Soft Matters, or the Recursions with Massive Spinors

We discuss recursion relations for scattering amplitudes with massive particles of any spin. They are derived via a two-parameter shift of momenta, combining a BCFW-type spinor shift with the soft limit of a massless particle involved in the process. The technical innovation is that spinors corresponding to massive momenta are also shifted. Our recursions lead to a reformulation of the soft theorems. The well-known Weinberg's soft factors are recovered and, in addition, the subleading factors appear reshaped such that they are directly applicable to massive amplitudes in the modern on-shell language. Moreover, we obtain new results in the context of non-minimal interactions of massive matter with photons and gravitons. These soft theorems are employed for practical calculations of Compton and higher-point scattering. As a by-product, we introduce a convenient representation of the Compton scattering amplitude for any mass and spin.

±h are called the soft factors. They are universal, in the sense that they do not depend on what other "hard" particles are involved in the scattering.
Since the seminal works of Low [1] and Weinberg [2], which explored the leading term in the soft momentum expansion, the importance of the soft theorems has been revealed through applications that span very different fields. In particular, soft theorems turn out to be intricately related to the consistency conditions of the S-matrix such as gauge invariance, locality and unitarity [3,4]. Furthermore, it was shown that they can be understood as Ward identities of asymptotic symmetries [5]. They have also been relevant to understanding the landscape of effective field theories (EFTs) (see e.g. [6][7][8][9][10]).
A new life into the study of soft theorems was brought by the emergence of on-shell methods for calculating scattering processes. These are founded on the spinor techniques, which make manifest the little group transformation properties of the amplitudes. An important step in this development was Ref. [11] by Cachazo and Strominger. Using the BCFW recursion relations [12] they were able to extend the graviton soft theorems up to sub-subleading order at tree level. 1 Subsequently, the work by Elvang et al. [16] extended on-shell soft theorems to a broad class of theories with massless particles. This is achieved thanks to a modified recursion relation that combines double BCFW and soft shifts. This approach leads to an elegant proof that various S-matrix consistency conditions, such as charge conservation and the equivalence principle, follow directly from Poincaré invariance, locality, and unitarity, without ever mentioning gauge invariance. Moreover, Ref. [16] systematized the classes of effective operators that modify the soft theorems at the subleading order for photons and the sub-subleading order for gravitons.
Despite these rapid developments, all of the results derived with on-shell techniques are valid only for massless theories. 2 Meanwhile, a convenient spinor formalism for dealing with massive particles was introduced in [22]. There are several motivations to rewrite the soft theorems such that they can be directly applied to theories using the on-shell language and massive spinors. Perhaps the most pressing is the growing importance of the on-shell program in the context of black hole computations [23][24][25][26]. Another is the progress in on-shell formulation of massive EFTs, such as e.g. massive gravity [27][28][29][30], or the Standard Model EFT [31][32][33]. More generally, evolution of on-shell tools to deal with massive theories may reveal unexpected simplicity of their amplitudes, much as it happened for massless theories.
In this paper, we establish soft recursion relations that are valid for general theories with massive particles of any spin, extending the results of [16]. This is achieved by introducing a new kind of holomorphic spinor shift, where also massive spinors are shifted in addition to massless ones. This allows us to derive a general recursion relation that remains valid even when the soft particle is the only massless particle in the scattering process. Our formula can be applied to describe emission of a soft particle of any helicity |h| ≤ 2. In this paper we work out the resulting soft theorems for photons and gravitons. These have the same general structure as in the standard derivation, however they can be directly applied to amplitudes written in term of massive and massless spinors. In particular, the subleading soft factors become differential operators acting on the massive spinors and on their little group indices. We explicitly connect the subleading soft theorems to multipole expansion of the 3-point interaction between massive matter and photons/gravitons. As a corollary, we obtain a compact and elegant proof that gravitational dipole interactions of fundamental particles of any spin are inconsistent with the assumption of Poincaré invariance, locality, and unitarity.
We also discuss a couple of concrete applications of the soft recursion relations in massive theories. One is for the simple 4-body process of Compton scattering of photons and gravitons. For any spin of the matter particle, the recursion correctly captures the IR part of the amplitude, that is the one containing the physical poles in the Mandelstam variables. The result agrees with the one in Ref. [22] obtained by requiring that the residues of the physical poles factorize into 3-point amplitudes. As discussed in [22,34], for higher spins the IR part inevitably develops a spurious pole that is not associated with an exchange of a physical particle. Therefore, to make it consistent with unitarity, the Compton amplitude has to be augmented by a UV term that cancels the spurious pole without affecting the physical ones. In our derivation, this UV term arises as a boundary term in the recursion relation. As an aside, we present a systematic algorithm to build the UV term in Compton scattering, alternative to the one in [34]. We also make some baby steps toward higher-point amplitudes of any spin, by calculating specific 5-point amplitudes with 3 photons or 3 gravitons.
Finally, we touch on the subject of exponential representation of soft amplitudes. Exponentiated soft factors play an important role in the matching between quantum and classical limits. In particular, they are instrumental in certain calculations of black hole observables [24,35]. They can also be used to write scattering amplitudes in the Mellin space, which can be connected with CFT correlators in the celestial sphere (see e.g. [36][37][38]). Our recursion relations lead to a simple proof of the exponentiation of Compton amplitudes [24] for any mass and spin. We also remark that the photon and graviton soft factors can be written in terms of an exponential operator in the scattering of massive particles of generic spin, which previously was only done for the minimal-helicity violating (MHV) sector of massless gravity/Yang-Mills [14].
Our paper is organized as follows. The (massless and massive) spinor formalism is briefly reviewed in Section 2. The new soft recursions, valid for particles of any mass and spin, are derived in Section 3, from which the soft theorems in Section 4 directly follow. Some applications for calculating concrete amplitudes are presented in Section 5. Section 6 discusses the exponential representation of soft amplitudes and Compton scattering. Section 7 is reserved for conclusions and future directions. Appendix A lists the possible spinor shifts realizing our momentum shift, some of which are not employed in this paper but may be useful for other applications. Appendix B describes in detail our construction of UV/boundary terms in Compton scattering.
Massless momentum can be represented by a pair of (commuting) spinors λ α ,λβ: such that p 2 = 0 is automatic for any λ,λ thanks to the identity λ α λ α = 0 =λαλα. It follows that the massless spinors satisfy the Weyl equations: λ(pσ) = (pσ)λ = 0, (pσ)λ =λ(pσ) = 0. For real p µ we have the additional constraintλ =λ, but in general momenta are allowed to be complex, in which caseλ and λ are independent. The little group transformations (i.e. for a fixed p µ , the subset of Lorentz transformations leaving p µ invariant) correspond to U (1) acting as λ → t −1 λ,λ → tλ. The Lorentz-invariant and little group covariant building blocks are commonly represented by the bra-ket notation: 4) and the Lorentz contraction of momenta can be written as 2p i p j = ij [ji]. 4 The bra-ket notation can be naturally extended to more complicated contractions, e.g. (λ i p k σλ j ) ≡ ip k j], (λ i p k σp lσλj ) ≡ ip k p l j], etc. In the following, we choose to write uncontracted spinors explicitly as in Eq. (2.1), and reserve the bra-ket notation for Lorentz contractions only; e.g. we write ij λ k ≡ (λ i λ j )λ k . The massless spinors can be related to polarization tensors in the standard Lagrangian formalism. In particular, the polarization vectors of a massless spin-1 particle can be written in terms of the spinors as whereζ and ζ are arbitrary reference spinors representing the gauge freedom. Similarly, for a massless spin-2 particle, the polarization tensors are ± µν = ± µ ± ν , where now the freedom of choosing the reference spinors represents general coordinate invariance.
Massive momentum satisfying the on-shell condition p 2 = m 2 can be represented by four spinors χ 1 α , χ 2 α ,χβ 1 ,χβ 2 . They can be collected into χ J andχ J , where J = 1, 2 is identified with the SU (2) little group index which, in complete analogy to spinor indices, can be raised and lowered by epsilon tensors. The spinors are related to the momentum by the formula where summation over the little group indices is implicit. This is a natural and convenient generalization of the spinor helicity formalism to massive particles [22] (for earlier works, see e.g. [40,41]). The massive spinors are normalized as from which follow the Dirac equations These allow one to trade χ forχ and vice-versa. An amplitude describing scattering of n massive incoming particles with spin S i , i = 1 . . . n, for each i will contain exactly 2S i spinors χ J k i orχ J k i with uncontracted little group indices. Massive outgoing particles can be represented by spinors χ i J k orχ i J k with lowered little group indices, however in this paper we treat all external particles as incoming. The little group indices corresponding to the same particle are always implicitly symmetrized over. To reduce clutter, in the following we will often omit the little group indices whenever it does not lead to ambiguities. Lorentz contractions of massive spinors will be shortened via the bra-ket notation: where we adhere to the bold notation introduced in Ref. [22].

Soft Recursions
In this section we derive a formula connecting an n + 1-point amplitude with at least one massless particle to n-point amplitudes. The formula picks up the leading terms in the limit where the momentum of a massless particle is continuously taken to zero. That massless particle is called soft, and the formula is referred to as the soft recursion. The derivation follows similar steps as the one in Ref. [16], however it is adapted to allow for massive particles in the scattering process.

Momentum shift
Consider a set of n + 1 particles with momenta p 0 . . . p n satisfying n i=0 p i = 0 and the on-shell conditions p 2 i = m 2 i . The particle labeled as 0 is always massless, p 2 0 = 0, and it will be our soft particle. We perform a two-parameter deformation of this kinematics. One parameter z controls complex deformation of the momenta of the particle 0 and of two other particles, below labeled as j and k. The other parameter controls the soft limit of the particle 0. Specifically, the shift is defined as 5 where q 0 , q j , q k are four-vectors satisfying The first set of equations in Eq. (3.2) ensures that the shift does not change the particles' masses, p 2 i = m 2 i , while the role of the second is to ensure shifted momentum conservation, n i=0p i = 0, for any z and . One advantage of the two-parameter shift in Eq. (3.1) compared to simpler BCFW-like one-parameter shifts is that it will directly lead to soft expansion of recursion relations in powers of 1/ . As a bonus, it will allow for more control over the recursions' boundary terms, as the latter will start at higher orders in 1/ .

Recursion
Using the momentum shift in Eq. (3.1) one can relate, in a quite model-independent way, n + 1point amplitudes in the soft limit to n-point amplitudes. Consider an n + 1 point amplitude M n+1 ≡ M(1 . . . n0). We denoteM z, n+1 its shifted version, which is M n+1 evaluated for the "hatted" kinematics in Eq. (3.1). Similarly, we denote M n ≡ M(1 . . . n), and its shifted version byM z, n . Note thatM 0,1 = M. At tree levelM z, is a meromorphic function of z, with all singularities given by a finite set of simple poles at z = z i . Applying the Cauchy formula, where the boundary term is z , and C ∞ is the circle at infinity. IfM z, n+1 goes to zero as z → ∞ then B ∞ = 0. The recursion relation is most powerful when B ∞ = 0, but this will not always be true in the cases of interest.
We only focus on a subset of the poles related to the emission of the particle zero from an external leg the n-point amplitude M n . In the limit → 0, singular 1/ n terms inM z, n+1 can only come from these residues. There is n such poles z l related to the solutions of the equationŝ P 2 0l | z=z l = m 2 l , whereP 0l ≡p 0 +p l and it is understood thatp l = p l for l = j, k. The poles are located at and one can rewriteP 2 0l − m 2 l = −2q 0 p l (z − z l ). This shows that the poles atP 2 0l → m 2 l are in one-to-one correspondence with the poles at z → z l . Unitarity requires that at every pole the amplitude factorizes: Plugging this back into the Cauchy formula in Eq. (3.3) on obtains the soft recursion: This formula will be the starting point for deriving a new on-shell incarnation of soft theorems which is valid in the presence of massive particles. We will often refer to the n-point amplitude above as the hard factor. The boundary term in Eq. (3.3) is absorbed in O( 0 ) above, as it cannot produce singular terms in the → 0 limit.

Spinor shift
We turn to discussing spinor shifts that realize the momentum shift in Eq. (3.1). The discussion depends on whether the shifted particles j and k are massive or massless. In the following we first work out the case when m j > 0 and m k > 0, and later comment on other possible configurations. Note that m j and m k can be different. Recall also that particle 0 is always massless in this discussion, however other particles with l = 0, j, k can be massive or massless, as one pleases. One possible choice of the shift vectors q in Eq. (3.1) is where y is an arbitrary spinor satisfying y0 = 0. One can verify that these satisfy the conditions in Eq. (3.2). We can decompose the shifted momenta into spinors:p 0 σ = λ z 0λ 0 ,p j σ = χ J jχ z j J , p k σ = χ J kχ z k J . At the spinor level, the shift is realized as 6 {0mm} : while all other spinors remain unshifted. Since we shift the holomorphic spinor corresponding to the particle 0 and the antiholomorphic spinors corresponding to the particles j, k, we label this shift as {0mm}. The parity-reversed version, labelled as {0mm}, is defined in Appendix A. Note that our shift is non-trivial for z = = 0; in this limit the spinorsχ j,k must shift to absorb the original momentum of the particle 0. This "zero-order" shift will play some role in the following. On the other hand, the shift is trivial for z = 0 and = 1, but that is away from the soft limit → 0 on which we are focused in this paper. We end this section with a number of identities that will be useful in the following (impatient readers are encouraged to skip to the next section). For the {0mm} shift, the location of the poles in Eq. (3.4) can be recast as At the poles, the shifted spinors can be expressed as These identities allow us to rewriteP 0l at the poles. For l = j, k,P 0l =p 0 + p l , thuŝ SinceP 0l | z=z l is on shell by definition, we can factorize it into spinors: Similarly at z j and z k we have the decomposition In our story it will be important that these are independent of the soft parameter . Similarly, the shiftsχ z k j andχ z j k and are independent of . The case where one of the j, k particles is massive and the other is massless is covered in Appendix A. We also comment on the limit where all involved particles are massless. This can be easily obtained from Eq. (3.9) by "unbolding" the massive spinors, which corresponds to replacing χ J i → λ i ,χ J i →λ i while dropping the SU (2) little group index. Simplifying the resulting expression one obtains This reproduces the shift in Ref. [16] up to the replacement → − 1 which is due to the different prescription for taking the soft limit.

Soft Theorems
In this section we apply the recursion formula in Eq. (3.7) to derive the soft factors for amplitudes containing particles of any mass and spin. This generalizes the discussion in Ref. [16]. To our knowledge, this is the first derivation using the on-shell methods with massive spinors. The soft factors that we obtain are applicable to more general physical situations than those found in the previous literature. In the following we will deal with the soft particle of spin 2 (graviton) and spin 1 (photon), however Eq. (3.7) goes just as well with massless particles of spin 0 (Goldstone bosons), spin 1/2 (Goldstone fermions), or spin 3/2 (gravitinos). We restrict to single soft particle emission, leaving multiple soft limits for future publications.

Minimal gravity
We begin with a gravity theory where the massless spin-2 graviton is minimally coupled to matter and to itself. The soft theorems in this case have been studied back and forth, both in the standard Lagrangian and in the on-shell formalisms. Nevertheless, here we provide the soft theorems in a novel form that will be particularly convenient for practical calculations with massive spinors.
In the on-shell formalism, the minimal coupling of the graviton h to a matter particle X of mass m > 0 and any spin S corresponds to the on-shell 3-point amplitudes [22] GeV is the Planck constant, and ζ,ζ are arbitrary reference spinors satisfying [3ζ] = 0 and 3ζ = 0 (the amplitude is independent of their precise choice). In the standard QFT language, the on-shell amplitude in Eq. (4.1) corresponds to graviton coupling to matter via the energy-momentum tensor. Meanwhile, the minimal self-coupling of the graviton in the standard Einstein-Hilbert Lagrangian corresponds in the on-shell language to We focus on emission of a soft graviton with plus helicity. The starting point is the recursion in Eq. (3.7), with the amplitudes deformed by the {0mm} spinor shift in Eq. (3.9). We first need an expression for the shifted 3-point amplitude at the z l -poles: This expression is independent of the identity, mass, or spin of the l-th particle (except that for massive particles the identity operator in the little group indices is implicit), which is a realization of the equivalence principle. Plugging this back into Eq. (3.7) we obtain a raw version of the soft theorem:M To proceed we need to expand the n-point amplitude above in powers of . Using the identities in Eq. (3.11) we find where the differential operatorD l is for a massive particle l, where for once we explicitly displayed the SU (2) little group index L, which is implicitly summed over. For a massless particle l one should instead use [0l][0∂ l ] ≡ λ 0λl λ 0 ∂ ∂λ l . Plugging the expandedM z l , back into Eq. (4.4) we obtain the soft theorem in the standard form: The leading soft factor is given by In the second we step we rewrote the soft factor in the more familiar form as it appears in Ref. [2], with the polarization tensor of the soft graviton defined below Eq. (2.5) and the gauge parameter ζ is identified with the shift spinor y. This leading soft factor is independent of y as a consequence of the equivalence principle and momentum conservation [43]: Conversely, the condition that S +2 must be independent of the shift spinor y can be employed to prove the equivalence principle without any reference to general coordinate invariance. 7 Indeed, in our derivation, the soft theorems are a direct consequence of Poincaré invariance, locality, and unitarity. In a consistent theory they must lead to unambiguous soft factors for any choice of y. However, that would not be the case if the graviton coupling strength to matter in Eq. (4.1) were not universal and precisely correlated with the graviton self-coupling in Eq. (4.2). The subleading soft factor in Eq. (4.7) is a differential operator: It is independent of the shift spinor y as a consequence of angular momentum conservation. Indeed, (4.11) Now, n l=1D l can be related to the angular momentum operator J µν introduced in [44] and generalized to massive particles in [24]: For a massless particle l one should replace χ → λ. TradingD l for J µν we find Finally, the sub-subleading soft factor in Eq. (4.7) is a double-differential operator: (4.14) This one is manifestly independent of the shift spinor y. Following the same steps, but starting instead with the {0mm} shift defined in Eq. (A.12), one can obtain the soft factors for emission of a minus helicity graviton: Let us briefly comment on the relation of our results with the previous literature. It is a simple exercise to show that in the limit where all particles (also j and k) are massless, the soft factors we derived reduce to those found in [16]. In particular, the differential operatorD l reduces tõ where ∇ 0l is defined in Eq. (19) of [16]. On the other hand, our soft theorems do not exactly reduce to those in [11], the obstruction being the last two terms in the curly bracket in Eq. (4.17). As discussed in [16], this difference is due to working with "stripped" amplitudes M, rather than with S matrix elements T ∼ δ 4 (p 0 + · · · + p N )M as in [11]. The price to pay for us is that the soft theorems look somewhat more complicated, and that they depend on the exact manner in which the soft limit is taken. The advantage is that for practical purposes one works with the stripped amplitudes M. The expressions we provide are ready to use out of the box with massive amplitudes written using the modern on-shell spinor formalism.

Non-minimal gravity
We move to consider theories where gravitational interactions of matter are non-minimal. Let us start by assuming that a massive particle X with spin S > 0 and mass m > 0 has an anomalous gravitomagnetic dipole moment: It was argued in Ref. [34] that general coordinate invariance forbids d X = 0. But let us press on to see what happens. It turns out that the dipole affects the subleading soft factor, which is modified as The first term in the bracket comes from the minimal coupling, and is the same as in Eq. (4.10), except this time we explicitly display the identity operator acting on the little group indices of the l-th particle. The second factor is due to the dipole, where, as always, symmetrization of J 1 . . . J 2S l and L 1 . . . L 2S l is implicit. The problem is that, for d l = 0, this soft factor is not independent of the shift spinor: This provides a simple and general proof that, for a particle of any mass and spin, the presence of an anomalous gravitomagnetic dipole moment is in conflict with the assumptions of Poincaré invariance, locality and unitarity.
Given the dipole is forbidden, for S ≥ 1 the leading deformation of the minimal coupling is the quadrupole: where the dots stand for eventual higher multipoles, which do not affect the soft theorems. 8 The quadrupole does arise from local Lagrangians, roughly speaking, when gravity interacts with matter via the Riemann tensor; for example, for S = 1 it corresponds to the coupling L ∼ QX µν X αβ R µναβ .
We find that the quadrupole enters into the sub-subleading soft factor: where, as always, symmetrization over the little group indices is understood. The double-differential operator due to the minimal coupling is now supplemented by an algebraic operator originating from the quadrupole. Note that the latter acts non-trivially on the little group indices: the l-th term in the sum acts on the upper indices L ≡ L 1 . . . L 2S l of the l-th particle inM 0,0 (1 . . . l L . . . n).

Photons
Photons can be tackled by exactly the same methods as gravitons, except that the calculations are simpler. Below we summarize the soft theorems for photon emission. Electromagnetic interactions of a particle X with mass m and spin S are described by the following 3-point amplitudes [22,34]: where e is the electromagnetic coupling constant, q is the electric charge, (a+ā)/2 = (g−2)/2 is the anomalous magnetic moment in the standard normalization, a =ā gives the electric dipole moment, and the dots stand for eventual higher multipoles which are irrelevant for the soft theorems. The first term in each bracket describes the minimal coupling, while the second describes non-minimal dipole interactions for S ≥ 1/2. In the Lagrangian parlance, the former arises when photon interacts via D µ = ∂ µ −ieqA µ in the kinetic term, while the higher multipoles correspond to photon interacting via F µν . For example, for real a, the S = 1/2 Lagrangian would be L ⊃ eqψγ µ ψA µ + Plugging the plus helicity amplitude in Eq. (4.23) into the recursion in Eq. (3.7) one finds the raw version of the soft theorem: The first term in the bracket originates from the minimal coupling, while the second describes a correction from the dipole. The latter has a non-trivial action on the little group indices ofM. Using the expansion ofM in powers of , cf. Eq. (4.5), one derives the soft theorem: The leading soft factor for emission of a plus helicity photon is or the other way around: given that the soft recursion must be valid for any y proves charge conservation in any Poincaré invariant, local, and unitary theory. The subleading soft factor is given by whereD l is defined in Eq. (4.6). This one is manifestly independent of y, which in particular confirms what everybody knows that arbitrary electromagnetic dipoles are allowed. Here, we display the action on little group indices of a massive l-th particle, which is non-trivial when dipoles are involved. If on the other hand the l-th particle is massless, we should unbold the l-th term above, that is trade χ L l → λ l , dropping the SU (4.29)

Applications
The goal of this section is to demonstrate that the recursion relations and soft theorems we obtained previously allow one to efficiently calculate amplitudes with massive particles of arbitrary spin. Below we will reproduce some known results, as well as derive new ones not encountered in the literature so far.

Compton scattering
We start with the Compton scattering. We are interested in the amplitude M(1 X 2X3 ± γ 0 + γ ), where 0 γ is a soft photon 9 of plus helicity, and X is a particle of mass m, unit charge, and arbitrary spin S. We will use the {0mm} shift in Eq. (3.9) with j = 1 and k = 2. We first assume minimal coupling, that is the absence of the dipole and higher multipoles in Eq. (4.23) (the dipole case will be discussed shortly). The soft theorem in Eq. (4.25) reduces tô In this case the subleading soft factor S (1) +1 is void for the simple reason that, for Compton scattering, we only have two factorization poles z 1,2 , andD l in Eq. (4.6) is zero for l = j, k. Somewhat pedantically, we replaced O( 0 ) in Eq. (4.25) by the boundary term B ∞ , which is also O( 0 ). The point is that, for 4-body processes, photon emission from external legs accounts for all factorization poles. Therefore, the recursion relation in Eq. (3.7) should reproduce the full Compton amplitude (not just the leading terms for → 0) up to the boundary term introduced in Eq. (3.3). The leading soft factor for Compton scattering is easily calculated as For minimal coupling, the hard 3-point amplitude on the right-hand side of Eq. (5.1) evaluates tô It is important to stress that the shift is not void inM 0,0 . That is because the shift we defined in Eq. (3.9) is non-trivial for z = = 0 (it is trivial if = 1 instead). For this reasonM 0,0 (1 X 2X3 ± γ ) above is not simply a copy of M(1 X 2X3 ± γ ) in Eq. (5.1). This zero-order shift is absolutely crucial to recover the correct result. In particular, the non-trivial spinor structure inM 0,0 (1 X 2X3 − γ ) appears as a consequence of the identity (χ z . All in all we reconstruct On general grounds, this correctly describes leading terms of the Compton amplitude in the limit when the momentum of 0 + γ is soft. The full amplitude valid for any momentum of 0 + γ is recovered by setting = 1. In this limit, the same-helicity amplitude above is a fully consistent expression for any S, also when B ∞ = 0. 10 A non-zero B ∞ corresponds to a contact term (that is to say, without any singularities in the kinematic variables); for example we can have B On the other hand, the opposite-helicity amplitude is healthy as it stands only for S ≤ 1. As discussed in [22], for S > 1 the first term in Eq. (5.4) develops an unphysical pole at 3p 1 0] → 0. That means the boundary term cannot vanish for S > 1; it must be non-zero to cancel the unphysical pole, and at the same time it cannot have any singularities as p 0 p 1 → 0 or p 0 p 2 → 0. Therefore it must be of the form B ∞ = 2S−2 k=1 C k 3p 1 0] k , where C k are pure contact terms. One possible form of the boundary term is where {Y||Z} = Y for even k, and {Y||Z} = Z for odd k. Any other legal boundary term will differ from the one above only by contact terms. The absence of physical poles in Eq. (5.5) is not manifest, but it immediately follows from Y 2n − Z 2n = (Y 2 − Z 2 )(Y 2n−2 + · · · + Z 2n−2 ), given that (Y 2 − Z 2 ) = 16(p 1 p 0 )(p 2 p 0 ) 13 [20] 23 [10]. A rationale for this boundary term is postponed to Appendix B. For the present discussion we notice that B ∞ is a polynomial in z after applying our {0mm} shift, which is due to the fact that the shift enters only via the p 1 p 0 and p 2 p 0 factors in Y and Z. More precisely, the boundary term is of the form B ∞ = 2S−2 n=1 Pn(z) where P n (z) is the n-th order polynomial in z. Furthermore, P n (z) ∼ n in the soft limit. Thus the boundary term does not show up in the soft theorems, but still it contributes to the recursion in Eq. (3.7) via the integral over the circle at infinity. Overall B ∞ ∼ O( 1 ), better than O( 0 ) deduced on general grounds. The boundary term we propose is arguably more compact than the one quoted in [34]. It also makes manifest that the UV behavior of Compton scattering for S > 1 can be O(E 4S /m 4S ). In fact, one can further soften it down to O(E 4S−2 /m 4S−2 ) by subtracting from it a judiciously chosen contact term. See Appendix B for more details.
We turn to another example where the subleading soft factor is no longer moot. We consider Compton scattering for arbitrary S ≥ 1/2 when the dipole a in Eq. (4.23) is switched on. In order to keep this example simple, we restrict to the case where q = 0, however a more general formula can be easily worked out by the same methods. The hard amplitude with a minus helicity photon is given byM The subleading soft factor for a plus helicity photon takes the form Only those little group indices that act non-trivially on the hard factor are displayed.  This describes the leading effect of the dipole in the limit where the momentum p 0 is soft. Away from this limit, the expression makes sense for B ∞ = 0 as long as S ≤ 1, whereas for S ≥ 3/2 one needs to adjust B ∞ so as to cancel the unphysical pole at 3p 1 0] → 0. A possible boundary term can be constructed using the techniques discussed in Appendix B, and is displayed in full glory in Eq. (B.11).
The absence of subleading soft factors for minimal coupling is just a tad less trivial to demonstrate than in the photon case. Starting from the expression in Eq. (4.10) one notes that for our shift D 1 =D 2 = 0, while the remaining one reduces tõ Thus,D 3 is proportional to the total angular momentum operator, and it annihilates 3-point amplitudes. As in the photon case, the IR behavior of gravitational Compton amplitudes is fully determined by the leading soft factor, as long as the graviton is minimally coupled to matter. Given Eq. (4.1) and the zero-order shift in Eq. (3.9), the relevant hard factors are evaluated aŝ Thus, the gravitational Compton scattering amplitudes are reconstructed aŝ The IR part agrees with the expression derived in [22] by matching the residues at the kinematic poles. The amplitude away from the soft limit is obtained using M =M 0,1 . Much as for photons, it is consistent to set B ∞ = 0 for any spin, however for S > 2 the boundary term B ∞ has to be non-zero so as to cancel the unphysical 3p 1 0] pole, see Appendix B.2 for an explicit expression. Let us also work out a simple example beyond the leading soft factor. Consider a massive particle with spin S ≥ 1 and non-minimal quadrupole interactions with the graviton, corresponding to Q = 0 in Eq. (4.21). In this case, the plus-helicity sub-subleading soft factor in Eq. (4.22) for gravitational Compton scattering reduces to where we displayed the little group indices acting non-trivially on the hard factor. The latter for plus helicity photons is generalized aŝ Applying this to derive, for example, the same-helicity Compton amplitude we find Unlike in the minimal case, in the presence of the quadrupole the unphysical 3p 1 0] pole appears also in the same-helicity amplitude, and it is present already for S = 1. If one uses this formula away from the soft limit, one needs to include an appropriate boundary term B ∞ to cancel this pole for any S ≥ 1

Higher-point
To conclude this section, we discuss a couple of examples of 5-point amplitude calculations using the soft theorems of Section 4. The goal is to provide an illustration how these formulas should be employed in the case of higher-point amplitudes. The examples involve a matter particle X with mass m, arbitrary spin S, and the minimal coupling to photons and gravitons. Our first example is the amplitude describing the production of three photons in the XX annihilation: M(1 X 2X3 γ 4 γ 0 γ ). We use the {0mm} ({0mm}) shift on the 012 legs to calculate emission of a soft photon 0 γ with plus (minus) helcity. The leading soft factors for emission of 0 γ are given by .
Putting it all together, The similar approach can be used when photons are replaced with gravitons: M(1 X 2 X 3 h 4 h 0 h ). Below we sketch the derivation, however the final formulas are too lengthy to quote here. The leading soft factor for emission of a plus helicity graviton is Here we set y = λ 3 in Eq. (4.8), since the expression is true for an arbitrary spinor y. A major difference compared to the photon calculation before is that now the subleading soft factors are non-trivial. Indeed, picking again y = λ 3 in Eq. (4.10), the subleading soft factor can be written as The soft expansion of the 5-point amplitude M 0, (1 X 2 X 3 + h 4 + h 0 + h ) is calculated using Eq. (4.7) with the soft factors above acting on M 0,0 (1 X 2 X 3 + h 4 + h ), which in turn can be obtained from Eq. (5.13). One can explicitly verify thatD 4 M 0,0 (1 X 2 X 3 + h 4 + h ) = 0, thus the subleading soft factor is not moot in this case.

Exponential representation
It has been known for a long time that the multipole expansion of classical spinning bodies, such as a Kerr black hole, can be resumed in a simple exponential form [45]. However, the corresponding description in terms of on-shell amplitudes was obtained just recently in [24]. That reference noticed that the on-shell 3-point amplitude describing minimal graviton interactions with matter of any spin can be recast in a compact exponential form, with the angular momentum operator in the exponent. Subsequently, they demonstrated that also the 4-point gravitational Compton scattering amplitude can be recast in an exponential form. This formal rephrasing turns out to be useful to compute the scattering angle of two spinning black holes [24]. As we are going to show, our soft recursion leads to a simple proof of the exponentiation of the Compton amplitudes for arbitrary spin of the matter particle. In the following, we demonstrate it for the photon amplitudes, however the derivation for gravity is totally analogous. We also discuss the exponential representation of the soft theorems for any (integer) helicity of the soft particle.

Photon exponentiation
Let us first rewrite the photon's minimal coupling as This is obtained from the leading term in Eq. (4.23) by replacing the ζ terms with the polarization vectors defined in Eq. (2.5). 11 Eq. (6.1) can be equivalently recast as We move to the calculation of the opposite helicity Compton amplitude using the soft recursion in Eq. (3.7) with the {0mm} shift on the 012 legs. For the soft 3-point amplitude we use the exponentiated form in Eq. (6.2), while for the hard factor we use the minimal form in Eq. (6.1). Then the recursion reduces tô where we already contracted the χ 0l spinors using Eq. (2.7). The crucial identities for this calculation are One should stress that these are valid for any reference spinors ζ andζ; no "gauge-fixing" was 11 While polarization vectors themselves depends on the gauge (or the reference spinor ζ), the 3-point function is gauge-independent because p 1 p 3 = 0 on the 3-body kinematics.
performed along the way. Then Eq. (6.4) can be written aŝ In the exponentials, the numerators do not depend on the reference spinors thanks to the antisymmetry of J µν . On the other hand, in the denominators, (p i − 3 ) do depend on the reference spinors. Our calculation unambiguously dictates that these have to be evaluated at a particular reference spinor,ζ =λ 0 . 12 The two exponentials are in fact equal, so we can pull them in front of the curly bracket. Then, summing the t-and u-poles, the dependence on the shift spinor y cancels out and we obtain the formula for the Compton amplitude in the exponential form [26]: This proves exponentiation of IR terms in Compton scattering for any matter spin. Recall that B ∞ starts at O( 1 ), and thus qualifies as a UV term. Given its explicit form in Eq. (5.5), it does not appear to have a simple exponential representation. Finally, notice that the exponential can be rewritten in a more explicit form aŝ

Soft exponentiation
Ref. [14] showed that the soft theorem for gravitons and gluons could be written in terms of an exponential operator acting on lower point amplitudes. However, the proof is only valid for the minimal-helicity-violating (MHV) sector, as it relies on the vanishing of the boundary terms discussed in [11]. Using the soft recursion we can show that the exponential form is actually valid beyond the MHV sector. In fact, the exponential operator can be extended for amplitudes with massive particles of any spin and with a soft particle of any integer helicity |h| ≤ 2. We begin by rewriting the minimal 3-point amplitude for h ≥ 0 as where g X = √ 2eq for |h| = 1, and g X = m/ M Pl for |h| = 2. The soft expansion in Eq. (4.4) and Eq. (4.24) can be collectively written as: This way, for minimal coupling, we obtain a similar form of the soft theorem as the one in Ref. [14]: where the differential operatorD l is given in Eq. (4.6). This form may be useful to study amplitudes in the Mellin space along the lines of Ref. [46]. Moreover, our results in Eq. (4.28) for the electromagnetic dipole suggest that a generalization (for soft photons) consists in replacing

Conclusions
This paper provides a new toolbox to study theories with particles of arbitrary mass and spin. We introduced a class of complex momentum shifts combining a BCFW-like spinor shift with a soft limit of a massless particle present in the process. The important technical novelty is that our shifts are compatible with the massive spinors introduced by Arkani-Hamed, Huang, and Huang in Ref. [22]. Via the usual road of the Cauchy theorem, these shifts allow one to derive novel recursion relations for scattering amplitudes, even when all but one particles involved are massive. This formal development will find multiple practical applications, we hope. In this paper we applied it to derive a new incarnation of the soft theorems for emission of a soft photon or a graviton. They function, out of the box, with amplitudes expressed in the on-shell spinor formalism, also with massive spinors, thus generalizing the massless results of Ref. [16]. This allows us to systematically explore soft theorems for general theories, and make a direct connection with the multipole expansion of the on-shell 3-point interaction between matter and photons/gravitons. In particular, our soft theorems lead to an elegant proof that gravitational dipole interactions are incompatible with the assumption of locality, unitarity and Poincaré invariance. We also provide a concise expression for the modification of the soft theorems due to the quadrupole (for gravitons) and dipole (for photons) non-minimal interactions with matter.
Generally, soft theorems only capture the leading terms in the limit where the massless particle momentum goes to zero, because they take into account only the soft particle emission from external legs. However, for Compton scattering, emission from external legs accounts for all factorization channels. Therefore, Compton amplitudes derived via soft theorems reflect the complete structure of physical poles required by unitarity. A subtlety appears for spin S ≥ 3/2 (for photons) and S ≥ 5/2 (for gravitons) when these amplitudes develop a spurious pole, which has to be subtracted by adding a judiciously designed UV term. Our approach leads to a new perspective on these UV terms: they arise as boundary terms in our recursion relations. We laid out an algorithm for constructing the UV term for Compton scattering of photons, which leads to a more compact and transparent formula than the one in Ref. [34]. We also derived the UV term for Compton scattering via the electromagnetic dipole. These results may be useful for practical applications, for example in nuclear or atomic physics. Furthermore, our soft recursion leads by the shortest route to recasting the Compton amplitude into a form where the increasing matter spin corresponds to an action of an exponentiated angular momentum operator [24,26]. We were also able to rewrite more general soft recursions in the exponentiated form, extending the results of Ref. [14]. Our soft recursions may help with calculation of multi-leg amplitudes -the topic we only fleetingly touched in this paper. That could in turn boost the amplitude methods of classical calculations of electromagnetic and gravitational radiation along the lines of Ref. [23]. These recursions should also be useful to derive more general soft theorems for Goldstone fermions [47] and gravitinos [48]. Furthermore, it would be interesting to generalize our recursions to multiple soft particles [49] (as recently explored with the CHY formalism [50]). More speculatively, the tools we provide may open up new areas of exploration of massive amplitudes. One direction to mention in this context is the link with the asymptotic symmetries [5,51], another is the double copy structure of the soft theorems for massive particles [52].

A Summary of spinor shifts
In this appendix we list distinct spinor shifts realizing the momentum shift in Eq. (3.1). We label them as {0ĀB} or {0AB}, where A(B) = 0 when the j-th (k-th) particle is massless, and A(B) = m when the j-th (k-th) particle is massive. The label {0ĀB} ({0AB}) corresponds to shifting the holomorphic(antiholomorphic) spinor of the massless particle 0, and the antiholomorphic(holomorphic) spinors of the j-th and k-th particles.

B UV terms in Compton scattering
We have seen in Section 5 that, when spin S of the matter particle is larger than one, the IR part of the Compton amplitude obtained via soft theorems develops an unphysical pole at 3p 1 4] → 0. The same happens for S > 2 in gravitational Compton scattering. The unphysical pole has to be canceled by the boundary term in the recursion relation, which we refer to as the UV term in the discussion below. In this appendix we present an algorithm to construct a correct UV term, alternative to the one in Ref. [34]. For this discussion we find it convenient to introduce the auxiliary objects: A ≡ 13 [24], B ≡ 23 [14], They satisfy the important identities

B.1 Electromagnetic Compton scattering
The opposite helicity Compton scattering amplitude for matter particle X of mass m and spin S can be written as M(1 X 2X3 − γ 4 + γ ) = M IR + M UV . The IR term takes the form [22]: where t = (p 1 +p 3 ) 2 , u = (p 1 +p 4 ) 2 . As discussed in [22], the IR term fully accounts for the kinematic t-and u-channel poles required by unitarity. The UV term can be zero or a pure contact term for S ≤ 1. However, for S > 1 the IR term develops an unphysical pole at 3p 1 4] → 0. Therefore the UV term must be necessarily non-zero to cancel the unphysical pole, without screwing up the physical t-and u-poles.
To understand the general formula for arbitrary spin S, it is useful start with a simpler example of S = 2. The first identity in Eq. (B.2) allows one to reshuffle the IR term, so as to rewrite the residue of the unphysical pole in a more convenient form [34]. Plugging it into Eq. (B.3), the IR term for S = 2 can be rewritten as where the dots stand for terms with no unphysical poles. One can cancel the double pole in 3p 1 4] by including in the M UV the term − This would however tamper with the t-and u-poles, which we want to avoid. The trick is to compensate powers of Y with powers of Z which, by the second identity in Eq. (B.2), do not affect the residue at 3p 1 4] → 0. The crucial observation is that does not have a pole at t → m 2 or u → m 2 . Therefore, a UV term proportional to will affect the 3p 1 4] poles without destroying the correct structure of physical poles provided by the IR term. In our case, the double pole at 3p 1 In exactly the same way we can deal with the single 3p 1 4] pole in Eq. (B.4). All in all, a legal UV term canceling all 3p 1 4] poles in the S = 2 Compton amplitude is given by where C is any pure contact term. This construction guarantees that M IR +M UV has no unphysical poles, and that it has the same physical poles as M IR . The UV term is ultraviolet in two ways. One, it scales as O( 1 ) in the (complex kinematics) limit where t → m 2 , u → m 2 , and s → 0 which is the limit where the photons become soft. This is unlike the IR term which scales as O( −2 ) in the soft limit. Two, M UV grows for E m as O(E 8 /m 8 ), unlike the IR term, which is regular at high energies. This growth leads to a loss of perturbative unitarity at the scale Λ close to the mass m of the particle X, which is what ultimately prevents the possibility of fundamental spin-2 massive particles. In fact, we can further massage Eq. (B.6) so as to improve the UV behavior by one notch, so as to (slightly) delay the onset of strong coupling. Using the first identity in Eq. (B.2) to eliminate Y, and absorbing the resulting X 2 terms into C, we end up with This has a better UV behavior, M UV ∼ O(E 6 /m 6 ). Since the UV terms in Eq. (B.6) and Eq. (B.7) differ only by a contact term, they have exactly the same pole structure, and they both cancel the unphysical pole in M IR . Following exactly the same steps one can construct M UV for arbitrary S > 1: where {Y||Z} = Y(Z) for even (odd) k. Note that X and Z are odd under 1 ↔ 2, while Y is even. Therefore M UV has the same symmetry under 1 ↔ 2 as the IR term: it is even (odd) for integer (half-integer) spin. The absence of physical poles in M UV follows from Y n − Z n = (Y 2 − Z 2 )(Y n−2 + Y n−4 Z 2 + · · · + Z n−2 ) for any even n ≥ 2. This is the most compact form of the UV term, but not the one with the best possible UV behavior. As in the case of S = 2, we can always soften the UV behavior from O(E 4S /m 4S ) down to O(E 4S−2 /m 4S−2 ) by using the first identity in Eq. (B.2) and isolating the resulting O(E 4S /m 4S ) contact term. The consequence is that the maximum cutoff scale of an effective theory describing a spin-S > 1 particle with mass m and electric charge q satisfies Λ max 4π |q|e 1 2S−1 . (B.9) This bound was obtained by other methods in Ref. [53].
For Compton scattering via the electromagnetic dipole the IR amplitude can be read off from Eq. (B.10) The corresponding UV term can be easily constructed by the same techniques: (B.11)

B.2 Gravitational Compton scattering
We turn to the analogue of the Compton process in gravity, where photons are replaced by massless spin-2 gravitons. We assume minimal gravitational interactions of a spin-S massive particle X.
We focus on the opposite helicity Compton amplitude and again separate the UV and IR terms: The IR term takes the form [22]: where we abbreviated and A, B, X , Y are defined in Eq. (B.1). For S > 2 the IR term in Eq. (B.12) develops a pole at 3p 1 4] → 0, therefore M UV has to be non-zero for S > 2 so as to cancel the unphysical pole. Below we construct a befitting UV term. We will first find a set of functions U k such that U k + I k is regular at 3p 1 4] → 0 and has the same kinematic poles as I k . Given such a set, a legal UV term for S > 2 is simply (B.14) We will construct U k recursively. In this endeavour the following identity is instrumental: where once again we introduced some auxiliary notation: (B.17) We can rearrange c k as We can handle d k in a similar way (B.20) The solution is d k = 4m 2 (k−2)/2 l=1 I k−2l − X c k−2l X 2(l−1) + X k−2 d 2 , even k, Plugging the solution back into the formula for I k we obtain where the dots stand for d 2,3 and c 2 terms, which do not have poles at 3p 1 4]. We are ready to write down a recursive equation for U k : where {Z||Y } = Z(Y) for even (odd) k, and {2||3} = 2(3) for even (odd) l. An implicit convention above is that, if the upper limit of a summation is smaller than the lower limit, then the whole sum is void. The recursive equation (B.23) supplemented by the boundary condition U 0 = U 1 = 0 defines U k for arbitrary k. Via Eq. (B.14), this allows one to construct an appropriate UV term for any spin S of the matter particle. Let us see how it works in some simple examples. For k = 2 we get