aa r X i v : . [ h e p - ph ] J a n Bases of massless EFTs via momentum twistors
Adam Falkowski Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
I present a novel method of deriving a basis of contact terms in massless effective field theories(EFTs). It relies on the parametrization of N -body kinematics via the so-called momentum twistors.A basis is constructed directly at the amplitude level, without using fields or Lagrangians. Themethod consists in recasting any local contact term as a sum of rational functions built from Lorentz-invariant contractions of momentum twistors. The end result is equivalent to constructing a basis ofhigher-dimensional operators in an EFT Lagrangian, however it is considerably simpler, especiallyfor theories with higher-spin particles. The method is applied to contact terms in 4-point amplitudes.I provide a compact algebraic formula for basis elements for any helicity configuration of the externalparticles, and I illustrate its usage with several physically relevant examples. I. INTRODUCTION
In a relativistic quantum theory, amplitudes can be cal-culated using Feynman rules derived from a Lagrangian.The alternative is to directly construct on-shell ampli-tudes using the basic principles of Poincar´e symmetry, lo-cality, and unitarity, without using the crutches of fieldsand Lagrangians [1–4]. This approach is based on thefact that, on kinematic poles, residues of an N -point am-plitude factorize into products of lower-point amplitudes.For example, a tree-level amplitude with 4 massless par-ticles on the external legs can be represented as A (1234) = − A (12ˆ p s ) A (34 p s ) s − A (13ˆ p t ) A (24 p t ) t − A (14ˆ p u ) A (23 p u ) u + C (1234) , (1)where 1 . . . p s ≡ p + p , p t ≡ p + p , p u ≡ p + p , the Mandelstam invariants are x ≡ p x for x = s, t, u , and the hats denote outgoing par-ticles. The last piece stands for contact terms , which areregular functions of s, t, u without poles or other singu-larities, therefore they are not connected to lower-pointamplitudes by unitarity. All in all, an on-shell 4-pointamplitude can be bootstrapped from 3-point ones, up tocontact terms. The latter are the focus of this paper.One can fix the contact terms through physically mo-tivated assumptions about the high-energy or analyticbehavior of the amplitudes. This path is relevant forcertain important theories, such as QCD or general rela-tivity (GR), which are completely fixed by their 3-pointamplitudes and do not admit any free parameters enter-ing at N > C (1234) = ∞ X D =4 X k c D,k O D,k (1234)Λ D − , (2)where O D,k are basis elements spanning the space of allpossible Lorentz-invariant contact terms. Each O D,k is aregular function of Mandelstam variables with mass di-mension [mass] D − , while c D,k are free parameters called the Wilson coefficients. Formally, the contact terms de-pend on an infinite number of Wilson coefficients. How-ever, for processes with a characteristic energy scale E ≪ Λ only O D,k with low enough D are numericallyimportant, and the amplitude can be well approximatedusing a finite number of c D,k and O D,k .There remains the highly non-trivial issue of writingdown all possible O D,k , given the quantum numbers (inparticular spin and helicity) of the external particles. Inthe Lagrangian language, the parallel problem is con-structing all independent Lorentz-invariant local opera-tors of canonical dimension D . That task can be sys-tematically organized thanks to the Hilbert series tech-niques [7–9]. Alternatively, for massless EFTs one canbypass Lagrangians and construct contact terms directlyat the amplitude level [10–12] using the spinor helicityvariables ( helicity spinors , in short). This method maybe simpler, especially for higher spins, as spinors en-code helicity information in a transparent way. Recently,Ref. [13] proposed an algorithm for constructing a ba-sis O D,k as harmonic modes of the physical manifold ofhelicity spinors parametrizing N -point amplitudes. Seealso Refs. [14, 15] for other amplitude-based approachesto constructing contact terms.In this paper I propose a novel way of constructinga basis of contact terms in massless EFTs. The depar-ture point is the parameterization of N-point amplitudesusing the momentum twistor variables [2, 16]. Namely,the kinematic data can be encoded in N spinor pairs Z i = ( λ i , ˜ µ i ). From these, the four-momenta p i of allexternal particles can be reconstructed, and they auto-matically satisfy the on-shell condition ( p i = 0) and mo-mentum conservation ( P i p i = 0). The basic facts aboutmomentum twistors are summarized in Section II. Theseare very convenient and natural variables on the space of N -body kinematics. In particular, they greatly simplifythe task of constructing independent Lorentz invariantsfrom the kinematic data. Section III shows how to tradelocal contact terms for rational functions of Lorentz in-variant ( λ i λ j ) and (˜ µ i ˜ µ j ) spinor contractions. Amongthese, a set of independent basis elements is identified,from which any contact term can be constructed at agiven order in the EFT expansion and for a given he-licity configuration of external particles. In this paperI focus on 4-point amplitudes, however the method canbe generalized to higher N . For N = 4, the candidatebasis elements are characterized by a compact formulain Eq. (16). At a fixed EFT order they are labeled byone integer bounded to a finite range, thus they can beenumerated order by order in the EFT expansion. Sec-tion IV illustrates this method with a number of sim-ple examples rooted in physically relevant theories. Aslightly more involved example of constructing contactterms and scattering amplitudes in the EFT extension ofGR is relegated to Appendix A.I work in 4 spacetime dimensions with the mostly mi-nus metric: η µν = diag(1 , − , − , − SU (2) × SU (2), with holomorphicand anti-holomporphic spinors λ α and ˜ λ ˙ α transformingunder the respective SU (2) factors. For the spinors Iadopt the conventions of Ref. [17]. Spinor indices areraised with the antisymmetric tensor ǫ αβ and loweredwith ǫ αβ : λ α = ǫ αβ λ β , λ α = ǫ αβ λ β , and idem for ˜ λ ,with the convention ǫ = − ǫ = 1. The Lorentz in-variant spinor contractions are λ αi λ j α ≡ ( λ i λ j ) ≡ h ij i ,˜ λ i ˙ α ˜ λ ˙ αj ≡ (˜ λ i ˜ λ j ) ≡ [ ij ]. Vector and spinor Lorentz in-dices can be traded with the help of the sigma matrices[ σ µ ] α ˙ β = ( , ~σ ), where ~σ are the Pauli matrices. I abbre-viate p · σ ≡ p µ σ µ . II. FLASH REVIEW OF MOMENTUMTWISTORS
I start by reviewing the momentum twistor descrip-tion of massless N -body kinematics in four spacetimedimensions, following closely the presentation in Ref. [2].Consider a set of four-momenta p i , i = 1 . . . N , subjectto the on-shell conditions p i = 0 and momentum con-servation P Ni =1 p i = 0. The restrictions on p i may beinconvenient to work with, and for many applications itis beneficial to introduce different variables that trivializethe constraints. The on-shell conditions are dealt with byintroducing the helicity spinors, that is N holomorphicand anti-holomorphic 2-component spinors λ i , ˜ λ i relatedto the four-momenta by p i · σ = λ i ˜ λ i . This trivializes theon-shell constraints, in the sense that an arbitrary pair( λ i , ˜ λ i ) defines a (possibly complex) p i that automati-cally satisfies p i = 0. As a bonus, the transformation λ i → t − i λ i , ˜ λ i → t i ˜ λ i does not change p i therefore itrepresents the little group action on the particle i . How-ever, momentum conservation is not automatic in thesevariables, and implies one non-linear constraint on the N spinors λ i and ˜ λ i . The idea behind the momentumtwistors is to trade the helicity spinors into a different setof variables so as to trivialize the momentum conserva-tion as well. To this end, one defines the dual coordinates y i via the relation p i = y i − y i − ⇒ y i = y N + i X j =1 p j , (3)with the cyclic identification y = y N . One can think of y i as vertices of a polygon in the dual coordinate space,and of p i as the sides of that polygon. Note that y i have units of [mass] , unlike the spacetime coordinates.The four-momenta p i do not depend on y N : they do notchange when the polygon is moved around in the dualspacetime. This is just the translation invariance in thedual coordinate space. One can always gauge-fix it, e.g.by setting y N = 0.Introducing y i via Eq. (3) trivializes the momentumconservation. However the goal is to construct variablesthat trivialize both the momentum conservation and theon-shell conditions. This is achieved by introducing a setof N anti-holomorphic spinors ˜ µ i defined as˜ µ i = λ i σ · y i = λ i σ · y i − . (4)Given that λ i have dimension [mass] / , ˜ µ i have dimen-sion [mass] / . The spinor pairs Z i = ( λ i , ˜ µ i ) are calledthe momentum twistors. Any set of N such pairs auto-matically defines an N -body kinematics with p i = 0 and P Ni =1 p i = 0. In order to see this, note that Eq. (4) canbe solved for the dual coordinate: y i · σ = λ i +1 ˜ µ i − λ i ˜ µ i +1 ( λ i λ i +1 ) . (5)Therefore, starting from Z i one can reconstruct all y i ,and thus p i via Eq. (3), Furthermore, one can show thatthese four-momenta can be decomposed as p i · σ = λ i ˜ λ i ,where˜ λ i = − ˜ µ i − ( λ i λ i +1 ) + ˜ µ i ( λ i +1 λ i − ) + ˜ µ i +1 ( λ i − λ i )( λ i − λ i )( λ i λ i +1 ) . (6)From Eq. (5) and Eq. (6) it follows that λ i and ˜ µ i trans-form in the same way under the little group: Z i → t − i Z i .In other words, Z i are defined projectively: an indepen-dent rescaling of each Z i does not change p i . In the lit-erature, momentum twistors are most often used in su-perconformal frameworks (see e.g. Ref. [18]), in whichcase Z i are unbreakable building blocks of the ampli-tudes. In this paper however I am interested in moredown-to-earth theories. I will treat λ i and ˜ µ i as separatebuilding blocks, from which I construct Lorentz-invariantholomorphic ( λ i λ j ) and anti-holomorphic (˜ µ i ˜ µ j ) spinorcontractions.Let us compare the number of degrees of freedom onthe four-momentum and momentum twistor sides. Incomplex kinematics, each p i has 6 real degrees of freedomafter imposing the on-shell condition. Momentum conser-vation fixes 8 degrees of freedom, thus the manifold of N -body kinematics is (6 N − y N = 0corresponds via Eq. (4) to setting ˜ µ = ˜ µ N = 0. Thisleaves N spinors λ i , and N − µ i . Each spinor has4 real degrees of freedom, however 2 degrees of freedomin each λ i can be removed by little group transformations Z i → t − i Z i . This leaves 2 N + 4( N −
2) = 6 N − N momentum twistors Z i is in 1-to-1correspondence (after gauge-fixing the translations) withthe space of all independent massless N -body kinematics.Instead of Z i = ( λ i , ˜ µ i ), one could represent the kine-matic data by momentum anti-twistors ˜ Z i ≡ ( µ i , ˜ λ i ) re-lated to the dual coordinates by µ i = y i · σ ˜ λ i = y i − · σ ˜ λ i .In the spinor helicity variables, parity exchanges λ i ↔ ˜ λ i .The binary choice of either Z i or ˜ Z i to represent the kine-matic data is not parity invariant, therefore parity is notmanifest in momentum twistor variables. III. CONTACT TERMS VIA MOMENTUMTWISTORS
Contact terms are functions of the helicity spinors thatcan be written in the form O = Π i I start with Eq. (7) for N = 4. The (˜ λ i ˜ λ j ) spinorcontractions can be traded for momentum twistors using Eq. (6):[12] = (˜ µ ˜ µ ) h ih i , [13] = − (˜ µ ˜ µ ) h ih ih ih i , [14] = (˜ µ ˜ µ ) h ih i , [23] = (˜ µ ˜ µ ) h ih ih ih i , [24] = − (˜ µ ˜ µ ) h ih ih ih i , [34] = (˜ µ ˜ µ ) h ih i . (9)The fact that on the momentum twistor side there is onlyone possible anti-holomorphic contraction (˜ µ ˜ µ ) greatlysimplifies the proceedings for N = 4. Plugging Eq. (9)into Eq. (7) yields a representation of any 4-point localterm as a rational function of (˜ µ ˜ µ ) and ( λ i λ j ) contrac-tions: O = (˜ µ ˜ µ ) n h i n +˜ n − n h i n +˜ n h i n +˜ n × h i n +˜ n − n h i n +˜ n h i n +˜ n − n , (10)where n = P i 0. One spinorcontraction, e.g. h i , can be eliminated via the Schoutenidentity h ih i − h ih i + h ih i = 0. Then O isrepresented as a sum of rational functions of momentumtwistor contractions: O = (˜ µ ˜ µ ) n h i n h i n − n +˜ n − ˜ n h i n + n +˜ n +˜ n − n − α × h i n +˜ n + α h i n +˜ n + α h i n +˜ n − n , (11)and α is an integer in the range [0 , n + ˜ n ]. In thelast step I traded n ij and ˜ n ij for external helicities usingEq. (8). All in all, any contact term can be expressed asa sum of basis elements: O h h h h n,k = (cid:18) (˜ µ ˜ µ ) h i (cid:19) n (cid:18) h ih ih ih i (cid:19) k h i h − h − h + h × h i − h h i h − h + h − h h i − h + h − h − h , (12)where k = n + ˜ n + α ≥ 0. For a given helicity configu-ration, the candidate basis elements are parametrized bytwo integers: n and k . The former controls the canonicaldimension, which is related to n by D = 2 n + 4 − X i h i . (13)The latter labels basis elements for each canonical dimen-sion, that is at each EFT order. An important point isthat, for n fixed, k is constrained to a finite range for O h h h h n,k to possibly be a local term. This can be seenby looking at the scaling: (˜ µ ˜ µ ) ∼ s √ u , h i ∼ √ u , h i ∼ √ s , h i ∼ √ s , h i ∼ √ t , h i ∼ √ t . A contactterm must be non-singular when s or t go to zero, whichleads to the necessary condition: k min ≤ k ≤ ¯ k + n,k min = max (cid:18) , − h + h − h + h (cid:19) , ¯ k = − h + h − h − h . (14) In this discussion signs and numerical factors are irrelevant, andthey are dropped between equations. For a given n it selects a finite (or empty) set of possiblechoices of k . For the allowed range of k to be non-empty, n is bounded by n ≥ n min , n min = max (cid:0) , h + h , − ¯ k (cid:1) . (15)I stress that Eq. (14) and Eq. (15) are merely necessaryconditions. The sufficient condition for O h h h h n,k to be alocal contact term is that it can be written in the form ofEq. (7). This requires the existence of a solution to theequations relating h i , n and k to the non-negative, inte-ger exponents n ij and ˜ n ij . That is often more restrictive,thus the allowed range of k, n can be smaller (but neverlarger) than the one suggested by the necessary condi-tions. In the EFT approach one is usually interested inlowest dimension terms in the 1 / Λ expansion. Then itsuffices to inspect locality of O h h h h n,k for a few valuesof n close to n min and for the corresponding range of k .The formulas in Eqs. (12)-(15) do not treat all incom-ing particles in the same way. This is because arbitrarychoices that break the interchange symmetry have beenmade along the way: gauge-fixing µ and µ , and elim-inating h i via the Schouten identity. One could ofcourse alter these choices to arrive at a different fam-ily of basis candidates. Note that Eq. (12) is not man-ifestly parity-invariant because the momentum twistorformalism is not. Furthermore, the scaling h i ∼ √ u implies that Eq. (12) has a singularity in the u -variablefor h − h − h + h < 0. For such helicity configurationsa local term may be a linear combination of O h h h h n,k with different k . This may be cumbersome in practice,therefore it is easier to work with the configurations sat-isfying h − h − h + h ≥ 0, and obtain basis elements for h − h − h + h < P acting as h i → − h i , λ i ↔ ˜ λ i . The final comment is that Eq. (12) isnot automatically symmetric under permutations of ex-ternal particles i , j , even when h i = h j . For identicalparticles, symmetrization or anti-symmetrization of thebasis elements has to be performed a posteriori.To summarize the algorithm, the candidate basis ele-ments to span the contact terms of massless 4-point am-plitudes are given in Eq. (12). They are parametrized bytwo integers: n and k . The former is constrained by theinequality in Eq. (15) and controls the EFT expansion,with increasing n corresponding to increasing canonicaldimensions. For n fixed, k is constrained to a finite rangeby Eq. (14). The fact that momentum twistors are un-constrained variables on the manifold of 4-body kinemat-ics ensures the independence of O h h h h n,k for different n and k . The elements of this set that are local, meaningthey can be written as in Eq. (7), form a basis of con-tact terms for a given helicity configuration h , , , andcanonical dimension D = 2 n − P i h i + 4. This prescrip-tion is a purely algebraic algorithm to write down a basisof 4-point contact terms in any massless theory.For practical applications it is more convenient to trademomentum twistors for the standard helicity spinors us-ing the identity (˜ µ ˜ µ ) = − s h i = h i [23], which fol-lows directly from Eq. (4). Furthermore, one can simplify spinor expressions using t h ih i = − s h ih i . All inall, Eq. (12) can be recast as O h h h h n,k = s n − k t k h i − h h i h − h − h + h × h i h − h + h − h h i − h + h − h − h . (16)This is the central result of this paper. B. Higher-point A similar algorithm as in Section III A can be workedout for higher N . Any N -point contact term can berecast as a sum of rational function of Lorentz-invariantcontractions of N holomorphic spinors λ . . . λ N and N − µ . . . ˜ µ N − : O = Π N − k This section provides some applications of the masterformula Eq. (16) to construct bases of 4-point contactterms in concrete physical theories. A. Scalar I start with a trivial example of scattering of 4 distinctscalars, h i = 0. Eq. (16) reduces to O n,k = s n − k t k , (18)where n ≥ ≤ k ≤ n from Eq. (14).In this case O n,k simply generates all independent kine-matic invariants at a given order in the EFT expansion.For n = 0 ( D =4, or O (Λ )) the only option is k = 0, thatis a constant contact term O , = 1. For n = 1 ( D =6or O (Λ − )) there are 2 options: k = 0 and k = 1, whichcorrespond to the 2 independent Mandelstam invariants O , = s and O , = t . For n = 2 ( O (Λ − )) there are 3 in-dependent invariants O , = s , O , = st , and O , = t .And so on... For 4 identical scalars one needs to constructlinear combinations of the basis elements that are invari-ant under the S permutation symmetry. O , is triviallyinvariant. For n = 1 no invariant combination exists. For n = 2 the unique permutation-invariant combination is O , + O , + O , = ( s + t + u ) / 2. And so on... B. Scalar and Spin-h A bit less trivial exercise is to determine 4-point con-tact terms for 2 scalars and 2 identical spin-h particles.I take h , = 0 and h , = ± h . For the both-minus he-licity configuration, Eq. (14) and Eq. (15) imply n ≥ ≤ k ≤ n . I am interested in the lowest possible n , corresponding to the lowest canonical dimension viaEq. (13). For n = 0 and k = 0 Eq. (16) yields: O −− ≡ O −− , = h i h , (19)which is manifestly local. Thus the lowest order contactterm in this helicity sector is unique and is O (Λ − h ). Themirror term in the both-plus sector can be immediatelyobtained via the parity transformation: O ++ ≡ P · O −− = [23] h . (20)Obtaining the same result directly from Eq. (16) is moretricky, as O ++ is a linear combination P k a k O ++2 h,k withthe coefficients a k chosen such that the singularity in the u variable cancels out. This illustrates the fact that themethod is not manifestly parity invariant. Note that,for any integer or half-integer spin h , O −− and O ++ au-tomatically have the correct (anti-)symmetry propertiesunder the exchange 2 ↔ ≤ k ≤ n − h and n ≥ h Thismay suggest that the lowest order contact term corre-sponds to n = h . However, upon inspection of Eq. (16)for h = − h = − h , O − + n,k = s n (cid:18) h ih ih ih i (cid:19) k h i h h i − h , (21)it is clear that at least n = 2 h is needed to cancel thesingularity in h i . This example illustrates the fact thatEq. (14) and Eq. (15) are necessary but not sufficientconditions. In the end, the lowest order contact term inthe − + sector is O (Λ − h ): O − + ≡ O − +2 h, = (cid:18) s h ih i (cid:19) h = ( λ p σ ˜ λ ) h . (22)This is higher order in the EFT expansion than O −− and O ++ for any h > 0. The parity mirror is O + − = P · O − + = ( λ p σ ˜ λ ) h . The same conclusion can be reached by using the formula inRef. [15] for the minimal dimension of a contact term for a givenhelicity configuration. In summary, at the leading order in the EFT expan-sion a basis of contact terms for interactions of 2 scalarand 2 spin-h particles is 2-dimensional, consisting of O −− and O ++ . These correspond to operators of canonicaldimension D = 2 h + 4. In the language of Lagrangians,the case h = 1 / φ ( cψψ + ¯ c ¯ ψ ¯ ψ ), h = 1 to the dimension-6 operators φ ( cF µν F µν + ˜ cF µν ˜ F µν ), and h = 2 to the dimension-8operators φ ( cC µναβ C µναβ +˜ cC µναβ ˜ C µναβ ) where C µναβ is the Weyl tensor [19]. The expressions in Eqs. (19)-(22)can also be used for h > 2. Of course, massless par-ticles with spin higher than two cannot be consistentlycoupled to gravity [1, 20], thus for strictly massless par-ticles this only makes sense as an academic exercise inthe limit M Pl → ∞ . Nevertheless, the structure of thecontact terms in the massless limit may give some guid-ance for constructing a basis of contact terms for massive higher-spin particles, for which consistent theories existas EFTs. C. Euler-Heisenberg The final example is derivation of the leading ordercontact terms for spin-1 particles, h i = ± 1. I first dis-cuss an academic theory of 4 distinguishable masslessspin-1 particles, and then restrict to 4 identical particles,aka photons. The latter case corresponds to the Euler-Heisenberg Lagrangian.Starting with all-minus helicities, Eq. (14) and Eq. (15)reduce to 0 ≤ k ≤ n , and n ≥ 0. At the leading EFTorder, n = 0 or O (Λ − ), there are 3 independent contactterms corresponding to k = 0 , , O −−−− = h i h i ,O −−−− = st h i h i = −h ih ih ih i ,O −−−− = s t h i h i = h i h i . (23)where O −−−− i ≡ O −−−− ,i − . These are all manifestly lo-cal, thus O −−−− , , span a basis of contact terms in theall-minus sector. Exactly the same basis would be ob-tained via the harmonics method of Ref. [13]. A basis inthe all-plus sector is most easily obtained by the parityoperation, O ++++ i = P · O −−−− i : O ++++1 = [13] [24] , O ++++2 = − [12][13][24][34] ,O ++++3 = [12] [34] . (24)Equivalently, these elements could be obtained fromEq. (16) for n = 4 and k = 0 , , h , = − h , = +1,which implies n ≥ 2, 0 ≤ k ≤ n − 2. The leading contactterms corresponds to n = 2, k = 0: O + −− + ≡ O + −− +2 , = s h i h i h i = h i [14] , (25)and is also O (Λ − ). For the other 2-plus-2-minus con-figurations the leading contact terms can be trivially ob-tained by permutations: O −− ++ = h i [34] , O − + − + = h i [24] , etc.For configurations with a single minus or a single plushelicity Eq. (16) does not generate any O (Λ − ) local con-tact terms (they first appear at O (Λ − )). All in all, foramplitudes with 4 distinct spin-1 particles, all possiblecontact terms are spanned by the following 12 basis ele-ments: O −−−− , , , O ++++1 , , , O −− ++ + permutations . (26)In the Lagrangian parlance the the corresponding objectis a basis of dimension-8 operators, e.g. F µν F µν F αβ F αβ , F µν F µν F αβ F αβ , F µν F µν F αβ F αβ , and 9 analogous onewith one or two field strengths replaced by the dual fieldstrength: F → ˜ F . One could easily continue this exer-cise into higher orders in the EFT expansion, simply byincrementing n in the master formula Eq. (16). ThenEq. (14) suggests that, for every helicity configuration,the number of basis elements increases by one at eachconsecutive EFT order.For photons, a basis of contact terms can be con-structed as linear combinations of the elements inEq. (26) that are invariant under permutations of identi-cal particles. In the all-minus and all-plus sectors theseare O − = O −−−− + O −−−− + O −−−− ,O + = O ++++1 + O ++++2 + O ++++3 . (27)The four-photon helicity amplitudes are given by A ++++ = C + O + + O (Λ − ), A −−−− = C − O − + O (Λ − ), A −− ++ = C O −− ++ + O (Λ − ). At the leading orderin the EFT they are characterized by 3-independent pa-rameters: C − , C + , and C . These correspond to the 3independent dimension-8 four-photon operators : L = 1Λ h aF µν F αβ + bF µν ˜ F µν F αβ ˜ F αβ + cF µν F αβ ˜ F αβ i , (28)where the map is C − = 8( a − b + ic ), C + = 8( a − b − ic ), C = 8( a + b ). If parity is conserved (as in QED) then C − = C + , or equivalently c = 0, reducing the number ofparameters to two. V. SUMMARY AND DISCUSSION This paper brings you an algorithm for constructing abasis of contact terms of 4-point amplitudes in masslessEFTs. The master formula is Eq. (16). It gives a compactexpression for candidate basis elements for any helicityconfiguration h , , , of the external particles. The can-didates O n,k are labeled by two integers n and k . Theformer fixes the canonical dimension of the contact term,thus its order in the EFT expansion. At any fixed orderthere is a finite number of candidates counted by k . All O n,k in Eq. (16) are independent, but they are not guar-anteed to be local. Selecting local expressions from thisset yields a basis of 4-point contact terms. Formally, onecan test for locality by searching for solutions to a set oflinear equations relating h i , n , and k to the exponents n ij , ˜ n ij in Eq. (7), which must be non-negative integers.In the 4-point case I find it more practical to pursue a lesssystematic approach, where first a finite range of allowed n and k is identified by general arguments, for which lo-cality of O n,k can then be quickly assessed by eye.It is worth noting that this method directly constructs basis elements, although it does not a priori count them.In this sense it is orthogonal to the Hilbert series method.Indeed, the two can be used together, with the Hilbertseries method providing a useful guidance about the EFTorder where local contact terms appear and the numberof them. I verified that for all examples studied in thispaper the number of local contact terms generated byEq. (16) agrees with that predicted by the Hilbert seriesmethod [21].It is straightforward to generalize this method tohigher-point contact terms, but I leave the details to a fu-ture publication. Less straightforward is to imagine gen-eralization to EFTs in different spacetime dimensions,or with massive particles. Indeed, momentum twistorsare only defined for massless particles in four dimen-sions. New clever variables to encode kinematic dataseem necessary to perform a similar program beyond 4-dimensional massless EFTs.The final comment is that Eq. (16) contains more in-formation than just contact terms. In particular, it alsogenerates all independent terms with a single pole in theMandelstam variables. This may provide a shortcut toconstructing full tree-level amplitudes (rather than justcontact terms), which may be useful especially for higher-point amplitudes. ACKNOWLEDGMENTS I would like to thank Brian Henning, Tom Melia andIgor Prlina for useful discussions. AF is partially sup-ported by the European Union’s Horizon 2020 researchand innovation programme under the Marie Sk lodowska-Curie grant agreements No 690575 and No 674896. Appendix A: GR EFT