Double Soft Graviton Theorem

• Double Soft Graviton Theorem is a fundamental result that characterizes gravitational amplitudes when two external gravitons become soft, leading to factorized products of single soft factors.
• It reveals distinct contributions at leading and subleading orders, with contact terms and nested Ward identities reflecting the underlying BMS symmetry structure.
• The theorem is validated through diverse methods including diagrammatics, the CHY representation, and Wilson line formalism, offering insights into infrared dynamics and gravitational memory effects.

The double soft graviton theorem concerns the universal behavior of gravitational scattering amplitudes in the simultaneous limit where two external gravitons become soft. This universality emerges both at leading order, where the amplitude factorizes into products of single soft factors, and at subleading order, where novel structures such as contact terms and nested asymptotic symmetry Ward identities become manifest. The theorem has implications for infrared dynamics, the algebraic structure of the gravitational S-matrix, and asymptotic symmetries, particularly the Bondi-Metzner-Sachs (BMS) algebra and its extensions.

1. Precise Statement and Factorization Formulae

Let $$M_{n+2}(q_1, \epsilon_1; q_2, \epsilon_2; \{k_i, h_i\})$$ denote the tree-level amplitude for $n$ hard particles of momenta $k_i$ and helicities $h_i$, plus two gravitons with momenta $q_1, q_2 \to 0$ and polarization tensors $\epsilon_1, \epsilon_2$. The amplitude expands as

$$M_{n+2}(q_1, \epsilon_1; q_2, \epsilon_2; \{k_i\}) = [S^{(0)}(q_1, q_2) + S^{(1)}(q_1, q_2) + O(E_{q_1}, E_{q_2})] M_n(\{k_i\}),$$

with $E_{q_a} = |q_a|$ the energy of graviton $a$. The soft expansion contains:

• Leading double soft factor $S^{(0)}(q_1, q_2)$ scaling as $1/(E_{q_1} E_{q_2})$.
• Subleading terms $S^{(1)}(q_1, q_2)$, scaling as $1/E_{q_1}$ or $1/E_{q_2}$.

Explicitly, for the leading factor,

$$S^{(0)}(q_1, q_2) = \sum_{i=1}^n S^{(0)}(q_1; k_i) \sum_{j=1}^n S^{(0)}(q_2; k_j)$$

with single-soft factor

$$S^{(0)}(q; k_i) = \frac{\epsilon^{\mu\nu}(q) k_{i\mu} k_{i\nu}}{q \cdot k_i}.$$

At subleading order, the consecutive double soft theorem takes the form

$$S^{(1)}_{\mathrm{cons}}(q_1, q_2) = \sum_{i=1}^n S^{(0)}(q_1; k_i) \sum_{j=1}^n S^{(1)}(q_2; k_j) + \sum_{i=1}^n \left[ \text{contact}_{i}(q_1, q_2) \right],$$

where $S^{(1)}(q; k_i)$ is the subleading single-soft factor,

$$S^{(1)}(q; k_i) = \frac{\epsilon^{\mu\nu}(q) q_\rho J_i^\rho{}_\mu k_{i\nu}}{q \cdot k_i},$$

and $J_i^{\rho\mu}$ acts with total angular momentum on hard leg $i$. The "contact" term arises when both soft gravitons couple to the same hard particle.

2. Simultaneous vs Consecutive Double Soft Limits

Double-soft limits are defined in two principal ways:

• Consecutive soft limit (CSL): One soft momentum is taken to zero, followed by the other. This produces factorized expressions involving products of single-soft theorems, with ordering ambiguities at the subleading level.
• Simultaneous soft limit (DSL): Both momenta $q_1, q_2$ scale to zero together, often parameterized by $q_r = \lambda k_r$ with $\lambda \rightarrow 0$. This yields manifestly symmetric factorization up to contact terms.

In both approaches, the leading term $S^{(0)}$ always factorizes as a product of single soft factors, but at subleading order, additional terms—especially contact terms for mixed helicities and simultaneous limits—arise. For instance, the DSL in spinor-helicity notation for two same-helicity gravitons lacks contact terms, whereas mixed-helicity double soft limits include explicit contact contributions (Klose et al., 2015).

3. Diagrammatic and CHY Derivations

Feynman diagrammatic analysis demonstrates that the double-pole diagrams (where two soft gravitons attach to the same external line) generate the product of single soft factors. Subleading contributions come from diagrams in which soft insertions interact closely (contact terms) or from next-to-soft insertions on external lines. These results have been confirmed via the Cachazo-He-Yuan (CHY) representation for gravity tree amplitudes, which treats punctures on the Riemann sphere and splits solutions into nondegenerate (factorized) and degenerate (contact) sectors. The residues at nondegenerate punctures reproduce the factorized product $S^{(0)}(q_1) S^{(0)}(q_2)$, while degenerate solutions yield the irreducible contact terms required by the full theorem (Chakrabarti et al., 2017, Saha, 2017). Inclusion of the overall sign factor $(-1)^n$ in the CHY integrand matches convention and resolves discrepancy in the soft limit signs.

4. Nested BMS Ward Identities and Asymptotic Symmetries

The BMS algebra, governing asymptotic symmetries at null infinity, decomposes into supertranslation (labelled by $f(z, \bar{z})$) and superrotation (labelled by $V^A(z, \bar{z}$) charges. Single-soft graviton theorems have been established as equivalent to Ward identities associated with these symmetries. For double-soft gravitons, nested Ward identities involving two BMS charges yield factorization theorems:

• The nested commutator $[Q_f, [Q_g, S]]$ for two supertranslations leads to the product $S^{(0)}(q_1) S^{(0)}(q_2)$.
• The mixed commutator $[Q_f, [Q_V, S]]$ produces a leading × subleading factorization, matching consecutive double-soft results (H et al., 2018).

These Ward identities must be evaluated on out-states defined around non-Fock vacua, reflecting the action of BMS charges—the soft supertranslation does not annihilate the Fock vacuum but produces a Goldstone (zero-frequency) graviton mode, yielding a supertranslated vacuum $|f\rangle$.

5. Subleading Structures and Contact Terms

Subleading order introduces universal contact terms when both soft gravitons couple to a single hard particle. Explicit forms arise, for example,

$$M_{n+2}^{(-1)} = \sum_{i=1}^n \left\{ S_i^{(0)}(q_1, \epsilon^{(1)}) S_i^{(1)}(q_2, \epsilon^{(2)}) + S_i^{(0)}(q_2, \epsilon^{(2)}) S_i^{(1)}(q_1, \epsilon^{(1)}) \right\} M_n + \sum_{i=1}^n \frac{1}{p_i \cdot (q_1 + q_2)} M_{gg}(p_i; \epsilon^{(1)}, q_1; \epsilon^{(2)}, q_2) M_n,$$

with $M_{gg}$ accounting for two-graviton emissions from the same hard leg and its detailed tensor structure appears in (Bhatkar et al., 2018) and (Chakrabarti et al., 2017).

In spinor helicity representation for four dimensions, the simultaneous double-soft limit for mixed helicities includes

$$C_{\mathrm{mixed}} = \frac{1}{q_{12}^2} \sum_{a=3}^{n+2} \frac{[1a]^3 [2a]^3}{[1a][2a]} \frac{1}{2 k_a \cdot q_{12}},$$

with $q_{12} = q_1 + q_2$ (Klose et al., 2015).

6. Generalized Wilson Line Formalism and Infrared Dynamics

The generalized Wilson line (GWL) formalism provides an alternative perspective for double soft graviton insertions. In this approach, amplitudes are expressed as vacuum matrix elements of Wilson line operators,

$$\tilde{S}_n = \langle 0 | \prod_{i=1}^n \Phi_i(0, \infty) | 0 \rangle,$$

with $\Phi_i$ generating eikonal factors along each hard line. Next-to-eikonal corrections yield generalized Wilson lines encapsulating subleading and double soft effects. This translates into an exponentiated dressing operator acting on the hard amplitude,

$$\mathcal{A} = \exp \left\{ \int_k h S^{(1)} + \frac{1}{2} \int_{k, l} h h S^{(2)} + \cdots \right\} \mathcal{H}_n,$$

where $S^{(2)}$ contains the double soft kernels. The operator structure describes squeezed coherent states of soft gravitons, with oscillator form

$$\tilde{S}_n = \exp \left\{ \kappa \int_{\vec{k}} \alpha_i^*(k) a_i(k) + \frac{\kappa^2}{2} \sum_{i, j} \int_{\vec{k}, \vec{l}} \beta^*_{ij}(k, l) a_i(k) a_j(l) \right\},$$

providing a compact resummation of double soft effects (Fernandes et al., 7 Nov 2025).

Applications include:

• Controlling infrared divergences in post-Minkowskian (PM) binary scattering (Fernandes et al., 7 Nov 2025).
• Generating nonlinear gravitational memory effects via exponentiation.
• Building effective field theories of massive worldlines interacting with soft radiation.

7. Algebraic Structure, Symmetries, and Implications

The double soft graviton theorems generalize the algebraic structure of asymptotic symmetries in gravity. Leading soft terms map to supertranslation generators in BMS, while subleading terms are linked to superrotation generators. Nested and mixed commutators encode deeper algebraic structures, possibly revealing non-Abelian extensions and hidden symmetries (H et al., 2018). For consecutive soft limits, the ordering reflects noncommuting generators; for simultaneous soft limits, the contact terms probe central extensions and nonlinearities.

The emergence of Goldstone modes in non-Fock asymptotic vacua, the direct correspondence between double soft limits and nested BMS Ward identities, and the universality across different derivations (BCFW recursion, CHY contour integration, diagrammatics, and Wilson lines) collectively confirm that double soft graviton insertions are governed by the infinite-dimensional BMS algebra acting on the gravitational S-matrix and its nontrivial vacuum structure (H et al., 2018, Klose et al., 2015, Chakrabarti et al., 2017, Fernandes et al., 7 Nov 2025).

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Summary Table: Double Soft Limit Structures

Order	Term Structure	Universal Property
Leading	$S^{(0)}(q_1) S^{(0)}(q_2)$	Product of single soft
Subleading	$$S^{(0)}(q_1) S^{(1)}(q_2) + S^{(1)}(q_1) S^{(0)}(q_2)$$ + contact	Ward identity structure
Contact term	Local emission of two gravitons from same leg	Required by symmetry

The double soft graviton theorem fully characterizes the infrared structure of quantum gravity amplitudes in the soft sector and intertwines with symmetry constraints and memory effects, establishing a foundation for further exploration in classical and quantum gravitational theories.