Two-Loop Gravity amplitudes from four dimensional Unitarity

TL;DR

This paper calculates specific two-loop four- and five-graviton scattering amplitudes in four-dimensional gravity, providing compact analytic expressions and confirming the consistency of counterterms through logarithmic term analysis.

Contribution

It presents new compact analytic formulas for two-loop graviton amplitudes and verifies the matching of counterterms via logarithmic term extraction.

Findings

Derived simple analytic forms for two-loop graviton amplitudes

Confirmed the consistency of the $R^3$ counterterm across different amplitudes

Matched $ ext{ln}( ext{ extmu}^2)$ terms with theoretical expectations

Abstract

We compute the polylogarithmic parts of the two-loop four and five-graviton amplitudes where the external helicities are positive and express these in a simple compact analytic form. We use these to extract the $\ln(\mu^2)$ terms from both the four and five-point amplitudes and show that these match the same $R^3$ counterterm.

Equations

$$\sim \frac{209}{80ϵ}\times A^R$$

$$\left(\frac{a_1}{ϵ}+a_2\ln ({\mu }^2)\right)\times A^R$$

$${\mathcal{M}}_n^{(L)}(1,\cdots ,n)=\frac{i(\kappa /2{)}^{n-2+2L}({\mathrm{r}}_{\Gamma }{)}^L}{(4\pi {)}^{L(2-ϵ)}}{M}_n^{(L)}(1,\cdots ,n)$$

$$M_n^{(2)}=$$

$$I_n^{(1)}=$$

$$\frac{1}{{ϵ}^2}\left(\sum _{i<j}^n{s}_{ij}\right)=\frac{1}{{ϵ}^2}\times 0$$

$$\frac{1}{ϵ}\times \left(\sum _{i<j}^n{s}_{ij}\ln (-s_{ij}/{\mu }^2)\right)\times M_n^{(1)}.$$

$$G_n^{(2)}=\sum _{i<j}^n{c}_{ij}\frac{1}{ϵ}{\left(\frac{{\mu }^2}{-s_{ij}}\right)}^{ϵ}.$$

$$\left(\sum _{i<j}{c}_{ij}\right)\times \left(\frac{1}{ϵ}+\ln ({\mu }^2)\right).$$

$$M_n^{(0)}(1^{+},2^{+},\cdots ,n^{+})=0$$

$$M_4^{(1)}(1^{+},2^{+},3^{+},4^{+})=-{\left(\frac{st}{⟨12⟩⟨23⟩⟨34⟩⟨41⟩}\right)}^2\frac{(s^2+t^2+u^2)}{120}+O(ϵ)$$

$$M_n^{(1)}(1^{+},2^{+},3^{+},\cdots ,n^{+})=\frac{(-1{)}^n}{960}\sum _{a,b,M,N}h(a,M,b)h(b,N,a){\mathrm{tr}}^3(aMbN)+O(ϵ)$$

$$h(a,\{c\},b)=\frac{1}{{⟨ac⟩}^2{⟨cb⟩}^2},h(a,\{c,d\},b)=\frac{1}{⟨ac⟩⟨ad⟩}\frac{\left[cd\right]}{⟨cd⟩}\frac{1}{⟨cb⟩⟨db⟩}.$$

$$M_4^{(1)}(1^{+},2^{+},3^{+},4^{+})$$

$$M_5^{(1)}(1^{+},2^{+},3^{+},4^{+},5^{+})$$

$${\beta }_{123(45)}=-{\left[12\right]}^2{\left[23\right]}^{2}h(1,\{4,5\},3)$$

$$I_m[{\mu }_0^{2r}]=-ϵ(1-ϵ)\cdots (r-1-ϵ)(4\pi {)}^r{I}_m^{D=4+2r-2ϵ}.$$

$$M_4^{(2)}(1^{+},$$

$$-\frac{2s(3u^2+3t^2-2s^2)}{(s^2+t^2+u^2)}\frac{(-s/{\mu }^2{)}^{-ϵ}}{ϵ}-\frac{2t(3s^2+3u^2-2t^2)}{(s^2+t^2+u^2)}\frac{(-t/{\mu }^2{)}^{-ϵ}}{ϵ}$$

$$\displaystyle-{2u(3t^{2}+3s^{2}-2u^{2})\over(s^{2}+t^{2}+u^{2})}{(-u/\mu^{2})^{-\epsilon}\over\epsilon}+\hbox{\rm rational terms}\biggr{)}\,.$$

$$M_4^{(1)}(1^{+},2^{+},3^{+},4^{+})\times \frac{30stu}{(s^2+t^2+u^2)}\times \left(\frac{1}{ϵ}+\ln ({\mu }^2)\right)$$

$$=$$

$$-\frac{1}{4}⟶-\frac{N_B-N_F}{8}$$

$${\ell }_1=\frac{⟨cd⟩}{⟨ec⟩}{\bar{\lambda }}_d{\lambda }_e$$

$${\mathcal{C}}_{\{a,b\},c,d,e}=$$

$$=$$

$$I_4^{1m}(S,T,M^2)=$$

$$+$$

$$I_{4}^{\rm 1m:IR}(S,T,M^{2})\equiv-{2\over ST}\Biggl{[}-{(\mu^{2\epsilon})\over\epsilon^{2}}\Bigl{[}(-S)^{-\epsilon}+(-T)^{-\epsilon}-(-M^{2})^{-\epsilon}\Bigr{]}\Biggr{]}\;.$$

$$I_3^{1\mathrm{m}}(K^2)=\frac{({\mu }^{2ϵ})}{{ϵ}^2}(-K^2{)}^{-1-ϵ}.$$

$$\displaystyle I_{3}^{2\rm m}\bigl{(}K_{1}^{2},K_{2}^{2}\bigr{)}={(\mu^{2\epsilon})\over\epsilon^{2}}{(-K_{1}^{2})^{-\epsilon}-(-K_{2}^{2})^{-\epsilon}\over(-K_{1}^{2})-(-K_{2}^{2})}\ .$$

$$\frac{({\mu }^{2ϵ})}{{ϵ}^2}(-K^2{)}^{-ϵ}$$

$${\left(\sum {\mathcal{C}}_{\{a,b\},c,d,e}{I}_4^{1\mathrm{m}}+\sum {\mathcal{C}}_{\{a,b,c\},d,e}{I}_3^{1\mathrm{m}}+\sum {\mathcal{C}}_{\{a,b\},c,\{d,e\}}{I}_3^{2\mathrm{m}}\right)}_{IR}$$

Full text

I Introduction

Computing the scattering amplitudes of a quantum theory using its singular and analytic structure has a long history with many notable sucesses Eden . Recently the five-point all-plus helicity amplitude has been computed in QCD: first the integrands were determined using the method of maximal cuts Badger:2013gxa and $D$-dimensional unitarity and then the integrals were evaluated yielding a compact analytic form Gehrmann:2015bfy . In $D$-dimensional unitarity the cuts are computed in $D=4-2\epsilon$ dimensions where, typically, the components of the cuts are considerably more complicated than in four dimensions. In Dunbar:2016aux it was demonstrated that four-dimensional unitarity techniques Bern:1994zx ; Bern:1994cg blended with a knowledge of the singular structure of the amplitude could reproduce this form in a straightforward way. The four-dimensional approach was also used to calculate the six-point amplitude Dunbar:2016cxp ; Dunbar:2016gjb which was subsequently verified Badger:2016ozq .

Here we apply these techniques to gravity amplitudes with a particular emphasis on their Ultra-Violet (UV) behaviour. Understanding the UV structure of quantum gravity necessitates studying the two-loop amplitude. ’t Hooft and Veltman OneLoopFinite demonstrated the one-loop finiteness of on-shell amplitudes in quantum gravity by showing the available counterterms in four dimensions made no contribution to perturbative amplitudes. This cancellation does not persist to two loop and Goroff and Sagnotti Goroff:1985th ; vandeVen:1991gw in a landmark calculation were able to compute a UV infinity of the form

[TABLE]

where $A^{R}$ is the functional form generated by an $R^{3}$ counterterm. A feature of the computation was the appearance of sub-divergences which were cancelled diagram by diagram. In Bern:2015xsa this computation was revisited using evanescent operators which arise at one-loop to remove the sub-divergences in the four-point two-loop amplitude. For gravity coupled to various matter multiplets they found UV terms

[TABLE]

and noted that the coefficient of $\ln(\mu^{2})$, $a_{2}$, was more robust and simpler than $a_{1}$. Specifically, when coupled to $n_{3}$ (non-propagating) three form fields $a_{1}$ obtained a contribution proportional to $n_{3}$ but $a_{2}$ did not. Additionally when coupled to scalars and vectors $a_{2}$ was simply proportional to the the difference between the number of bosonic and fermionic degrees of freedom. As was argued in ref Bern:2015xsa , the coefficient $a_{2}$ has physical content since after renormalisation the amplitude depends upon $a_{2}$ but not $a_{1}$.

In this article we explore and compute, up to rational terms, the four and five-point two-loop amplitude in quantum gravity where all the external gravitons have positive helicity. We use the techniques which have proven successful for the gluon amplitude: the amplitude is organised using a knowledge of its singularity structure, then four-dimensional unitarity is used to determine the logarithmic and dilogarithmic parts. Since $\ln(\mu^{2})$ only appears in the combination $\ln(K^{2}/\mu^{2})$ we can extract the coefficient of $\ln(\mu^{2})$ using unitarity. We obtain the same coefficient of $\ln(\mu^{2})$ for the four-point amplitude as in ref. Bern:2015xsa . We also obtain the coefficient of $\ln(\mu^{2})$ for the five-point amplitude and show that this matches to the same $R^{3}$ counterterm.

II Structure of the Amplitude

As a convention we remove the coupling constant factors from the full $n$-point $L$-loop amplitude, ${\cal M}_{n}^{(L)}$ using

[TABLE]

where ${\rm r}_{\Gamma}=\Gamma^{2}(1-\epsilon)\Gamma(1+\epsilon)/\Gamma(1-2\epsilon)$.

We then organise the amplitude according to its singularity structure. The amplitude has both Infra-Red (IR) and UV singularities in the dimensional regulation parameter $\epsilon$. The all-plus amplitudes, which are finite at one-loop, can be divided as:

[TABLE]

where the first term contains the IR singularities of the amplitude. The function $I^{(1)}_{n}$ is Weinberg:1965nx ; Dunbar:1995ed ; Naculich:2011ry ; Akhoury:2011kq

[TABLE]

and $M^{(1)}_{n}$ is the all-$\epsilon$ form of the one-loop amplitude (with $s_{ij}=(k_{i}+k_{j})^{2}$). The leading singularity of $I^{(1)}_{n}$ is only $\epsilon^{-1}$ since

[TABLE]

by momentum conservation and the leading singularity is then

[TABLE]

The amplitude also has finite logarithmic terms. In our four dimensional formulation these arise in two-particle cuts and have the form of one-loop bubble integral functions,

[TABLE]

where $c_{ij}$ are rational functions of $\lambda_{k}$ and $\bar{\lambda}_{k}$.111As usual, a null momentum is represented as a pair of two component spinors $p^{\mu}=\sigma^{\mu}_{\alpha\dot{\alpha}}\lambda^{\alpha}\bar{\lambda}^{\dot{\alpha}}$. For real momenta $\lambda=\pm\bar{\lambda}^{*}$ but for complex momenta $\lambda$ and $\bar{\lambda}$ are independent Witten:2003nn . We are using a spinor helicity formalism with the usual spinor products $\left\langle a\,b\right\rangle=\epsilon_{\alpha\beta}\lambda_{a}^{\alpha}\lambda_{b}^{\beta}$ and $\left[a\,b\right]=-\epsilon_{\dot{\alpha}\dot{\beta}}\bar{\lambda}_{a}^{\dot{\alpha}}\bar{\lambda}_{b}^{\dot{\beta}}$.

The all-plus two-loop amplitude in QCD does not contain this term Catani:1998bh . This give rise to $\epsilon^{-1}$ and $\ln(\mu^{2})$ terms in the combination

[TABLE]

There may be other sources of $\epsilon^{-1}$ terms not directly determined by unitarity. The function $F^{(2)}_{n}$ contains the remaining polylogarithms of the amplitude and $R^{(2)}_{n}$ contains the remaining rational terms. In dimensional regularisation the internal momenta lie in $D=4-2\epsilon$ and it is really $D$-dimensional unitarity which should be used to reconstruct the amplitude. Consequently, four dimensional unitarity is not sensitive to the rational terms, however it does give considerable simplification. The rational terms may be computed by complementary methods such as recursion. For the case of QCD, the rational terms for $n=5,6$ were computed by recursion starting from the four point amplitude. We will not compute $R^{(2)}_{5}$ here: it not being necessary for our analysis and $R^{(2)}_{4}$ not being available.

The all-plus two-loop amplitude is a particularly simple amplitude. The all-plus helicity tree amplitude vanishes,

[TABLE]

This can be seen as a consequence of supersymmetric Ward identities SWI . These imply that this amplitude vanishes to all orders in perturbation theory in supersymmetric theories. Since the $n$-graviton amplitudes for pure gravity coincide with those for supersymmetric theories at tree level then the gravity tree amplitude also vanishes.

The one-loop four-point amplitude for pure gravity is Dunbar:1994bn 222We use for four point kinematics $s\equiv s_{12}$, $t\equiv s_{14}$ and $u\equiv s_{13}$.

[TABLE]

and the $n$-point amplitude can be expressed as Bern:1998sv

[TABLE]

where $h(a,M,b)$ are the “half-soft” functions of ref. Bern:1998sv . The summation is over pairs of legs $(a,b)$ and partitions $(M,N)$ of the remaining legs where both the sets $M$ and $N$ have at least one element. The half-soft functions we need for the five-point amplitude are

[TABLE]

When coupled minimally to additional bosons and fermions these amplitudes are multiplied by a factor $(N_{B}-N_{F})/2$ where $N_{B/F}$ is the total number of bosonic/fermionic degrees of freedom. A key feature of the one-loop amplitudes is that they are, to order $\epsilon^{0}$, rational functions and as such have no cuts in four dimensions. Thus if computing amplitudes using cuts in four dimensions they are indivisible and can be treated as a vertex. The only four dimensional cuts of the $n$-point all-plus amplitude are shown in fig. 1. For this helicity amplitude the only tree amplitudes necessary to compute the cuts are the three point amplitudes and the Maximally-Helicity-Violating (MHV) tree amplitudes Berends:1988zp .

The four-dimensional calculation gives the coefficient of $I_{n}^{(1)}$ to be $M_{n}^{(1)}$ to leading order in $\epsilon$. As in the QCD case, we promote this to the all-$\epsilon$ form. The all-plus one-loop amplitudes in eq. (4) for four and five-points are known to all orders in $\epsilon$ Bern:1996ja ; Bern:1998sv

[TABLE]

where

[TABLE]

and $I_{n}[\mu_{0}^{2m}]$ are the $n$-point integral functions with $\mu_{0}^{2m}$ inserted where $\mu_{0}$ are the $-2\epsilon$ coordinates in dimensional regularisation, $\int d^{D}x=\int d^{4}xd^{-2\epsilon}\mu_{0}$. The superscripts denote the ordering and clustering of the external legs. These are related to scalar integrals in higher dimensions Bern:1995db ; Bern:1996ja ,

[TABLE]

III The Four-Point All-Plus Helicity Amplitude

The four point all-plus helicity amplitude has some significant simplifications. Specifically, the quadruple cuts vanish333In computing the quadruple cuts for a four-point amplitude the only non-vanishing product of cut amplitudes has alternating $MHV$ and $\overline{MHV}$ three-point vertices at the corners. This precludes any box functions for the four-point all-plus amplitude. and there are only one-mass triangle and bubble contributions. In fact this amplitude is sufficiently simple that using the one loop amplitude as a vertex both the triangle and bubble functions can be obtained simply from the two-particle cuts Dunbar:1994bn with the result,

[TABLE]

This expression contains $\epsilon^{-1}$ and the $\ln(\mu^{2})$ terms in the combination,

[TABLE]

If gravity is coupled to matter with $N_{B}-2$ additional bosonic degrees of freedom and $N_{F}$ fermionic degrees of freedom, the one-loop all-plus amplitude is multiplied by a factor of $(N_{B}-N_{F})/2$ and the calculation follows through as above. The all-plus amplitude in this theory is thus the expression above with the replacement

[TABLE]

which matches the result of Bern:2015xsa . Consequently, we note that four dimensional unitarity gives the correct ${\it renormalised}$ amplitude up to rational terms although, as was known in ref Dunbar:1994bn , the coefficient of $\epsilon^{-1}$ does not match the field theory calculation.

IV The Five Point All-Plus Amplitude

This amplitude contains functions, particularly dilogarithms, that are not present in the four-point amplitude. These are contained in the box contributions shown in fig. (2). The box contribution is readily evaluated using a quadruple cut BrittoUnitarity .

With the labelling of fig. 2 the cut momenta are

[TABLE]

giving the coefficient of the box function

[TABLE]

This is the coefficient of the integral function $I_{4}^{\rm 1m}(s_{cd},s_{de},s_{ab})$ where Bern:1993kr

[TABLE]

and overall factors have been removed according to the normalisation of eq. (3).

This integral function splits into singular terms plus a remainder $I_{4}^{\rm 1m}=I_{4}^{\rm 1m:IR}+I_{4}^{\rm 1m:F}$ where

[TABLE]

The one-mass integral function is

[TABLE]

and the two-mass triangle function is,

[TABLE]

The boxes, one-mass and two-mass triangles all have IR infinite terms of the form

[TABLE]

The coefficients of the triangle contributions can be evaluated using triple cuts Bidder:2005ri ; Darren ; BjerrumBohr:2007vu ; Mastrolia:2006ki and a canonical basis Dunbar:2009ax . Summation over the box and triangle contributions gives an overall coefficient of $M^{(1),\epsilon^{0}}_{5}(a^{+},b^{+},c^{+},d^{+},e^{+})$, i.e

[TABLE]

where $M^{(1),\epsilon^{0}}_{5}(a^{+},b^{+},c^{+},d^{+},e^{+})$ is the order $\epsilon^{0}$ truncation of the one-loop amplitude. A key step is to promote the coefficient of these terms to the all-$\epsilon$ form of the one-loop amplitude which then gives the correct singular structure of the amplitude. We have confirmed relationship (28) for the $n$-point amplitude by computing the triple and quadruple cuts at specific kinematic points up to $n=10$.

Consequently, we can obtain the compact explicit analytic form for the dilogarithmic remainder part of the amplitude

[TABLE]

where the permutation sum is over the 30 independent permutations of the legs $(\{a,b\},c,d,e)$ after factoring out for the symmetries $(\{a,b\},c,d,e)\equiv(\{b,a\},c,d,e)\equiv(\{a,b\},e,d,c)$.

Note that the coefficients in the $F_{5}^{(2)}$ term contain $\left\langle e\,c\right\rangle^{-4}$ singularities. On this singularity the integral function vanishes and $F_{5}^{(2)}$ has $\left\langle e\,c\right\rangle^{-3}$ singularities. These are spurious and not present in the full amplitude. They cancel against the $G_{5}^{(2)}$ terms as we will discuss at the end of the next section.

V Coefficient of $\ln(s/\mu^{2})$

We determine the presence of the $\ln(s_{ab}/\mu^{2})$ functions using two-particle cuts. The coefficient has two contributions as shown in fig.3.

We determine these using canonical forms. The canonical basis approach Dunbar:2009ax is a systematic method to determine the coefficients of triangle and bubble integral functions in a one-loop amplitude from the three and two-particle cuts. A two particle cut is of the form

[TABLE]

The product of tree amplitudes appearing in the two-particle cut can be decomposed in terms of canonical forms ${\cal H}_{i}$,

[TABLE]

where the $e_{i}$ are coefficients independent of $\ell_{j}$. We then use substitution rules to replace the ${\cal H}_{i}(\rho_{k},{\ell_{j}})$ by rational functions $H_{i}[\rho_{k},P]$ to obtain the coefficient of the bubble integral function as

[TABLE]

Here we use this technique treating the one-loop all-plus vertex as a tree amplitude.

The canonical forms we need and their substitutions are

[TABLE]

where $P$ is the cut-momentum $P=k_{a}+\cdots k_{b}$. We take the product of amplitudes in the cut and split them into a sum of terms of type in eqn.(33) using partial fractioning

[TABLE]

For the five-point amplitude the cuts have $P=k_{a}+k_{b}$. Working specifically with $s_{ab}=s_{45}$, the two-particle cut has two contributions

[TABLE]

From the term A we obtain a contribution to the coefficient of

[TABLE]

with

[TABLE]

where

[TABLE]

and

[TABLE]

Contributions from the second configuration are more complicated, but using relationships between the different terms to simplify the final expression we obtain

[TABLE]

with

[TABLE]

where

[TABLE]

and

[TABLE]

Thus the amplitude contains

[TABLE]

The full amplitude thus has $\epsilon^{-1}$ and $\ln(\mu^{2})$ terms of

[TABLE]

We have determined the above expression using four-dimensional unitarity which isolates the coefficient of $\ln(s_{ij})$. The value of $\ln(\mu^{2})$ then follows. The coefficient of $\epsilon^{-1}$ in this is tied to the $\ln(\mu^{2})$ but is, presumably, not the value which would be obtained from a field theory calculation.

The bubble coefficients contain spurious $\left\langle e\,c\right\rangle^{-3}$ singularities which must cancel Dunbar:2011xw against the singularities in the $F_{5}^{(2)}$ term. Specifically the singularities are of the form

[TABLE]

where the permutations are of of $a,b,d$. The $F_{5}^{(2)}$ term contains dilogarithms but near the point $\left\langle e\,c\right\rangle=0$ these simplify and

[TABLE]

We have explicitly checked that within $F_{5}^{(2)}+G_{5}^{(2)}$ both the $\left\langle e\,c\right\rangle^{-3}$ *and * $\left\langle e\,c\right\rangle^{-2}$ singularities cancel, leaving the full amplitude free of spurious singularities: this is a strong consistency check. We have also checked the collinear limit of the five-point amplitude.

VI Counterterm Lagrangian

In this section we enumerate the possible independent counterterms for pure gravity in four dimensions. In general, graviton scattering amplitudes, in $D$ dimensions at $L$ loops, require the introduction of counterterms of the form

[TABLE]

where $n+2m=(D-2)L+2$ and we have suppressed the indices on $R$. $R$ may stand either for the Riemann tensor, $R_{abcd}$, the Ricci tensor $R_{ab}\equiv g^{cd}R_{acbd}$ or the curvature scalar $R\equiv g^{ab}R_{ab}$. Although, there are a large number of tensor structures which may appear, fortunately, the symmetries of the Riemann tensor reduce these considerably. Firstly, there are the basic symmetries of $R_{abcd}$

[TABLE]

and the cyclic symmetry,

[TABLE]

Secondly, we have the Bianchi identity for $\nabla_{e}R_{abcd}$,

[TABLE]

There are also “derivative identities” which involve two covariant derivatives,

[TABLE]

These symmetries will be used to determine the minimal set of inequivalent counterterms.

From power counting the possible two-loop counterterms in $D=4$ are of the form $R^{3}$ or $\nabla^{2}R^{2}$. The independent terms involving $R_{abcd}$, $R_{ab}$ and $R$ are R3D4 ; Fulling:1992vm ,

[TABLE]

For the case of pure gravity, the counterterm structure can be represented as a single counterterm with a numerical coefficient. We review the argument leading to the conclusion that a single counterterm is sufficient. (When matter is coupled to gravity this is no longer the case.)

For pure gravity the equation of motion is

[TABLE]

Hence terms involving the Ricci tensor or curvature scalar will not contribute to the $S$-matrix and such terms can be discarded when calculating the counterterms. (If calculating an off-shell object, such counterterms can, and do, appear.) Ignoring such terms leaves us with three tensors - $T_{3}$, $T_{11}$ and $T_{12}$. The term $T_{3}$

[TABLE]

can be rearranged using the identity in eq. (54) into terms involving the Ricci tensor plus cubic terms in the Riemann tensor. Thus for pure gravity this term is equivalent to a combination of $T_{11}$ and $T_{12}$ and can be eliminated from the list of inequivalent counterterms.

Finally, in six dimensions the scalar topological density can be written

[TABLE]

which implies that the combination

[TABLE]

is topological for some coefficients $a_{i}$ in dimensions $D\leq 6$. Hence for pure gravity amplitudes we can replace $T_{12}$ with $T_{11}$ (or vice versa). Thus we are led to the fact that the counterterm can be taken as a single tensor with a coefficient. Thus the counterterm can be chosen to be

[TABLE]

with the free coefficient $C_{R^{3}}$.

Computing with this Lagrangian, the parts of the four-point amplitudes proportional to $C_{R^{3}}$ are Dunbar:2002jx

[TABLE]

Comparing this to the coefficient of $\ln(\mu^{2})$ in eqn. (18) we find

[TABLE]

We also require the five point amplitude computed using the above Lagrangian. Perturbative gravity calculations based upon Feynman diagrams are notoriously difficult, however we can compute the higher point functions using recursion.

The original BCFW shift Britto:2005fq ,

[TABLE]

does not lead to an expression with the correct symmetry, however the shift Risager:2005vk ; BjerrumBohr:2005jr

[TABLE]

where $\lambda_{\eta}$ is an arbitrary spinor does. Using this shift we can obtain an expression for the five-point amplitude. The amplitude has two factorisations,

[TABLE]

with six terms of the first type and three of the second. With $(a,b,c,d,e)=(1,2,3,4,5)$ we find that the first term gives

[TABLE]

which is symmetric under $2\leftrightarrow 3$. The second is

[TABLE]

which is symmetric under $2\leftrightarrow 3$ and $4\leftrightarrow 5$. The normalisation is

[TABLE]

The resultant contribution to the amplitude is

[TABLE]

where the summation is over the six independent $T^{A}$ and the three independent $T^{B}$.

The expression for ${\cal M}^{R^{3}}_{5}(1^{+},2^{+},3^{+},4^{+},5^{+})$ is

$\bullet$ Fully crossing symmetric between external legs

$\bullet$ Independent of the spinor $\eta$

These are strong indicators that we have computed the correct expression. We have checked this construction for the $n$-point all-plus amplitude up to $n=8$.

Additionally

$\bullet$ As $z\longrightarrow\infty$ for the BCFW shift the amplitude does not vanish but behaves as $z^{2}$. This is the reason why the shift (68) does not generate this amplitude.

$\bullet$ The expression has soft limits with

[TABLE]

where $S^{(i)}$ are the leading, sub-leading and sub-sub-leading soft operators SoftTheorems . As a two-loop amplitude, there is a possibility that the sub-sub-leading would not satisfy this so we regard this as a feature of the constructed amplitudes rather than as a necessary constraint.

Comparing expression (74) with (45) we find

[TABLE]

and we therefore obtain $C_{R^{3}}=-1/4$. The counterterm is thus consistent with that required for the four point amplitude.

VII Conclusion

Computing quantum gravity amplitudes is notoriously difficult. Only a small number of on-shell scattering amplitudes have been computed analytically. For pure gravity only the four and five-point one-loop amplitudes have been presented for all helicity configurations with all-$n$ expressions for the all-plus and single-minus amplitudes. Progress beyond one-loop has been confined to theories which are supersymmetric where the enhanced symmetries significantly simplify the amplitudes.

In this article we have shown how four dimensional cutting techniques allow us to compute large and interesting parts of two-loop pure gravity amplitudes and have obtained the (poly)logarithmic parts of the all-plus helicity amplitude for four and five-points in compact analytic forms. We also obtain the associated $\ln(\mu^{2})$ terms which as argued in ref. Bern:2015xsa determine the non-renormalisability of the amplitudes. We have matched these to the same $R^{3}$ counterterm for both the four and five-point amplitude. Given that the $\ln(\mu^{2})$ terms are key to renormalisability, this technique provides a straightforward method to study the UV behaviour of gravity theories.

Our approach has been entirely based upon physical on-shell amplitudes and is very different from a field theory approach where the one-loop renormalisation uses “evanescent” operators Bern:2015xsa . We do not obtain the $\epsilon^{-1}$ term found there, but do reproduce the four-point renormalised amplitude and present a five-point amplitude correctly renormalised.

Note added As this article was been prepared ref. Bern:2017puu appeared where the four-point two-loop amplitude is studied using similar techniques.

VIII Acknowledgements

This work was supported by Science and Technology Facilities Council (STFC) grant ST/L000369/1.