1 Introduction

Unimodular gravity is an interesting truncation of General Relativity, where the spacetime metric is restricted to be unimodular,

$$\begin{aligned} g\equiv \text {det}~g_{\mu \nu }=-1. \end{aligned}$$
(1)

It is convenient to implement the truncation through the (non invertible) map

$$\begin{aligned} g_{\mu \nu }\longrightarrow |g|^{-1/n}~g_{\mu \nu }. \end{aligned}$$
(2)

The resulting theory is not Diff invariant anymore, but only TDiff invariant. Transverse diffeomorphisms are those such that their generator is transverse, that is,

$$\begin{aligned} \partial _\mu \xi ^\mu =0. \end{aligned}$$
(3)

The ensuing action of Unimodular Gravity (cf. [1] for a recent review with references to previous literature), reads

$$\begin{aligned} S_{UG}\equiv & {} \int \mathrm{d}^n x~\mathcal{L}_{UG}\\\equiv & {} -M_P^{n-2}\int |g|^{1/n}\left( R+{(n-1)(n-2)\over 4 n^2}{g^{\mu \nu }\nabla _\mu g\nabla _\nu g\over g^2}\right) .\nonumber \end{aligned}$$
(4)

It can easily be shown using the Bianchi identities that the classical equations of motion (EM) of Unimodular Gravity coincide with those of General Relativity with an arbitrary cosmological constant. The main difference at this level between the two theories is that a constant value for the matter potential energy does not weight at all, which solves part of the cosmological constant problem (namely why the cosmological constant is not much bigger that observed). This property is preserved by quantum corrections [2].

A natural question to ask at this stage is whether the S-matrix would be the same for Unimodular Gravity as for General Relativity. Although the S-matrix elements have been studied by several authors in the case of General Relativity [4–8], we are not aware of any results concerning the computation of S-matrix elements in Unimodular Gravity. The propagators as well as the vertices are quite different in the two theories, so that the answer to the question we asked at the beginning of this paragraph is not immediate.

In the present paper we shall carry out the calculation of the maximally helicity violating three-, four-, and five-graviton amplitudes at tree level and find complete agreement between the two theories, a fact that we find remarkable.

2 Feynman rules

The graviton propagator in Unimodular Gravity (cf. Appendix A) reads

$$\begin{aligned} P^\text {UG}_{ \mu \nu , \rho \sigma }= & {} \frac{i}{2k^2}\left( \eta _{ \mu \sigma }\eta _{ \nu \rho }+\eta _{ \mu \rho }\eta _{ \nu \sigma } \right) -\frac{i}{k^2}\frac{ \alpha ^2n^2-n+2}{{ \alpha ^{2}} {n^{2}}(n-2)}\eta _{{ \mu } { \nu }}\eta _{{ \rho } \sigma }\nonumber \\&+\frac{2i}{n-2}\left( \frac{k_\rho k_\sigma \eta _{ \mu \nu }}{k^4}+\frac{k_\mu k_\nu \eta _{\rho \sigma }}{k^4}\right) -\frac{2in}{n-2}\frac{k_{ \mu } k_{ \nu }k_{ \rho }k_{ \sigma }}{k^6}\nonumber \\ \end{aligned}$$
(5)

for the gauge choice of [1].

Recall that the usual General Relativity graviton propagator in the de Donder gauge,

$$\begin{aligned} P_{\mu \nu \rho \sigma }^{\text {GR}}=\dfrac{i}{2k^2}\left( \eta _{\mu \sigma }\eta _{\nu \rho }+\eta _{\mu \rho }\eta _{\nu \sigma }-\dfrac{2}{n-2}\eta _{\mu \nu }\eta _{\rho \sigma }\right) , \end{aligned}$$
(6)

has only simple poles at \(k^2=0\). In the unimodular propagator, by contrast, there appear double and triple poles in addition to the simple ones. This is a technical complication and the main reason why we cannot, a priori, apply some of the recent useful techniques [19] to reduce the computation of the diagrams. In Appendix B we shall show that no gauge choice in Unimodular Gravity can yield a propagator of the form

$$\begin{aligned} P_{\mu \nu \rho \sigma }= & {} \dfrac{i}{2k^2}\left( \eta _{\mu \sigma }\eta _{\nu \rho }+\eta _{\mu \rho }\eta _{\nu \sigma }-f_1(k^2)\,\eta _{\mu \nu }\eta _{\rho \sigma }\right) \nonumber \\&+f_2(k^2) (k_\rho k_\sigma \eta _{\mu \nu }+k_\mu k_\nu \eta _{\rho \sigma })+f_3(k^2)\,k_{\mu }k_{\nu }k_{\rho }k_{\sigma },\nonumber \\ \end{aligned}$$
(7)

\(f_3(k^2)\) having no pole at \(k^2=0\), if the Newtonian potential is to be obtained in the nonrelativistic limit. Actually, we shall see that the triple pole term in (5) is needed to retrieve the correct nonrelativistic static limit.

Since we are going to focus on the three-, four-, and five-point amplitudes, we also need the three- and four-graviton vertices. These are obtained from the second and third order expansion of the Lagrangian around flat space (cf. Appendix C) and can be expressed in a condensed form, with a parameter n that gives the General Relativity vertex for \(n=2\) and the Unimodular Gravity one for \(n=4\). With the convention of all momenta being incoming the expression for the three-graviton vertex reads

$$\begin{aligned} V^{\mu \nu ,\rho \sigma ,\alpha \beta }_{(p1,p2,p3)}= & {} i \kappa ~\mathcal{S}\left\{ - \frac{(2 + n) (p_1.p_2) \eta ^{\alpha \rho } \eta ^{\beta \sigma } \eta ^{\mu \nu }}{n^2} \right. \nonumber \\&\left. - \frac{(p_1.p_2) \eta ^{\alpha \beta } \eta ^{\mu \rho } \eta ^{\nu \sigma }}{2 n}\right. \nonumber \\&\left. + \frac{(2 + n) (p_1.p_2) \eta ^{\alpha \beta } \eta ^{\mu \nu } \eta ^{\rho \sigma }}{2 n^3} \right. \nonumber \\&\left. + \frac{2 \eta ^{\beta \nu } \eta ^{\rho \sigma } p_1^{m} p_2^{\alpha }}{n} + \tfrac{1}{2} \eta ^{mr} \eta ^{\nu \sigma } p_1^{\alpha } p_2^{\beta } \right. \nonumber \\&\left. -\frac{(2 + n) \eta ^{\mu \nu } \eta ^{\rho \sigma } p_1^{\alpha } p_2^{\beta }}{2 n^2}\right. \nonumber \\&\left. - 2 \eta ^{\beta \sigma } \eta ^{\nu \rho } p_1^{\alpha } p_2^{\mu } - \eta ^{\alpha \nu } \eta ^{\beta \sigma } p_1^{\rho } p_2^{\mu } \right. \nonumber \\&\left. + \frac{\eta ^{\alpha \beta } \eta ^{\nu \sigma } p_1^{\rho } p_2^{\mu }}{n} + \frac{2 \eta ^{\beta \mu } \eta ^{\rho \sigma } p_1^{\alpha } p_2^{\nu }}{n}\right. \nonumber \\&\left. - \frac{2 \eta ^{\alpha \beta } \eta ^{\rho \sigma } p_1^{\mu } p_2^{\nu }}{n^2} + \frac{2 \eta ^{\alpha \mu } \eta ^{\beta \nu } p_1^{\sigma } p_2^{\rho }}{n}\right. \nonumber \\&\left. + (p_1.p_2) \eta ^{\alpha \nu } \eta ^{\beta \sigma } \eta ^{\mu \rho }\right\} . \end{aligned}$$
(8)

The four-graviton vertex, in turn, is given by

$$\begin{aligned} V^{\mu \nu ,\rho \sigma ,\alpha \beta ,\eta \lambda }_{(p1,p2,p3,p4)}= & {} i \kappa ^2~ \mathcal{S} \left\{ \frac{\Bigl (2 + n\Bigr ) (p_3.p_4) g^{\mu \nu } g^{\rho \sigma } g^{\alpha \beta } g^{\eta \lambda }}{4 n^4}\right. \nonumber \\&- \frac{(2 + n) (p_3.p_4) g^{\mu \rho } g^{\alpha \beta } g^{\eta \lambda } g^{\nu \sigma }}{4 n^3} \nonumber \\&+ \frac{(2 + n) (p_3.p_4) g^{\mu \eta } g^{\rho \alpha } g^{\nu \lambda } g^{\sigma \beta }}{2 n^2} \nonumber \\&- \frac{(2 + n) (p_3.p_4) g^{\mu \nu } g^{\rho \eta } g^{\alpha \beta } g^{\sigma \lambda }}{ n^3} \nonumber \\&+ \frac{(2 + n) (p_3.p_4) g^{\mu \rho } g^{\alpha \beta } g^{\eta \sigma } g^{\nu \lambda }}{ n^2}\nonumber \\&- \frac{(p_3.p_4) g^{\mu \nu } g^{\rho \sigma } g^{\alpha \eta } g^{\beta \lambda }}{4 n^2} \nonumber \\&+ \frac{(p_3.p_4) g^{\mu \nu } g^{\rho \eta } g^{\alpha \sigma } g^{\beta \lambda }}{ n} \nonumber \\&+ g^{\mu \eta } g^{\alpha \sigma } g^{\beta \lambda } p_3^{\nu } p_4^{\rho } \nonumber \\&+\frac{(2 + n) g^{\mu \rho } g^{\alpha \beta } g^{\eta \lambda } p_3^{\sigma } p_4^{\nu }}{2 n^2} \nonumber \\&- \tfrac{1}{2} g^{\mu \rho } g^{\alpha \eta } g^{\beta \lambda } p_3^{\sigma } p_4^{\nu } \nonumber \\&+ \frac{(2 + n) g^{\mu \alpha } g^{\eta \lambda } g^{\nu \beta } p_3^{\rho } p_4^{\sigma }}{ n^2} \nonumber \\&+ \frac{g^{\mu \nu } g^{\alpha \eta } g^{\beta \lambda } p_3^{\rho } p_4^{\sigma }}{2 n} - g^{\mu \alpha } g^{\eta \nu } g^{\beta \lambda } p_3^{\rho } p_4^{\sigma } \nonumber \\&- 2\frac{g^{\mu \alpha } g^{\rho \beta } g^{\eta \lambda } p_3^{\nu } p_4^{\sigma }}{n} \nonumber \\&- \frac{(2 + n) g^{\mu \nu } g^{\alpha \beta } g^{\eta \lambda } p_3^{\rho } p_4^{\sigma }}{2 n^3}\nonumber \\&- 2\frac{g^{\mu \alpha } g^{\eta \sigma } g^{\nu \beta } p_3^{\rho } p_4^{\lambda }}{n}\nonumber \\&+ 2\frac{g^{\mu \nu } g^{\rho \alpha } g^{\eta \lambda } p_3^{\sigma } p_4^{\beta }}{n^2} - \nonumber \\&- 2\frac{g^{\mu \rho } g^{\alpha \sigma } g^{\eta \lambda } p_3^{\nu } p_4^{\beta }}{n}\nonumber \\&+ 2g^{\mu \eta } g^{\rho \lambda } g^{\alpha \nu } p_3^{\sigma } p_4^{\beta } \nonumber \\&-2\frac{g^{\mu \nu } g^{\rho \eta } g^{\alpha \lambda } p_3^{\sigma } p_4^{\beta }}{n} \nonumber \\&-2\frac{g^{\mu \eta } g^{\rho \alpha } g^{\nu \lambda } p_3^{\sigma } p_4^{\beta }}{n} \nonumber \\&+ 2g^{\mu \rho } g^{\alpha \lambda } g^{\eta \sigma } p_3^{\nu } p_4^{\beta }\nonumber \\&-2\frac{g^{\mu \rho } g^{\alpha \beta } g^{\eta \sigma } p_3^{\nu } p_4^{\lambda }}{n}\nonumber \\&+2\frac{g^{\mu \nu } g^{\rho \eta } g^{\alpha \beta } p_3^{\sigma } p_4^{\lambda }}{n^2}\nonumber \\&+ \frac{g^{\mu \nu } g^{\rho \sigma } g^{\alpha \eta } p_3^{\lambda } p_4^{\beta }}{2 n^2} \nonumber \\&- \frac{g^{\mu \nu } g^{\rho \eta } g^{\alpha \sigma } p_3^{\lambda } p_4^{\beta }}{ n} \nonumber \\&+ g^{\mu \rho } g^{\alpha \nu } g^{\eta \sigma } p_3^{\lambda } p_4^{\beta } \nonumber \\&-\frac{g^{\mu \rho } g^{\alpha \eta } g^{\nu \sigma } p_3^{\lambda } p_4^{\beta }}{2 n} \nonumber \\&- \frac{g^{\mu \nu } g^{\rho \sigma } g^{\alpha \beta } p_3^{\eta } p_4^{\lambda }}{ n^3} \nonumber \\&+ \frac{g^{\mu \rho } g^{\alpha \beta } g^{\nu \sigma } p_3^{\eta } p_4^{\lambda }}{ n^2} \nonumber \\&- 2\frac{g^{\mu \rho } g^{\alpha \sigma } g^{\nu \beta } p_3^{\eta } p_4^{\lambda }}{n} \nonumber \\&+ 2\frac{g^{\mu \nu } g^{\rho \alpha } g^{\sigma \beta } p_3^{\eta } p_4^{\lambda }}{n^2} \nonumber \\&- \tfrac{1}{2} (p_3.p_4) g^{\mu \eta } g^{\rho \lambda } g^{\alpha \nu } g^{\sigma \beta }\nonumber \\&- (p_3.p_4) g^{\mu \rho } g^{\alpha \nu } g^{\eta \sigma } g^{\beta \lambda } \nonumber \\&\left. +\frac{(p_3.p_4) g^{\mu \rho } g^{\alpha \eta } g^{\nu \sigma } g^{\beta \lambda }}{4 n}\right\} \end{aligned}$$
(9)

where \(\mathcal{S}\) is a shorthand for a double symmetrization, namely:

  1. 1.

    A summation over all momentum-index combinations (\(p_1,{\mu \nu }\); \(~p_2,\rho \sigma \); \(~;p_3,\alpha \beta \); \(~p_4,\eta \lambda \)).

  2. 2.

    A symmetrization of each pair on indices \(\mu \nu \), \(\rho \sigma \), \(\alpha \beta \), \(\eta \lambda \).Footnote 1

3 Spinor helicity formalism for massless particles

Although we are not using the spinor helicity formalism explicitly, we can take advantage of some useful relationships that can be derived from it and will greatly simplify the calculations.

The four momentum \(p^\mu \) for an on-shell particle is written in terms of two commuting Weyl spinors as

$$\begin{aligned} p_{\alpha \dot{\alpha }}=\bar{\sigma }_{\mu ,\alpha \dot{\alpha }}p^\mu =\lambda _{\alpha }\widetilde{\lambda }_{\dot{\alpha }}+\mu _\alpha \widetilde{\mu }_{\dot{\alpha }}. \end{aligned}$$
(10)

In the case of a massless particle, the condition \(\text {det}(p_{\alpha \dot{\alpha }})=0\) implies

$$\begin{aligned} p_{\alpha \dot{\alpha }}=\bar{\sigma }_{\mu ,\alpha \dot{\alpha }}p^\mu =\lambda _{\alpha }\widetilde{\lambda }_{\dot{\alpha }}. \end{aligned}$$
(11)

On the other hand, the polarization tensor of the graviton can be written in terms of the gluon ones as

$$\begin{aligned} \epsilon ^-_{\mu \nu }=\epsilon ^-_\mu \epsilon ^-_\nu \longrightarrow \epsilon ^-_{a\dot{a},b\dot{b}}=\epsilon ^-_{a\dot{a}}\epsilon ^-_{b\dot{b}}\quad \text {and}\quad \epsilon ^+_{a\dot{a},b\dot{b}}=\epsilon ^+_{a\dot{a}}\epsilon ^+_{b\dot{b}}. \end{aligned}$$
(12)

The gluon polarization vector depends on the momentum of the given gluon and an arbitrary reference momentum \(\epsilon ^-_{i,\mu }\equiv \epsilon ^-_\mu (p_i,r_i)\) where, following the conventions of [9], the gluon polarization spinors are given by

$$\begin{aligned} \epsilon ^-_{a\dot{a}}=\sqrt{2}\dfrac{\lambda _\alpha \widetilde{\mu }_{\dot{a}}}{[\lambda \mu ]},\qquad \epsilon ^+_{a\dot{a}}=-\sqrt{2}\dfrac{\widetilde{\lambda }_{\dot{\alpha }} {\mu }_\alpha }{\langle \lambda \mu \rangle } \end{aligned}$$
(13)

with \(\mu \) and \(\widetilde{\mu }\) the reference spinors which are related with the freedom to perform a gauge transformation. Therefore, they can be chosen in such a way as to simplify the computations as much as possible: this is achieved by choosing the so-called “minimal gauge”—see [10–15]—as displayed next.

Altogether, this implies that, for any given particle,

$$\begin{aligned} \epsilon ^+_i\cdot \epsilon ^-_i=-1,\qquad \epsilon ^+_i\cdot \epsilon ^+_i=\epsilon ^-_i\cdot \epsilon ^-_i =0,\qquad \epsilon _i^{\pm }\cdot p_i=\epsilon _i^{\pm }\cdot r_i=0. \end{aligned}$$
(14)

Henceforth, with the appropriate choice of the reference spinors we get the following rules:

  1. 1.

    For the four-graviton amplitudes, by choosing \(r_1=r_2=p_4\) and \(r_3=r_4=p_1\) we get the extra relations

    $$\begin{aligned}&\epsilon _1^-\cdot p_4=0,\end{aligned}$$
    (15)
    $$\begin{aligned}&\epsilon _2^-\cdot p_4=0,\end{aligned}$$
    (16)
    $$\begin{aligned}&\epsilon _3^+\cdot p_1=0,\end{aligned}$$
    (17)
    $$\begin{aligned}&\epsilon _4^+\cdot p_1=0,\end{aligned}$$
    (18)
    $$\begin{aligned}&\epsilon _i^{\pm }\cdot \epsilon _j^{\pm }=0~~\text {except for}~~ \epsilon _2\cdot \epsilon _3 \end{aligned}$$
    (19)
  2. 2.

    For the five-graviton amplitudes, we choose now \(r_1=r_2=p_5\) and \(r_3=r_4=r_5=p_1\) and we get

    $$\begin{aligned}&\epsilon _1^-\cdot p_5=0,\end{aligned}$$
    (20)
    $$\begin{aligned}&\epsilon _2^-\cdot p_5=0,\end{aligned}$$
    (21)
    $$\begin{aligned}&\epsilon _3^+\cdot p_1=0,\end{aligned}$$
    (22)
    $$\begin{aligned}&\epsilon _4^+\cdot p_1=0,\end{aligned}$$
    (23)
    $$\begin{aligned}&\epsilon _5^+\cdot p_1=0,\end{aligned}$$
    (24)
    $$\begin{aligned}&\epsilon _i^{\pm }\cdot \epsilon _j^{\pm }=0~~\text {except for}~~ \epsilon _2^{\pm }\cdot \epsilon _3^{\pm }~~\text {and}~~\epsilon _2^{\pm }\cdot \epsilon _4^{\pm }. \end{aligned}$$
    (25)

4 Three-graviton amplitudes

The fact that Unimodular Gravity perturbatively expanded around Minkowski spacetime is Lorentz invariant and that the graviton polarizations are the same as in General Relativity leads, by repeating the standard analysis [3], to the conclusion that the on-shell three-point amplitudes vanish on-shell for real momenta. Now, let us stress that little group scaling operates in Unimodular Gravity exactly in the manner as in General Relativity. Hence, it is plain that for conserved complex momenta the on-shell nonvanishing three-point amplitudes are the same in Unimodular Gravity as in General Relativity but, perhaps, for a global constant. By explicit computation of the corresponding Feynman diagrams we have found that the constant in question is the same in the two theories, as in fact the classical Newton constant is indeed the same in the two theories. Let us notice that the on-shell three-point functions for complex momenta are the elementary objects in the recursive construction of the amplitudes in theories like Yang–Mills and General Relativity with or without SUSY.

5 Four-graviton tree amplitudes

Let us recall that our goal is to compute the tree diagrams both in Unimodular Gravity and General Relativity in order to see whether there is any difference between the two theories. This is relevant for the physical content of the theories because these amplitudes give us information on the tree-level S-matrix.

We shall focus on the maximally helicity violating (MHV) diagrams with three, four and five external gravitons because they are the simplest nontrivial ones.

There are only three types of diagrams—which correspond to the well-known s, t, and u channels, respectively—that involve four external gravitons to be worked out explicitly. The diagram that is a pure four vertex vanishes because no nonvanishing contribution to the amplitude diagram can be constructed out of two momenta entering the vertex and the four graviton polarizations satisfying the equations displayed in Sect. 3. The s-, t-, and u-channel diagrams are shown in the next figures where all gravitons are outgoing (Figs. 1, 2, 3).

Fig. 1
figure 1

s-channel

Full size image
Fig. 2
figure 2

t-channel

Full size image
Fig. 3
figure 3

u-channel

Full size image

The explicit result is

$$\begin{aligned} \mathcal{A}_s(1^-2^-;3^+4^+)= & {} \epsilon _1^{-\mu _1}\epsilon _1^{-\nu _1}\epsilon _2^{-\mu _2}\epsilon _2^{-\nu _2}V^{\mu _1\nu _1,\mu _2\nu _2,\alpha \beta }_{(p1,p2,q)}\nonumber \\&P_{\alpha ,\beta ,\rho ,\sigma }V^{\rho \sigma ,\mu _3\nu _3,\mu _4\nu _4}_{(p,p3,p4)}\epsilon _3^{-\mu _3}\epsilon _3^{-\nu _3}\epsilon _4^{-\mu _4}\epsilon _4^{-\nu _4}\nonumber \\= & {} - \frac{i \kappa ^2 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3)^2 (\epsilon _4.p_2)^2}{s^2}\nonumber \\= & {} i\frac{\kappa ^2}{4}\frac{\langle 12\rangle ^5[34]^2}{[12]\langle 23\rangle ^2\langle 14\rangle ^2},\end{aligned}$$
(26)
$$\begin{aligned} \mathcal{A}_t(1^-2^-;3^+4^+)= & {} \epsilon _1^{-\mu _1}\epsilon _1^{-\nu _1}\epsilon _3^{+\mu _3}\epsilon _3^{+\nu _3}V^{\mu _1\nu _1,\mu _3\nu _3,\alpha \beta }_{(p1,p3,q)}\nonumber \\&P_{\alpha ,\beta ,\rho ,\sigma }V^{\rho \sigma ,\mu _2\nu _2,\mu _4\nu _4}_{(p,p2,p4)}\epsilon _2^{-\mu _2}\epsilon _2^{-\nu _2}\epsilon _4^{+\mu _4}\epsilon _4^{+\nu _4}\nonumber \\= & {} 0,\end{aligned}$$
(27)
$$\begin{aligned} \mathcal{A}_u(1^-2^-;3^+4^+)= & {} \epsilon _1^{-\mu _1}\epsilon _1^{-\nu _1}\epsilon _4^{+\mu _4}\epsilon _4^{+\nu _4}V^{\mu _1\nu _1,\mu _4\nu _4,\alpha \beta }_{(p1,p4,q)}\nonumber \\&P_{\alpha ,\beta ,\rho ,\sigma }V^{\rho \sigma ,\mu _2\nu _2,\mu _3\nu _3}_{(p,p2,p3)}\epsilon _2^{-\mu _2}\epsilon _2^{-\nu _2}\epsilon _3^{+\mu _3}\epsilon _3^{+\nu _3}\nonumber \\= & {} \frac{i \kappa ^2 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3)^2 (\epsilon _4.p_2)^2}{u^2}\nonumber \\= & {} i\dfrac{\kappa ^2}{4}\dfrac{\langle 12\rangle ^8[24]^2}{\langle 13\rangle ^3[31]\langle 23\rangle ^2\langle 14\rangle ^2}, \end{aligned}$$
(28)

where as usual \(s=p_1+p_2\) and \(u=p_1+p_3\).

These amplitudes are diagram to diagram exactly the same that the ones for General Relativity. The complete amplitude is therefore

$$\begin{aligned} \mathcal{A}(1^-2^-;3^+4^+)=i\dfrac{\kappa ^2}{4}\dfrac{\langle 12\rangle ^8[12]}{\langle 12\rangle \langle 13\rangle \langle 14\rangle \langle 23\rangle \langle 24\rangle \langle 34\rangle ^2}, \end{aligned}$$
(29)

in agreement with the result presented for General Relativity in [7].

6 Five-point diagrams

When computing the diagrams with five external gravitons there are three sets of diagrams. The one that is purely a five vertex vanishes identically. Indeed, no nonvanishing contribution to the amplitude diagram can be built from two momenta entering the vertex and the five graviton polarizations introduced in Sect. 3. Let us consider the others in turn

6.1 Three vertices

There are 15 different diagrams that involve three three vertices of the type shown in Fig. 4; this we shall denote by \(\mathcal{A}(1^-,2^-;3^+;4^+,5^+)\), the others will be analogously represented by using the obvious notation.

Let us write this one as an example; the full set of amplitudes can be found in Appendix D.

Fig. 4
figure 4

\(\mathcal{A}(1^-,2^-;3^+;4^+,5^+)\)

Full size image

We have

$$\begin{aligned}&\mathcal{A}(1^-,2^-;3^+;4^+,5^+)\\&\quad =- \frac{i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _4)^2 (\epsilon _3.p_2)^2 (\epsilon _5.p_2)^2}{(p_1+p_2)^2 (p_4+p_5)^2}\nonumber \\&\qquad + \frac{2i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3)^2 (\epsilon _4.p_2) (\epsilon _4.p_3) (\epsilon _5.p_2) (\epsilon _5.p_3)}{(p_1+p_2)^2 (p_4+p_5)^2}\nonumber \\&\qquad - \frac{i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3)^2 (\epsilon _4.p_3)^2 (\epsilon _5.p_2)^2}{(p_1+p_2)^2 (p_4+p_5)^2}\nonumber \\&\qquad +\frac{2i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3) (\epsilon _2.\epsilon _4) (\epsilon _3.p_2) (\epsilon _4.p_2) (\epsilon _5.p_2) (\epsilon _5.p_3)}{(p_1+p_2)^2 (p_4+p_5)^2}\nonumber \\&\qquad - \frac{2i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _4)^2 (\epsilon _3.p_2)^2 (\epsilon _5.p_2) (\epsilon _5.p_3)}{(p_1+p_2)^2 (p_4+p_5)^2} \nonumber \\&\qquad - \frac{i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _4)^2 (\epsilon _3.p_2)^2 (\epsilon _5.p_3)^2}{(p_1+p_2)^2 (p_4+p_5)^2} \nonumber \\&\qquad - \frac{2i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3) (\epsilon _2.\epsilon _4) (\epsilon _3.p_2) (\epsilon _4.p_3) (\epsilon _5.p_2) (\epsilon _5.p_3)}{(p_1+p_2)^2 (p_4+p_5)^2} \nonumber \\&\qquad -\frac{i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3)^2 (\epsilon _4.p_2)^2 (\epsilon _5.p_3)^2}{(p_1+p_2)^2 (p_4+p_5)^2} \nonumber \\&\qquad + \frac{2i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3) (\epsilon _2.\epsilon _4) (\epsilon _3.p_2) (\epsilon _4.p_2) (\epsilon _5.p_3)^2}{(p_1+p_2)^2 (p_4+p_5)^2} \nonumber \\&\qquad -\frac{2i \kappa ^3 (\epsilon _1.p_2)^2 (\epsilon _2.\epsilon _3) (\epsilon _2.\epsilon _4) (\epsilon _3.p_2) (\epsilon _4.p_3) (\epsilon _5.p_2)^2}{(p_1+p_2)^2 (p_4+p_5)^2}. \end{aligned}$$

6.2 The four vertex

The rest of the a priori nonvanishing diagrams are those that involve one three vertex and one four vertex \(\mathcal{A}(1^-,2^-;3^+,4^+,5^+)\) as shown in Fig. 5.

Fig. 5
figure 5

\(\mathcal{A}(1^-,2^-;3^+,4^+,5^+)\)

Full size image

Explicit computation shows that all the 10 different diagrams do vanish.

7 Conclusions

It has been shown that the MHV three-, four-, and five-graviton tree amplitudes give the same contribution both in General Relativity and Unimodular Gravity. This result holds for each diagram independently and not only for the whole amplitude. Therefore we can conclude that, at least at tree level and for three, four or five external legs, the MHV contribution to the S-matrix for pure Unimodular Gravity without coupling to other fields is the same in the two theories.

A remarkable fact is that all the terms that involve the double and triple poles in the propagator of Unimodular Gravity (5) do not contribute to any diagram we have computed in pure Unimodular Gravity. We have explicitly checked this by introducing an arbitrary coefficient in front of each piece and then verify that the final result is independent of the arbitrary coefficient we have introduced. That the contributions coming from those higher order poles go away is not trivial and we did not find any reason to expect it before computing the diagrams. Indeed, on the one hand, the triple pole summand in the propagator is needed to recover the Newtonian potential–see Appendix B—and, on the other hand, in Unimodular Gravity, one obtains the following nonzero result:

$$\begin{aligned}&k_{\alpha } k_{\beta } V^{\mu \nu ,\rho \sigma ,\alpha \beta }_{(p,q,k)}\, \epsilon _{1\,\mu \nu }(p)\,\epsilon _{2\,\rho \sigma }(q) \nonumber \\&\quad = i\kappa \,(p\cdot q)\,(p\cdot \epsilon _{2})(q\cdot \epsilon _{1})\, (\epsilon _{1}\cdot \epsilon _{2}) \end{aligned}$$
(30)

when \(k=-p-q\) is off-shell and the polarizations with well-defined helicity \(\epsilon _{1\,\mu \nu }(p)=\epsilon _{1\,\mu }(p)\epsilon _{1\,\nu }(p)\) and \(\epsilon _{2\,\rho \sigma }(q)=\epsilon _{2\,\rho }(q)\epsilon _{2\,\sigma }(q)\) are arbitrary. This is in contrast with the fact that the computation of the corresponding object in General Relativity yields a vanishing result as a consequence of invariance under the full Diffeomorphism group.

As a straightforward consequence, and since the BFCW recursion relations [19] can be applied to the diagrams of General Relativity [4], our results suggest that BFCW (or a similar recurrence) can be applied to Unimodular Gravity as well. This would be remarkable because of the existence of higher order poles in the propagator. Work on these issues is ongoing, and we expect to report on this soon.