Scattering Amplitude Recursion Relations in BV Quantisable Theories

Tree-level scattering amplitudes in Yang-Mills theory satisfy a recursion relation due to Berends and Giele which yields e.g. the famous Parke-Taylor formula for MHV amplitudes. We show that the origin of this recursion relation becomes clear in the BV formalism, which encodes a field theory in an $L_\infty$-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that $L_\infty$-algebra, known as a minimal model. In fact, the quasi-isomorphism contains all the information about the scattering theory. As we explain, the computation of such a minimal model is readily performed in any BV quantisable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.


Introduction and results
Whilst string theory has not yet fulfilled its initial promise of a complete and unified description of nature, it has certainly become a successful way of thinking about quantum field theories. Here, we would like to adopt a perspective which has its origin in string field theory and which was suggested e.g. in [1][2][3][4]. The structures of the Hilbert spaces of string field theories are encoded in homotopy algebras, and in the case of closed string field theory in terms of L 8 -algebras [5]. The relevant classical action is simply the canonical action associated with an L 8 -algebra and which is known as the homotopy Maurer-Cartan action. Homotopy Maurer-Cartan theory can be thought of as a vast generalisation of Chern-Simons theory. One might hope that this rich structure is somehow reflected in ordinary field theory, which one could then exploit, e.g. in often cumbersome computations of scattering amplitudes. As we shall see, this is indeed the case and the place to look for L 8 -algebras is the Batalin-Vilkovisky (BV) formalism [6][7][8][9][10].
In this paper, we shall combine the following three facts, which should be familiar to any expert on BV quantisation and which are explained in detail e.g. in [3,4]: i) The BV formalism assigns to any field theory it can treat an L 8 -algebra describing its symmetries, field contents, equations of motion, and Noether currents; ii) Quasi-isomorphic L 8 -algebras describe physically equivalent field theories [3] (see also [11]); iii) Any L 8 -algebra comes with a minimal model which is a smallest representative in its quasi-isomorphism class and whose propagators vanish.
Given an L 8 -algebra of a classical field theory, we are thus led to conclude that the npoint vertices of the field theory described by its minimal model should be the tree-level scattering amplitudes of the original field theory. An obvious candidate for investigating the validity of our conclusion is the famous Parke-Taylor formula, which describes a huge simplification in adding up the Feynman diagrams contributing to maximally helicity violating (MHV) gluon scattering amplitudes. After its conjecture in [12], this formula was proved by Berends and Giele in [13] using a recursion relation for particular currents. We shall show that this recursion relation is nothing but the explicit formula for computing a minimal model in the concrete case of Yang-Mills theory.
We begin with a very concise review of L 8 -algebras, quasi-isomorphisms, minimal models, and their appearance in the BV formalism in Section 2. We then discuss our formalism in Section 3 for the example of scalar field theory in which the relevant structures and their interpretation become very obvious. Full Yang-Mills theory and the gluon current recursion relations are then discussed in Section 4. We explain that the Berends-Giele recursion relations for tree-level scattering amplitudes in Yang-Mills theory arise as recursion relations of an underlying quasi-isomorphism of L 8 -algebras.
Let us also summarise a number of general important observations that follow from our constructions. Any BV quantisable field theory gives rise to an L 8 -algebra L. By the strictification theorem for L 8 -algebras, there is a strict L 8 -algebraL that is quasiisomorphic to L. In other words, any BV quantisable field theory can be cast into an equivalent field theory which has only propagators and cubic interaction vertices. Notice, however, whilst always guaranteed in theory, finding the explicit strict version of a theory might be difficult in practice. Furthermore, the minimal model L˝of the L 8 -algebra L can always be constructed recursively, and this construction involves a particular map, taking the role of a contracting homotopy, which can be chosen to include the Feynman propagator of the theory. This explains that the minimal model describes the tree-level scattering amplitudes of the original field theory. Other choices of the contracting homotopy, however, are possible, and we expect interesting results for perturbation theory to emerge also from those.
There are clearly many avenues for further study of the structures we discussed. Most important is certainly the development of the full quantum picture, going beyond the tree level. We note that besides deeper insights into the symmetries and structures of Feynman diagrams, this research may also lead to a formulation of quantum field theory in purely algebraic terms, which should be much more accessible to mathematicians than the standard textbook presentation.
Whilst finishing this paper, we received the announcement of the forthcoming paper [14] by Alexandros Arvanitakis in which he also discusses the S-matrix in terms of minimal models. At the same time, we became aware of the preprint [15] in which a mathematical explanation of the on-shell Britto-Cachazo-Feng-Witten (BCFW) recursion relations [16,17] via minimal models of L 8 -algebras was given. Contrary to our general constructions in which the L 8 -algebra of a field theory always arises from the BV formalism, the discussion in [15] relies on the explicit strict models of the L 8 -algebras for the considered field theories. We also note that the idea of obtaining scattering amplitudes from minimal models is not new and can be traced back, at least, to [1,2]. The latter references also developed much of the technology we are using in the following.
2. L 8 -algebras of field theories L 8 -algebras are generalisations of Lie algebras to differential graded Lie algebras and beyond. We will review the relevant definitions and refer the interested reader to [3,4] for all the details.
2.1. L 8 -algebras and quasi-isomorphisms L 8 -algebras. To begin with, let L :" À kPZ L k be a Z-graded vector space. Elements of L k are said to be homogeneous and of degree k, and we shall denote the degree of a homogeneous element P L by | | L P Z. Suppose there is a differential µ 1 : L Ñ L of degree 1. This allows us to consider the chain compleẍ¨¨Ý Next, we equip this complex with products µ i : Lˆ¨¨¨ˆL Ñ L of degree 2´i for i P N which are i-linear and totally graded antisymmetric and subject to the higher or homotopy Jacobi identity 1 ÿ j`k"i ÿ σPShpj;iq χpσ; 1 , . . . , i qp´1q k µ k`1 pµ j p σp1q , . . . , σpjq q, σpj`1q , . . . , σpiq q " 0 (2.1b) for 1 , . . . , i P L. The sum over σ is taken over all pj; iq shuffles which consist of permutations σ of t1, . . . , iu such that the first j and the last i´j images of σ are ordered: σp1q ă¨¨¨ă σpjq and σpj`1q ă¨¨¨ă σpiq. Furthermore, the sign χpσ; 1 , . . . , i q is called the graded Koszul sign and defined by means of Specifically, when i " 1, the homotopy Jacobi identity (2.1b) just says that µ 1 is a differential. For i " 2, it says that µ 1 is a derivation with respect to µ 2 and for i " 3, it says that the binary product µ 2 satisfies a generalisation of the standard Jacobi identity, and so on.
A Z-graded vector space with such products µ i is called an L 8 -algebra [18][19][20]. Particular examples of L 8 -algebras include the trivial L 8 -algebra L " À kPZ L k with L " t0u, ordinary Lie algebras with L " L 0 and the only non-vanishing product being µ 2 , as well as differential graded Lie algebras with general L for which µ i " 0 for i ě 3. The latter are also called strict L 8 -algebras. L 8 -morphisms. Morphisms between Lie algebras are maps preserving the Lie bracket. The higher categorical nature of L 8 -algebras now leads to a vast generalisation of Lie algebra morphisms to L 8 -morphisms. Explicitly, an L 8 -morphism φ : pL, µ i q Ñ pL 1 , µ 1 i q between two L 8 -algebras pL, µ i q and pL 1 , µ 1 i q is a collection of i-linear totally graded antisymmetric maps φ i : Lˆ¨¨¨ˆL Ñ L 1 of degree 1´i such that with χpσ; 1 , . . . , i q the Koszul sign and ζpσ; 1 , . . . , i q given by (2.2b) Note that for Lie algebras, this definition just reduces to the standard definition of a Lie algebra morphism.
Since µ 1 is a differential, we may study the cohomology ring H ‚ µ 1 pLq of the chain complex (2.1a). If the map φ 1 of an L 8 -morphism φ : pL, µ i q Ñ pL 1 , µ 1 i q induces an isomorphism on the cohomology rings H ‚ µ 1 pLq -H ‚ µ 1 pL 1 q, then φ is called a quasi-isomorphism, generalising quasi-isomorphisms of chain complexes. Quasi-isomorphisms are, in most cases, the appropriate notion of isomorphisms for L 8 -algebras.
Strictification theorem. General strictification theorems for homotopy algebras [21,22] specialise to L 8 -algebras and state that any L 8 -algebra is quasi-isomorphic to a strict L 8 -algebra. The latter is then called a strict model for the former. Recall that a strict L 8 -algebra is a differential graded Lie algebra because only the differential and the binary product are non-vanishing.
We shall see explicit examples of strictifications of L 8 -algebras in Sections 3.2 and 4.2. In practice, however, it often turns out that the transition to the strict model of an L 8algebra is either hard to begin with or not very convenient and too restrictive for further computations.
Minimal model theorem. A companion theorem to the above one is the minimal model theorem: any L 8 -algebra pL, µ i q is is quasi-isomorphic to an L 8 -algebra pL˝, µi q for which µ1 " 0. The latter is indeed a minimal model, i.e. a minimal representative of the quasi-isomorphism class of L, since the graded vector space underlying L˝is its own cohomology ring H ‚ µ 1 pLq. We note that minimal models are unique up to L 8 -isomorphisms (i.e. L 8 -morphisms φ with φ 1 invertible).
The construction of a minimal model for an L 8 -algebra L means to compute the L 8structure given by brackets µi on the cohomology ring H ‚ µ 1 pLq ": L˝. Since this will be central to our discussion, we give some more details. We start from a choice of projection p : L L˝together with an embedding e : L˝ãÑ L which are both degree 0 chain maps and satisfy p˝e " id L˝. We then always have a degree´1 chain map h : L Ñ L such that Such a map h is called a contracting homotopy, and we summarise this pictorially as (2.4) Evidently, since pL, µ 1 q is a complex and p and e are chain maps, we have µ 1˝µ1 " 0 , µ 1˝e " 0 , and p˝µ 1 " 0 . (2.5) Upon combining this with (2.3), we obtain The map e˝p in (2.3) is a projector onto a subspace of L, however, the maps h˝µ 1 and µ 1˝h on the right-hand side are not, in general. We can always rectify this by redefining the contracting homotopy according to Indeed, with the help of (2.3) and (2.5), one may check that h satisfies In particular, h is again a contracting homotopy. Moreover, we now have a decomposition 2 L -L harm ' L ex ' L coex , L harm :" impe˝pq , L ex :" impµ 1˝h q , L coex :" imph˝µ 1 q (2.9) with L harm -L˝. This is known as the abstract Hodge-Kodaira decomposition. It is rather straightforward to verify that The quasi-isomorphism between pL, µ i q and pL˝, µi q is now determined by the maps φ i : L˝ˆ¨¨¨ˆL˝Ñ L which are constructed recursively as [2] φ 1 p 1 q :" ep 1 q , φ 2 p 1 , 2 q :"´ph˝µ 2 qpφ 1 p 1 q, φ 1 p 2 qq , . . .
Cyclic L 8 -algebras. The appropriate notion of a metric (or indefinite inner product) on an L 8 -algebra is the following one. A cyclic structure on an L 8 -algebra pL, µ i q is a non-degenerate bilinear graded symmetric pairing x´,´y L : LˆL Ñ R of degree k which is cyclic in the sense of for 1 , . . . , i`1 P L. An L 8 -algebra quipped with an inner product is called a cyclic L 8algebra.

Homotopy Maurer-Cartan theory
The BV formalism can be seen as a reformulation of a Lagrangian field theory as a homotopy Maurer-Cartan theory, which is a generalised form of a Chern-Simons theory. In the following, we concisely recall the basic facts and refer again to [3,4] for more details.
Homotopy Maurer-Cartan equation. Given an L 8 -algebra pL, µ i q, we call an element of degree 1, a P L 1 , a gauge potential and define its curvature by (2.14) Due to the higher Jacobi identities (2.1b), the curvature f obeys the Bianchi identity Infinitesimal gauge transformations are mediated by degree 0 elements c 0 P L 0 and are given by a Þ Ñ a`δ c 0 a with δ c 0 a :" Using the higher Jacobi identities (2.1b), one may show that Thus, the gauge transformations always close for strict L 8 -algebras, for which only the differential and the 2-product are non-trivial. In the general case, however, a restriction of the gauge potential is required to ensure closure, and a sufficient condition is It is important to stress that the gauge parameters c 0 P L may enjoy gauge freedom themselves which is mediated by next-to-lowest gauge parameters c´1 P L of degree´1. In turn, the next-to-lowest gauge parameters c´1 P L enjoy gauge freedom that is mediated by next-to-next-to-lowest gauge parameters c´2 P L for degree´2, and so on. These are known as the higher gauge transformations and they are given by with c´k P L of degree´k. Note that f " 0 is also a sufficient condition for the higher gauge transformations to close.
Homotopy Maurer-Cartan action. The homotopy Maurer-Cartan equation is variational whenever pL, µ i , x´,´yq is a cyclic L 8 -algebra with an inner product x´,´y of degree´3. Indeed, the gauge invariant action functional known as the homotopy Maurer-Cartan action, has the homotopy Maurer-Cartan equation as its stationary locus.
Homotopy Maurer-Cartan elements and L 8 -morphisms. Let us now briefly explain how Maurer-Cartan elements transform under L 8 -morphisms. For any L 8 -morphism φ between two L 8 -algebras pL, µ i q and pL 1 , µ 1 i q, there is a natural morphism of gauge potentials, a Þ Ñ a 1 :" which thus maps Maurer-Cartan elements to Maurer-Cartan elements. Furthermore, a gauge transformation a Þ Ñ a`δ c 0 a with gauge parameter c 0 P L 0 of a Maurer-Cartan element a P L 1 is transformed under an L 8 -morphism to a 1 Þ Ñ a 1`δ c 1 0 a 1 with a 1 P L 1 given by (2.21) and Consequently, gauge equivalence classes of Maurer-Cartan elements are mapped to gauge equivalence classes of Maurer-Cartan elements. Note that whenever φ is a quasi-isomorphism, the moduli space of Maurer-Cartan elements for pL, µ i q (that is, the space of solutions to the homotopy Maurer-Cartan equation modulo gauge transformations) is isomorphic to the moduli space of Maurer-Cartan elements for pL 1 , µ 1 i q.

Field theory and underlying L 8 -structures
Classical observables. The most general approach to the quantisation of gauge theories is certainly the BV formalism. To prepare the field theory for quantisation, the BV formalism constructs a modern description of the space of classical observables, which are the functionals on the space of solutions to the field equations modulo gauge symmetries. This space is now described as (part of) the cohomology ring of a differential complex known as the BV complex,¨¨Q where C 8 i pFq denote functionals of degree i on the Z-graded vector space of BV fields F, which is parametrised by fields, ghosts (of positive degree), and anti-fields (of negative degree). The BV complex contains the classical observables in C 8 0 pFq and the kernel of the BV differential Q BV : C 8 0 pFq Ñ C 8 1 pFq consists of functionals of degree 0, restricted to the solutions of the classical field equations. Gauge trivial observables are given by the image of Q BV in C 8 0 pFq. The elements of degree 0 in the cohomology ring thus contain indeed the space of classical observables. L 8 -algebra structure. Now the BV complex is evidently a differential graded commutative algebra and its dual is a differential graded commutative coalgebra, which is nothing but an L 8 -algebra. Explicitly, the action of the BV differential Q BV on the coordinate functions on F is written as a polynomial in the fields, ghosts, antighosts, and their derivatives, which is simply the dual of the sum over all higher products µ i on the graded vector space F. Schematically, we can write where ξ is the sum over all coordinate functions on F parametrising fields, ghosts, and antighosts. Given the µ i , we can construct Q BV and knowing Q BV , we can reconstruct the µ i as higher products from this equation.
Recall furthermore that the BV formalism comes with a Poisson bracket, sometimes called the antibracket, which is induced by a canonical symplectic form of degree´1 on F. This form induces a cyclic structure on the L 8 -algebra on F, which is of degree´3.
Altogether, we conclude that any field theory has an associated cyclic L 8 -algebra structure on its BV field space F, and this structure is recovered from the BV differential and the BV antibracket. For more details and the precise formulation of equation (2.24), see [3,4].
The fact that any classical Lagrangian field theory comes with a L 8 -algebra is certainly well-known by experts on the BV formalism. It has been rediscovered several times, see e.g. [23] or [24] and also [3,4] for more historical references. The structural advantages of this description, however, have not been fully exploited in our opinion, and this is what we set out to do in this paper.

Scalar field theory
As an introductory example illustrating the construction of an L 8 -algebra for a classical field theory, the computation of its minimal model and the recursion relations, we consider scalar field theory on four-dimensional Minkowski space R 1,3 :" pR 4 , ηq with η the Minkowski metric. In the following, µ, ν, . . . " 0, . . . , 3, and we shall write x¨y :" η µν x µ y ν " x µ y µ and l :" B µ B µ . All of our constructions in this section generalise rather evidently to arbitrary field theories admitting a (classical) BV formulation.

L 8 -algebra formulation of scalar field theory
Instead of plain ϕ 4 -theory, we start from the action which will demonstrate the relation between the minimal model and tree level amplitudes more clearly.
Scalar L 8 -algebra. The associated L 8 -algebra of this field theory is obtained as usual from the BV formalism. 3 Here, we merely note that in a field theory without (gauge) symmetry to be factored out, the BV action agrees with the classical action. The homological vector field Q BV therefore acts only non-trivially on the anti-field ϕ`, and we have The resulting L 8 -algebra is therefore 3 See also [25] for pure ϕ 4 -theory and [3] for a discussion closer to ours.
Cyclic structure. This L 8 -algebra, however, is too general. In particular, we cannot extend it to a cyclic one with the cyclic structure given by the integral, Firstly, finiteness of the integral is not guaranteed and secondly, boundary terms arising when partially integrating the Laplace operator may violate cyclicity. We are thus led to restricting the function space to the Schwartz functions S pR 1,3 q Ď C 8 pR 1,3 q, i.e. functions which are rapidly decreasing towards the boundary of Minkowski space R 1,3 . This restriction, however, is too harsh: the kinematical operator is invertible as a map µ 1 : S pR 1,3 q Ñ S pR 1,3 q, as we will explain in more detail below. Therefore, the cohomology of L would be trivial and the L 8 -algebra would be quasi-isomorphic to the trivial one.
To fix this issue, we should also include the solutions to the classical Klein-Gordon equation, kerpµ 1 q Ď C 8 pR 1,3 q, in our field space. More precisely, we should restrict ourselves to those solutions with compactly supported Cauchy data, ker c pµ 1 q Ď kerpµ 1 q. Our total field space is then F :" ker c pµ 1 q ' S pR 1,3 q . Note that both subspaces are vector spaces and their intersection is empty, since there are no solutions to the Klein-Gordon equation in S pR 1,3 q, which is a simple consequence of energy conservation. This, however, requires an adjustment of the higher products, since F is not closed under multiplication: the product of two elements in ker c pµ 1 q is neither in S pR 1,3 q nor in ker c pµ 1 q. The standard procedure here is to replace the coupling constants with bump functions, which reflects the fact that interactions should be turned off at asymptotic times. On the other hand, we require that the product of two Schwartz-type functions can have an overlap with ker c pµ 1 q, which we extract by restricting the Fourier transform to its on-shell modes. Explicitly, we have a map to on-shell states where δ is the Dirac delta and Θ the Heaviside distributions and the hat indicates the Fourier transform. Compactly supported Cauchy data of the image of ℘ is guaranteed since the Fourier transform maps S pR 1,3 q to S pR 1,3 q. Finally, we regularise the kinematical operator µ 1 already now to obtain well-defined Green's functions later on. Altogether, we thus define for ϕ 1,2,3 P C 8 pR 1,3 q and δ, ε P R`. Here, p1`℘qpϕq is shorthand for ϕ`℘pϕq. Clearly, these products break Lorentz invariance, but we can eventually consider the limit δ Ñ`0 to restore Lorentz invariance in all our final results. Also, note that the homotopy Jacobi identities are trivially satisfied since the only non-trivial µ i map L 1ˆ¨¨¨ˆL1 to L 2 and thus nested expressions of µ i vanish trivially.
As cyclic structure, we clearly want to use the L 2 -inner product on S pR 1,3 q Ď L 2 pR 1,3 q, and we extend it as follows to F: for ϕ 0 , ψ 0 P ker c pµ 1 q and ϕ i , ψ i P S pR 1,3 q. Here, C is a Cauchy surface of constant time in R 1,3 and the inner product is independent of the choice. This inner product is indeed cyclic with respect to the higher products (3.7). We thus complete the construction of a cyclic L 8 -algebra structure on the complex lo omo on It should be rather obvious that the above construction of a cyclic L 8 -algebra can be performed for any field theory to which we can apply the BV formalism; in particular, the specialisation of the field space to Schwartz-type functions and on-shell modes readily generalises. We also note that the technicalities of choosing the appropriate field space mostly arose because we insisted on a consistent cyclic structure on the L 8 -algebra. If one is happy to do without a precise cyclic structure, one can essentially neglect this issue, cf. also [15].

Strictification of scalar field theory
Let us briefly discuss the strictification of the L 8 -algebra L, which consists of an L 8 -algebrã L which is quasi-isomorphic to L and for whichμ i " 0 for i ě 3. Equivalently, we find an actionS which yields a field theory that is classically equivalent to that of the action (3.1) but whose interaction terms are at most cubic in the fields. This is done by introducing auxiliary fields, and the strictification for pure ϕ 4 -theory with κ " 0 was already given in [3]. For simplicity, we will work with the naive, unregularised L 8 -algebra (3.3).
Differential graded Lie algebra structure. Constructing an equivalent actionS is straightforward, and one possible form is where X and Y are two additional auxiliary scalar fields X, Y P C 8 pRq. The corresponding L 8 -algebraL reads as lo omo on 11a) and has non-trivial higher products A possible quasi-isomorphism ϕ :L Ñ L has non-trivial maps (3.12) As one readily checks, these maps satisfy the non-trivial relations in (2.2), where Φ i " pϕ i , X i , Y i q PL 1 . Furthermore, the homotopy Maurer-Cartan action (2.20) for L is indeedS.

Scattering amplitudes and recursion relations
Minimal model. To compute the minimal model L˝of L, we need to find a contracting homotopy (3.14) We start by noticing that µ 1 is a map µ 1 : F Ñ S pR 1,3 q and it is invertible as a map µ 1 : S pR 1,3 q Ñ S pR 1,3 q. Its inverse is the Feynman propagator G F , which is defined for functions ϕ P S pR 1,3 q by where the integral kernel is It satisfies`´l´m

16a)
for ϕ P S pR 1,3 q or, more formally, For more and precise details, see e.g. [26,Chapter 14]. We trivially extend G F to a linear functionG F on all of F with kerpG F q " ker c pµ 1 q.
Because the invertibility of µ 1 on S pR 1,3 q implies surjectivitiy on S pR 1,3 q, it is now clear that the graded vector space of the minimal model L˝of L is With the help ofG F , we can define the projections The embeddings e 1,2 : ker c pµ 1 qãÑF are simply the trivial ones. The maps we introduced so far satisfy the relations and we have the following picture: S pR 1,3 q

(3.25)
Cyclic structure. We also stress that the compatibility of the L 8 -algebra morphism φ : L˝Ñ L with the cyclic structure x´,´y L on L is trivially satisfied. First of all, we have xφ 1 p¨¨¨q, φ i p¨¨¨qy L " xep¨¨¨q, hp¨¨¨qy L " 0 for i ě 2 since impeqker c pR 1,3 q, imphq -S pR 1,3 q and the direct sum F " ker c pR 1,3 q ' S pR 1,3 q is an orthogonal decomposition with respect to the cyclic structure x´,´y L . Then, we have xφ i p¨¨¨q, φ j p¨¨¨qy L " xhp¨¨¨q, hp¨¨¨qy L " 0 for i, j ě 2, since imphq Ď L 2 and the cyclic structure x´,´y L necessarily vanishes between two elements of L 2 .
At a more abstract level and for a general field theory, it is rather evident that we will always have a Feynman propagator serving as a contracting homotopy, which allows us to construct the minimal model of the field theory's L 8 -algebra explicitly. Also, the graded vector space L 1 will consist of the on-shell modes, L 2 will be isomorphic to L 1 and the symplectic form on field space provides a non-degenerate pairing of both spaces. Using the usual trick of polarisation, we can introduce the pi`1q-point functions xϕ 1 , µi pϕ 2 , . . . , ϕ i`1 qy L˝. As we shall see now, these are indeed the tree-level pi`1q-point functions. Let us consider the 3-point function 1 3! xϕ 1 , µ2pϕ 2 , ϕ 3 qy˝for ϕ 1,2,3 plane waves with on-shell momenta k 1 , k 2 , and k 3 , respectively. We have which yields the coupling constant κ together with the usual Dirac distribution, enforcing energy-momentum conservation: For the 4-point function we now get additional contributions from the 3-point vertices: 1 pk σp2q`kσp3q q 2´m2`i ε`λ‚ .
(3.29) In terms of Feynman diagrams, we have the 4-point function Let us now discuss the general interpretation of the φ i and the µi arising in the quasiisomorphism. While φ 1 is simply the embedding of on-shell modes into the original L 8algebra L, the higher φ i take on-shell (i.e. elements of ker c pµ 1 q) or off-shell modes (i.e. elements of S pR 1,3 q), combine them with a pj`1q-point vertex encoded in µ j and propagate the resulting state to an off-shell mode or current in S pR 1,3 q. For example, The µi , on the other hand, take either on-shell states or the currents arising from the φ j with j ě 2, combine them with a pj`1q-point vertex encoded in µ j and project the result back to an on-shell state. For example, Scattering amplitude recursion relations. The tree level pi`1q-point functions are now obtained from the inner product of one external state with µi of the remaining i external states, so their information is encoded in the higher products of the minimal model L˝. The tree-level corrections to the classical n`1-point vertex are now evidently constructed from diagrams involving the currents φ j with j ă n, which are recursively constructed from currents φ k with k ă j. This is obvious from formulas (2.11) and in terms of Feynman diagrams, we have for example

3.33)
In most interesting quantum field theories, the non-trivial higher products are very restricted, and the recursive computation of the quasi-isomorphism to the minimal model (2.11a) simplifies to interesting recursion relations for the currents φ i . These, in turn, may be solved in particular examples, yielding vast simplifications in the evaluation of the treelevel pi`1q-point functions. We shall discuss an important example in the next section but, again, it should be clear that our discussion applies to an arbitrary BV quantisable field theory.

L 8 -algebra formulation of Yang-Mills theory
We now study supN q Yang-Mills theory on four-dimensional Minkowski space R 1,3 . Let Ω p pR 1,3 , supN qq be the supN q-valued differential p-forms on R 1,3 . Furthermore, we let d be the exterior derivative and set d : :" ‹d‹ for ‹ the Hodge star operator with respect to the Minkowski metric. To keep our discussion clear, we shall neglect the intricacies of fall-off conditions on our function spaces; the details developed in the example of scalar field theory can be translated to Yang-Mills theory.

Strictification of Yang-Mills theory
First-order formulation. It is well-known that four dimensions Yang-Mills theory admits an alternative first-order formulation [31], given by the action functional Here, B`P Ω 2 pR 4 , supN qq is an supN q-valued self-dual 2-form on R 4 for ε P R`and we switched to Euclidean space to allow for real self-dual 2-forms. As we are only concerned with scattering amplitudes, which depend holomorphically on the kinematic variables, this switch in signature is largely irrelevant.
Integrating out B`, we find tr pF^F q , (4.8) where F`:" 1 2 pF`‹F q. Hence, we recover the standard Yang-Mills action (4.4) plus a topological term, which is irrelevant for perturbation theory.
Quasi-isomorphism. Whilst we have already seen that the actions (4.4) and (4.7) are equivalent by integrating out the self-dual 2-form, it is instructive to give the explicit quasiisomorphism between pL YM 2 , µ i , x´,´y L YM 2 q and pL YM 1 , µ i , x´,´y L YM 1 q. 4 In particular, we have ' where we have combined the two complexes (4.1a) and (4.9a). The maps φ 1 are given by As one may check, all square-subdiagrams of (4.11a) are commutative, and, consequently, we have obtained a chain map between the underlying complexes of L YM 2 and L YM 1 . In fact, it is a quasi-isomorphism of complexes since this chain map reduces to the identity (modulo constant pre-factors) on the cohomologies. Moreover, the set of maps φ 1 can be enlarged to include maps φ i : L YM 2ˆ¨¨¨ˆL YM 2 Ñ L YM 1 to obtain a fully-fledged quasi-isomorphism (2.2) between the L 8 -algebras L YM 2 and L YM 1 . Indeed, the only non-vanishing higher map φ i is given by the polarisation of φ 2 pA, Aq :"´1 ε P`rA, As . (4.12) In [33,3], this quasi-isomorphism was given in the Q-manifold language which is somewhat more transparent. Altogether, we conclude that the L 8 -algebra pL YM 2 , µ i , x´,´y L YM 2 q is indeed the strictification of pL YM 1 , µ i , x´,´y L YM 1 q

Scattering amplitudes and recursion relations
Minimal model from the second-order formulation. The cohomology of the L 8algebra (4.1) reads as LY M 2 " LM axwell 2 b supN q with LM axwell 2 :" p R Ý ÝÝ Ñ kerpd : dq{impdq Ý ÝÝ Ñ kerpd : dq{impdq Ý ÝÝ Ñ R q . (4.13) We choose the projectors p k to be the evident L 2 -projectors onto the subspaces LY M 2 ,k Ď L YM 2 ,k and we have the trivial embeddings e k . To compute the L 8 -structure on LY M 2 , we need also a contracting homotopy h " ph k q with h k : L k Ñ L k´1 which satisfies (2.8). Some algebra shows that 5 is a possible choice. Here, G F is the Green operator (3.15) and P ex is the projector onto the exact part under the abstract Hodge-Kodaira decomposition as discussed in Section 2.1 i.e. onto the image of d : d. Explicitly, in momentum space and suppressing the gauge algebra for the moment, we havê Recall that our formulas (2.11) were derived under the assumption that h 1 pAq " 0, cf. (A.4).
Here, this implies that we work in Lorenz gauge d : A " 0, and the propagator G F P ex is indeed the corresponding gluon propagator. It remains to insert the projectors and contracting homotopies into (2.11) to write down the quasi-isomorphism as well as the higher products for the minimal model.
Minimal model from the strictification. We could have also constructed a minimal model and corresponding recursion relations for tree-level scattering amplitudes from the strictified L 8 -algebra pL YM 1 , µ i , x´,´y L YM 1 q. See [3] for the construction of the contracting homotopy in this case. Any resulting minimal model LY M 1 is certainly L 8 -isomorphic to LY M 2 but the shape of the recursion relation is particularly suited for discussing the BCFW recursion relations [16,17] as shown in [15], because only trivalent vertices are present in pL YM 1 , µ i , x´,´y L YM 1 q. In addition, this also simplifies the off-shell recursion relations (4.29).

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A. Proof of the minimal model recursion relations
To derive the recursion relations (2.11), we need to construct a quasi-isomorphism φ : L˝Ñ L that allows us to pull back the higher products on L to L˝via formula (2.2). Our construction of φ follows the idea of [2], where essentially the same construction was given in the case of A 8 -algebras. In particular, we assume that we have a Maurer-Cartan element a˝in L˝and map it to an element a in L. The fact that Maurer-Cartan elements are mapped to Maurer-Cartan elements under quasi-isomorphisms, cf. (2.21), together with the assumption that a˝(and therefore a) is small, will give us enough constraints to determine the quasi-isomorphisms and the higher products on L˝.
Proof. We start from the contracting homotopy where we can assume that h 2 " 0 and e˝p, µ 1˝h and h˝µ 1 are projectors onto L harm , L ex , and L coex , respectively. Moreover, let a˝P L1 be a Maurer-Cartan element. Under a quasi-isomorphism φ, a˝is mapped to A convenient choice is φ 1 " e, and it remains to identify φ i for i ą 1. We will do this by fixing a as a function of a˝.
Recall that (2.9) yields the unique decomposition a " a harm`aex`acoex , with a harm, ex, coex P L harm, ex, coex .
There is some freedom in the choice of φ and without of loss of generality, we may impose the gauge fixing condition hpaq " 0 .
This is, in fact, a generalisation of the Lorenz gauge fixing condition from ordinary gauge theory. Consequently, a ex " pµ 1˝h qpaq " 0. Moreover, the fact that µ 1 is a chain map implies that µ 1 pa harm q " pµ 1˝e˝p qpaq " 0 so that the homotopy Maurer-Cartan equation for a becomes If we now assume that a˝is small, say of order Opgq with g ! 1 for g a formal parameter, we may rewrite (A.2) as We can then compute the solution a of the homotopy Maurer-Cartan equation order by order in g using (A.6). In this process, we can choose to put a piq harm " 0 for i ą 1 so that a " g a p1q harm loomoon " a harm`ÿ iě2 Substituting this expansion into (A.6), we arrive at the recursion relation for a coex . Comparison with (A.2) then yields the quasi-isomorphism (2.11) when evaluated at degree 1 elements.
To recover also the brackets µi on L˝listed in (2.11) by pullback, we note that upon applying the projector p to (A.5) and using the fact that p is a chain map, we immediately find that ÿ iě2 1 i! pp˝µ i qpa harm`acoex , . . . , a harm`acoex q " 0 . (A.9) Hence, after substituting the expansion (A.7), we recover the brackets (2.11) for degree 1 elements. Our derivation above is strictly speaking only applicable to Maurer-Cartan elements, which are elements of the L 8 -algebra of degree 1. As noted in [3], however, we may enlarge every L 8 -algebra L to the L 8 -algebra L C :" C 8 pLr1sq b L where C 8 pLr1sq are the smooth functions on the grade-shifted vector space Lr1s. Then, every element in L gives rise to a degree 1 element in L C , and, applying the above construction to L C yields the full L 8 -quasi-isomorphism and brackets listed in (2.11).
Cyclic L 8 -algebras. Finally, we note that the above construction also extends to the cyclic case. For this, we need h chosen such that xL coex , L coex y L " 0 .
(A.10) This is always possible since cyclicity (2.12) for µ 1 implies in general that xL ex , L ex y L " xL harm , L ex y L " 0 . (A.11) The remaining freedom in the choice of h can therefore be used to ensure that the only non-vanishing entries of the underlying metric are xL harm , L harm y L , xL ex , L coex y L , and xL coex , L ex y L . (A.12) If we now pull-back the cyclic structure from L to L˝and define x 1 , 2 y L˝: " xφ 1 p 1 q, φ 1 p 2 qy L , (A. 13) we have satisfied the first condition in (2.13) on a morphism of cyclic L 8 -algebras. The second condition is automatically satisfied, because (A.10) together with impφq Ď L coex implies the second condition in (2.13).

B. Dynkin-Specht-Wever lemma
Statement. For simplicity, let a be a matrix algebra and l be the Lie subalgebra generated by the elements that generate a, that is, the free Lie algebra over a. Consider the Dynkin map D : a Ñ l defined by a Q ÿ σPS i λ σ X σp1q¨¨¨Xσpiq Þ Ñ ÿ σPS i λ σ rX σp1q , rX σp2q , . . . rX σpi´1q , X σpiq s¨¨¨ss P l , (B.1) where X 1 , . . . , X i P a and the coefficients λ σ are some numbers. The Dynkin-Specht-Wever lemma then asserts that if ppXq :" ř σPS ip λ σ X σp1q¨¨¨Xσpipq P l then Hence, for any homogeneous polynomial ppXq P a of degree i p , we obtain pD˝DqpppXqq " i p DpppXqq.
Proof. To prove (B.2), we follow [34]. Firstly, we set adpXqpY q :" rX, Y s. Then, one can show by induction on the degree of the polynomial ppXq that if ppXq P l then adpppXqq " ppadpXqq (B.3a) with ppadpXqq :" ÿ σPS ip λ ppq σ adpX σp1q q˝¨¨¨˝adpX σpipq q . (B.3b) Secondly, (B.2) is certainly true for i p " 1 so let us assume it is true for i p ą 1 and prove the statement by induction. To this end, let ppXq P l and qpXq P l be homogeneous polynomials of degrees i p and i q , respectively. Then, DpppXqqpXqq " ÿ σPS ip λ ppq σ rX σp1q , rX σp2q , . . . rX σpip´1q , rX σpipq , DpqpXqqss¨¨¨ss where in the third step we have used (B.3a) since qpXq P l and in the fifth step the induction hypothesis. Thus, DprppXq, qpXqsq " pi p`iq qrppXq, qpXqs .

C. Gluon recursion for general Lie groups
Let us present a derivation of the Berends-Giele recursion from the quasi-isomorphism (4.25) in the case of a general gauge group not necessarily simple and compact, and which uses the Dynkin-Specht-Wever lemma discussed in the previous section. We again consider plane waves of the form (4.18) and make the ansatz φ i pAp1q, . . . , Apiqq "´p´1 q i i ÿ σPS i J µ pσp1q, . . . , σpiqq e ipk σp1q`¨¨¨`kσpiq q¨xr X σp1q , rX σp2q , r. . . , rX σpi´2q , rX σpi´1q , X σpiq ss¨¨¨ss dx µ .