive-spinor identities. Here too, the massless limit provides a useful starting point, but finding the mass corrections to the massless identities requires a case-by-case, and often tedious, analysis. It would be interesting to examine whether a more direct and algorithmic construction could be achieved. Bypassing the intermediate construction of stripped-contact-term bases would be particularly useful, perhaps by extending the massless harmonics approach of refs. [12,13] to massive amplitudes. Streamlining the (anti)symmetrization required in the presence of identical particles, preferably so this is built-in in the construction of the contact terms, would also be useful. The results presented here already allow for a wide range of applications. The ingredients we provide, in the form of independent spinor structures, can be used to derive the most general four-point EFT amplitudes for spin ≤ 1 particles, up to any dimension, simply by appending Lorentz invariants. These basic spinor structures, which carry all the spin information, are furthermore given by very compact expressions. Thus, various properties of the amplitudes, such as the leading mass effects in their interference with the standard-model amplitudes, can be easily read-off.
For the electroweak bosons and fermions, we also perform the required (anti)symmetrization of all contact terms up to dimension six, including also few notable higher-dimensional ones. This results in interesting patterns, with some structures only appearing at very high dimensions. It would be interesting to see how these patterns are further restricted by the SU(2) L ×U(1) Y symmetry, and to identify sensitive probes to differentiate between these different EFTs. Since they furthermore encode the full v/Λ expansion, the amplitudes we derived are interesting starting points for the exploration of general EFT extensions of the standard model.  Table 3. Spinor structures forming bases of massless four-point stripped contact terms for spins 0, 1/2, 1, and obtained by the algorithmic procedure described in ref. [16] and section 4.1. Only structures with non-negative total helicity are displayed. Others are obtained by parity conjugation. Massless four-point bases for amplitudes involving spin-3/2 and 2 are respectively provided in Table 4 Table 4. Spinor structures forming bases of massless four-point stripped contact terms for particles of spin 3/2 at most. At least a particle of spin 3/2 must be involved. For spin 1 at most, see Table 3.  Table 5. Spinor structures forming bases of massless four-point stripped contact terms for particles of spin 2 at most. At least a particle of spin 2 must be involved. For spin 1 and spin 3/2 at most, see Table 3 and 4.