All-But-One-Plus Graviton Current

• The all-but-one-plus graviton current is defined as an off-shell Berends–Giele current with one negative-helicity graviton among otherwise positive-helicity gravitons, enabling the derivation of tree-level MHV amplitudes.
• It employs a recursive spanning tree method that gathers contributions from subgraphs, and utilizes celestial current algebras to articulate positive-helicity soft limits and null state dynamics.
• The discussion extends to EFT contexts where maximal biform symmetries and anomaly protection enforce a massless spin-2 mode, linking dual gravity formulations with emergent linearized gravity.

Searching arXiv for recent and directly relevant papers on the all-but-one-plus graviton current, including Berends–Giele, celestial MHV, and biform-symmetry contexts. arXiv search query: all-but-one-plus graviton current Berends-Giele gravity MHV The all-but-one-plus graviton current denotes, in the literature considered here, a graviton current associated with configurations in which one graviton is distinguished from an otherwise positive-helicity set, and also, in a separate EFT usage, a maximal mixed-symmetry current of type $(2|2)$ relevant to linearized gravity. In the Berends–Giele formulation, it is an off-shell current with one negative-helicity graviton and positive-helicity gravitons $2,\ldots,n$, whose scalar part obeys a recursion over partitions of the positive-helicity set and whose explicit solution is a sum over spanning trees; amputating the final propagator recovers the tree-level gravity MHV formula (Hasuwannakit et al., 18 Jul 2025). In celestial gravity, the same positive-helicity-dominated sector is closed under positive-helicity soft limits, and leading plus subleading soft graviton symmetries already determine the tree-level MHV amplitudes, while the subsubleading theorem yields descendant and null-state relations rather than new independent constraints (Banerjee et al., 2021). In the mixed-symmetry EFT setting, the relevant object is the $(2|2)$-biform current $J_{\mu_1\mu_2|\nu_1\nu_2}$, whose mixed anomaly with its magnetic dual fixes the nonlocal current-current correlator, forces a massless spin-2 pole in the Källén–Lehmann decomposition, and organizes the infrared EFT of linearized gravity (Hinterbichler et al., 2022).

1. Helicity configuration and scope of the term

In the helicity-amplitude setting, the all-but-one-plus graviton current is the off-shell Berends–Giele current with one negative-helicity graviton labeled $1$ and positive-helicity gravitons labeled by a set $\mathcal K=\{2,\dots,n\}$. The current is written in a scalarized form: the tensor structure is fixed by helicity and reference spinors, and the nontrivial content is carried by a scalar factor $J(1|\mathcal K)$. The corresponding all-plus current is denoted $J(\mathcal K)$. The ansatz works after choosing the positive-helicity reference spinor equal to the negative-helicity momentum spinor, $q=1$, because all contributing diagrams preserve the same external tensor structure (Hasuwannakit et al., 18 Jul 2025).

In the MHV amplitude context this current is tied to the sector with exactly two negative-helicity gravitons and the rest positive. That sector is special because positive-helicity soft limits map an MHV amplitude to a lower-point MHV amplitude, whereas negative-helicity soft limits would produce an amplitude with only one negative-helicity graviton and the rest positive, and such amplitudes vanish at tree level in Einstein gravity. The phrase “all-but-one-plus graviton current” is therefore naturally associated with a positive-helicity-dominated subsector in which a small number of negative-helicity insertions are treated as distinguished legs (Banerjee et al., 2021).

A distinct but technically important use of the terminology appears in linearized gravity. There the relevant current is the $(2|2)$-biform current $2,\ldots,n$0, the $2,\ldots,n$1 member of a maximal $2,\ldots,n$2 family. In the deep infrared it is essentially the linearized Riemann/Weyl tensor, up to traces and improvements, and its magnetic dual is obtained by double Hodge dualization (Hinterbichler et al., 2022).

2. Berends–Giele formulation in gravity

The gravitational Berends–Giele construction parallels the Yang–Mills one but with a more intricate combinatorics. For the all-plus graviton current the tensor structure is

$2,\ldots,n$3

and for one negative-helicity graviton $2,\ldots,n$4 and positive-helicity gravitons $2,\ldots,n$5,

$2,\ldots,n$6

The scalar current satisfies the recursion

$2,\ldots,n$7

with $2,\ldots,n$8 and $2,\ldots,n$9 the reference spinor of the negative-helicity graviton (Hasuwannakit et al., 18 Jul 2025).

The derivation uses the chiral Einstein–Cartan Feynman rules. For this helicity configuration, only the $(2|2)$0 cubic vertex contributes; the other cubic and quartic contributions vanish because they force contractions of too many $(2|2)$1’s. This yields the characteristic “gravity $(2|2)$2 YM$(2|2)$3” pattern: the kinematic factor $(2|2)$4 is squared in gravity, and the sum runs over all partitions of the set $(2|2)$5, not just ordered splits (Hasuwannakit et al., 18 Jul 2025).

The resulting current is more complicated than the Yang–Mills single-minus current, but the complication is combinatorial rather than structural. The all-plus current already carries the essential graph-theoretic data, while the all-but-one-plus current is a deformation of it by factors that are themselves built recursively from lower substructures.

3. Spanning-tree solution and MHV extraction

The all-plus gravity current is a sum over spanning trees of the complete graph on $(2|2)$6,

$(2|2)$7

where each tree contribution carries valency-dependent factors $(2|2)$8 and edge factors $(2|2)$9. The all-but-one-plus current admits a tree-by-tree solution of the form

$J_{\mu_1\mu_2|\nu_1\nu_2}$0

so each spanning tree contributes its all-plus weight multiplied by a correction factor $J_{\mu_1\mu_2|\nu_1\nu_2}$1 (Hasuwannakit et al., 18 Jul 2025).

The factor $J_{\mu_1\mu_2|\nu_1\nu_2}$2 is itself a sum over subgraphs,

$J_{\mu_1\mu_2|\nu_1\nu_2}$3

and the explicit formula for $J_{\mu_1\mu_2|\nu_1\nu_2}$4 is organized by the ends $J_{\mu_1\mu_2|\nu_1\nu_2}$5, the stem $J_{\mu_1\mu_2|\nu_1\nu_2}$6, and the higher-valency vertices $J_{\mu_1\mu_2|\nu_1\nu_2}$7. Each tree contribution has $J_{\mu_1\mu_2|\nu_1\nu_2}$8 numerator factors, $J_{\mu_1\mu_2|\nu_1\nu_2}$9 propagator-like squared-momentum denominators, and a sum over all $1$0 permutations of the leaf-addition order. Low-point examples already show the pattern: $1$1 and

$1$2

Only a small subset of these terms contributes to the on-shell amplitude (Hasuwannakit et al., 18 Jul 2025).

The amplitude is obtained by putting the off-shell leg on shell and contracting with the negative-helicity polarization tensor,

$1$3

After using momentum conservation, the crucial result is that only the highest-order term in $1$4, namely $1$5, has the pole that cancels the amputating factor $1$6. All lower-subgraph terms are regular and vanish after the on-shell extraction. The final tree-sum formula is

$1$7

which recovers the standard gravity MHV formula directly from the gravitational Feynman rules (Hasuwannakit et al., 18 Jul 2025).

4. Celestial current algebra and the MHV sector

In celestial gravity, the same positive-helicity-dominated sector is characterized by tree-level MHV graviton scattering amplitudes in Einstein gravity, with exactly two negative-helicity gravitons and the rest positive helicity. The sector is autonomous under positive-helicity soft limits, while negative-helicity soft limits decouple because the resulting single-minus amplitudes vanish at tree level. This closure is the reason current-algebra methods work particularly cleanly in the MHV sector (Banerjee et al., 2021).

The relevant soft operators are the conformally soft positive-helicity graviton operators. The leading soft operator is

$1$8

and its modes generate commuting supertranslations. The subleading soft operator is

$1$9

and the associated currents $\mathcal K=\{2,\dots,n\}$0, $\mathcal K=\{2,\dots,n\}$1, satisfy the $\mathcal K=\{2,\dots,n\}$2 current algebra

$\mathcal K=\{2,\dots,n\}$3

A key result is that supertranslations plus the $\mathcal K=\{2,\dots,n\}$4 current algebra are already enough to reconstruct the full tree-level MHV amplitude (Banerjee et al., 2021).

The decisive ingredients are the null states

$\mathcal K=\{2,\dots,n\}$5

and

$\mathcal K=\{2,\dots,n\}$6

Their decoupling yields linear differential equations for the MHV amplitudes, one for each positive-helicity leg. This is the technical reason the MHV sector can be fixed without invoking additional independent symmetry data (Banerjee et al., 2021).

The subsubleading soft graviton theorem enters through

$\mathcal K=\{2,\dots,n\}$7

but it does not add new independent constraints. Instead, it generates descendant relations such as

$\mathcal K=\{2,\dots,n\}$8

and

$\mathcal K=\{2,\dots,n\}$9

which follow algebraically from the existing null-state structure. The celestial OPE of positive-helicity gravitons is invariant under the global subsubleading symmetry up to null states. A recurrent misconception is therefore excluded: within the tree-level MHV sector, the subsubleading theorem is a consistency condition rather than an independent amplitude-determining principle (Banerjee et al., 2021).

5. Maximal biform current and anomaly protection of a massless spin-2 mode

A different but structurally related notion of all-but-one-plus graviton current arises in EFTs with maximal biform symmetries. The electric current is a $J(1|\mathcal K)$0-biform transforming in the irreducible $J(1|\mathcal K)$1 representation with two equal columns of length $J(1|\mathcal K)$2. For $J(1|\mathcal K)$3, the current is the $J(1|\mathcal K)$4-biform

$J(1|\mathcal K)$5

antisymmetric within each column and subject to the Young symmetry constraint that antisymmetrizing all indices in the first column with any one from the second vanishes. Its magnetic dual is defined by double Hodge dualization,

$J(1|\mathcal K)$6

Because the columns have equal length, dualizing both columns preserves the class of maximal biforms (Hinterbichler et al., 2022).

The defining feature of the phase is a mixed ’t Hooft anomaly between the electric and magnetic conservation laws. At separated points the current obeys

$J(1|\mathcal K)$7

while in the preferred presentation the anomalous magnetic conservation law reads

$J(1|\mathcal K)$8

with $J(1|\mathcal K)$9 the background-field strength built from the source $J(\mathcal K)$0. For gravity, the anomaly is between the $J(\mathcal K)$1 electric symmetry and the $J(\mathcal K)$2 magnetic symmetry, and it cannot be removed by field redefinitions; local contact terms can only move it between the various conservation conditions (Hinterbichler et al., 2022).

The anomaly fixes the nonlocal part of the electric–magnetic current correlator up to contact terms. In the gravity case the correlator is proportional to the spin-2 projector structure, and its Källén–Lehmann decomposition forces a delta-function at $J(\mathcal K)$3 in the spin-2 compatible channel: $J(\mathcal K)$4 The spectrum therefore necessarily contains a massless mode. Since local counterterms change only contact terms and do not affect the spectral density, the $J(\mathcal K)$5 at $J(\mathcal K)$6 is robust. In this sense the masslessness of the graviton is protected by the anomalous biform symmetries (Hinterbichler et al., 2022).

6. Emergent linearized gravity from biform symmetry

The same symmetry data yields an EFT whose most relevant term resembles the linear Einstein action. The local fields are a symmetric tensor $J(\mathcal K)$7, interpreted as the graviton potential, and an auxiliary $J(\mathcal K)$8-biform field $J(\mathcal K)$9. The invariant curvatures are

$q=1$0

and

$q=1$1

They are invariant under linearized diffeomorphisms,

$q=1$2

The current $q=1$3 is the mixed-symmetry curvature built from the emergent connection-like field (Hinterbichler et al., 2022).

The action is

$q=1$4

After integrating out $q=1$5, this is exactly equivalent to the linearized Einstein/Fierz–Pauli action

$q=1$6

The equations of motion are the partial flatness condition

$q=1$7

which is the biform analogue of the Einstein equation (Hinterbichler et al., 2022).

This formulation assigns a precise status to linearized diffeomorphism invariance. It is not imposed as a fundamental gauge redundancy; rather, it emerges from the way the biform symmetry and its anomaly are realized in a local EFT. The same work also states the limitations of the construction: these theories are not ultraviolet-complete, the relevant symmetries can be viewed as emergent, and the formalism does not include the nonlinearities necessary to make the theory fully diffeomorphism invariant. Consequently, there is no contradiction with the expectation that quantum gravity cannot have global symmetries (Hinterbichler et al., 2022).

7. Dual-gravity and generalized charge perspectives

A further extension of the current-like viewpoint appears in the non-linear realization of

$q=1$8

where $q=1$9 is the vector representation. The generalized translation generators begin with

$(2|2)$0

and the corresponding generalized coordinates begin with

$(2|2)$1

At low levels the field content includes the graviton $(2|2)$2, the dual graviton $(2|2)$3, and higher mixed-symmetry fields. Truncating to the ordinary coordinates $(2|2)$4 and the lowest-level graviton field yields ordinary Einstein gravity; the invariant equation $(2|2)$5 reduces to $(2|2)$6 in vacuum (Glennon et al., 2020).

The formalism also provides a gravity–dual gravity duality relation. The spin connection

$(2|2)$7

is incorporated into a modified object with $(2|2)$8-derivative terms so that it transforms properly under the local $(2|2)$9 symmetry. The resulting duality equation $2,\ldots,n$00 links the gravity Cartan form and the dual graviton Cartan form by a first-order relation. The dual graviton also obeys a second-order symmetric equation of motion derived from the variation of the gravity equation under the local level-one symmetry (Glennon et al., 2020).

This paper does not use the exact phrase “all-but-one-plus graviton current,” but it makes the generalized-current interpretation explicit. The lowest component $2,\ldots,n$01 is the momentum current tied to ordinary spacetime translations and hence to the graviton. The next component $2,\ldots,n$02 is associated with the dual graviton and with the Taub–NUT charge. Higher $2,\ldots,n$03-type generators correspond to further generalized brane charges. A bilinear invariant,

$2,\ldots,n$04

reinforces the interpretation of the $2,\ldots,n$05 tower as a hierarchy of generalized conserved quantities. In that sense, the graviton and dual graviton occupy the first two slots in a broader current/charge multiplet (Glennon et al., 2020).