Asymptotic charges from soft scalars in even dimensions

We study asymptotic charges associated to a spin-zero analogue of Weinberg's soft photon and graviton theorems in even dimensions. Simple spacetime expressions for the charges are given, but unlike gravity or electrodynamics, the symmetry interpretation for the charges remains elusive. This work is a higher dimensional extension of the four dimensional case studied in arXiv:1703.07885 .

Soft factorization theorems in scattering amplitudes can be understood as conservation laws of asymptotic charges, see [2] for a recent review. When the soft particle has spin 1 or 2, the asymptotic charges are associated to gauge symmetries that are non-trivial at infinity [2]. For spin zero soft particles one can still construct conserved asymptotic charges [1,3], but their interpretation in terms of symmetries remains incomplete. The factorization studied in [1] is a 'leading' soft theorem (a spin-zero analogue of Weinberg's soft theorem [4]), whereas [3] deals with factorization at subleading order in the soft scalar energy.
In the hope to improve the understanding of spin-zero asymptotic charges, here we extend the results of [1] to higher even dimensions. One of the advantages is that we can arXiv:1711.05773v1 [hep-th] 15 Nov 2017 study massless ϕ 3 theory (which does not have a good null-infinity description in dimension four, see section III A). This is perhaps the simplest context where to explore the question of asymptotic symmetries associated to soft scalars. Furthermore, it includes (upon incorporating internal indices) the case of the bi-adjoint scalar theory [6], whose asymptotic symmetries one hopes [1] could be related to those in Yang-Mills through a 'double-copy' [7]. As in [1] our discussion will be restricted to tree-level amplitudes. The study of loop corrections would require to work in a theory where masslessness of the scalar is protected (as for instance the model studied in [3]).
The organization of this paper is as follows. In section II we recall the charge associated to the soft-scalar theorem in [1]. In section III we interpret this charge in terms of asymptotic fields at null infinity for massless ϕ 3 theory. In section IV we extend the analysis to include massive fields ψ coupled to the massless scalar. In section V we discuss smeared version of the charges and give a simple expression for a spacetime current. Some of the calculations are given in appendices.
Our conventions are as follows. We take spacetime dimension to be D = 2m + 2 with m = 2, 3, . . .. We use mostly plus signature spacetime metric. For simplicity in our discussion we focus on future null and time infinities; a parallel discussion applies to the past infinities. We also omit a discussion of fields at spacelike infinity. This shares many features with the study of fields at time-infinity, except that differential equations become hyperbolic. In particular, it allows one to understand the conservation of asymptotic charges from a dynamical perspective [1,5].

II. SOFT THEOREM CHARGE
We consider a theory of a self-interacting scalar field ϕ coupled to a massive scalar ψ, 1 At tree level, one can obtain a 'soft theorem' that relates an (n + 1)-particle amplitude with an n-particle amplitude, where the (n + 1)−th particle is a soft ϕ. This soft theorem can be written as matrix elements of the identity [2] Q(q)S = SQ(q), where S is the S-Matrix,q the direction of the soft ϕ-particle and [1] Here a and b are the Fock operators associated with the fields ϕ and ψ respectively, dp the standard Lorentz invariant measure and a null, future pointing four-momentum associated toq.

III. CHARGES IN MASSLESS ϕ 3 THEORY
In this section we set g = 0 in (2.1) and restrict attention to massless ϕ 3 theory. We remind the reader that we work in even spacetime dimension grater than four: D = 2m + 2 with m = 2, 3, . . ..

A. Field expansion near null infinity
We assume the standard 1/r expansion off null infinity, As we shall see, the leading term φ 0 represents the 'free data' in terms of which the remaining φ n 's can be expressed by recursively solving the field equations. The field equation for ϕ is: where, in retarded coordinates, the wave operator takes the form with D 2 the Laplacian on the 2m-dimensional sphere. Plugging the expansion (3.1) in (3.2) one finds: For higher n one gets O(g) contributions from the second term in (3.2). For n = m − 1 this gives: We will see that φ m−1 plays a role analogous to 'Bondi mass-aspect' in gravity or 'charge aspect' in Maxwell theory. In particular, its u → −∞ asymptotic value defines an angular charge density σ(x), which, as we shall see, is conserved and encodes the same information as the charge Q(q) defined in Eq. (2.3). Before moving on, we need to specify the u → ±∞ fall-offs implicitly assumed in (3.6) and consistent with (3.5). We require: as u → ±∞ for some > 0. These conditions may be thought of as real-space version of the condition that ωa(ω,q) has a well-defined ω → 0 limit (see Eq. (2.3) and the next subsection).
With these conditions we can now use Eqns. (3.4), (3.5) to write ∂ u φ m−1 in terms of the free-data φ 0 . Introducing the notation:

B. Relation between Q(q) and σ(x)
In this section we use the sphere-differential operator appearing in (3.9) to relate the charge Q(q) (2.3) with σ(x) (3.6). Define the 2(m − 1)−th differential operator: In appendix A it is shown that K is invertible, with inverse given by the Green's function To establish these relations we compare the 'soft' and 'hard' parts of Q(q) and σ(x) separately. The soft and hard parts of Q(q) are: where in writing Q hard (q) we expressed the momentum integral in spherical coordinates p = Ep. For σ(x) we start with definition (3.6) to obtain: As we shall see in the next section, the term φ m−1 (u = ∞,x) accounts for massive particles but is zero in the massless theory. The term ∂ u φ m−1 (u,x) can be expressed in terms of the free-data φ 0 by means of Eq. (3.9), yielding 'soft' (linear in φ 0 ) and 'hard' (quadratic in φ 0 ) pieces, To compare with Q(q) we express σ soft/hard in terms of the Fourier transform of φ 0 . Defining We finally expressφ 0 (E,x) in terms of the Fock operator a(Ep). In the asymptotic future, ϕ is described by the free-field expression ϕ(x) ≈ dp a( p)e ip·x + c.c., where dp ≡ d 2m+1 p (2π) 2m+1 2| p| . The integral over the spherep = p/| p| can be evaluated via saddle point in the r → ∞, u =const. limit, yielding an expansion of the type (3.1) with We thus obtain,φ Using (3.11) it is straightforward to verify This concludes the proof of Eq. (3.12). The inverse relation (3.13) follows automatically.

IV. MASSIVE FIELDS
In this section we study contribution from massive fields. We thus take g = 0 in (2.1). For simplicity we set g = 0 so that the field equation for ϕ is Our aim is to show that relations (3.12) and (3.13) still hold in this case. The soft part of the charges is the same as before, so we focus on the hard part of the charges. For Q(q) this is given by the last term in (2.3), whereas for σ(x) it is given by (recall Eq. (3.16)) As in [1] we proceed by studying the field equations at time-like infinity and then compare the null and time-infinity expansion in their common range of validity. To this end, we switch to hyperbolic coordinates where y ≡ y α , α = 1, . . . , 2m + 1 parametrize the unit hyperboloid H. In these coordinates the Minkowski metric takes the form with h αβ the unit hyperboloid metric. We start by studying the τ → ∞, Y µ =const. behavior of the massive field ψ. At late times the field is described by the usual Fourier expansion ψ(x) ≈ dp b( p)e ip·x + c.c., (4.6) where dp = d 2m+1 p (2π) 2m+1 2 √ p 2 +µ 2 . In the τ → ∞ limit the momentum integral can be evaluated by a saddle-point yielding 5 From here we conclude that the leading τ → ∞ asymptotics of the RHS term in (4.1) is given by (4.9) The field equation (4.1) together with (4.8) implies a τ 2m−1 asymptotic fall-off for ϕ ϕ(τ, y) = ϕ H (y) τ 2m−1 + . . . where the relevant Green's function is with Vol(S 2m ) = 2 2m+1 π m m! (2m)! (4.14) the volume (or area) of the unit 2m−sphere. We now use (4.12) to express σ hard (x) (4.3) in terms of j(y). We first choose coordinates y α = (ρ,x) such that Y µ = ( 1 + ρ 2 , ρx) (4.15) with ρ > 0 andx a unit vector parametrizing points on S 2m . The conformal boundary of H is obtained by taking ρ → ∞ with fixedx. In appendix B it is shown that consistency of the null and time-infinity expansions for ϕ implies The right hand side of (4.16) can be evaluated from expression (4.12) yielding where Y · (1,x) ≡ −Y 0 + Y ·x is the Minkowski inner product between Y µ and the null vector (1,x). We finally wish to compare (4.17) with Q hard (q). To this end we first express the momentum integral in (4.2) as an integral over H by doing the change of variable p = µ Y (y), resulting in The relation between (4.17) and (4.18) can then be established in the 'inverse' form (3.13), by means of a 'shadow transform' identity 6 [13] which in our conventions reads: (4.20) The converse relation σ hard (x) = K Q hard (x) then follows automatically.

V. SMEARED CHARGES AND SPACETIME CURRENT
Given a function λ on the sphere we define the smeared charge by The role of the shadow transform in the context of spin 1 and 2 soft theorems was recently discussed in [14].
(In particular, the smearing λ(x) = (−1 +q ·x) −1 yields the charge Q(q).) As in [1], we will show that the smeared charge can be written in terms of a spacetime current j a = ∂ b k ab , where X a ∂ a = x µ ∂ µ is the dilatation vector field and Λ a space-time scalar determined by λ according to: a normalization constant. The field (5.4) satisfies the free wave equation with certain fall-off properties we now describe.
In the r → ∞, u =const null infinity limit, one can verify (5.4) implies On the other hand, in hyperbolic coordinates adapted to time-like infinity one has (5.10) Expression (5.10) can alternatively be interpreted as the solution to the boundary-value problem (see appendix C for further details). We now show that the charge (5.2) can be understood as arising from the spacetime current j a = ∂ b k ab . We follow the same strategy as in [8] in which one integrates the current over a t =const spatial slice and evaluates the integral in the t → ∞ limit. Two contributions arise in this limit, depending on whether the limit is taken along null or time-like directions. The null infinity contribution is given by where u = t − r and To evaluate (5.12) we first substitute the 1/r expansions for ϕ and Λ in (5.13). This leads to a 1/r expansion for k ru of the form k ru = r m m n=1 r −n k ru n + O(r −1 ). (5.14) We note that k ru diverges in the null infinity limit. However the potentially divergent terms in (5.12) integrate to zero due to the u → ±∞ fall-offs. Indeed, from (3.7) and ( We now evaluate the contribution from time-infinity. To this end we switch to hyperbolic coordinates adapted to time-infinity. From the τ → ∞ fall-offs for ϕ and Λ one finds leading to a time-infinity contribution to the charge: where in the last equality we used the field equations (4.11) and (5.11). On the other hand, according to Eq. (4.16) the 'time-infinity' contribution to (5.2) should be given by: That (5.20) coincides with (5.19) can be verified by writing ϕ H in terms of j according to (4.12), noting that Following upon the work [1], in this paper we studied the asymptotic charges associated to a spin zero analogue of Weinberg's soft theorem in higher even dimensions. Whereas it is relatively straightforward to recast the soft theorem as a conservation of charges Q(q) in the sense of Eq. (2.2), it is less obvious how to interpret these charges in terms of asymptotic fields. Here we provided such interpretation, for the case of both massless and massive 'hard' particles.
The interpretation of Q(q) in terms of asymptotic fields allows one to obtain a spacetime expression for smeared charges. As in gravity and electrodynamics, these smeared charges can be written as integrals of total-derivative currents, j a = ∂ b k ab . But unlike those cases, there is so far no symmetry understanding for the tensor k ab : In gravity and electrodynamics k ab are Noether forms of local symmetries [11]; but there are no local symmetries associated to scalars! The question remains as to what is the appropriate symmetry interpretation for the 'soft scalar' asymptotic charges. where is the volume of the unit S 2m−1 sphere. We now discuss the m = 2 and m > 2 cases separately. For m = 2, K = −(4π) −2 ∆ 1 and so K −1 = −(4π) 2 G 1 = 2(1 −q ·x) −1 which coincides with expression (3.11). For m > 2 we use the identity [9], on H. On the other hand, Eq. (C4) implies [12,13] lim ρ→∞ ρ h Λ H (ρ,x) = λ(q).
In other words, G(y;q) is the Green's function for the boundary value problem given by Eqns. (C6), (C7). At time-infinity there are also fields ϕ H that capture the contribution to charges due to massive particles [1,5,8,15]. These fields are in a sense 'dual' to Λ H and satisfy Poisson-type equations (D 2 + h(d − h))ϕ H = j; ϕ H (ρ,x) where j is a source term that is quadratic in the massive particle Fock operators. If G(y; y ) is the Green's function for the elliptic problem (C8) then its ρ → ∞ asymptotic value is related to the Green's function (C5) by [16]: a well-known fact in the AdS/CFT literature. For the case of interest in the present paper we set h = 1 and d = 2m. In this case G is given by Eq. (4.13), G can be read-off from Eq. (5.10) (or alternatively from Eq. (C5)), and relation (C9) becomes lim ρ→∞ ρ 2m−1 G(ρ,x; y ) = − 1 2(m − 1) G(y ;x), which corresponds to Eq. (5.21) in the body of the paper.