---
title: Gravity MHV Formula for Graviton Amplitudes
url: https://www.emergentmind.com/topics/gravity-mhv-formula
type: topic
---

# Gravity MHV Formula for Graviton Amplitudes

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 [1204.1930] [1603.08158].

## 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 [2506.05460] [2507.13943] [1611.03242].

The Hodges matrix \(\Phi\) is an \(n\times n\) symmetric matrix with off-diagonal entries
\[
\Phi_{ij}=\frac{[ij]}{\langle ij\rangle}\qquad (i\neq j),
\]
and diagonal entries fixed by reference spinors \(|x\rangle,|y\rangle\),
\[
\Phi_{ii}
=
-\sum_{k\neq i}\frac{[ik]}{\langle ik\rangle}
\frac{\langle kx\rangle\langle ky\rangle}{\langle ix\rangle\langle iy\rangle}.
\]
Equivalently, one may use
\[
\det{}'\Phi\equiv c_{abc}^2\,|\Phi|^{abc}_{abc},
\qquad
c_{abc}=\frac{1}{\langle ab\rangle\langle bc\rangle\langle ca\rangle},
\]
with \(|\Phi|^{abc}_{abc}\) the minor obtained by deleting rows and columns \(a,b,c\). With momentum conservation imposed, \(\det{}'\Phi\) is independent of \(|x\rangle,|y\rangle\) and of the deleted rows and columns, reflecting the corank-three structure of \(\Phi\) [2506.05460] [1204.1930].

An equivalent notation often used in CHY-based treatments is
\[
\mathcal M_n(1^+,\ldots,x^-,\ldots,y^-,\ldots,n^+)
=
\langle xy\rangle^8\,\bar M_n(12\cdots n),
\]
where \(\bar M_n\) is the reduced amplitude built from the Hodges matrix. This form isolates the helicity weight \(\langle xy\rangle^8\) from the permutation-symmetric determinant structure [1603.08158].

## 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 \(n-1\) and \(n\), the MHV amplitude is
\[
M_n^{\mathrm{MHV}}
=
\frac{1}{\langle n-1,n\rangle^2\prod_{a=1}^{n-2}(\langle a,n-1\rangle\langle a,n\rangle)^2}
\sum_{T}
\prod_{(ab)\in E(T)}
\left(
\frac{[ab]}{\langle ab\rangle}
\langle a,n-1\rangle\langle b,n-1\rangle\langle a,n\rangle\langle b,n\rangle
\right),
\]
where the sum runs over spanning trees on the \(n-2\) positive-helicity legs. This is the NSVW/BDPR tree formula. By the matrix-tree theorem, it is equivalent to the reduced determinant form [2112.15029].

A closely related half-soft form obtained from Berends–Giele currents reorganizes the same amplitude as
\[
M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+)
=
\langle12\rangle^8\;
\frac{1}{\prod_{i=3}^n\langle1i\rangle\langle2i\rangle}
\sum_{\text{trees on }\{3,\ldots,n\}}
\prod_{\langle ij\rangle}\frac{[ij]}{\langle ij\rangle}.
\]
In pure-connection variables, the same object is written as
\[
M^{\mathrm{MHV}}(x^+,y^+;1^-,\dots,n^-)
=
2i\,(xy)^6\,h(x,K,y),
\]
with \(h(x,K,y)\) the half-soft function defined as a weighted sum over spanning trees on the set \(K\) [2507.13943] [1410.5647].

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 [2112.15029] [1410.5647].

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 \(n-2\) 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
\[
\mathcal M_n^{\mathrm{tree}}
=
\int d\mu_n\,\big[\mathrm{Pf}'\Psi\big]^2
=
\sum_{\{\sigma\}}
\frac{\big[\mathrm{Pf}'\Psi(\sigma)\big]^2}{\det{}'(\Phi(\sigma))},
\]
with \(d\mu_n\) the Möbius-invariant scattering-equation measure, \(\Psi\) the \(2n\times2n\) CHY matrix, and \(\det'(\Phi)\) the reduced Jacobian determinant [1603.08158].

In four dimensions, the decisive statement is that only one special rational solution of the scattering equations supports the MHV amplitude. After fixing \(\sigma_{n-2}=0\) and \(\sigma_{n-1}=1\), Weinzierl’s MHV solution is
\[
\sigma_a^{(1)}
=
\frac{\langle a,n-2\rangle\langle n-1,\chi\rangle}
{\langle a,\chi\rangle\langle n-1,n-2\rangle},
\]
with \(|\chi\rangle\) parametrizing the residual \(\mathrm{SL}(2,\mathbb C)\) freedom. At this solution the three essential ingredients factorize as
\[
\det{}'(\Phi)=(F_\chi)^{2n}(P_\chi)^4\,\bar M_n,
\]
\[
\mathrm{Pf}'(\Psi)
=
(-1)^{s(n)}(\sqrt2)^n(F_\chi)^n(P_\chi)^2\langle12\rangle^4\,\bar M_n,
\]
\[
\sigma_{12}\sigma_{23}\cdots\sigma_{n1}
=
(1/F_\chi)^n D_n (P_\chi)^2.
\]
Combining these yields
\[
M_n^{\mathrm{grav}}
\longrightarrow
2^n\langle12\rangle^8\,\bar M_n,
\]
which reproduces the Hodges formula up to the overall \(2^n\) normalization associated with the Pfaffian convention for polarizations. The auxiliary factors \(F_\chi\) and \(P_\chi\) cancel in the final amplitude, leaving a manifestly \(\mathrm{SL}(2,\mathbb C)\)-invariant result [1603.08158].

The conjugate special rational solution,
\[
\sigma_a^{(2)}
=
\frac{[a,n-2][n-1,\tilde\chi]}
{[a,\tilde\chi][n-1,n-2]},
\]
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 \(\mathfrak C_\pm\): only solutions with \(\mathrm{rank}(\mathfrak C_-)=k+1\) contribute to \(N^k\)MHV amplitudes, so MHV selects precisely the special MHV solution [1603.08158] [1611.03242].

## 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 \(\mathcal K\), the all-plus graviton current has the scalar form
\[
J(\mathcal K)
=
\sum_{\Gamma_{\mathcal K}}
\prod_{i\in\mathcal K}(qi)^{2\alpha_i-4}
\prod_{\langle jk\rangle\in\Gamma_{\mathcal K}}\frac{[jk]}{(jk)},
\]
a sum over spanning trees of the complete graph on \(\mathcal K\). The all-but-one-plus current \(J(1|\mathcal K)\) is solved by the same tree sum multiplied by a factor \(\Phi(\Gamma_{\mathcal K})\) 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 [2507.13943].

In the pure-connection formalism, the relevant cubic vertex simplifies enough that an all-minus current can be computed recursively. Its scalar coefficient \(A(K)\) is a sum over Cayley trees,
\[
A(K)
=
\sum_{T\in\mathcal T(K)}
\left[\prod_{(jk)\in E(T)}\frac{[jk]}{(jk)}\right]
\left[\prod_{i\in K}\big((iq)^2\big)^{d_i(T)-2}\right].
\]
This immediately suggests the MHV amplitude after the symmetric replacement \(q\to x,y\), yielding again the half-soft/BGK formula
\[
M^{\mathrm{MHV}}(x^+,y^+;1^-,\dots,n^-)
=
2i\,(xy)^6
\sum_{T\in\mathcal T(K)}
\left[\prod_{(jk)\in E(T)}\frac{[jk]}{(jk)}\right]
\left[\prod_{i\in K}\big((ix)(iy)\big)^{d_i(T)-2}\right].
\]
The same tree sum admits a Hodges-like determinant representation via a weighted Laplacian [1410.5647].

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”,
\[
\Phi_0(\phi_1,\ldots,\phi_N)=\sum_{t\in\mathcal T_N}\phi_t,
\]
with each edge carrying the operator \(D^{ij}/z_{ij}\). 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 [2408.11139].

## 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 \(\mathcal N=4\) and \(\mathcal N=8\) 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,
\[
\mathcal M_{n,0}(1^-,2^-,3^+,\dots,n^+;\Lambda)
=
\frac{1}{\Lambda}
\int\frac{d^{8|8}X}{\mathrm{vol}\,\mathrm{GL}(2,\mathbb C)}
\left[
(X^2)^2\left|\mathcal H^{12}_{\ 12}\right|
+
\sum_{i,j,k,l}\omega^1_{ij}\omega^2_{kl}
\left|\mathcal H^{12ijkl}_{\ 12ijkl}\right|
\right]
\prod_m h_m\,D\sigma_m
+(1\leftrightarrow2).
\]
The flat-space limit \(\Lambda\to0\) is smooth and reproduces Hodges’ formula nontrivially [1307.5043]. A related rational-curve formula for gauged \(\mathcal N=8\) supergravity in \(\mathrm{AdS}_4\) specializes in the MHV sector to
\[
\mathcal M_{n,1}
=
\int
\frac{\prod_{r=0,1}d^{4|8}\mathcal Z_r}
{\mathrm{vol}\,\mathrm{GL}(2,\mathbb C)}
\,
\det{}'(\mathcal H)\,
\det{}'(\mathcal H^\vee)\,
\prod_{i=1}^n h_i(Z(\sigma_i)),
\]
and reduces in the flat-space limit to the usual MHV supergravity amplitude [1508.02554].

The determinant itself admits a hidden \(\mathrm{GL}(n,\mathbb C)\) reformulation in an auxiliary \(n\)-space whose indices are the particle labels. In that framework the amplitude is written as
\[
M_n^{\mathrm{MHV}}
=
\frac{1}{\langle12\rangle^2\langle23\rangle^2\langle31\rangle^2}
\det\!\left(\phi_{ab}+\sum_{i=1}^3 e_{L\,a}{}^i e_{R\,b}{}^i\right),
\]
or, with arbitrary reference 3-planes \(L\) and \(R\),
\[
M_n^{\mathrm{MHV}}
=
\frac{\det(\phi+LR^T)}
{\left(\sum_{abc}L_{abc}V_{abc}/3\right)\left(\sum_{def}R_{def}V_{def}/3\right)}.
\]
Different gauge fixings reproduce the \(S_{n-3}\)-symmetric Hodges form, the \(S_{n-2}\)-symmetric tree form, or new fully \(S_n\)-symmetric expressions [1207.4458].

A more recent celestial reformulation derives Hodges’ determinant from an \(Lw_{1+\infty}\) Ward identity. The resulting one-leg recursion,
\[
M_n
=
\sum_{i=1}^{n-1}
\frac{[ni]}{\langle ni\rangle}
\frac{\langle \alpha i\rangle^2}{\langle \alpha n\rangle^2}
\,
M_{n-1}\!\left(\ldots,|i]+\frac{\langle \alpha n\rangle}{\langle \alpha i\rangle}|n],\ldots\right),
\]
with seed
\[
M_3=
\left(\frac{\langle12\rangle^4}{\langle12\rangle\langle23\rangle\langle31\rangle}\right)^2,
\]
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 [2506.05460] [2008.04330].

## 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:
\[
\mathcal M_n^{\mathrm{MHV}}(1^-,2^-,3^+,\ldots,n^+)
=
\sum_{t=0}^{\lfloor\frac{n-3}{2}\rfloor}
\int dx^-
\sum_{a_1,\ldots,a_t}
\sum_{i\in\bar a}
\left|\mathcal H^i_{\ i}[a_1,\ldots,a_t]\right|
\frac{\langle12\rangle^5[\tilde\iota i]}
{\langle i1\rangle\langle i2\rangle^2[\tilde\iota2]}
\,i^{\,t-|a|}
\,e^{iF_n(x^-)}
\prod_{m=1}^t \partial_-^{|a_m|-1}f(x^-).
\]
The trivial-background limit removes the tail terms and restores the Hodges determinant [2003.13501].

At one loop, maximally supersymmetric supergravity admits an integrand-level MHV formula obtained by double copying BCJ-satisfying \(\mathcal N=4\) SYM numerators:
\[
\mathcal M^{\text{1-loop}}_{N,\mathrm{MHV}}
=
\delta^{16}(Q)\,
\frac{1}{\prod_{i=2}^{N}\langle1i\rangle^4}
\int\frac{d^D\ell}{(2\pi)^D}
\sum_{m=4}^{N}
\sum_{A_2\cup\cdots\cup A_m=\{2,\dots,N\}}
\frac{\big(\bar n_{1|A_2|\cdots|A_m}(\ell)\big)^2}{\prod_{\alpha}p_\alpha^2},
\]
where the stripped numerators \(\bar n\) are generated by a deletion operator \(\mathfrak D\) acting on self-dual \(X\)-chains. This realizes the MHV gravity integrand as a square of Yang–Mills-like building blocks [1507.06288].

For \(\mathcal N=4\) supergravity, the one-loop \(n\)-graviton MHV amplitude is expressed instead as a sum of scalar box and bubble functions plus a rational remainder,
\[
M_n^{\text{1-loop}}
=
\sum_{\alpha\in\text{boxes}} a_\alpha\, I^{\text{tc}}_\alpha
+
\sum_{\beta\in\text{bubbles}} c_\beta\,I_2^\beta
+
R_n,
\]
with \(R_n\) written in terms of cycle sums \(C_r\) and soft-lifting polynomials \(S_m\). 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 [1111.1153].

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 \(\langle xy\rangle^8\) from a permutation-symmetric reduced amplitude.

Source: https://www.emergentmind.com/topics/gravity-mhv-formula