MHV amplitudes and BCFW recursion for Yang-Mills theory in the de Sitter static patch

We study the scattering problem in the static patch of de Sitter space, i.e. the problem of field evolution between the past and future horizons of a de Sitter observer. We formulate the problem in terms of off-shell fields in Poincare coordinates. This is especially convenient for conformal theories, where the static patch can be viewed as a flat causal diamond, with one tip at the origin and the other at timelike infinity. As an important example, we consider Yang-Mills theory at tree level. We find that static-patch scattering for Yang-Mills is subject to BCFW-like recursion relations. These can reduce any static-patch amplitude to one with N−1MHV helicity structure, dressed by ordinary Minkowski amplitudes. We derive all the N−1MHV static-patch amplitudes from self-dual Yang-Mills field solutions. Using the recursion relations, we then derive from these an infinite set of MHV amplitudes, with arbitrary number of external legs.


I. INTRODUCTION A. Scattering in finite regions of Minkowski and de Sitter space
In the last decades, theoretical physics got progressively better at calculating observables defined on the boundary of spacetime.These include scattering amplitudes in Minkowski space, as well as correlators at conformal infinity of dS or AdS.There are good reasons for focusing on such observables.First, they're relatively easy to calculate.Second, they're observationally relevant: the flat S-matrix describes collider experiments, whereas correlations at dS future infinity encode the consequences of the conjectured inflationary epoch [1].
Finally, in quantum gravity, observables at infinity are the only ones we know how to make sense of (with the AdS case the best-understood, via AdS/CFT [2][3][4][5]).
For all these reasons, the evolution of systems confined to finite regions of space has received less attention within fundamental theory.As a simple example, consider a spherical region in flat spacetime.The causal development of such a region is a causal diamond, bounded by the lightcone of one point in the past, and the lightcone of another point in the future.One can then define a "scattering" problem for this finite region: to calculate the fields (or the quantum state) on the final lightcone in terms of those on the initial lightcone.
Almost no work on this problem exists.
Is this just the proper state of affairs?Should we dismiss such finite-region observables as some combination of complicated, pointless and (in quantum gravity) ill-defined?Tempting though this may be, we have an observational fact to contend with -the accelerated expansion of our Universe, which appears consistent with a positive cosmological constant, and thus a de Sitter asymptotic future.This implies a cosmological horizon that asymptotes to a finite size, with no access -even in principle -to spatial infinity.De Sitter space does have a future conformal boundary, but that only becomes observable if the accelerated expansion eventually ends, as in inflation.If we believe that we are truly stuck in an asymptotically dS world, we must come to term with physics without observables at infinity.For quantum gravity, this is a tall order indeed.However, we can take baby steps, by familiarizing ourselves with field-theory questions that are natural for an observer inside a de Sitter cosmological horizon.
For simplicity, we now leave real-world cosmology aside, and consider pure de Sitter space dS 4 -the simplest spacetime in which every observer is trapped inside a spherical horizon of finite size.The largest spacetime region available to such an observer is a static patch of dS 4 ; see discussions in e.g.[6,7].The static patch is bounded by a past horizon and a future horizon -the lightcones of the past and future endpoints of the observer's worldline.Confined to such a region, the closest thing we have to an asymptotic observable is the "S-matrix" encoding field evolution from the past horizon to the future one.This general problem -the problem of static-patch scattering -was posed and studied by us in [8,9].One of our long-term goals is to work out this scattering problem within the context of higher-spin gravity [10,11] -a gravity-like theory of massless interacting fields with all spins, whose holographic description [12][13][14][15] appears to carry over from AdS to dS [16], and which apparently can be formulated on a pure, non-fluctuating dS 4 geometry [17].
On the route to higher-spin theory, the more ordinary theories of interacting massless fields form natural stepping stones.We studied free massless fields of all spins in [8], and a scalar with cubic interaction in [9].The natural next case is interacting spin-1, i.e.Yang-Mills theory.This will be our subject in the present paper.We will restrict our analysis to tree-level, where YM theory enjoys the simplifying property of conformal symmetry.
As a result, it doesn't actually see the curvature of de Sitter space.All that remains of the de Sitter static patch is its conformal structure, which is identical to that of a causal diamond in Minkowski (the two are also conformal to the Rindler wedge of Minkowski, and to the static hyperbolic space R × H 3 ).Thus, in this case, the cosmologically motivated problem of static-patch scattering is actually equivalent to the flat causal-diamond problem we mentioned before.
We are thus pursuing two goals.The first is just to study finite-region scattering, using the simple case of a conformal theory in a (conformally) flat causal diamond.The second is to study the de Sitter static patch specifically, in the hope that the simple case of YM theory will provide a useful stepping stone towards perturbative GR, and ultimately higher-spin gravity.

B. Outline of the paper
As always, it is crucial to set up the calculation in a way that makes best use of available symmetries.A priori, the Minkowski causal diamond is quite challenging in this regard, since its only isometries are SO(3) rotations.The dS 4 static patch is only slightly better, with a symmetry of R × SO (3), where the R describes time translations.In [9], we proposed a general strategy for the static-patch problem, which makes use of the larger symmetry of dS 4 as a whole.This involves artificially extending the static patch's boundaries into geodesically complete cosmological horizons, each one defining a Poincare patch.These are endowed with spatial translation symmetry, making it possible to work in momentum space.
The static-patch problem was then decomposed into a pair of Poincare-patch evolutions, sewn together by a coordinate inversion at the conformal boundary of dS 4 .Under such an inversion, the spatial translations of one Poincare patch become the conformal boosts of the other, and vice versa.
In the present paper, we take a simpler approach, taking advantage of the fact that we are dealing with a conformal theory.The conformal symmetry of a Minkowski causal diamond, or a dS 4 static patch, is R × SO (1,3).This is better than the non-conformal case, but not good enough on its own.Just as in [9], we can gain translation symmetry (in this case, full 4d Minkowski translations) by artificially extending our scope outside the causal diamond.
The first step is to choose a flat conformal frame in which the causal diamond's tips are fixed at (past timelike) infinity and the origin.In this frame, the diamond's past boundary (or the past horizon of the de Sitter observer) becomes a portion of past null infinity I − , while its future boundary (the de Sitter observer's future horizon) becomes the origin's past lightcone.We can then artificially extend the initial data to all of I − , and decompose it into plane waves.As we will see in more detail below, this reduces our scattering problem to a calculation of bulk YM fields in a lightcone gauge, out of plane-wave initial data.In this setup, the analog of an n-point scattering amplitude is encoded in the bulk field at (n − 1)'st order in the initial data.This quantity is sometimes called a Berends-Giele current, after the work [18].Thus, the static-patch problem is reduced to a fairly standard Minkowski calculation, involving n − 1 ingoing on-shell legs, and one outgoing off-shell leg (there is no loss of generality in having one outgoing leg, since we'll be working at the level of field operators rather than Fock states).
Having thus framed the scattering problem, we will proceed to tackle it at tree level.Our main results are as follows: 1.The simplest non-vanishing static-patch amplitudes are those with N -1 MHV helicity structure, i.e. those in which all but one external leg have the same helicity.These are closely related to the Parke-Taylor MHV amplitudes [19] of the Minkowski Smatrix.We will derive these N -1 MHV static-patch amplitudes from perturbative selfdual solutions of the Yang-Mills equations [20][21][22][23], closely following [23].
2. All other static-patch amplitudes can be reduced to these N -1 MHV ones, using an appropriately modified version of the BCFW recursion relations [24,25].This constitutes a modern upgrade over Berends and Giele's original recursion relations [18] for the Berends-Giele current: while those relations basically translate into a sum over all Feynman diagrams, our BCFW-type relations will generally involve fewer summands.
We emphasize that this is more than what's been achieved for standard (A)dS boundary correlators.Just like our static-patch problem, the (A)dS boundary problem for tree-level Yang-Mills can be conformally transformed into flat spacetime, where the (A)dS boundary becomes just a flat hypersurface z = 0.And yet, this problem is not so easy!In particular, already N −2 MHV correlators are non-zero, and already for them there isn't a known formula for general n.Correlator formulas in spinor-helicity language are only known for n = 3 [26] and n = 4 [27] (see also [28][29][30]), with results [31] outside the spinor-helicity formalism for n = 5, 6.Some recursion relations have also been developed in this context [32][33][34], but these either leave a z integration to be performed [32,33], or else pertain only to one Witten diagram at a time [34].Thus, for Yang-Mills theory, the static-patch scattering problem, while less trivial than the Minkowski S-matrix, is easier than standard (A)dS boundary correlators, despite having nominally lower symmetry.The simplification can be ascribed to the lightlike nature of the static patch's boundary.This allows helicity to be represented naturally in terms of "little-group" SO(2) rotations around the lightrays, as opposed to the SO(3) rotational structure on the non-lightlike conformal boundary of (A)dS.As for the static patch's lower overall symmetry, it is in large part compensated by the fact that the static-patch problem has a square root, as in [9]: the past and future horizons can be regarded separately, which gives us access to the symmetry of the Poincare patch -or, in our case, due to YM theory being tree-level conformal, to the full symmetry of Minkowski space.
The rest of the paper is structured as follows.In section II, we define the static-patch scattering problem natively in de Sitter space, and then gradually transform it into a moreor-less standard calculation in Minkowski, in a spinor-helicity formalism.The final results of that section are given in eqs.( 58)-(59).In section III, we derive all the tree-level N -1 MHV static-patch amplitudes from a self-dual Yang-Mills solution and an anti-self-dual perturbation over it.The results for these amplitudes are given in eqs.(74),(84),(90),(93).In section IV, we discuss the pole structure of tree-level static-patch amplitudes, relating them to the standard Minkowski S-matrix, and proving a BCFW-type recursion relation.The final form of this recursion relation is given in eqs.( 112)-(113).In section V, we apply the recursion to compute a class of MHV static-patch amplitudes.These are given in eqs.( 124)-(125).
Section VI is devoted to discussion and outlook.

II. GEOMETRY AND KINEMATICS
In this section, we set up the geometry and kinematics of the static-patch problem and its Minkowski reformulation.In section II A, we describe the dS 4 static patch, and define the initial and final field data on its past and future horizons.In section II C, we introduce Poincare coordinates adapted to the past horizon, and the associated conformal transformation into Minkowski space.In section II D, we construct linearized Yang-Mills solutions in Minkowski, and relate them to initial data on the past horizon.Finally, in section II E, we discuss non-linear bulk fields, and show how certain components of them in a certain gauge correspond to final data on the future horizon.
A. The static-patch problem, formulated in embedding space De Sitter space dS 4 is the hyperboloid of unit spacelike radius inside the flat 5d embedding space R 1,4 .We use lightcone coordinates r I = (u, v, r) for R 1,4 , where boldface indicates a 3d Euclidean vector.The R 1,4 metric reads: and the dS 4 hyperboloid is given by: The curved metric of dS 4 is just the flat 5d metric (1), restricted to the hyperboloid (2).
Each horizon is a lightlike cylinder, consisting of a spatial unit sphere r 2 = 1, multiplied by the lightlike u or v axis.See the Penrose diagram in Figure 1(a).
We will be dealing with a YM field on dS 4 , which can be written as a 1-form ÂI = ( Âu , Âv , Â).The hats are to distinguish these components in the (u, v, r) basis from the ones we'll introduce in a Poincare-patch basis below.We set to zero the component ÂI r I that points outside the dS 4 .We will not explicitly write color indices; instead, we understand ÂI to take values in the gauge algebra.While final answers can only depend on the Lie bracket, i.e. on the commutators of gauge algebra elements, it will be very convenient to work as if they have an associative product.This can be made concrete by defining the ÂI as matrices over the gauge group's fundamental representation.We will not make assumptions about the gauge group itself and its structure constants, and we'll simply keep different product orderings as distinct terms.This attitude, taken from [23], is analogous to the now standard decomposition of the YM S-matrix into separately considered color-ordered pieces.
Our task will be to express the YM field on the future horizon in terms of that on the past horizon.As discussed in [9], this is a bit different from the usual scattering problem in Minkowski, where we are interested in the S-matrix relating Fock states on I − to Fock states on I + .We will consider this distinction in more detail in section II B.
Now, what should we take as the initial (final) field data on the past (future) horizon?
First of all, since the horizons are lightlike, it is sufficient to consider the value of ÂI on them: there is no need to separately include the normal derivative.Second, due to gauge freedom in the dS 4 bulk, it is sufficient to consider the components of ÂI along each horizon, i.e. ( Âv , Â⊥ ) on the past horizon and ( Âu , Â⊥ ) on the future one, where Â⊥ ≡ Â − ( Â • r)r denotes the components along the 2-sphere r 2 = 1.Finally, we fix the residual gauge freedom on each horizon by taking the potential along its lightrays to vanish.This sets Âv = 0 ( Âu = 0) on the past (future) horizon respectively, leaving just the spatial components Â⊥ along the 2-sphere.Equivalently, we can work with the derivative of Â⊥ along each horizon's lightrays, which in our chosen gauge encodes the field strength components Fv⊥ = ∂ v Â⊥ or Fu⊥ = ∂ u Â⊥ .In Maxwell theory, we could forget about gauge altogether and just focus on these field strength components.However, in YM theory, the field strength isn't gaugeinvariant, and so the gauge choice Âv = 0 or Âu = 0 remains important.
So far, then, our initial data on the past horizon is given by Â⊥ (0, v, r), and the final data on the future horizon -by Â⊥ (u, 0, r).For our scattering calculation, we will want to express the initial data as plane waves in the Poincare coordinates associated with the past horizon.As we will see below, this simply requires a Fourier transform with respect to v [8,9].In the interest of treating both horizons symmetrically, we'll Fourier-transform the final data with respect to u as well.Now, recall that the boundaries of our static patch are actually "half-horizons", confined to the lightlike coordinate ranges v < 0 and u > 0; the other half of each horizon is unobservable.When we Fourier-transform the initial (final) data with respect to v (u), we actually extend the original scattering problem.In the extended problem, we calculate the final fields on the entire horizon (u ∈ R, v = 0) -a lightlike initial data hypersurface for the entire dS 4 spacetime, as a functional of the initial fields on the entire horizon (u = 0, v ∈ R) -another such hypersurface.This extension of the problem, which will prove convenient, is quite harmless.Indeed, at the very end, we can always limit our attention to the observable final fields at u > 0. Since our field theory in dS 4 is causal, these can only depend on the initial fields in the observable range v < 0; this dependence then defines our sought-after static-patch scattering [43].
With this understood, we proceed to package the initial data into (gauge-algebra-valued) Fourier coefficients, as: As we will see, in the Poincare coordinates associated with the past horizon, the coefficients c in (ω, r) describe lightlike plane waves, with 4-momentum: where the minus sign stems from the fact that a wave traveling along k is coming from the direction of −k at past null infinity.
We can now introduce spinor-helicity variables, by taking the spinor square root of this momentum [8,26].We begin by introducing SO(3) spinors ψ α , whose indices are raised and lowered as ψ α = ǫ αβ ψ β and ψ α = ψ β ǫ βα , with spinor complex conjugation acting as ψα = (ψ α ) * , and with the Pauli matrices σ α β .Now, for each 4-momentum of the form (4), we define its spinor square root (λ α , λα ) via: where the factors of 2 are for later convenience.The reality of k µ implies that the spinors (λ α , λα ) are related by complex conjugation, up to the sign of the energy ω: Note also that, as usual, ( 5) defines (λ α , λα ) only up to multiplication by opposite complex phases: One advantage of spinor-helicity variables is that the polarizations of Â⊥ can be decomposed into two helicities, given by the null complex vectors λσλ and λσ λ = λσ λ .More precisely, we will use the following normalized versions of these vectors: Extracting the components of (3) along m and m, we obtain the initial mode coefficients as spinor-helicity functions: where the superscript ± denotes helicity.Under the phase rotation (7), the coefficients c ± in transform with weight ±2 respectively.The reality condition on the fields (which we will For the final data on the future horizon, we define spinor-helicity functions c ± out (µ α , μα ) in complete analogy with (9): Again, though we are ultimately interested in the observable half-horizon u > 0, it will be more convenient to work with Fourier coefficients on the entire u axis, as in (10).At the very end, we can restrict attention to the original static patch by simply throwing away the unobservable u < 0 portion of the final data.Just like the initial modes ( 9), the final modes (10) also describe lightlike plane waves, but in a different Poincare frame -the one adapted to the future horizon.In [9], we explicitly made use of both Poincare frames.However, in our present case of a conformal theory, it will be simpler to just stay in one of themfor concreteness, the one adapted to the past horizon [44].There is a price to be paid: as we'll see, in this approach, the gauge choice Âu = 0 on the final horizon is easy to enforce only on one lightray at a time.As a result, it will be better to work with the field strength components Fu⊥ = ∂ u Â⊥ , which, unlike Â⊥ itself, don't depend on the gauge choice on neighboring lightrays.With this in mind, let us use integration by parts w.r.t.u to rewrite (10) in terms of field strengths: The static-patch scattering problem now boils down to expressing the final mode coefficients c ± out (µ, μ) as functionals of the initial ones c ± in (λ, λ).

B. Working with fields vs. with Fock states over a vacuum
In this section, we take a brief digression to discuss the relationship between our fieldbased formulation of the scattering problem, and the more conventional one based on Fock states.From our point of view, the more fundamental objects are the fields operators on the past and future horizons.From these, one may construct Fock states by taking two additional steps.The first step is to make a distinction between positive-frequency and negative-frequency field modes.In the context of our lightlike horizons, this can be done by choosing a lightlike time coordinate along the lightrays, and then Fourier-transforming with respect to that coordinate.Once defined, the positive-frequency and negative-frequency modes can be thought of as annihilation and creation operators.This designation then defines a vacuum state, as the state that is annihilated by all the annihilation operators.
Other states can now be formed by acting on the vacuum with creation operators, via the usual Fock procedure.Thus, the second step in going from field operators to states is to restrict attention to one of the two frequency signs.Now, in principle, the two vacua as defined on the past and future horizons may or may not be the same.As we'll see below, in relevant cases for us, the two are the same.When this is true, positive-frequency modes on the past horizon can only evolve into positive-frequency modes on the future horizon, and vice versa.This then completes the usual picture of Fock states evolving into Fock states.
To recap, the entire vacuum/Fock-space structure follows from a choice of lightlike time coordinate on the horizons.There exist two particularly natural choices.The first is to use the coordinate τ = − ln(−v) on the past horizon, and τ = ln u on the future horizon.The range τ ∈ R then covers the static patch, while translations in τ simply describe the static patch's time-translation symmetry, i.e. their generator is the static-patch Hamiltonian.
Since these time translations are a spacetime symmetry, we are assured that the distinction into positive/negative frequencies, as well as the vacuum state, are consistent between the two horizons as anticipated above.With these structures, one can now define an S-matrix intrinsic to the static patch, which evolves Fock states into Fock states, and makes no reference to spacetime regions outside the patch.While this picture is conceptually simplest, it is technically inconvenient.The reason is the one mentioned in the Introduction: the static patch has no spatial translation symmetry, so one is forced to work instead with spherical harmonics.In [7], the static-patch S-matrix in this language was computed for a free conformally-massless scalar, and the resulting expression is already quite non-trivial; extending it to include interactions seems prohibitively challenging.
Because of this, our choice in this paper is to use the lightlike coordinates v and u themselves, instead of τ .As discussed above, this entails the cost of extending the past/future horizons to causally cover the entire dS 4 spacetime, for the benefit of gaining spatial translation symmetry.The vacuum state defined by v (u) on the past (future) horizon is the Bunch-Davies vacuum of global dS 4 (which will become the usual Minkowski vacuum upon conformal transformation to Minkowski spacetime).In particular, the modes (3) with ω > 0 define annihilation operators with respect to this vacuum, while the ones with ω < 0 define creation operators.While translations of u and v are not spacetime symmetries, the associated choice of vacuum and distinction into positive/negative frequencies is consistent between the two horizons.This is because the Bunch-Davies vacuum has a universal definition in terms of a Euclidean path integral.
Thus, in choosing v and u as the lightlike coordinates that get Fourier-transformed, we've effectively taken the first step on the way from field modes to Fock states: we made a choice of vacuum, which is conformal to the usual Minkowski one.As a result, our scattering formulas will look a lot like ordinary S-matrix amplitudes with respect to the usual Minkowski vacuum.However, this vacuum is a pure state of global dS 4 , rather than of the static patch, and the associated positive/negative frequency modes are only defined on the extended horizons, rather than on their observable "halves".As a result, we choose to not take the second step, of throwing away the field modes with the "wrong" frequency sign.
Instead, we consider arbitrary field modes on the observed half of e.g. the past horizon, and then extend them arbitrarily into the unobservable half.
We conclude this subsection with a final comment.The word "scattering" in Minkowski space is often associated with the notion of fields becoming free as one approaches I ± .This is often presented as a prerequisite for the Fock-space construction of particle states.On the other hand, near a de Sitter horizon, the fields are no more free than anywhere else in the spacetime.There is in fact no conflict here.For fields on a null hypersurface, one can always perform the Fock-space construction based on positive/negative frequencies with respect to a lightlike coordinate.Interactions do not affect this picture.From this point of view, the asymptotic boundary I ± of Minkowski space is just another null hypersurface.
However, since it's only defined asymptotically, one may worry whether the fields will have well-defined values on it.It is this requirement that translates into the need for Minkowski fields to be "sufficiently free" near I ± .For horizons in the bulk of dS 4 , the issue does not arise.

C. Flat coordinates adapted to past horizon
Having defined the static-patch problem in section II A, we will now set up a flat conformal frame in which it can be solved more easily.This frame will be based on the Poincare coordinates associated with the past horizon, breaking the symmetry between the two horizons which we've been careful to maintain so far.These Poincare coordinates x µ = (t, x) are related to the embedding-space coordinates r I = (u, v, r) of section II A as: The x µ coordinates define a flat metric η µν dx µ dx ν ≡ −dt 2 +dx 2 , which is conformally related to the dS 4 metric (1) via: The static patch, still in gray, is a subregion of both dS 4 and Minkowski.(c) The Minkowski picture, with dS 4 removed.The static patch is now the causal past of the origin, the past horizon is past lightlike infinity, and the future horizon is the origin's lightcone.This lightcone is again a geodesically complete "doubling" of the static patch's future boundary, but now the completion is into the future.
The YM gauge potential is conformally invariant.We may therefore disregard the conformal factor of t 2 , and work with the Minkowski metric η µν .The potential's components A µ = (A t , A) in the x µ basis are related to those in the embedding-space basis via A µ = (∂r I /∂x µ ) ÂI .Imposing the constraint r I ÂI = 0, the relation between the components becomes: The field strength in the intrinsic x µ coordinates is derived from the potential as: The coordinate range t < 0 spans the expanding Poincare patch u > 0 of dS 4 , i.e. the half of dS 4 that lies to the future of the past horizon.The past horizon itself is given by the limit t → −∞, |x| → ∞ with t + |x| finite.In particular, the coordinates (v, r) on the initial horizon are given in this limit by: Thus, in the coordinates x µ , the past horizon becomes past lightlike infinity I − .As discussed in section II A, the actual past boundary of the static patch is restricted to v < 0, but we extend the initial data to the entire range v ∈ R arbitrarily.
The future horizon of the dS 4 static patch becomes the origin's lightcone η µν x µ x ν = 0 in the x µ coordinates.The horizon coordinates (u, r) are then given in terms of x µ as: In particular, the observable half-horizon u > 0 is described by the past lightcone t = −|x|.
See the Penrose diagrams in Figure 1(b,c).
Note that as we switch conformal frames between dS 4 and the Minkowski space x µ , the spacetime's global structure changes.As the Poincare time t increases through the range t < 0, it reaches the future conformal boundary of dS 4 at t = 0 − .We then encounter a discontinuity: t = 0 + is at the past conformal boundary of dS 4 , and the range t > 0 spans a contracting Poincare patch that lies to the past of the past horizon.In contrast, from the point of view of the flat metric η µν , t = 0 is just a regular time slice, and the flat Minkowski space continues right through it.On the other hand, the flat metric treats the past horizon as I − , and doesn't see the complementary Poincare patch to its past.In particular, both conformal frames agree that the observable half u > 0 of the future horizon is geodesically incomplete, and that completing it involves extending u or t = −1/u to the entire real line.However, the two frames disagree on the direction of this extension.In dS 4 , the horizon wants to continue smoothly into the past, from u > 0 to u < 0, which for the t coordinate looks like a discontinuous jump through t = ±∞.In Minkowski, the picture is reversed: the horizon (or, rather, lightcone) wants to continue smoothly into the future, from t < 0 to t > 0, which is discontinuous for the u coordinate.Fundamentally, it doesn't matter which of the pictures we adopt, since they only disagree outside the observable static patch (though note that the term "observable" here still refers to the dS 4 metric).As a matter of convenience, we will adopt the Minkowski frame, and with it the extension of the future horizon into the future through t = 0, rather than into the past through u = 0.
We can continue using the formulas ( 10)-( 11) for the future horizon modes, but with the understanding that the u < 0 range refers to the future t > 0 half of Minkowski space, rather than to the contracting Poincare patch of dS 4 .
We now turn to introduce spinor notation for the Minkowski space x µ .This simply extends the 3d spinor notation from section II A. In particular, we introduce a distinction between left-handed (undotted) and right-handed (dotted) spinor indices.Index raising and lowering are defined as before: and we define shorthands for inner products as: Spinor complex conjugation is now defined by ψ α = (ψ α ) * .The 3d Pauli matrices become σ α α, and are incorporated with the identity matrix σ α α t into the 4d Pauli matrices σ α α µ , which satisfy: We use σ α α µ to translate between vector and spinor indices, via: The YM gauge potential ( 14) can now be written as A α α.Its field strength (15) decomposes as: F α β and F αβ encode the self-dual (right-handed) and anti-self-dual (left-handed) parts of F µν , respectively.In terms of A α α, they read: where ∂ α α ≡ σ µ α α∂ µ .

D. Lightlike plane waves and initial horizon data
A lightlike momentum k µ = (ω, k), either future-pointing or past-pointing, can be written in spinor notation as: This coincides with our previous SO(3)-spinor expression (5), given the componentwise equality of the 4d complex conjugate λ α and the 3d one λα .The momentum ( 25) is again invariant under the phase rotations: A lightlike wave with the momentum (25) takes the form: Adding appropriate polarization factors, we can construct purely right-handed or left-handed plane-wave solutions to the Maxwell equations: where q α and q α are arbitrary spinors encoding the gauge freedom (in particular, the field strength doesn't depend on them).The general linearized solution to the YM equations is obtained by integrating over such plane waves with gauge-algebra-valued coefficients c ± in (λ, λ): where the integration range is understood to include both positive-frequency and negativefrequency modes: The spinors (λ α , λ α) in the integrand of (30) are the square root of k µ , as in (25), with the phase freedom (26) fixed arbitrarily.For the result to not depend on the choice of phase, the mode coefficients c ± in must transform under the phase rotations (26) with the appropriate weights ±2: Note that we used the same notation c ± in for the plane-wave coefficients in (30) and for the mode coefficients (9) on the past horizon.Let us show that they are in fact equal, justifying our identification of (4) as a 4-momentum.To do this, we evaluate the potential (30) in the null-infinity limit ( 16) that describes the past horizon in the x µ coordinates.This is a standard calculation, in which we decompose the d 3 k integral into integrals over its magnitude k and its direction k/|k|.In the null-infinity limit, the integral over directions can be found by the stationary-phase method, with the two stationary points k = ±ωr.Of these, only the point k = −ωr survives; the other leads to a rapidly oscillating phase in the integral over the magnitude |k|.All in all, we get: where the spinors (λ, λ) are related to (ω, r) as in ( 4)-( 5), and t ≈ −|x| goes to −∞.Due to the t in the denominator, the components A µ on the past horizon all vanish.Transforming into the embedding-space basis via (14), we conclude that we are automatically in the gauge Âv = 0 in which the initial data ( 9) is defined.As for the spatial components Â⊥ along the 2-sphere in the embedding-space basis, they are given by −t times the corresponding components of (35), which is finite.Furthermore, these transverse components don't depend on the choice of gauge spinors (q, q), which can only shift A µ longitudinally, along λσ µ λ].
To make contact with the SO(3) formalism of section II A, it's convenient to choose: which is equivalent to fixing A t = 0 everywhere.We can now descend to 3d spinor notation, leaving only undotted spinor indices, treating σ α α t as the identity matrix, and identifying λ α with λα .The potential's spatial components on the past horizon then read: where we recognize the null polarization vectors from (8).Contracting with these vectors and Fourier-transforming with respect to v, we recover the initial-data expressions (9).

E. Non-linear corrections and final horizon data
Ultimately, we are interested in the final data c ± out (µ, μ) as a functional of the initial data c ± in (λ, λ).The Taylor coefficients of this functional define the static-patch "scattering amplitudes", which we'll denote as S(1 h 1 , . . ., n hn ; µ, μ, h).Here, n is the number of ingoing c ± in factors, and each argument i h i is a shorthand for a pair of spinor-helicity variables (λ i , λi ), along with a helicity sign h i = ±: Similarly, (µ, μ) are the spinor-helicity variables for the outgoing c ± out mode, and h = ± is its helicity sign.For brevity, we will sometimes omit the dependence on (µ, μ).Note that the order of the ingoing legs (1, . . ., n) is important, because the c ± in initial data are gauge-algebra-valued, and thus do not commute.As mentioned above, we follow here the "color ordering" convention, which is to treat each ordering of c ± in as a distinct term.With this convention, the group's structure constants never enter the calculation, and the colorordered "amplitudes" S (which themselves are gauge singlets) do not depend on the gauge group.Note that since the modes c ± in contain both positive and negative energies, they may also not commute as quantum operators; however, we will work at tree level, where this issue doesn't arise.Our expression for c ± out in terms of the "amplitudes" S is given below, in eq.(58).To motivate the prefactors there, we must first prepare some groundwork.
At tree level, the final data c ± out can be read off from a classical field solution A α α(x), determined by the initial data c ± in .We will therefore need the non-linear corrections to the linearized potential (30).From now on, we mostly specialize to a gauge in which the spinors (q, q) in (30) are constant, i.e. do not depend on (λ, λ).This amounts to the gauge condition: i.e. a lightcone gauge with respect to the constant null vector q α q α.We now apply the condition (39) to the entire non-linear field, thus gauge-fixing the non-linear corrections to (30).Making the dependence on (q, q) explicit, we Taylor-expand the potential A α α in powers of the initial data c ± in as: Here, we sum over all choices of the ingoing helicity signs (h 1 , . . ., h n ).We use 1...n as a shorthand for an n-fold integral over on-shell ingoing momenta as in ( 30)-( 32): K 1...n denotes the sum of these momenta: and we will similarly denote partial sums of consecutive momenta by K µ i...j .At n = 1, the coefficients a α α in (40) can be read off from the linearized potential (30) as: The coefficients with n ≥ 2 describe the non-linear corrections to the bulk field, which can be found by computing Feynman diagrams with n + 1 external legs, of which n are on-shell and 1 is off-shell.When the momentum (42) of the off-shell, "outgoing" leg goes on-shell, the coefficients a α α acquire poles, whose residues are the Minkowski scattering amplitudes.We will discuss these in section IV A. However, generally, we will need not only these residues, but the non-linear field itself, at general, off-shell momenta K µ .
As usual, to be uniquely defined, the non-linear corrections require boundary conditions, which amount to an iε prescription in the propagators.We fix these by demanding that the initial data c ± in (λ, λ) continues to describe the field at past null infinity, i.e. at the past horizon, in the sense of (9).This dictates that we should use retarded propagators, which can be encoded by adding an infinitesimal future-pointing imaginary part to each ingoing 4-momentum k µ .We will keep this understanding implicit, and omit iε's below.
In complete analogy with (40), we define expansions of the right-handed and left-handed field strengths: At n = 1, the field strength's coefficients can be read off from ( 29), (43) as: f α β (−; q, q) = f αβ (+; q, q) = 0 . (46) While these coefficients of the linearized field strength don't depend on the gauge spinors (q, q), this will not be the case for the non-linear corrections.
Let us now understand how the final data c ± out on the future horizon can be read off from the non-linear fields (40) or ( 44)-( 45).Unlike the initial data, which is unaffected by the non-linear corrections, the final data will receive contributions from both the linear and non-linear terms in (40).The linear contribution can be worked out using a spinor Fourier transform [8], as was done explicitly for a scalar field in [9].Here, we'll present an alternative derivation, which works equally well for the off-shell momenta of the non-linear corrections.
Consider the final data c ± out (µ, μ) on the final horizon, evaluated at some value of the spinorhelicity variables (µ, μ).In embedding-space coordinates, this is given by a Fourier transform ( 10)-( 11) with respect to the null time u, along the lightray r = − μσµ / μµ .In our flat frame, the future horizon corresponds to the origin's lightcone, as in eq. ( 17).Thus, in Minkowski coordinates, the lightray along which the Fourier transform (10)-( 11) is taken reads: As for our gauge choice Âu = 0 on the future horizon, it becomes simply µAμ] = 0.The simplest way to impose this gauge condition on the lightray (47) is to impose it everywhere, i.e. to adopt the final-horizon spinors (µ, μ) as our gauge spinors (q, q), defining the lightcone gauge (39).Thus, to evaluate the final data on each separate lightray of the future horizon, we will calculate the bulk field A α α(x) in a separate lightcone gauge.Note that this doesn't affect our encoding c ± in (λ, λ) of the initial data, since the latter doesn't depend on (q, q).However, we can no longer use the formula (10) for the final modes, because the potential A µ depends on the gauge choice not only on the chosen lightray, but also on the neighboring ones.Instead, we must use the formula (11), which uses the field strength.
Let us now see exactly how c ± out (µ, μ) can be read off from the non-linear bulk potential in the appropriate gauge, i.e. from A α α(x; µ, μ).For the moment, we can abstract away from the expansion (40), and simply consider A α α and its field strength in momentum space, i.e. decomposed into general plane waves: (48) Let's now plug this field into our definition (11) of the final data.Eq. ( 11) refers to the field strength in the embedding-space basis.The relevant components are related to those in the Minkowski basis via: These helicity components can also be expressed in terms of the (Minkowski, lightcone gauge) potential A α α.Indeed, in the gauge µAμ] = 0, we have µ α A α α ∼ μ α and μ αA α α ∼ µ α , which implies the vanishing of µ α µ β A (α αA β) α and μ α μ βA α( αA α β) .As a result, the field-strength components µ α µ β F αβ and μ α μ β F α β depend on the potential linearly: or, in momentum space: We are now ready to evaluate the Fourier integral (11) w.r.t. the null time u along the future horizon's lightray.The u dependence in the integral comes from the Fourier factor e i μµ u/2 in (11), from the u −3 scaling factor in (51), and from the e iK•x plane-wave factor in (48), where x µ depends on u as in (47).The u integral thus takes the form (taking into account also the energy factor μµ −2 from ( 11)): where we rescaled the integration variable as U ≡ 1 2 μµ u.We see that the dependence on energy μµ -a non-Lorentz-invariant vestige of the 3d formalism -has been reduced to a Lorentz-invariant sign.Let's now evaluate the integral by considering it in the complex U plane.At U → ∞, the integration contour can be closed from above.We must also deform the contour around the essential singularity at U = 0.The deformation that leads to a well-defined answer is the one for which Im( µK μ]/U) is negative.Thus, for µK μ] < 0, we must bypass U = 0 from below.The contour is then equivalent to a circle around U = 0, and the integral evaluates to the Bessel function of the first kind J 2 : The result ( 55) is reminiscent of the one found in [9] for an on-shell spin-0 field.We now turn to the case µK μ] > 0. Here, we must bypass U = 0 from above, resulting in a closed contour with no singularities inside, so the integral vanishes.This makes sense, since µK μ] > 0 implies that K α α (a 4-momentum in our flat frame adapted to the past horizon) and µ α μ α (a 4-momentum in a different flat frame, adapted to the future horizon) have energies of opposite sign with respect to the lightlike coordinate u.
Reinstating the polarization factors, we obtain the final data on the future horizon in terms of the non-linear bulk field as: or, in terms of the field strength: Plugging in the field's perturbative expansion (40) or ( 44)-( 45), the d 4 K integral goes away, because the momentum K µ in ( 40),( 44)-( 45) is always just a sum (42) of initial momenta k µ .Thus, our expression for c ± out as functionals of c ± in finally takes the form: where θ is the step function, and the "amplitudes" S are related to the perturbative expansions ( 40),( 44)-( 45) of the bulk potential and field strength via: The overall minus signs in ( 58)-( 59) are inserted for later convenience.
We have thus reduced the static-patch scattering problem to a Minkowski-space problem of calculating (certain components of) the non-linear field functional (40)  ) with a single off-shell leg.In our present formalism, the free-field propagation between the static-patch horizons, first considered in [8], is described by the trivial "2-point amplitudes": In this section, we present the results for tree-level static-patch scattering with N -1 MHV helicities, i.e. with one of the external leg's helicities negative and the rest positive.As we will see, the N -2 MHV amplitudes, with all helicities positive, vanish.The N -1 MHV amplitudes (and the vanishing of the N -2 MHV ones) can be obtained by plugging known classical solutions of Yang-Mills in Minkowski space [20,23] into our master kinematical prescription (58)-( 59).Here, we review these solutions, adding some clarifications to the original treatments.We begin in section III A with a purely self-dual solution; from this, we'll read off the N -1 MHV static-patch amplitude in which the negative helicity is on the outgoing leg.Then, in section III B, we write the linearized left-handed (i.e.anti-self-dual) field strength perturbation over this self-dual solution; from this, we'll read off the N -1 MHV static-patch amplitude in which the negative helicity is on one of the ingoing legs.

A. Self-dual solution
In this subsection, we focus on the c − in -independent piece of the non-linear field (40), i.e. the part of A α α that only depends on the right-handed initial data c + in (λ, λ).At tree level, this is given by a self-dual solution to the YM field equations, i.e. a solution with purely right-handed field strength, which will generate the N -1 MHV static patch amplitudes S(1 + , . . ., n + ; −).Let's now describe this self-dual solution, following [23].We will be somewhat less general than the authors of [23], by continuing to work with a constant, i.e.
The key to the construction of [23] is a gauge group element g(x; q, ρ), where q α and ρ α are two left-handed spinors (q α will end up assuming its role as gauge spinor in (40), while ρ α is a new spinor variable): Interchanging the spinors (q, ρ) inverts the group element: We can prove this by explicitly writing out all the terms in the product: We apply the Schouten identity to the numerators in the sum over i: which rearranges the corresponding fractions as: All the terms in the sum over n in (63) now cancel, thus proving eq.(62).
We can now use the group element (61) to define the self-dual non-linear solution, which we denote by A (0) α α: The highly non-trivial part of eq. ( 66) is that it is indeed linear in ρ α , so that A (0) α α does not depend on ρ α .This can be verified by direct computation, analogously to (63), with the gradient producing momentum factors as in (27): Applying again the Schouten identity as in (65), we find that most of the terms cancel, leaving just a single sum over momenta at each n, which rearranges as: where we denoted: Eq. ( 68) is linear in ρ α as promised, and we can read off the field A α α as: The n = 1 piece of (70) clearly coincides with the right-handed part of the linearized field (30), (43).It remains to show that the non-linear corrections make for a self-dual YM solution.This follows from the structure of eq. ( 66), which is similar to a flatness condition.
In particular, eq.(66) directly implies that the contraction of the left-handed field strength with ρ α ρ β vanishes: Since this holds for any value of ρ α , we conclude that F (0) α α describes a self-dual field as promised, and therefore automatically solves the YM field equations.As a corrolary, the N -2 MHV amplitudes all vanish: We can now read off from (70) the potential's Taylor coefficients a α α(1 + , . . ., n + ; q, q): Substituting q α = µ α and plugging into (59), we obtain the N -1 MHV static-patch amplitude: Some further properties of the self-dual solution (70) will be useful below.First, it does not depend on q α.As a result, the gauge condition qAq] = 0 extends into the stronger condition: which is trivial to check.Second, we can evaluate the (purely right-handed) field strength of A α α.This is easy, because the piece quadratic in A α α vanishes, so we only get the contribution from the derivative: In terms of Taylor coefficients, this corresponds to: Finally, we can find the gauge transformation that relates the gauge fields A (0) α α(x; q) with different values of q α .Denoting two such values by (q, q ′ ), it turns out that the necessary gauge parameter is simply g(x; q, q ′ ): Since the Weyl spinor space is 2-dimensional, it is enough to verify the contractions of this equation with q α and q ′α .These follow directly from eqs. (62), (66) and (75).
The entire derivation can of course be repeated with opposite chiralities.The anti-selfdual gauge potential reads: which corresponds to field coefficients: and leads to the anti-N -1 MHV amplitude with negative helicities on all ingoing legs (with the anti-N -2 MHV amplitude vanishing): B. Anti-self-dual field strength perturbation Having constructed the (perturbatively) most general self-dual field solution (70), we now turn to construct a linearized anti-self-dual perturbation over it.It will be sufficient for our purposes to consider just the left-handed field strength F αβ of this perturbation.By linearity, we can discuss separately the perturbations due to left-handed initial data c − in (λ, λ) with different values (λ, λ) = (λ ′ , λ′ ) of the spinor-helicity variables.Thus, we consider a perturbation which, at the non-interacting level, is given simply by: Here, the O(c + in ) corrections are due to the interaction with the self-dual background (70), and q α sets the gauge in which the latter is defined.This interaction is described by the left-handed half of the YM equations, at first order in the perturbation: βα ] = 0 .
Luckily, there is a gauge in which the interaction becomes trivial.Indeed, in the gauge q α = λ ′α , when we plug the linearized solution from (86) into the field equation (87), we find that the interaction term vanishes, thanks to (75).In this gauge, then, the solution is just the non-interacting one.To obtain the result at general q α , we just need to apply the gauge transformation (79): Plugging in our expressions (61)-(62) for g and g −1 , we can read off from (88) the field strength coefficients f αβ with one ingoing negative helicity on the i'th leg: Plugging into (59), we obtain the corresponding N -1 MHV static-patch amplitude: This is immediately recognizable as the Parke-Taylor formula for MHV scattering [19] in the context of the Minkowski S-matrix: However, the helicities in (90) are N -1 MHV, and µ α in our context is not the spinor square root of the final momentum K µ 1...n , which isn't even on-shell.Nevertheless, as we'll see in the next section, the similarity between these amplitudes is not a coincidence.
Once again, the entire analysis can be repeated with the opposite chiralities, yielding the field strength coefficients: and the amplitude:

IV. POLES, MINKOWSKI S-MATRIX AND BCFW-TYPE RECURSION
In this section, we zoom back out from calculating specific amplitudes to discussing general properties of the framework.In particular, we examine the pole behavior of the non-linear tree-level potential A α α, its field strength F αβ , F αβ , and the resulting tree-level static-patch amplitudes (59).In section IV A, we discuss how poles in the field strength, which encode the usual tree-level Minkowski S-matrix, are related to finite components of Taylor expansion w.r.t.c ± in of the residue term in (94) takes the form: . ., n hn ; q, q) = −λ α λ β M(1 h 1 , . . ., n hn , {λ, − λ, +}) ; lim where M denotes an (n+ 1)-point Minkowski S-matrix amplitude.The flipped sign on λ α in its argument simply reverses the final leg's 4-momentum, so as to treat ingoing and outgoing 4-momenta on an equal footing (here, by making them all ingoing).As a general reference on the relationship between tree-level S-matrix amplitudes and classical field solutions, see e.g.[35].
These are easy to evaluate, using the fact that, near the pole, we can approximate K 2 ≈ λK λ].We find: In particular, the RHS again does not depend on (q, q).We can now take the limit (q, q) → (λ, λ), in which we recognize the LHS of (97) as the generating functions for static-patch amplitudes (59).Taylor-expanding in c ± in , we conclude: where the amplitudes S on the RHS are evaluated at K α α 1...n = λ α λ α.Comparing with (95), we see that the Minkowski S-matrix amplitudes M are a special case of our static-patch amplitudes S, evaluated at on-shell outgoing momentum K α α 1...n = λ α λ α, and with an opposite helicity on the outgoing leg: As a special case, the N -1 MHV static-patch amplitude S(1 + , . . ., i − , . . ., n + ; +), when evaluated at K α α 1...n = λ α λ α, should reproduce the Parke-Taylor MHV formula (91).As we've seen in ( 90), the two in fact agree for general values of K α α 1...n .This stronger agreement is not a coincidence either: as we'll see below, the two amplitudes are governed by essentially the same BCFW recursion relations.
The relation (99) has one apparent exception: it suggests that the 3-point Minkowski Smatrix amplitude M(+, +, −) should equal the static patch amplitude S(+, +; +); however, the latter is equal to zero, according to (72).It turns out that this is an order-of-limits ambiguity: if we calculate S(+, +; +) within the same limiting procedure as the one that led to (99), we find a non-zero answer that agrees with M(+, +, −) (at necessarily complex momenta, as usual for the 3-point M amplitude).Indeed, consider the potential coefficients (73) with n = 2 ingoing legs: ], the contraction from (97) reads: If we now set q α = λ α , we'll get zero, as in (72).Instead, let us first take the limit of lightlike K µ 12 via λ 1 λ 2 → 0. This can be expressed as: where w is some scalar.In this limit, eq. ( 101) becomes: As in (97), the RHS is now (q, q)-independent, and we can trivially take the limit (q, q) → (λ, λ).We then recognize the two sides of eq. ( 103) as: in agreement with eq. ( 99) (since the amplitudes in this case are even in λ α, flipping its sign has no consequence).

B. BCFW-type recursion
In this section, we define a BCFW-type recursion relation for the tree-level static-patch amplitudes S.These recursion relations reduce a static-patch amplitude to a Minkowski S-matrix amplitude M of the same size, plus products of smaller amplitudes.The recursion can be applied whenever we can find an ingoing leg with the same helicity sign as the outgoing one; this is always the case, except for the amplitudes S(1 + , . . ., n + ; −) and S(1 − , . . ., n − ; +), which we already calculated in section III A from the self-dual solution (70).Thus, our recursion will reduce any amplitude S down to: 1.The Minkowski S-matrix amplitudes M (which in turn can be subjected to the usual BCFW recursion).
We now proceed to construct the recursion.We single out one of the ingoing legs, e.g.leg number i. This, together with the outgoing leg, will form the two external legs involved in the BCFW shift.We assume for concreteness that our singled-out ingoing leg has negative helicity.We then shift its right-handed spinor-helicity variable, as: Here, z is a complex variable, while (µ, μ) are the spinor-helicity variables on the future horizon (which, in our Minkowski treatment, are simply defining the lightcone gauge µAμ] = 0).The shift (105) changes the 4-momentum of our ingoing leg by zλ α i μ α.The same shift then applies to the outgoing 4-momentum K α α 1...n ; similarly, it applies to any internal leg that includes the i'th one as a summand, i.e. to the sum K α α j...l of any consecutive set of ingoing momenta with j ≤ i ≤ l.Now, the static-patch amplitude S will have a pole whenever the shift takes one of these momenta on-shell (not counting the i'th ingoing momentum itself, which is already on-shell).The key claim is then that the amplitude's original value at z = 0 can be recovered from the residues at these poles: This is equivalent to the statement that the contour integral S(z) z dz at infinity vanishes, for which it is sufficient that S(z) itself vanishes there: As we will prove in section IV D, this is indeed the case if the helicity of the outgoing leg is the same as of the shifted ingoing leg.For now, let us unpack the content of eq. ( 106).The value of z at which the 4-momentum K µ j...l becomes lightlike reads: At this value, the deformed momentum assumes an on-shell value defined by spinors (λ, λ) as follows: up to the freedom of rescaling λ α and λ α by opposite factors.Our distance from the pole can be parameterized by the contraction: Near the pole, the field on the newly on-shell leg is described by on-shell plane waves (94),(96), in general with both helicities, whose coefficients are given by Minkowski S-matrix amplitudes, as in (95).These on-shell waves then feed into our overall amplitude S(z) as a the sum will includes one or two poles.These poles reduce the (n + 1)-point static-patch amplitude S into an n-point amplitude of the same type, times a 3-point Minkowski S-matrix amplitude M (see Figure 2).This is completely analogous to how the usual BCFW recursion works on the MHV and anti-MHV M amplitudes.This explains the "coincidence" between the static-patch amplitudes (90),( 93) and the Parke-Taylor formula (91).

C. Comparison with scalar field theory
This is a good place to draw a comparison with scalar field theories.Consider a scalar theory that is conformal at tree-level, i.e. a conformally massless scalar with ϕ 4 interaction.
For such a theory, we can pose the static-patch scattering problem, and work out its kinematics, just like in section II, but without gauge choices or polarization factors.In particular, eqs.( 58)-(59) carry through: we just need to remove all helicity signs, all references to the (µ, μ)-dependent lightcone gauge, the µK 1...n μ] denominator in (58), and the µ, μ, K 1...n prefactors in (59).In this way, the problem of static-patch scattering for a conformal scalar theory can be reduced to that of calculating a non-linear bulk field in Minkowski space, as a functional of ingoing linearized plane waves.If we wish, we can of course also consider this Minkowski problem for more general, non-conformal scalar theories; however, we will then lose the original connection with static-patch scattering.
Consider, then a scalar theory in Minkowski.Unlike Yang-Mills, we know that its Minkowski S-matrix is not subject to BCFW recursion, because it doesn't vanish at z → ∞.
Here, the "current" J (1;m) µ denotes the interaction terms, which contain exactly one factor of A (1;m−1) µ , some factors of A (0) µ (either one or two), and at most one momentum factor.The gauge-algebra-valued function θ (1;m) (K) is arbitrary, and encodes the solution's gauge freedom at each order.
In our present formalism, the BCFW shift (105) consists in shifting the momentum of A (1;m) α α at every order m, as in (114).At large z, this implies the ∼ z behavior (115) for K 2 .Therefore, the solution (119) vanishes as ∼ 1/z (like in the case of scalar field field theory), if the expression in parentheses does not grow with z.This can be arranged by suitably tuning θ (1;m) (K), so long as J (1;m) α α grows with z at most linearly, and only along the shift vector λ ′ α μ α.Let us now show that this is indeed the case, assuming that the background field is given in the gauge λ ′ A (0) μ] = 0. First, note that if A (1;m) µ at some order m vanishes as ∼ 1/z, then J (1;m+1) µ doesn't grow with z at all.This is because positive powers of z can only arise from factors of the shifted momentum (114), but there's at most one such factor in every term in J α α is z-independent, there is a danger of positive powers of z from the terms in J (1;1) µ that contain a factor of the shifted momentum (114), or, equivalently, a spacetime gradient acting on A (1;1) µ .There are three such terms: where the dots denote terms factors of the shifted momentum.Let's now examine the terms one by one.The first term does not grow with z, thanks to our assumed gauge condition λ ′ A (0) μ] = 0 on the background field.Similarly, the second term doesn't grow with z, thanks to the property λ ′ A (1;0) μ] = 0 of the non-interacting perturbation (117).Finally, the third term in (120) does grow linearly with z, but only along the shift vector λ ′ α μ α, which means that the growth can be canceled by tuning the gauge function θ (1;1) .This completes our proof that, in a certain gauge, the corrections A (1;m) µ at all orders m > 0 vanish as 1/z.
Let us now transform into the gauge µAμ] = 0 that is relevant for the static-patch amplitudes.First, we apply a gauge transformation g (0) that brings the background field A (0) α α from the gauge λ ′ A (0) μ] = 0 into the desired one µA (0) μ] = 0.This transformation is z-beyond tree-level.For a theory that's conformal at the quantum level, such as N = 4 SYM, this should be relatively straightforward.Without supersymmetry, though, the conformal symmetry of YM theory is broken by loop corrections.Perhaps in perturbation theory, it's somehow possible to treat these violations systematically, and still apply Minkowski methods to static-patch scattering?Another obvious question is what about perturbative GR in the de Sitter static patch.
On one hand, gravity is not conformal even at tree level, and its perturbation theory in de Sitter space is rather painful.On the other hand, it is tempting to speculate that our static-patch results for Yang-Mills, such as the simple formulas for (N -1 )MHV scattering, or the validity of BCFW recursion, should somehow "square" into true statements about GR, along the lines of color-kinematics duality [27,36,37].Going yet further up in spin, it would be interesting to see if any insights from the Yang-Mills case may carry over to higher-spin gravity in de Sitter space.In particular, perhaps the N -1 MHV amplitudes (74), which are a product of self-dual Yang-Mills theory, can be uplifted into chiral higher-spin theory [38][39][40][41][42].

FIG. 1 :
FIG. 1: Penrose diagrams of the dS 4 static-patch problem and its Minkowski counterpart.(a) The static patch (in gray) inside dS 4 ; its past and future lightlike boundaries have been "doubled" into geodesically complete cosmological horizons.(b) The dS 4 Poincare patch associated with the past horizon, and its extension into a full Minkowski space through the future boundary of dS 4 .