Deriving interaction vertices in higher derivative theories

We derive cubic interaction vertices for a class of higher-derivative theories involving three arbitrary integer spin fields. This derivation uses the requirement of closure of the Poincar\`e algebra in four-dimensional flat spacetime. We find two varieties of permitted structures at the cubic level and eliminate one variety, which is proportional to the equations of motion, using suitable field redefinitions. We then consider soft theorems for field theories with higher-derivative interactions and construct amplitudes in these theories using the inverse-soft approach.


Introduction
The study of scattering amplitudes has revealed surprising simplicity in the mathematical structures underlying Yang-Mills theories.The past few decades have seen impressive progress in our understanding of amplitudes and our efficiency in computing them.Amplitudes exhibit a number of interesting properties and satisfy a variety of relations (KLT, BCJ, color-kinematics and so on).The light-cone gauge offers a not-so-mainstream perspective on scattering amplitudes: with both locality and covariance being non-manifest, this gauge eliminates unphysical degrees of freedom at the cost of making computations more technical.Importantly, spurious degrees of freedom and redundancies do not obscure symmetries in the theory -symmetries often being key to the search for simplicity (the compact spinor helicity variables also emerge naturally in this gauge).
The classic paper [1] presented the 'derivation' of consistent cubic interaction vertices using just two ingredients: physical fields (unphysical degrees of freedom having been eliminated) and the Poincaré algebra (which must close).However, this study did not include higherderivative corrections, which often appear in effective actions (and serve as potential counter terms in loop amplitudes).Such terms were precluded by the choice of length dimension L λ−1 for the coupling constant (λ being the helicity of the fields).There has been considerable work on constructing consistent interaction vertices in the light front approach using the Fock-space method [2][3][4][5][6][7] and in momentum space [8,9].This paper expands the framework of [1] to include, beyond the usual structures, higherderivative terms.The consequences of these terms and their implications for amplitude structures -which have close ties to the light-cone formalism [10,11] -are examined.The inverse-soft method [12][13][14] is then used to build higher-point amplitudes.
Since the light-cone formalism is not covariant, Lorentz invariance needs to be verified.The key idea is to convert this 'task' into a tool, using it to constrain and then determine the Hamiltonian entirely.This allows us to construct cubic interaction vertices for a class of higher-derivative theories.This approach is also generalized to higher-point vertices and as an example, the quartic vertex is constructed for the simplest possible higher derivative operator.We also invoke symmetry arguments to explain the permissible structures for n-point interaction vertices.
Scattering amplitudes for a large class of higher derivative operators have been studied in the literature previously [15][16][17][18][19] using methods like CSW, BCFW, CHY and colorkinematic duality.These operators are not generally constructible, because of the potential boundary term.However, there have been some attempts to recursively construct a class of amplitudes for higher derivative operators using BCFW or all-line shift method [17,18].But in general, higher derivative theories are not constructible.In this paper, we attempt to recursively construct scattering amplitude for higher derivative theories using inputs from soft theorems.We use the inverse soft method, a complementary technique to those mentioned above, to derive higher-point tree-level amplitudes [13,14].In this approach, lower-point amplitudes are multiplied by a universal soft factor with appropriate legs shifted.This method is equivalent to BCFW recursion relations.In fact for MHV amplitudes, the inverse soft method is much simpler than the other known recursion relation methods.This method can only be used to construct amplitudes if there is no pole at infinity.Starting with the derived cubic interaction vertex as a seed amplitude, we construct MHV amplitudes for a class of higher derivative theories.We then extend our construction to higher-point NMHV amplitudes by starting with the known seed amplitudes, using the inverse soft technique to recursively contruct higher-point NMHV amplitudes.
2 Construction of cubic interaction vertices for higherdimensional operators We define light-cone co-ordinates in (−, +, +, +) Minkowski space-time as with ∂ ± , ∂, ∂ being the corresponding derivatives and the operator 1 ∂ + defined following the prescription in [20].x + is chosen as the time coordinate so p − is the light-cone Hamiltonian.
The Poincaré algebra in these coordinates is realized on the two physical degrees of freedom φ and φ.The Poincaré generators split into two types: kinematical K which do not involve the time derivative ∂ + and dynamical D which do -and hence pick up non-linear contributions in the interacting theory [1].The generators are The algebraic structures are Here, we review key features of this formalism and refer the reader to appendix A for additional details.
The Hamiltonian for the free field theory is with φ i referring to a field of helicity λ i with i ∈ Z + .Upon switching 'on' interactions, the δ p − operator picks up corrections, order by order, in the coupling constant α.
Reference [1] focused on the case of interactions between fields, all having helicity λ fields with α having dimensions of L λ−1 .
In this paper, we consider instead the following two -most general -ansatze for cubic interactions (based on dimensional analysis and helicity counting) Type-2 : where µ, ρ, σ, a, b are integers and A and C are numerical factors.
The key departure from [1] for both types of ansatze being that the dimension of the coupling constant is [α] = L λ 2 +λ 3 +λ1 −1 .This choice will permit us to derive cubic interaction vertices through the algebra-closure method in a new class of theories -higher derivative theories, formulated in the light-cone gauge 1 .

Type-1 cubic interaction vertices
We first start with δ ′ α p − φ 1 and use the commutation relations and dimensional analysis to find the unknown parameters.The commutators imposes the following conditions on our ansatz Let λ = λ 2 + λ 3 + λ 1 so the first equation of ( 8) reads (c + d) − (a + b) = λ.The dimensional analysis of (5) gives us the following relation Adding the first equation of ( 8) and ( 9), we get As a, b, c, d > 0, this implies that a = b = 0. Therefore, mixed derivative terms are not allowed for type-1 vertices (5).As c + d = λ, there are λ + 1 possible values for a pair (c, d).
We rewrite the ansatz (5) as a sum of these λ + 1 terms The next commutator [ δj+ , δ The above condition is satisfied if the coefficients obey the following recursion relations To determine the exact values of ρ, µ, and σ, we need the dynamical commutators [ δ j − , δ The boost generators j − and j− also get corrected when interactions are turned on and are of the form The boost generators are determined if we know the spin parts δ α s φ , δ α s φ.For type-1 vertices they are structurally of the form Due to helicity, the transformations δ α s φ and δ α s φ do not exist.We now compute The solution of the above recursion relation for ρ, σ and µ subject to the boundary conditions Plugging the values of ρ, σ and µ in our ansatz, we find Since the interaction Hamiltonian is As is well known, for odd λ, non-trivial cubic vertices require the introduction of an antisymmetric structure constant f abc .

Amplitude structures
In momentum space, the cubic vertices (22) have the following structure (with measure and constants suppressed) The off-shell spinor products in this language are In terms of spinor helicity variables [11], the vertex reads This is consistent with the general result for three-point amplitudes derived in [22,23] using S-matrix arguments, and little group scaling and in [2,3] using a Fock-space approach.

Type-2 cubic interaction vertices
We start with (6) and compute the commutators to arrive at the following conditions We also have, from dimensional analysis, Adding ( 28) and ( 30) Note that if λ 1 = 0 then (26) becomes a type-1 vertex.We now encounter a double sum as opposed to the single sum in (12).We need an index n associated with the λ 2 + λ 3 + 1 possible values the pair (a, b) can take and an index m for the λ 1 + 1 values that the pair (c, d) run over.We rewrite our ansatz (6) as the double sum The detailed calculation for this variety of vertex is presented in appendix B.
We find, for the type-2 vertex, with where u, v are functions of the λ i .
Poincaré invariance is insufficient to uniquely fix the form of cubic interaction vertices of type-2.This is because the helicity constraints permit a non-zero term in the spin part of the boost generator, ie.δ sφ (see appendix B), which is disallowed for type-1 vertices.Physically this non-zero spin part can be thought of as a loop correction to the usual spin transformation (equation 3.26 of [1]).This was noted in [24] where three-point counterterms were constructed for gravity in the light-front formalism.In that work, it was suggested that additional symmetry is necessary to uniquely determine the exact form of the counterterms.In the case of gravity, the residual gauge symmetry was used to fix the exact form of the three-point counterterm.To determine the type-2 vertex uniquely, an analog of residual gauge symmetry is likely to be necessary.
Since the type-2 vertex contains both kinds of derivatives, it can be shown to be proportional to the free equations of motion [2,25] at cubic order.We rewrite (34) as where we have used the equation of motion ).So, type-2 cubic vertices are proportional to the free equations of motion.Therefore, for this class of higher derivative theories, with only type-1 cubic vertices (22), the Hamiltonian is For example, a R 2 type operator, based on dimensional analysis and helicity, can only produce a type-2 cubic interaction vertex [24].This being proportional to the equations of motion, may be removed by a suitable field redefinition (thus all n-point graviton amplitudes produced by the R 2 term vanish as expected [19]).
We can generalize this framework from cubic vertices to specific class of n-point interaction vertices as discussed below.We construct the simplest possible quartic vertex as a specific example.The details are presented in the appendix C.
Comments on n-point interaction vertices in higher derivative theories In this section, we first deduce the structure of interaction vertices at higher orders purely from dimensional and kinematical constraints and then prove that all n-point vertices containing purely one type of transverse derivatives can be uniquely fixed by the Poincaré algebra.
We work here with a special class of higher derivative theories where λ i = λ, and work out the structure of interaction vertices at higher orders purely from symmetry constraints.In a perturbative expansion, the dimension of the coupling for a n-point interaction vertex is where α is the 3−point coupling.The n-point Hamiltonian is of the form φ p φq .We start with the ansatz where p + q = n and the a i , c i are non-negative integers and the µ i are integers.The commutator [δ j , δ p − ] yields Using (38), (39) and the non-negativity of powers of transverse derivatives we obtain For λ = 1, we get p, q ≤ 3.At cubic order, note that the (p = 3, q = 0) A A A structure and (p = 0, q = 3) Ā Ā Ā structure follow from this.At the next order, two new structures A 3 Ā and Ā3 A are allowed as compared to the usual Yang-Mills quartic vertices.
We then consider λ = 2, and use q = n − p in (40) and ( 41) The interaction vertex may be odd or even.For an odd point vertex, n = 2m + 1 where m is a positive integer.This gives Thus, for a cubic vertex where m = 1, (43) allows helicity structures h h h and h h h.
For an even point vertex n = 2m we get Since p is an integer, the condition (44) is For a quartic vertex where m = 2, (45) allows terms of helicity structure h h3 and h h 3 .Thus, higher derivative operators produce new helicity configurations at each order.For example, in [26] it was shown for n = 6, that only h 3 h3 type of vertices occur.Here, additional vertices of type h 2 h4 and h2 h 4 appear at n = 6.
We now prove that: all npoint vertices containing purely one type of transverse derivatives can be uniquely fixed by the Poincaré algebra.
We start with an ansatz for the n-point interaction vertex (38) and plug a i = 0.It reads Consistency with the helicity generator j requires The commutator with the generators j +− , j + determines the vertex upto the powers of ∂ + s.The exact powers of ∂ + s are fixed by the dynamical generators.We argue that the spin transformation appearing in the dynamical generator j − at this order is trivial and hence can be used to uniquely determine the vertex.
The number of transverse derivatives in the spin transformation δ α n−2 s φ ∼ α n−2 φ 1 ...... φn−1 (derivatives suppressed) must be one less than that in δ α n−2 p − φ due to its dimensionality.The dynamical generator j − has helicity +1.In order for the spin transformation δ α n−2 s φ to have helicity +1, the number of transverse derivatives in it must be one greater than that in δ α n−2 p − φ.This proves that the spin transformation δ α n−2 s φ cannot be consistent with the helicity and dimensionality simultaneously and hence must vanish.This allows us to uniquely determine the vertex.Therefore, all n-point vertices containing purely one type of transverse derivatives can be uniquely fixed by the Poincaré algebra.

Vertices to Amplitudes : Motivation to use the inverse soft technique
We would now like to compute amplitudes using the interaction vertices derived previously.In principle, one can construct the n-point tree-level amplitudes.However, the calculation becomes mathematically tedious as the number of Feynman diagrams and the number of terms in the diagram increases exponentially.For example, to calculate a 4-point tree-level amplitude, we need to sum over contributions from the exchange diagrams and the contact diagram.The contact term may be derived by the closure of Poincaré algebra in special cases (as discussed above) which in itself requires a lot of work.We therefore use a complementary technique, the inverse soft method, in the next section to derive higher-point tree-level amplitudes for the higher derivative theories (see appendix C).

Inverse soft construction in higher derivative theories
Soft factors in the light-cone gauge and their use in the construction of higher-point interaction vertices were the focus of [27] (this included a review of the light-cone realization of many covariant results from [12,13,28]).In this section, we explore these methods in the context of the higher-derivative theories considered in this paper.
The idea that higher-point tree level amplitudes can be constructed from the lower-point ones by using a multiplicative universal factor, associated with the emission of a soft boson was first presented in [12] and then subsequently developed in [13,14].It was shown in [13], that inverse soft construction with BCFW [29,30] as a guide, can be used to construct gauge theory and gravity amplitudes.

The inverse soft recursion relation for a n-point tree-level gauge amplitude is
The prime above indicates that momentum conservation on the right-hand side requires a shift in the momenta of adjacent particles p n−1 and p 1 .For a positive helicity soft particle n, the shift is [13] Here only the neighbouring particles are shifted because only they are affected by the soft limit.For the case of gravity, the soft factor depends on all legs thus inverse soft expression (47) will involve sum over all particles.
We will now employ the inverse soft method to construct higher-point amplitudes for theories with higher-dimensional operators.We use the three-point amplitudes found in the previous section as seed amplitudes to systematically construct different classes of higherpoint amplitudes.

HF 2 operator
This is the simplest possible gauge invariant higher-dimensional operator involving spin-0 and spin-1 fields.This is a 5-dimensional operator given by where ν is the gluon field strength and f abc , the structure constant of the gauge group.H is the real scalar field.We consider the cubic interaction Hamiltonian by plugging λ 1 = 0 , λ 2 = λ 3 = 1 in (36) where φ is a complex scalar field and H = φ + φ † .
We can see that by working with the physical fields (using the light-cone gauge), the amplitude for the operator HF 2 naturally decomposes into a holomorphic and an antiholomorphic parts.This key idea was first presented in [15] due to the self-duality of the field tensor.The full amplitude for this operator (49) can be obtained as where A n is the partial color ordered amplitude.The color decomposition for tree-level amplitudes of this operator is similar to the Yang-Mills case [15].
We now construct higher-point amplitudes for this operator.We start with the three-point amplitudes and construct separately the holomorphic and anti-holomorphic amplitudes using the inverse soft and then add them to obtain the full amplitude.
The above formula is valid for both massive and massless scalars, and it reduces to the pure Yang-Mills amplitude when we take the momentum of the scalar to zero.
We now construct the non-MHV amplitudes using the inverse soft method.The four point anti-holomorphic amplitude is constructed by adding a soft gluon of positive helicity and using the appropriate shift defined in (48) One can now recursively construct this for n-point by multiplying with appropriate soft factor and deforming the adjacent legs The above amplitude under the shift [1 + , n + behaves well at large z and there is no pole at infinity so the inverse soft construction is valid for non-MHV amplitudes.As, A 3 (φ, 1 + , 2 + ) = 0 therefore all higher-point amplitudes will be zero.So the full n-point amplitude is Similarly A n (H, 1 − , 2 − , ....n − ) can be constructed using the inverse soft method.Our results match with the amplitudes derived using MHV vertex expansion method in the literature [15,31].

F 3 operator
We now consider a gauge invariant higher-dimensional operator involving purely spin-1 fields.The simplest of such operators is 6-dimensional given by where is the gluon field strength and f abc , the structure constant of the gauge group.
The cubic interaction vertex can be obtained from (36) by setting λ 1 = λ 2 = λ 3 = 1 so λ = 3, and the Hamiltonian is Similar to the previous case, we see that the decomposition of the operator (62) into holomorphic and anti-holomorphic parts is manifest in the light-cone gauge.This decomposition was first presented in [15,16].So the full F 3 amplitude can be constructed by taking the sum of the holomorphic and anti-holomorphic amplitude.
In this theory, holomorphic amplitudes with exactly three negative helicity gluons and an arbitrary number of positive helicity gluons are referred as MHV amplitudes and are denoted by A F + .The anti-holomorphic amplitudes with exactly three positive helicity gluons and an arbitrary number of negative helicity gluons are referred as anti-MHV and are denoted by A F − .
It was shown in [33,34] that for the higher derivative theory of massless particles in four dimensions, the tree-level soft photon and graviton theorems receive modifications at subleading and subsubleading orders.However, the leading soft factors are not altered for these theories because these interactions (F 3 , R 3 ) are generically suppressed in the soft-limit [34].Therefore, the leading soft factor for the Yang-Mills theory with F 3 correction in the soft gluon limit is same as (53).
The three-point amplitudes extracted from (63) are A four-point tree-level MHV amplitude corresponding to a single insertion of the F 3 operator can be constructed using this as follows (we attach a soft gluon between the adjacent legs 1 and 3).
The above construction is valid because under the following shift [1 − , 4 + , the amplitude has no pole at infinity.So the inverse soft construction is valid for this class of amplitudes.
The 5-point MHV amplitude can also be constructed similarly.At the 6-point level, the The inverse soft recursion relation for gravity, for a n-point amplitude, reads [13] M M HV (1, ......., n The prime indicates the momentum shift, mentioned earlier.For a positive helicity soft graviton k, the shift is Using the inverse soft method, the four-point tree-level amplitude is For the case of gravity and R 3 theories, the higher-point amplitudes are functions of both the holomorphic and anti-holomorphic spinors that makes the construction of amplitudes using the inverse soft approach more involved.We show below an example of the construction of the five-point amplitude in the R 3 theory.
[45] 14 24 54 15 25 Using the appropriate shifts, the amplitude reads where M R 5 is the five-point amplitude for pure gravity (the proportionality to gravity ensures 'constructibility').The n-point holomorphic MHV amplitude for R 3 operator can be written as The anti-holomorphic part corresponding to R 3 operator can be similarly obtained.This is consistent with the results derived in [16] using color-kinematic duality.Similar to the previous section, we can construct a class of higher-point NMHV amplitudes using the inverse soft technique.We present the construction of the five-point NMHV amplitude in appendix D.
As we turn on interactions, the other dynamical generators also pick up corrections where δ g s and δ g s represent the spin transformations.
The requirement of closure of the Poincaré algebra imposes various conditions on the integers introduced in (80).The result is From ( 78), the complete Hamiltonian to this order reads [10] If we set λ 1 = λ 2 = λ 3 = λ ′ in (83) with λ ′ odd, H g vanishes.Hence a self-interaction Hamiltonian for odd integer spins exists, if and only if we introduce a gauge group and it reads

B Derivation of cubic vertices with mixed derivatives
The refined ansatz for cubic interaction vertices with mixed derivatives reads To determine the vertex, we first use the kinematical commutators They give the following conditions These conditions are satisfied if the coefficients obey the following recursion relations The idea is to use the dynamical commutators [ δ j − , δ p − ] α φ 1 = 0 to fix ρ, µ, and σ.To solve these commutators we need the spin parts δ α s φ , δ α s φ.For type-2 vertices, these are structurally For type-2 vertices, both the spin parts have a non-trivial structure and closing the dynamical commutators [ δ j − , δ p − ] α φ 1 = 0 does not fix the values of ρ, µ, and σ.Therefore we are forced to introduce two functions u(λ i ) and v(λ i ) that capture this ambiguity in the powers of ∂ + .
The ansatz for the quaric vertex is where β is the coupling constant, X is a constant and f ijk is the structure constant.The commutator [ j , p − ] β produces The commutator [ j The solutions to the recursion relation fall in four independent classes.For this helicity configuration, δ β s φ does not exist because its consistency with the helicity generator j requires such a term to have four transverse derivatives rendering it inconsistent with the dimensionality.We then fix the values of µ , ρ , σ , δ using the commutator with dynamical generator j − for each class of solution.The final form of the quartic vertex is where C 1 , C 2 , C 3 and C 4 are numerical cofficients to be fixed.The Poincaré algebra uniquely fixes the structure of the vertex up to the overall numerical constant for each possible solution.These coefficients will be fixed using the fact that the vertex is antisymmetric under the exchange of gluon legs.
The form of the vertex (98) in real space is complicated, we write the vertex in the momentum space such that it is manifestly antisymmetric.In momentum space, the quartic vertex has the following structure (measure and constants suppressed) + C 2 q a q + (N −a) 1 p + k c k + (2−N −c) l (3−a−c) l + (a+c−1) + C 3 q a q + (N −a) p (N −a) p + (a−1) k c k + (2−N −c) l (3−N −c) l + (c−1) (99) + C 4 q (3−a−c) q + (a+c−1) 1 p + k a k + (2−N −a) l c l + (N −1−c) φ Āi Āj Āk + c.c. .We write the above expression in terms of spinor helicity variables using the binomial expansion.
V β = The vertex (101) matches with the known result in the literature [36].The construction for the case (0 + ++) follows in a similar manner.The vertices with helicity configurations (0++−) , (0−−+) contain mixed transverse derivatives.For such type of vertices, the spin transformations in both j − , j− are non-trivial.Hence, these cannot be uniquely fixed by the algebra.Moreover, at this order, these vertices are proportional to the free equations of motion and hence can be removed by suitable field redefinition.
D Five-point NMHV amplitude for R 3 operator The five-point NMHV amplitude can be constructed using the inverse soft method as shown below The soft factor S for graviton leg 5 of positive helicity is [13] S(1, 5 The first class of diagram has three terms which are related by the interchange of gravitons 2, 3, 4. We evaluate one of the terms