It is useful to view the unitarity-based technique as an alternate way of evaluating sets of ordinary Feynman diagrams by collecting together gauge-invariant sets of terms containing residues of poles in the integrands corresponding to those of the propagators of the cut lines. This gives a region of loop-momentum integration where the cut loop momenta go on shell and the corresponding internal lines become intermediate states in a unitarity relation. From this point of view, even more restricted regions of loop momentum integration may be considered, where additional internal lines go on mass shell. This amounts to imposing cut conditions on additional internal lines. In constructing the full amplitude from the cuts it is convenient to use unrestricted integrations over loop momenta, instead of phase space integrals, because in this way one can obtain simultaneously both the real and imaginary parts. The generalized cuts then allow one to obtain multi-loop amplitudes directly from combinations of tree amplitudes. As a first example, the generalized cut for a one-loop four-point amplitude in the channel carrying momentum $k_1+k_2$  , as shown in Fig.  9 , is given by

$$\begin{array}{c}{{\sum }_{states}\int \frac{d^D{L}_1}{(4\pi {)}^D}\frac{i}{L_1^2}{\mathcal{A}}_4^{tree}(-L_1,1,2,L_2)\frac{i}{L_2^2}{\mathcal{A}}_4^{tree}(-L_2,3,4,L_1)|}_{2-cut},\end{array}$$

(43)

where $L_2=L_1-k_1-k_2$  , and the sum runs over all physical states of the theory crossing the cut. In this generalized cut, the on-shell conditions $L_1^2=L_2^2=0$  are applied even though the loop momentum is unrestricted. In addition, any physical state conditions on the intermediate particles should also be included. The real and imaginary parts of the integral functions that do have cuts in this channel are reliably computed in this way. However, the use of the on-shell conditions inside the unrestricted loop momentum integrals does introduce an arbitrariness in functions that do not have cuts in this channel. Such integral functions should instead be obtained from cuts in the other two channels.

A less trivial two-loop example of a generalized “double” two particle cut is illustrated in panel (a) of Fig.  10 . The product of tree amplitudes appearing in this cut is

$$\begin{array}{ccc}{\sum }_{states}{\mathcal{A}}_4^{tree}(1,2,-L_2,-L_1)& \times & {\mathcal{A}}_4^{tree}(L_1,L_2,-L_3,-L_4)\end{array}$$

$$\begin{array}{ccc}& \times & {{\mathcal{A}}_4^{tree}(L_4,L_3,3,4)|}_{2\times 2-cut},\end{array}$$

(44)

where the loop integrals and cut propagators have been suppressed for convenience. In this expression the on-shell conditions $L_i^2=0$  are imposed on the $L_i$  , $i=1,2,3,4$  appearing on the right-hand side. This double cut may seem a bit odd from the traditional viewpoint in which each cut can be interpreted as the imaginary part of the integral. It should instead be understood as a means to obtain part of the information on the structure of the integrand of the two-loop amplitude. Namely, it contains the information on all integral functions where the cut propagators are not cancelled. There are, of course, other generalized cuts at two loops. For example, in panel (b) of Fig.  10 , a different arrangement of the cut trees is shown.

Complete amplitudes are found by combining the various cuts into a single function with the correct cuts in all channels. This method works for any theory where the particles can be taken to be massless and where the tree amplitudes are known as an analytic function of dimension. The restriction to massless amplitudes is irrelevant for the application of studying the ultra-violet divergences of gravity theories. In any case, gravitons and their associated superpartners in a supersymmetric theory are massless. (For the case with masses present the extra technical complication has to do with the appearance of functions such as $m^{-2\epsilon }$  which have no cuts in any channel. See Ref. [28] for a description and partial solution of this problem.) This method has been extensively applied to the case of oneand two-loop gauge theory amplitudes [16, 17, 20, 21, 12] and has been carefully cross-checked with Feynman diagram calculations. Here, the method is used to obtain loop amplitudes directly from the gravity tree amplitudes given by the KLT equations. In the next section an example of how the method works in practice for the case of gravity is provided.

6 Gravity Loop Amplitudes from Gauge Theory

The unitarity method provides a natural means for applying the KLT formula to obtain loop amplitudes in quantum gravity, since the only required inputs are tree-level amplitudes valid for $D$  -dimensions; this is precisely what the KLT relations provide.

Although Einstein gravity is almost certainly not a fundamental theory, there is no difficulty in using it as an effective field theory [140, 64, 51, 85, 101] , in order to calculate quantum loop corrections.

The particular examples discussed in this section are completely finite and therefore do not depend on a cutoff or on unknown coefficients of higher curvature terms in the low energy effective action.

They are therefore a definite low energy prediction of any fundamental theory of gravity whose low energy limit is Einstein gravity. (Although they are definite predictions, there is, of course, no practical means to experimentally verify them.) The issue of divergences is deferred to Section  7 .

6.1 One-loop four-point example

As a simple example of how the unitarity method gives loop amplitudes, consider the one-loop amplitude with four identical helicity gravitons and a scalar in the loop [22, 23] . The product of tree amplitudes appearing in the $s_{12}$  channel unitarity cut depicted in Fig.  9 is

$$\begin{array}{c}{M}_4^{tree}(-L_1^s,1_h^{+},2_h^{+},L_3^s)\times M_4^{tree}(-L_3^s,3_h^{+},4_h^{+},L_1^s),\end{array}$$

(45)

where the superscript $s$  indicates that the cut lines are scalars. The $h$  subscripts on legs $1\dots 4$  indicate that these are gravitons, while the “ $+$  ” superscripts indicate that they are of plus helicity.

From the KLT expressions ( 10 ) the gravity tree amplitudes appearing in the cuts may be replaced with products of gauge theory amplitudes. The required gauge theory tree amplitudes, with two external scalar legs and two gluons, may be obtained using color-ordered Feynman diagrams and are

$$\begin{array}{ccc}{A}_4^{tree}(-L_1^s,1_g^{+},2_g^{+},L_3^s)& =& i\frac{{\mu }^2}{(L_1-k_1{)}^2},\end{array}$$

(46)

$$\begin{array}{ccc}{A}_4^{tree}(-L_1^s,1_g^{+},L_3^s,2_g^{+})& =& -i{\mu }^2\left[\frac{1}{(L_1-k_1{)}^2}+\frac{1}{(L_1-k_2{)}^2}\right].\end{array}$$

(47)

The external gluon momenta are four-dimensional, but the scalar momenta $L_1$  and $L_3$  are $D$  -dimensional since they will form the loop momenta. In general, loop momenta will have a non-vanishing $(-2\epsilon )$  -dimensional component $\vec{\mu }$  , with $\vec{\mu }\cdot \vec{\mu }={\mu }^2>0$  . The factors of ${\mu }^2$  appearing in the numerators of these tree amplitudes causes them to vanish as the scalar momenta are taken to be four-dimensional, though they are non-vanishing away from four dimensions. For simplicity, overall phases have been removed from the amplitudes. After inserting these gauge theory amplitudes in the KLT relation ( 10 ), one of the propagators cancels, leaving

$$\begin{array}{c}{M}_4^{tree}(-L_1^s,1_h^{+},2_h^{+},L_3^s)=-i{\mu }^4\left[\frac{1}{(L_1-k_1{)}^2}+\frac{1}{(L_1-k_2{)}^2}\right].\end{array}$$

(48)

For this cut, one then obtains a sum of box integrals that can be expressed as

$$\begin{array}{ccc}& \int \frac{d^D{L}_1}{(2\pi {)}^D}{\mu }^8& \frac{i}{L_1^2}\left[\frac{i}{(L_1-k_1{)}^2}+\frac{i}{(L_1-k_2{)}^2}\right]\end{array}$$

$$\begin{array}{ccc}& & \times \frac{i}{(L_1-k_1-k_2{)}^2}\left[\frac{i}{(L_1+k_3{)}^2}+\frac{i}{(L_1+k_4{)}^2}\right].\end{array}$$

(49)

By symmetry, since the helicities of all the external gravitons are identical, the other two cuts also give the same combinations of box integrals, but with the legs permuted.

The three cuts can then be combined into a single function that has the correct cuts in all channels yielding

$$\begin{array}{ccc}{M}_4^{1loop}(1_h^{+},2_h^{+},3_h^{+},4_h^{+})& =& 2({\mathcal{ℐ}}_4^{1loop}\left[{\mu }^8\right](s_{12},s_{23})+{\mathcal{ℐ}}_4^{1loop}\left[{\mu }^8\right](s_{12},s_{13})\end{array}$$

$$\begin{array}{ccc}& & +{\mathcal{ℐ}}_4^{1loop}\left[{\mu }^8\right](s_{23},s_{13})),\end{array}$$

(50)

and where

$$\begin{array}{c}{\mathcal{ℐ}}_4^{1loop}[\mathcal{P}](s_{12},s_{23})=\int \frac{d^{D}L}{(2\pi {)}^D}\frac{\mathcal{P}}{L^2(L-k_1{)}^2(L-k_1-k_2{)}^2(L+k_4{)}^2}\end{array}$$

(51)

is the box integral depicted in Fig.  11 with the external legs arranged in the order 1234. In Eq. ( 50 ) $\mathcal{P}$  is ${\mu }^8$  . The two other integrals that appear correspond to the two other distinct orderings of the four external legs. The overall factor of 2 in Eq. ( 50 ) is a combinatoric factor due to taking the scalars to be complex with two physical states.

Since the factor of ${\mu }^8$  is of $\mathcal{O}\left(\epsilon \right)$  , the only non-vanishing contributions come where the $\epsilon$  from the ${\mu }^8$  interferes with a divergence in the loop integral. These divergent contributions are relatively simple to obtain. After extracting this contribution from the integral, the final $D=4$  result for a complex scalar loop, after reinserting the gravitational coupling, is

$$\begin{array}{c}{\mathcal{ℳ}}_4^{1loop}(1_h^{+},2_h^{+},3_h^{+},4_h^{+})=\frac{1}{(4\pi {)}^2}{\left(\frac{\kappa }{2}\right)}^4\frac{s_{12}^2+s_{23}^2+s_{13}^2}{120},\end{array}$$

(52)

in agreement with a calculation done by a different method relying directly on string theory [55] .

(As for the previous expressions, the overall phase has been suppressed.)

Figure 11 : The one-loop box integral. Each internal line in the box corresponds to one of the four Feynman propagators in Eq. ( 51 ).

This result generalizes very simply to the case of any particles in the loop. For any theory of gravity, with an arbitrary matter content one finds:

$$\begin{array}{c}{\mathcal{ℳ}}_4^{1loop}(1_h^{+},2_h^{+},3_h^{+},4_h^{+})=\frac{N_s}{2}\frac{1}{(4\pi {)}^2}{\left(\frac{\kappa }{2}\right)}^4\frac{s_{12}^2+s_{23}^2+s_{13}^2}{120},\end{array}$$

(53)

where $N_s$  is the number of physical bosonic states circulating in the loop minus the number of fermionic states. The simplest way to demonstrate this is by making use of supersymmetry Ward identities [72, 104, 20] , which provide a set of simple linear relations between the various contributions showing that they must be proportional to each other.

6.2 Arbitrary numbers of legs at one loop

Surprisingly, the above four-point results can be extended to an arbitrary number of external legs. Using the unitarity methods, the fiveand six-point amplitudes with all identical helicity have also been obtained by direct calculation [22, 23] . Then by demanding that the amplitudes have the properties described in Section  3.4 for momenta becoming either soft [139, 10] or collinear [22] , an ansatz for the one-loop maximally helicity-violating amplitudes for an arbitrary number of external legs has also been obtained. These amplitudes were constructed from a set of building blocks called “half-soft-function”, which have “half ” of the proper behavior as gravitons become soft. The details of this construction and the explicit forms of the amplitudes may be found in Refs. [22, 23] .

The all-plus helicity amplitudes turn out to be very closely related to the infinite sequence of one-loop maximally helicity-violating amplitudes in $N=8$  supergravity. The two sequences are related by a curious “dimension shifting formula.” In Ref. [23] , a known dimension shifting formula [18] between identical helicity QCD and $N=4$  super-Yang–Mills amplitudes was used to obtain the four-, five-, and six-point $N=8$  amplitudes from the identical helicity gravity amplitudes using the KLT relations in the unitarity cuts. Armed with these explicit results, the soft and collinear properties were then used to obtain an ansatz valid for an arbitrary number of external legs [23] . This provides a rather non-trivial illustration of how the KLT relations can be used to identify properties of gravity amplitudes using known properties of gauge theory amplitudes.

Interestingly, the all-plus helicity amplitudes are also connected to self-dual gravity [108, 52, 109] and self-dual Yang–Mills [143, 53, 94, 93, 4, 30, 33] , i.e. gravity and gauge theory restricted to self-dual configurations of the respective field strengths, $R_{\mu \nu \rho \sigma }=\frac{i}{2}{{\epsilon }_{\mu \nu }}^{\alpha \beta }{R}_{\alpha \beta \rho \sigma }$  and $F_{\mu \nu }=\frac{i}{2}{{\epsilon }_{\mu \nu }}^{\alpha \beta }{F}_{\alpha \beta }$  , with ${\epsilon }_{0123}=+1$  . This connection is simple to see at the linearized (free field theory) level since a superposition of plane waves of identical helicity satisfies the self-duality condition. The self-dual currents and amplitudes have been studied at tree and one-loop levels [53, 4, 30, 33] . In particular, Chalmers and Siegel [33] have presented self-dual actions for gauge theory (and gravity), which reproduce the all-plus helicity scattering amplitudes at both tree and one-loop levels.

The ability to obtain exact expressions for gravity loop amplitudes demonstrates the utility of this approach for investigating quantum properties of gravity theories. The next section describes how this can be used to study high energy divergence properties in quantum gravity.

7 Divergence Properties of Maximal Supergravity

In general, the larger the number of supersymmetries, the tamer the ultraviolet divergences because of the tendency for these to cancel between bosons and fermions in a supersymmetric theory. In four-dimensions maximal $N=8$  supergravity may therefore be expected to be the least divergent of all possible supergravity theories. Moreover, the maximally supersymmetric gauge theory, $N=4$  super-Yang–Mills, is completely finite [128, 98, 81] , leading one to suspect that the superb ultraviolet properties of $N=4$  super-Yang–Mills would then feed into improved ultra-violet properties for $N=8$  supergravity via its relation to gauge theory. This makes the ultraviolet properties of $N=8$  supergravity the ideal case to investigate first via the perturbative relationship to gauge theory.

7.1 One-loop cut construction

The maximal $N=8$  supergravity amplitudes can be obtained by applying the KLT equations to express them in terms of maximally supersymmetric $N=4$  gauge theory amplitudes. For $N=8$  supergravity, each of the states of the multiplet factorizes into a tensor product of $N=4$  super-Yang–Mills states, as illustrated in Eq. ( 9 ). Applying the KLT equation ( 10 ) to the product of tree amplitudes appearing in the $s_{12}$  channel two-particle cuts yields:

$$\begin{array}{ccc}{\sum }_{\genfrac{}{}{0ex}{}{N=8}{\text{states}}}& & M_4^{tree}(-L_1,1,2,L_3)\times M_4^{tree}(-L_3,3,4,L_1)\end{array}$$

$$\begin{array}{ccc}& =& -s_{12}^2{\sum }_{\genfrac{}{}{0ex}{}{N=4}{\text{states}}}{A}_4^{tree}(-L_1,1,2,L_3)\times A_4^{tree}(-L_3,3,4,L_1)\end{array}$$

$$\begin{array}{ccc}& & \times {\sum }_{\genfrac{}{}{0ex}{}{N=4}{\text{states}}}{A}_4^{tree}(L_3,1,2,-L_1)\times A_4^{tree}(L_1,3,4,-L_3),\end{array}$$

(54)

where the sum on the left-hand side runs over all 256 states in the $N=8$  supergravity multiplet.

On the right-hand side the two sums run over the 16 states (ignoring color degrees of freedom) of the $N=4$  super-Yang–Mills multiplet: a gluon, four Weyl fermions and six real scalars.

The $N=4$  super-Yang–Mills tree amplitudes turn out to have a particularly simple sewing formula [29] ,

$$\begin{array}{ccc}& & {\sum }_{\genfrac{}{}{0ex}{}{N=4}{\text{states}}}{A}_4^{tree}(-L_1,1,2,L_3)\times A_4^{tree}(-L_3,3,4,L_1)\end{array}$$

$$\begin{array}{ccc}& & =-i{s}_{12}{s}_{23}{A}_4^{tree}(1,2,3,4)\frac{1}{(L_1-k_1{)}^2}\frac{1}{(L_3-k_3{)}^2},\end{array}$$

(55)

which holds in any dimension (though some care is required to maintain the total number of physical states at their four-dimensional values so as to preserve the supersymmetric cancellations).

The simplicity of this result is due to the high degree of supersymmetry.

Using the gauge theory result ( 55 ), it is a simple matter to evaluate Eq. ( 54 ). This yields:

$$\begin{array}{ccc}{\sum }_{\genfrac{}{}{0ex}{}{N=8}{\text{states}}}& & M_4^{tree}(-L_1,1,2,L_3)\times M_4^{tree}(-L_3,3,4,L_1)\end{array}$$

$$\begin{array}{ccc}& =& i{s}_{12}{s}_{23}{s}_{13}{M}_4^{tree}(1,2,3,4)\left[\frac{1}{(L_1-k_1{)}^2}+\frac{1}{(L_1-k_2{)}^2}\right]\end{array}$$

$$\begin{array}{ccc}& & \times \left[\frac{1}{(L_3-k_3{)}^2}+\frac{1}{(L_3-k_4{)}^2}\right].\end{array}$$

(56)

The sewing equations for the $s_{23}$  and $s_{13}$  kinematic channels are similar to that of the $s_{12}$  channel.

Applying Eq. ( 56 ) at one loop to each of the three kinematic channels yields the one-loop four graviton amplitude of $N=8$  supergravity,

$$\begin{array}{ccc}{\mathcal{ℳ}}_4^{1loop}(1,2,3,4)& =& -i{\left(\frac{\kappa }{2}\right)}^4{s}_{12}{s}_{23}{s}_{13}{M}_4^{tree}(1,2,3,4)\end{array}$$

$$\begin{array}{ccc}& & \times \left({\mathcal{ℐ}}_4^{1loop}(s_{12},s_{23})+{\mathcal{ℐ}}_4^{1loop}(s_{12},s_{13})+{\mathcal{ℐ}}_4^{1loop}(s_{23},s_{13})\right),\end{array}$$

$$\begin{array}{ccc}& & \end{array}$$

(57)

in agreement with previous results [69] . The gravitational coupling $\kappa$  has been reinserted into this expression. The scalar integrals are defined in Eq. ( 51 ), inserting $\mathcal{P}=1$  . This is a standard integral appearing in massless field theories; the explicit value of this integral may be found in many articles, including Refs. [69, 27] . This result actually holds for any of the states of $N=8$  supergravity, not just external gravitons. It is also completely equivalent to the result one obtains with covariant Feynman diagrams including Fadeev–Popov [59] ghosts and using regularization by dimensional reduction [122] . The simplicity of this result is due to the high degree of supersymmetry.

A generic one-loop four-point gravity amplitude can have up to eight powers of loop momenta in the numerator of the integrand; the supersymmetry cancellations have reduced it to no powers.

7.2 Higher loops

At two loops, the two-particle cuts are obtained easily by iterating the one-loop calculation, since Eq. ( 56 ) returns a tree amplitude multiplied by some scalar factors. The three-particle cuts are more difficult to obtain, but again one can “recycle” the corresponding cuts used to obtain the two-loop $N=4$  super-Yang–Mills amplitudes [29] . It turns out that the three-particle cuts introduce no other functions than those already detected in the two-particle cuts. After all the cuts are combined into a single function with the correct cuts in all channels, the $N=8$  supergravity two-loop amplitude [19] is:

$$\begin{array}{ccc}{\mathcal{ℳ}}_4^{2loop}(1,2,3,4)& =& {\left(\frac{\kappa }{2}\right)}^6{s}_{12}{s}_{23}{s}_{13}{M}_4^{tree}(1,2,3,4)\end{array}$$

$$\begin{array}{ccc}& & \times (s_{12}^2{\mathcal{ℐ}}_4^{2loop,P}(s_{12},s_{23})+s_{12}^2{\mathcal{ℐ}}_4^{2loop,P}(s_{12},s_{13})\end{array}$$

$$\begin{array}{ccc}& & +s_{12}^2{\mathcal{ℐ}}_4^{2loop,NP}(s_{12},s_{23})+s_{12}^2{\mathcal{ℐ}}_4^{2loop,NP}(s_{12},s_{13})\end{array}$$

$$\begin{array}{ccc}& & +\text{cyclic}),\end{array}$$

(58)

where “ $+$  cyclic” instructs one to add the two cyclic permutations of legs (2, 3, 4). The scalar planar and non-planar loop momentum integrals, ${\mathcal{ℐ}}_4^{2loop,P}(s_{12},s_{23})$  and ${\mathcal{ℐ}}_4^{2loop,NP}(s_{12},s_{23})$  , are depicted in Fig.  12 . In this expression, all powers of loop momentum have cancelled from the numerator of each integrand in much the same way as at one loop, leaving behind only the Feynman propagator denominators. The explicit values of the two-loop scalar integrals in terms of polylogarithms may be found in Refs. [127, 133] .

Figure 12 : The planar and non-planar scalar integrals, appearing in the two-loop $N=8$  amplitudes. Each internal line represents a scalar propagator.

The two-loop amplitude ( 58 ) has been used by Green, Kwon, and Vanhove [68] to provide an explicit demonstration of the non-trivial M-theory duality between $D=11$  supergravity and type II string theory. In this case, the finite parts of the supergravity amplitudes are important, particularly the way they depend on the radii of compactified dimensions.

A remarkable feature of the two-particle cutting equation ( 56 ) is that it can be iterated to all loop orders because the tree amplitude (times some scalar denominators) reappears on the right-hand side. Although this iteration is insufficient to determine the complete multi-loop four-point amplitudes, it does provide a wealth of information. In particular, for planar integrals it leads to the simple insertion rule depicted in Fig.  13 for obtaining the higher loop contributions from lower loop ones [19] . This class includes the contribution in Fig.  4 , because it can be assembled entirely from two-particle cuts. According to the insertion rule, the contribution corresponding to Fig.  4 is given by loop integrals containing the propagators corresponding to all the internal lines multiplied by a numerator factor containing 8 powers of loop momentum. This is to be contrasted with the 24 powers of loop momentum in the numerator expected when there are no supersymmetric cancellations. This reduction in powers of loop momenta leads to improved divergence properties described in the next subsection.

Figure 13 : Starting from an $l$  -loop planar diagram representing an integral function, an extra line may be added to the inside using this rule. The two lines on the left represent two lines in some $l$  -loop diagram.

7.3 Divergence properties of $\mathit{N}\mathbf{=}8$  supergravity

Since the two-loop $N=8$  supergravity amplitude ( 58 ) has been expressed in terms of scalar integrals, it is straightforward to extract the divergence properties. The scalar integrals diverge only for dimension $D\ge 7$  ; hence the two-loop $N=8$  amplitude is manifestly finite in $D=5$  and 6, contrary to earlier expectations based on superspace power counting [81] . The discrepancy between the above explicit results and the earlier superspace power counting arguments may be understood in terms of an unaccounted higher-dimensional gauge symmetry [129] . Once this symmetry is accounted for, superspace power counting gives the same degree of divergence as the explicit calculation.

The cutting methods provide much more than just an indication of divergence; one can extract the explicit numerical coefficients of the divergences. For example, near $D=7$  the divergence of the amplitude ( 58 ) is

$$\begin{array}{c}{{\mathcal{ℳ}}_4^{two-loop,D=7-2\epsilon }|}_{div.}=\frac{1}{2\epsilon (4\pi {)}^7}\frac{\pi }{3}(s_{12}^2+s_{23}^2+s_{13}^2){\left(\frac{\kappa }{2}\right)}^6\times s_{12}{s}_{23}{s}_{13}{M}_4^{tree},\end{array}$$

(59)

which clearly diverges when the dimensional regularization parameter $\epsilon \to 0$  .

In all cases the linearized divergences take the form of derivatives acting on a particular contraction of Riemann tensors, which in four dimensions is equivalent to the square of the Bel–Robinson tensor [6, 37, 38] . This operator appears in the first set of corrections to the $N=8$  supergravity Lagrangian, in the inverse string-tension expansion of the effective field theory for the type II superstring [78] . Therefore, it has a completion into an $N=8$  supersymmetric multiplet of operators, even at the non-linear level. It also appears in the M-theory one-loop and two-loop effective actions [67, 116, 68] .

Interestingly, the manifest $D$  -independence of the cutting algebra allows the calculation to be extended to $D=11$  , even though there is no corresponding $D=11$  super-Yang–Mills theory. The result ( 58 ) then explicitly demonstrates that $N=1$  , $D=11$  supergravity diverges. In dimensional regularization there are no one-loop divergences so the first potential divergence is at two loops. (In a momentum cutoff scheme the divergences actually begin at one loop [116] .) Further work on the structure of the $D=11$  two-loop divergences in dimensional regularization has been carried out in Ref. [40, 41] . The explicit form of the linearized $N=1$  , $D=11$  counterterm expressed as derivatives acting on Riemann tensors along with a more general discussion of supergravity divergences may be found in Ref. [15] .

Using the insertion rule of Fig.  13 , and counting the powers of loop momenta in these contributions leads to the simple finiteness condition

$$\begin{array}{c}l<\frac{10}{D-2}\end{array}$$

(60)

(with $l>1$  ), where $l$  is the number of loops. This formula indicates that $N=8$  supergravity is finite in some other cases where the previous superspace bounds suggest divergences [81] , e.g. $D=4$  , $l=3$  : The first $D=4$  counterterm detected via the two-particle cuts of four-point amplitudes occurs at five, not three loops. Further evidence that the finiteness formula is correct stems from the maximally helicity violating contributions to $m$  -particle cuts, in which the same supersymmetry cancellations occur as for the two-particle cuts [19] . Moreover, a recent improved superspace power count [129] , taking into account a higher-dimensional gauge symmetry, is in agreement with the finiteness formula ( 60 ). Further work would be required to prove that other contributions do not alter the two-particle cut power counting. A related open question is whether one can prove that the five-loop $D=4$  divergence encountered in the two-particle cuts does not somehow cancel against other contributions [32] because of some additional symmetry. It would also be interesting to explicitly demonstrate the non-existence of divergences after including all contributions to the three-loop amplitude. In any case, the explicit calculations using cutting methods do establish that at two loops maximally supersymmetric supergravity does not diverge in $D=5$  [19] , contrary to earlier expectations from superspace power counting [81] .

8 Conclusions

This review described how the notion that gravity $\sim$  (gauge theory) $\times$  (gauge theory) can be exploited to develop a better understanding of perturbative quantum gravity. The Kawai–Lewellen–Tye (KLT) string theory relations [86] give this notion a precise meaning at the semi-classical or tree-level.

Quantum loop effects may then be obtained by using $D$  -dimensional unitarity [16, 17, 28, 20] . In a sense, this provides an alternative method for quantizing gravity, at least in the context of perturbative expansions around flat space. With this method, gauge theory tree amplitudes are converted into gravity tree amplitudes which are then used to obtain gravity loop amplitudes. The ability to carry this out implies that gravity and gauge theory are much more closely related than one might have deduced by an inspection of the respective Lagrangians. Some concrete applications were also described, including the computation of the two-loop four-point amplitude in maximally supersymmetric supergravity. The result of this and related computations is that maximal supergravity is less divergent in the ultraviolet than had previously been deduced from superspace power counting arguments [19, 129] . For the case of $D=4$  , maximal supergravity appears to diverge at five instead of three loops. Another example for which the relation is useful is for understanding the behavior of gravitons as their momenta become either soft or collinear with the momenta of other gravitons. The soft behavior was known long ago [139] , but the collinear behavior is new. The KLT relations provide a means for expressing the graviton soft and collinear functions directly in terms of the corresponding ones for gluons in quantum chromodynamics. Using the soft and collinear properties of gravitons, infinite sequences of maximally helicity violating gravity amplitudes with a single quantum loop were obtained by bootstrapping [22, 23] from the four-, five-, and six-point amplitudes obtained by direct calculation using the unitarity method together with the KLT relations. Interestingly, for the case of identical helicity, the sequences of amplitudes turn out to be the same as one gets from self-dual gravity [108, 52, 109] .

There are a number of interesting open questions. Using the relationship of gravity to gauge theory one should be able to systematically re-examine the divergence structure of non-maximal theories. Some salient work in this direction may be found in Ref. [54] , where the divergences of Type I supergravity in $D=8,10$  were shown to split into products of gauge theory factors. More generally, it should be possible to systematically re-examine finiteness conditions order-by-order in the loop expansion to more thoroughly understand the divergences and associated non-renormalizability of quantum gravity.

An important outstanding problem is the lack of a direct derivation of the KLT relations between gravity and gauge theory tree amplitudes starting from their respective Lagrangians. As yet, there is only a partial understanding in terms of a “left-right” factorization of space-time indices [124, 123, 125, 26] , which is a necessary condition for the KLT relations to hold. A more complete understanding may lead to a useful reformulation of gravity where properties of gauge theories can be used to systematically understand properties of gravity theories and vice versa. Connected with this is the question of whether the heuristic notion that gravity is a product of gauge theories can be given meaning outside of perturbation theory.

In summary, the perturbative relations between gravity and gauge theory provide a new tool for understanding non-trivial properties of quantum gravity. However, further work will be required to unravel fully the intriguing relationship between the two theories.

9 Acknowledgments

The author thanks Abilio De Freitas, Aaron Grant, David Dunbar, David Kosower, Maxim Perelstein, Joel Rozowsky, Henry Wong, and especially Lance Dixon for collaboration on work described here and for sharing their insight into quantum gravity. The author also thanks Eduardo Guendelman for a number of interesting discussions on the Einstein–Hilbert action and its relation to gauge theory. This work was supported by the US Department of Energy under grant DE-FG03-91ER40662. References

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