Gravity MHV Formula for Graviton Amplitudes

• Gravity MHV Formula is a set of compact expressions for computing four-dimensional graviton amplitudes in the maximal-helicity-violating sector.
• It leverages diverse representations such as Hodges’ determinant, spanning-tree sums, and CHY localization to exploit underlying matrix-tree identities.
• The formulation extends to twistor, celestial, and loop-level domains while consistently isolating the universal helicity factor.

The gravity MHV formula is the family of compact closed expressions for four-dimensional graviton amplitudes in the maximal-helicity-violating sector, namely amplitudes with two gravitons of one helicity and the remaining $n-2$ of the opposite helicity. In standard spinor-helicity conventions this is usually the two-negative-helicity sector, and its canonical tree-level form is Hodges’ reduced determinant; equivalent formulations include BGK and NSVW spanning-tree sums, CHY localizations on special solutions of the scattering equations, Berends–Giele current constructions, and twistor-space rational-curve formulae. The subject is also tied to matrix-tree theorems, hidden auxiliary-space symmetries, celestial Ward identities, and several controlled generalizations away from flat-space tree level (Hodges, 2012, Du et al., 2016).

1. Standard tree-level form

In spinor-helicity variables, with $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$, the basic brackets are

$$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$

For the MHV configuration with negative-helicity legs $1$ and $2$, one standard stripped form is

$M_n=\langle12\rangle^8\,\det{}'\Phi,$

and the full amplitude can be written as

$$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$

in the normalization $\kappa=2$, or as

$$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$

when the conventional gravitational coupling is restored. Several derivations suppress overall factors or state that overall normalizations are neglected, so the kinematic determinant formula is the invariant core across conventions (Guevara et al., 5 Jun 2025, Hasuwannakit et al., 18 Jul 2025, Du et al., 2016).

The Hodges matrix $\Phi$ is an $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$0 symmetric matrix with off-diagonal entries

$p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$1

and diagonal entries fixed by reference spinors $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$2,

$p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$3

Equivalently, one may use

$p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$4

with $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$5 the minor obtained by deleting rows and columns $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$6. With momentum conservation imposed, $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$7 is independent of $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$8 and of the deleted rows and columns, reflecting the corank-three structure of $p_i^{A\dot A}=\lambda_i^A\tilde\lambda_i^{\dot A}$9 (Guevara et al., 5 Jun 2025, Hodges, 2012).

An equivalent notation often used in CHY-based treatments is

$$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$0

where $$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$1 is the reduced amplitude built from the Hodges matrix. This form isolates the helicity weight $$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$2 from the permutation-symmetric determinant structure (Du et al., 2016).

2. Tree sums, half-soft functions, and equivalent representations

The determinant formula is equivalent to a spanning-tree formula. In one NSVW form, with negative-helicity legs $$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$3 and $$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$4, the MHV amplitude is

$$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$5

where the sum runs over spanning trees on the $$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$6 positive-helicity legs. This is the NSVW/BDPR tree formula. By the matrix-tree theorem, it is equivalent to the reduced determinant form (Cao et al., 2021).

A closely related half-soft form obtained from Berends–Giele currents reorganizes the same amplitude as

$$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$7

In pure-connection variables, the same object is written as

$$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$8

with $$\langle ij\rangle=\epsilon_{AB}\lambda_i^A\lambda_j^B, \qquad [ij]=\epsilon_{\dot A\dot B}\tilde\lambda_i^{\dot A}\tilde\lambda_j^{\dot B}.$$9 the half-soft function defined as a weighted sum over spanning trees on the set $1$0 (Hasuwannakit et al., 18 Jul 2025, Delfino et al., 2014).

A recurrent misconception is that these are different amplitudes. They are instead different presentations of the same tree-level MHV object: the reduced determinant, the half-soft function, and the tree sum are related by Laplacian identities and the matrix-tree theorem. In this sense, the gravity MHV formula is less a single expression than an equivalence class of determinant and graph representations (Cao et al., 2021, Delfino et al., 2014).

Another source of notational ambiguity is helicity convention. In Penrose/twistor conventions adapted to anti-self-dual geometry, amplitudes supported on a degree-1 twistor curve are “mostly minus,” so the gravitational MHV amplitudes are written with two positive-helicity gravitons and $1$1 negative-helicity ones; the standard two-negative-helicity form is recovered by parity, i.e. by interchanging angle and square brackets (0808.3907).

3. CHY localization and four-dimensional scattering equations

The CHY representation gives a direct route from scattering equations to the gravity MHV formula. The tree amplitude is written as

$1$2

with $1$3 the Möbius-invariant scattering-equation measure, $1$4 the $1$5 CHY matrix, and $1$6 the reduced Jacobian determinant (Du et al., 2016).

In four dimensions, the decisive statement is that only one special rational solution of the scattering equations supports the MHV amplitude. After fixing $1$7 and $1$8, Weinzierl’s MHV solution is

$1$9

with $2$0 parametrizing the residual $2$1 freedom. At this solution the three essential ingredients factorize as

$2$2

$2$3

$2$4

Combining these yields

$2$5

which reproduces the Hodges formula up to the overall $2$6 normalization associated with the Pfaffian convention for polarizations. The auxiliary factors $2$7 and $2$8 cancel in the final amplitude, leaving a manifestly $2$9-invariant result (Du et al., 2016).

The conjugate special rational solution,

$M_n=\langle12\rangle^8\,\det{}'\Phi,$0

does not support MHV amplitudes; it supports anti-MHV amplitudes instead. Moreover, subsequent four-dimensional analysis characterizes contributing solutions by the rank of discriminant matrices $M_n=\langle12\rangle^8\,\det{}'\Phi,$1: only solutions with $M_n=\langle12\rangle^8\,\det{}'\Phi,$2 contribute to $M_n=\langle12\rangle^8\,\det{}'\Phi,$3MHV amplitudes, so MHV selects precisely the special MHV solution (Du et al., 2016, Du et al., 2016).

4. Berends–Giele currents, pure-connection recursions, and heavenly-equation proofs

The Berends–Giele program gives a direct Feynman-diagram derivation of the gravity MHV formula. For positive-helicity external set $M_n=\langle12\rangle^8\,\det{}'\Phi,$4, the all-plus graviton current has the scalar form

$M_n=\langle12\rangle^8\,\det{}'\Phi,$5

a sum over spanning trees of the complete graph on $M_n=\langle12\rangle^8\,\det{}'\Phi,$6. The all-but-one-plus current $M_n=\langle12\rangle^8\,\det{}'\Phi,$7 is solved by the same tree sum multiplied by a factor $M_n=\langle12\rangle^8\,\det{}'\Phi,$8 that itself is a sum over subtrees. When the off-shell leg is amputated and contracted with the final negative-helicity polarization, only the top-level subtree contribution survives. The result is the BGK/half-soft MHV tree formula, and via the matrix-tree theorem it coincides with Hodges’ determinant (Hasuwannakit et al., 18 Jul 2025).

In the pure-connection formalism, the relevant cubic vertex simplifies enough that an all-minus current can be computed recursively. Its scalar coefficient $M_n=\langle12\rangle^8\,\det{}'\Phi,$9 is a sum over Cayley trees,

$$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$0

This immediately suggests the MHV amplitude after the symmetric replacement $$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$1, yielding again the half-soft/BGK formula

$$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$2

The same tree sum admits a Hodges-like determinant representation via a weighted Laplacian (Delfino et al., 2014).

A conceptually different proof starts from Plebański’s second heavenly equation. There the self-dual background generated by arbitrarily many positive-helicity gravitons is expanded as a sum over “marked tree graphs”,

$$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$3

with each edge carrying the operator $$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$4. Two negative-helicity gravitons are then inserted as anti-self-dual linear perturbations on that self-dual background. Evaluating the on-shell gravitational action shows that the MHV amplitude comes entirely from the boundary term of the self-dual Plebański action plus boundary term, and the resulting expression is precisely the NSVW tree formula. This gives a first-principles derivation that does not use BCFW recursion or twistor theory (Miller, 2024).

5. Twistor, cosmological-constant, auxiliary-space, and celestial formulations

Twistor theory supplies several of the most geometric versions of the gravity MHV formula. One derivation starts from a background-field calculation on an anti-self-dual spacetime, rewrites the scattering problem in twistor variables, and obtains the symmetric BGK formula directly. The same analysis yields a twistor action for the MHV diagram formalism of gravity and extends to $$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$5 and $$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$6 supergravity (0808.3907).

For nonzero cosmological constant, the conformal-gravity twistor action in axial gauge leads to an Einstein-gravity MHV formula on degree-1 curves,

$$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$7

The flat-space limit $$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$8 is smooth and reproduces Hodges’ formula nontrivially (Adamo et al., 2013). A related rational-curve formula for gauged $$\mathcal M_n^{\mathrm{MHV}} = i(2\pi)^4\,\delta^{(4)}\!\Big(\sum_{i=1}^n \lambda_i\tilde\lambda_i\Big)\, \langle12\rangle^8\,\det{}'\Phi$$9 supergravity in $\kappa=2$0 specializes in the MHV sector to

$\kappa=2$1

and reduces in the flat-space limit to the usual MHV supergravity amplitude (Adamo, 2015).

The determinant itself admits a hidden $\kappa=2$2 reformulation in an auxiliary $\kappa=2$3-space whose indices are the particle labels. In that framework the amplitude is written as

$\kappa=2$4

or, with arbitrary reference 3-planes $\kappa=2$5 and $\kappa=2$6,

$\kappa=2$7

Different gauge fixings reproduce the $\kappa=2$8-symmetric Hodges form, the $\kappa=2$9-symmetric tree form, or new fully $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$0-symmetric expressions (Cheung, 2012).

A more recent celestial reformulation derives Hodges’ determinant from an $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$1 Ward identity. The resulting one-leg recursion,

$$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$2

with seed

$$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$3

is unrelated to BCFW and is proved by the matrix-tree theorem to reproduce Hodges’ formula. In parallel, celestial current algebra methods show that tree-level MHV graviton amplitudes satisfy linear first-order PDE systems associated with positive-helicity null states; these have been checked against Hodges’ determinant, with one system verified generally and another checked up to six points (Guevara et al., 5 Jun 2025, Banerjee et al., 2020).

6. Extensions beyond the flat-space tree formula

Away from flat-space tree level, the gravity MHV formula persists in modified forms rather than as a single universal reduced determinant. On self-dual gravitational plane-wave backgrounds, the all-multiplicity MHV amplitude acquires dressed dotted spinors, a single residual light-front integral, and explicit graviton-tail terms: $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$4 The trivial-background limit removes the tail terms and restores the Hodges determinant (Adamo et al., 2020).

At one loop, maximally supersymmetric supergravity admits an integrand-level MHV formula obtained by double copying BCJ-satisfying $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$5 SYM numerators: $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$6 where the stripped numerators $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$7 are generated by a deletion operator $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$8 acting on self-dual $$\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+) = i\Big(\frac{\kappa}{2}\Big)^{n-2}\langle12\rangle^8\,\det{}'\Phi$$9-chains. This realizes the MHV gravity integrand as a square of Yang–Mills-like building blocks (He et al., 2015).

For $\Phi$0 supergravity, the one-loop $\Phi$1-graviton MHV amplitude is expressed instead as a sum of scalar box and bubble functions plus a rational remainder,

$\Phi$2

with $\Phi$3 written in terms of cycle sums $\Phi$4 and soft-lifting polynomials $\Phi$5. This suggests that beyond tree level the exceptional compactness of Hodges’ determinant is replaced by integrand- or basis-dependent structures tailored to the theory and regularization scheme (Dunbar et al., 2011).

The gravity MHV formula is therefore best understood as a central flat-space tree-level object—equivalently a reduced determinant, a tree sum, a CHY localization, a current recursion, or a twistor correlator—from which a broad network of extensions radiates. Across these formulations, the persistent structural themes are rank-deficient Laplacians, matrix-tree identities, self-dual building blocks, and the separation of the universal helicity factor $\Phi$6 from a permutation-symmetric reduced amplitude.