Enumerating higher-dimensional operators with on-shell amplitudes

We establish a simple formula for the minimal dimension of operators leading to any helicity amplitude. It eases the systematic enumeration of independent operators from the construction of massless non-factorizable on-shell amplitudes. Little-group constraints can then be solved algorithmically for each helicity configuration to extract a complete set of spinor structures with lowest dimension. Occasionally, further reduction using momentum conservation, on-shell conditions and Schouten identities is required. A systematic procedure to account for the latter is presented. The dressing of spinor structures with dot products of momenta finally yields the independent Lorentz structures for each helicity amplitude. We apply this procedures to amplitudes involving particles of spins 0,1/2,1,2. Spin statistics and elementary selection rules due to gauge symmetry lead to an enumeration of operators involving gravitons and standard-model particles, in the effective field theory denoted GRSMEFT. We also list the independent spinor structures generated by operators involving standard-model particles only. In both cases, we cover operators of dimension up to eight.


I. INTRODUCTION
Even before being known as such, effective field theories (EFTs) were employed in the early 1930s to model light-by-light scattering and nuclear beta decay. Today, among a wide range of applications, EFTs are notably exploited to parametrize new physics hiding beyond the scales directly probed experimentally. The lack of clear indications about the nature of physics beyond the standard model (SM) motivates the use of such modelindependent approaches and the development of efficient methods to study the standard-model effective field theory (SMEFT).
Extremely successful in renormalizable theories, onshell amplitude methods have also been applied to EFTs. Theories with enhanced soft limits have been the subject of various studies [1][2][3][4]. Non-renormalizations [5,6] and non-interferences [7] that are obscured by the Lagrangian formalism were exposed in the SMEFT. The construction of a complete set of independent operators (i.e. a basis) can also be rather cumbersome in the usual Lagrangian approach. Besides Hilbert series [8,9] and harmonics techniques applying to distinguishable particles [10], onshell methods have also been employed in this task. They have the advantage of avoiding the operator redundancies arising from field redefinitions. The counting of operators relevant to amplitudes involving one colour singlet and up to three gluons was presented in ref. [11]. The enumeration of amplitudes corresponding to the full set of dimension-six SMEFT operators was carried out in ref. [12]. The case of massive SM amplitudes was also addressed both at the renormalizable [13][14][15] and nonrenormalizable [16,17] levels. All massive three-point and one four-point electroweak amplitudes were bootstrapped and their high-energy limits studied [17]; factorizable bosonic [15,16] and fermionic [17] four-point amplitudes were constructed; constraints due to electroweak symmetry breaking were observed to emerge from perturbative unitarity [15][16][17]; a matching to the SMEFT was carried out in the broken phase [16,17].
In this paper, we further advance the program of operator enumeration using massless on-shell amplitudes. The case of interactions between gravitons and SM particles is examined as showcase example. The Hilbert-series enumeration of operators of dimension eight at most [18] is reproduced with on-shell methods. Our technique also allows for a straightforward listing of independent spinor structures generated by SMEFT operators. We also limit ourselves to operators of dimension eight at most in this case.
Massless on-shell amplitudes are reviewed in section II. An algorithmic procedure is notably described to construct spinor structures for a given helicity amplitude. A systematic means of exhausting relations deriving from Schouten identities is also presented. We establish a simple formula for the minimal dimension of operators leading to any helicity amplitude in section III. Applications to the GRSMEFT and SMEFT cases are treated in section IV. Helicity amplitudes generated by operators of dimension eight at most are listed. Minimal spinor structures compatible with little-group constraints are constructed systematically. Their reduction to an independent set is carried out explicitly, before dressing them with additional dot products of momenta. Constraints deriving from gauge invariance and spin statistics are accounted for in the GRSMEFT case. Our results are summarized in table I and table II.

II. MASSLESS AMPLITUDES
In the on-shell formalism, massless amplitudes with n external legs are constructed from (linear combinations of) square [ij] and angle ij spinor brackets (see e.g. refs. [19][20][21]) raised to integer powers that we denote a ij arXiv:1912.08827v1 [hep-ph] 18 Dec 2019 and b ij , respectively: Spinor brackets have mass dimension 1. The n(n − 1)/2 brackets of each type involve spinors of each possible pair of particles: 12, 13, 14, . . . , 23, 24, . . . , 34, . . .. All momenta are conventionally taken to be either incoming or outgoing. Little-group covariance requires that the helicity weights of the spinor structure matches the helicity h i of the external particles. This leads to n constraints: Each a ij , b ij power appears in exactly two of those equations. Summing them, one therefore obtains that: where we have defined h tot ≡ i h i , the total helicity. On the other hand, the mass dimension of a spinor structure is given by the sum of all spinor bracket powers: In massless three-point amplitudes (which can be written down for complex momenta), the square and angle spinors for each particle are all proportional to each other. Given that all dot products of momenta vanish, [ij] ji = 0, either square or angle brackets must vanish. Massless three points amplitudes can thus be constructed from a single type of brackets. The constraints of eq. (2) then form a linear system of three equations (n = 3) with three unknowns (n(n − 1)/2 = 3). A well-known unique solution is thus found for either square or angle brackets: Given that a massless three-point amplitude only involves either square or angle brackets, eq. (3) and eq. (4) can be combined to obtain that dim{spinors} = h tot when square brackets are employed and dim{spinors} = −h tot with angle brackets. To preserve locality, a positive mass dimension is required. One is therefore forced to respectively use square and angle brackets for positive h tot > 0 and negative h tot < 0 total helicities. In this paper, we address the construction of nonfactorizable amplitudes (having a trivial analytic structure with neither kinematic poles nor branch cuts) generated by contact operators. Being local, they can always be written in a form that does not involve any negative power of spinor bilinears: 1 a ij , b ij ≥ 0 ∀ij, for non-factorizable amplitudes. (6) We moreover target the construction of spinor structures that cannot be reduced to simpler ones, involving fewer spinors. Spinor structures involving positive powers of both [ij] and ij are not, in this sense, minimal. The [ij] ji = 2 p i ·p j ≡ s ij equality indeed allows us to reduce the number of spinors they involve. Solving little-group constraints for each a ij − b ij difference, one would use a positive a ij power of the ij square bracket for a ij −b ij > 0 solutions (setting b ij = 0) and a positive b ij power of the ij angle bracket for a ij − b ij < 0 solutions (setting a ij = 0): for each ij pair, in minimal spinor structures.
For the minimal spinor structures of non-factorizable amplitudes involving n legs, little-group constraints form a linear system of n equations with n(n − 1)/2 unknowns. There are thus n(n − 3)/2 parameters left undetermined: none for n = 3, two for n = 4, five for n = 5, etc. One can then for instance fix such a number of a ij − b ij differences (spanning small integer is sufficient to obtain spinor structures of lowest dimension) and solve the linear system of little-group constraints for the others. The spinor structures generated at this point are not necessarily independent. Momentum conservation, onshell conditions, and Schouten identities can be used to reduce them to an independent set in which no single spinor structure can be expressed as a linear combination of others. In an amplitude with n external legs, one Schouten identity can be written down for each possible set of four spinors (square or angle). There are thus n!/4!(n − 4)! of them. Each can be used to eliminate a fixed product of spinor bilinear. Performing such eliminations recursively reduces spinor structures to an independent set. In particular, in an amplitude featuring four legs (or four particles of non-vanishing helicity), one of the three [12][34], [13] [24], [14][23] product can always be eliminated to lead to an independent set. An application of this procedure to a five-point amplitude is discussed in section IV B.
To obtain more symmetric combinations in four-point amplitudes, one may however wish to perform the spinor structure reduction in more tailored way. For this purpose, we find it convenient to introduce the following shorthand notation: We also define the order of a (lmn) spinor structure as max{l, m, n}. In this notation, the Schouten identity writes (100)−(010)+(001) = 0. Each of these three terms carries helicity weight 1/2 for each of the 1, 2, 3, 4 particles. In various cases, minimal spinor structures compatible with little-group constraints can be expressed as a prefactor multiplying the various (lmn) structure satisfying the l + m + n = o constraint, for a positive integer o (which is their maximal order). Finally, minimal spinor structures can be complemented by scalar functions of momentum dot products s ij ≡ 2p i · p j . In local non-factorizable amplitudes, they only appear in numerators and increase the mass dimension by at least to units. All dot products however vanish in massless three-point amplitudes. There are (n − 1)(n − 2)/2 available dot products in amplitudes with n external legs, once one particle momentum is eliminated using momentum conservation. One onshell condition reduces this number to n(n − 3)/2 and Gram determinant relations reduce it further to 3n − 10 (for n > 3, see e.g. [23]). A systematic way to build a set of independent dot products based on the kinematic polynomial rings can be found in ref. [24].

III. MINIMAL OPERATOR DIMENSION
The dimension of an amplitude with n external legs is 4 − n. It receives contributions from the dimension of the spinor structure dim{spinors} as well as from the coefficient of the operator that generated it, which has dimension 4 − dim{operator}. From the operator coefficient dimension, one can also extract a factor of the Planck mass for each external graviton. One thus have dim{operator} = n − n tensors + dim{spinors} (9) where n tensors is the number of external Lorentz tensor legs (i.e. gravitons). Determining the minimal operator dimension at which a given non-factorizable helicity amplitude arises thus requires a determination of the dimension of the minimal spinor structure allowed for that amplitude.
Forbidding spinor brackets in denominators, the dimension of the spinor structure yielding a given helicity amplitude is at least that of the spinors needed to generate the suitable helicity weights. One spinor of dimension 1/2 is required for each half unit of helicity. The spinor structure required for a non-factorizable amplitude featuring external legs of helicities h i should thus have dimension at least equal to i |h i |: dim{spinors} ≥ i |h i |. This inequality is saturated when the i |2h i | spinors required to generate the correct helicity weights are all contracted in bilinears. This may however not always be possible. Additional momentum insertions are otherwise required and further increase the dimension of the spinor structure leading to Momentum insertions 2 allow to contract together angle and square brackets as in 1|p 2 |3] ≡ 123], or spinors of the same particle as in [1|p 2 p 3 |1] ≡ [1231]. In three-point amplitudes, such structures featuring additional momentum insertions however vanish due to momentum conservation and on-shell conditions. To determine the number of required momentum insertions, one starts by considering separately particles carrying positive and negative helicities, which respectively demand square and angle brackets. The crucial point is that, if the largest positive helicity exceeds the sum of all other positive ones, its square brackets cannot all be contracted in bilinears. The same holds for the largest negative helicity. In the example of a four-point amplitude with (+2, +1, +1/2, +1/2) positive helicities, the +2 helicity requires four square brackets which can all be contracted with the four square brackets required for the other three particles. In a (+2, +1, +1/2, −1/2) helicity amplitude, however, particles of helicity +1 and +1/2 only provide three square brackets which is insufficient to contract all the ones requires by the particle of helicity +2. One momentum insertion is required to contract its fourth square bracket with the angle bracket needed for the particle of helicity −1/2.
In general, one remains with 2 max hi>0 {|2h i |} − hi>0 |2h i | uncontracted square spinors and 2 max hi<0 {|2h i |} − hi<0 |2h i | angle spinors, if those numbers are positive. (The factor of two in front of the maximal helicity in the first term is required if the sum of the second term also includes it.) One can then for instance form 123] structures, with one momentum insertion, for each available pair of square and angle brackets. If they remain in unequal number, one would also need to form [1231] or 1231 structures with two momentum insertions. Forming preferentially trilinears or quadrilinears yields the same counting, as one can for instance split the product of a quadrilinear and a bilinear into two trilinears. Overall, the number of required momentum insertions is equal to the maximum between the numbers of remaining square and angle brackets: Our final expression for the minimal operator dimension at which a given non-factorizable amplitude is gen-erated thus takes the form dim{operator} ≥ n − n tensors (12) where again n is the number of external legs, n tensors the number of external tensors (i.e. gravitons), and h i are the helicities of each leg. This formula is only applicable for n > 3. If the number of required momentum insertions is non-vanishing, no local non-renormalizable three-point amplitude can be written down.
In the applications considered below, we find that the inequality of eq. (12) can always been saturated, before other constraints such as gauge invariance or spin statistics are imposed.

IV. OPERATOR ENUMERATIONS
We generically denote as s, f, v, t massless particles of spin 0, 1/2, 1, 2 and indicate the sign of their helicity with a ± superscript. Helicity amplitudes with more and more legs are successively considered. For simplicity, we restrict ourselves to amplitudes of non-negative total helicity. The others can be obtained by trading square for angle brackets, and vice versa (i.e. by flipping parity). The counting rule of eq. (12) is employed to determine the minimal dimension of operators giving rise to each of these helicity amplitudes. Focusing on those generated by operators of dimension eight at most, one then solves the little-group constraints discussed in section II to generate the corresponding minimal spinor structures. Straightforward in most cases, the reduction of spinor structures to an independent set relies on momentum conservation, on-shell conditions, and Schouten identities. Dot products of momenta can eventually be appended to minimal independent spinor structures. Re-considering the SM diversity of fermions and gauge bosons, as well as non-trivial gauge transformation properties, is then required. Bose and Fermi statistics also need to be imposed in the presence of identical fields. An enumeration of independent operators is finally obtained.
We consider below the case of operators involving at least a graviton in addition to SM particles. Additionally, for pure SM operators, we focus on the enumeration of independent spinor structures without accounting for gauge ones and momentum dot products. For a single generation, 993 independent dimension-eight operators would be generated in this SMEFT case [8].

A. GRSMEFT
The (positive) helicity amplitudes generated by operators of dimension eight at most and featuring at least a Lorentz tensor (i.e. graviton) are listed in table I.

Gauge symmetry
The SM gauge symmetries translates into simple selection rules: -forcing pairs same-helicity fermions to appear together with a Higgs field, or with another such pair, -forbidding odd numbers of Higgs fields in the absence of a same-helicity fermion pair, They exclude all amplitudes corresponding to operators of odd dimensions in table I. Demanding that the colour, weak isospin and hypercharge of gauge bosons, fermions, and Higgs fields compensate each other is eventually required once the nature of s, f, v, t are specified.

Explicit reduction and spin statistics
Let us reduce the spinor structures compatible with little-group constraints to independent sets, and impose the symmetry required by spin-statistics under the exchange of gravitons carrying identical helicities.
As discussed in section II, three-point amplitudes have a unique form in terms of powers of [12], [23], [13] spinor bilinears (for positive total helicity). No spinor structure reduction thus needs to be carried out in this case. The t + t + v + amplitude has a [12] 3 [23][13] spinor structure antisymmetric under the exchange of the two Lorentz tensors and is therefore forbidden when these represent identical gravitons. The same conclusion also holds for the five point t + t + v + ss spinor structure.
One scalar can always be ignored in the construction of higher-point amplitudes. It only manifests itself in the form of insertions of its momentum which can be traded for others using momentum conservation. One single minimal spinor structure, corresponding to the three-point amplitude solution, is thus found for the following four-point amplitude: t + t + ss, t + t + t + s, t + t + v + s, t + v + v + s, t + v + ss, t + f + f + s. The higher-point amplitudes, like t + t + sss, t + t + t + ss, t + t + v + ss and t + v + v + ss, featuring at most three particles of non-vanishing helicities and requiring no momentum insertion also have one single independent minimal spinor structure matching that of the corresponding three-point amplitude.
Among amplitudes featuring only particles of positive helicities, the t + v + f + f + (s) and t + f + f + f + f + ones only have one single minimal spinor structure: [12] 2 [13] [14] and [12][13] [14][15], respectively. No reduction is therefore needed in these cases. The latter is symmetric under the exchange of any pair of fermion. After combination with colour and SU (2) L structures, this will imply that the C R Q † Q † Q † L † operator (in the notation of ref. [18]) vanishes for a single generation.
The t + t + f + f + (s) amplitudes have three minimal structures compatible with little-group constraints: [12] [23]) (even). Due to gauge invariance in the SM, the t + v + v + v + amplitude necessarily involves two identical vectors and must therefore be symmetric under their exchange. One therefore remains with the latter spinor structure as single allowed one.
Finally, the t + t + t + t + amplitudes has fifteen minimal spinor structures satisfying little-group covariance. Employing again the (lmn) notation of eq. One thus finally obtains five independent spinor structures. The same counting is for instance also obtained by eliminating all (lmn) structures with l > 0 by using the (100) = (010) − (001) Schouten identity recursively, following the systematic procedure described in section II. When all four tensors represent identical gravitons, one can form two (400) + (040) + (004), (220) + (022) + (202) fully symmetric combinations. They are however proportional to each other, as just demonstrated. There is thus one single independent minimal spinor structure for the amplitude with four identical tensors of same helicity.
Momentum dot products Dot products of momenta can be added to minimal spinor structures and would increase the corresponding operator dimension by at least two units. Limiting ourselves to operators of dimension eight at most, one thus only needs to consider adding such s ij 's to structures already leading to operators of dimension six at most. In massless three-point amplitudes, all dot products of momenta vanish. Examining table I, there is thus only the dimension-six fourpoint t + t + ss amplitude with minimal [12] 4 spinor structure to which one single momentum dot product can be appended to reach dimension eight. In such a massless four-point amplitude, there are three possible products summing to zero: s 12 + s 13 + s 23 = 0. One can thus for instance form two linear combinations of definite symmetry transformation properties under the exchange of the two tensors: s 12 (even) and s 13 − s 23 (odd). When they represent identical gravitons, Bose statistics only permits the former one (given that the spinor structure is already symmetric).

B. SMEFT
Considering the enumeration of pure SMEFT operators, we focus on their spinor structures and do not account for gauge and flavour transformation properties. Factors of momentum dot products are not explicitly specified either. We still only construct explicitly amplitudes having non-negative total helicity, as the others can be obtained by exchanging square and angle brackets. The helicity amplitudes generated by operators of dimension eight at most are displayed in table II. For many helicity amplitudes, little-group constraints only allow for one single spinor structure corresponding to an operator of dimension eight at most. Using momentum conservation and Schouten identities, the reduction of the obtained minimal spinor structures is straightfor- We thus only discuss below the reductions of spinor structures for the v Note that the f + f + sss five-point amplitude receives not only a [12] contribution at dimension six, but also a [1342] one at dimension eight. Including contributions of dimension eight at most, we have not found other helicity amplitudes receiving contributions from spinor structures of different dimension.
In the (lmn) notation introduced in eq. (8), the minimal spinor structures compatible with littlegroup constraints for the v + v + v + v + amplitude are (200), (020), (002) of order two and (110), (101), (011) of order one. As seen before, squaring Schouten identities leads to equalities such as (011) = [(020) + (002) − (200)]/2 which allow us to eliminate order-one structures in favour of order-two ones. One is therefore left with three independent structures. Imposing symmetry under the exchange of the first two vectors, one remains with the (200) and (020) + (002) structures. Incidentally, these are simultaneously symmetric under the exchanges of the last two vectors. The single structure that is symmetric under the exchange of all four vectors is (200) + (020) + (002).
The six minimal structures compatible with littlegroup constraints for the v  [23] (not involving [5).

Four-point amplitudes beyond dimension eight
Focusing on four-point amplitudes, it is remarkable that only three helicity amplitudes are not generated by operators of dimension eight or lower. They are v + v + v − s, v + v − f + f + , and v + v + v + v − . Let us also reduce their minimal spinor structures to independent sets.
Only [12] 2 3123 is available for the v + v + v − s amplitude, and is antisymmetric under the exchange of the two vectors of positive helicity.
Five minimal structures are compatible with littlegroup constraints for the v + v − f + f + amplitude: [14] With momentum conservation and on-shell conditions, they can be reduced to a single independent structure: [12] 2 [3(1 − 2)4 2 which is symmetric under the exchange of the first two vectors.

V. CONCLUSIONS
We advanced the program of systematic nonfactorizable amplitude construction, establishing a simple formula for the minimal dimension of operators generating any massless helicity amplitude. Solving littlegroup constraints then yielded minimal spinor structures, which could subsequently be reduced to independent sets using momentum conservation, on-shell conditions, and Schouten identities. A systematic procedure to apply Schouten identities was presented. Dressing these minimal spinor structures with dot products of momenta led to the independent non-factorizable Lorentz structures. The enumeration of operators of dimension eight at most was provided for operators involving particles of spins 0, 1/2, 1, 2. Simple selection rules were sufficient to impose the constraints due to standard-model gauge symmetry, and symmetrization of the Lorentz structures under particle permutations provided compliance with Bose statistics for gravitons of same helicity and identical vector bosons. The minimal spinor structures generated by operators of dimension eight at most were also provided for the pure standard-model case, featuring only particles of spins 0, 1/2, 1. The combination of these with the full richness of standard-model gauge and flavour structures was not attempted. Such a combination would be interesting to perform in the future. Tools and techniques developed for operator constructions could then be exploited [25][26][27]. The systematic construction of massive non-factorizable amplitudes is also a task remaining to be performed. A counting rule for the number of independent minimal spinor structures and an algorithmic reduction procedure would also be useful to derive in the future.
Note added Reference [28] which appeared on the day of our submission to the arXiv employed momentum twistors to trivialize the constraint arising from momentum conservation, in the enumeration of operators from on-shell amplitudes.