Perturbative Quantum Gravity and its Relation to Gauge Theory

Since its inception, it has been clear that General Relativity has many striking similarities to gauge theories. Both are based on the idea of local symmetry and therefore share a number of formal properties. Nevertheless, their dynamical behavior can be quite different. While Maxwell electrodynamics describes a long-range force similar to the situation with gravity, the non-Abelian gauge theories used to describe the weak and strong nuclear forces have rather different behaviors. Quantum chromodynamics, which describes the strong nuclear forces, for example, exhibits confinement of particles carrying the non-Abelian gauge charges. Certainly, there is no obvious corresponding property for gravity. Moreover, consistent quantum gauge theories have existed for a half century, but as yet no satisfactory quantum field theory of gravity has been constructed; indeed, there are good arguments suggesting that it is not possible to do so. The structures of the Lagrangians are also rather different: The non-Abelian Yang Mills Lagrangian contains only up to four-point interactions while the Einstein-Hilbert Lagrangian contains infinitely many.

Despite these differences, string theory teaches us that gravity and gauge theories can, in fact, be unified. The Maldacena conjecture [94, 2], for example, relates the weak coupling limit of a gravity theory on an anti-de Sitter backround to a strong coupling limit of a special supersymmetric gauge field theory. There is also a long history of papers noting that gravity can be expressed as a gauging of Lorentz symmetry [135, 82, 78], as well as examples of non-trivial similarities between classical solutions of gravity and non-Abelian gauge theo ries [126]. In this review a different, but very general, relationship between the weak coupling limits of both gravity and gauge theories will be described. This relationship allows gauge theories to be used directly as an aid for computations in perturbative quantum gravity.

The KLT relations hold at the semi-classical level, i. e. with no quantum loops. In order to exploit the KLT relations in quantum gravity, one needs to completely reformulate the quantization process; the standard methods starting either from a Hamiltonian or a Lagrangian provide no obvious means of exploiting the KLT relations. There is, however, an alternative approach based on obtaining the quantum loop contributions directly from the semi-classical tree-level amplitudes by using D-dimensional unitarity [15, 16, 28, 20, 115]. These same methods have also been applied to non-trivial calculations in quantum chromodynamics (see e.g. Refs. [28, 21, 12]) and in supersymmetric gauge theories (see e.g. Refs. [15, 16, 29, 19]). In a sense, they provide a means for obtaining collections of quantum loop-level Feynman diagrams without direct reference to the underlying Lagrangian or Hamiltonian. The only inputs with this method are the D-dimensional tree-level scattering amplitudes. This makes the unitarity method ideally suited for exploiting the KLT relations.

An interesting application of this method of perturbatively quantizing gravity is as a tool for investigating the ultra-violet behavior of gravity field theories. Ultraviolet properties are one of the central issues of perturbative quantum gravity. The conventional wisdom that quantum field theories of gravity cannot possibly be fundamental rests on the apparent non-renormalizability of these theories. Simple power counting arguments strongly suggest that Einstein gravity is not renormalizable and therefore can be viewed only as a low energy effective field theory. Indeed, explicit calculations have established that non-supersymmetric theories of gravity with matter generically diverge at one loop [132, 43, 42], and pure gravity diverges at two loops [66, 136]. Supersymmetric theories are better behaved with the first potential divergence occurring at three loops [39, 81, 80]. However, no explicit calculations have as yet been performed to directly verify the existence of the three-loop supergravity divergences.

The method described here for quantizing gravity is well suited for addressing the issue of the ultraviolet properties of gravity because it relates overwhelmingly complicated calculations in quantum gravity to much simpler (though still complicated) ones in gauge theories. The first application was for the case of maximally supersymmetric gravity, which is expected to have the best ultraviolet properties of any theory of gravity. This analysis led to the surprising result that maximally supersymmetric gravity is less divergent [19] than previously believed based on power counting arguments [39, 81, 80]. This lessening of the power counting degree of divergence may be interpreted as an additional symmetry unaccounted for in the original analysis [128]. (The results are inconsistent, however, with an earlier suggestion [73] based on the speculated existence of an unconstrained covariant off-shell superspace for N = 8 supergravity, which in D = 4 implies finiteness up to seven loops. The non-existence of such a superspace was already noted a while ago [80].) The method also led to the explicit construction of the two-loop divergence in eleven-dimensional supergravity [19, 40, 41, 17]. More recently, it aided the study of divergences in type I supergravity theories [54] where it was noted that they factorize into products of gauge theory factors.

Other applications include the construction of infinite sequences of amplitudes in gravity theories. Given the complexity of gravity perturbation theory, it is rather surprising that one can obtain compact expressions for an arbitrary number of external legs, even for restricted helicity or spin configurations of the particles. The key for this construction is to make use of previously known sequences in quantum chromodynamics. At tree-level, infinite sequences of maximally helicity violating amplitudes have been obtained by directly using the KLT relations [10,14] and analogous quantum chromodynamics sequences. At one loop, by combining the KLT relations with the unitarity method, additional infinite sequences of gravity and super-gravity amplitudes have also been obtained [22, 23]. They are completely analogous to and rely on the previously obtained infinite sequences of one-loop gauge theory amplitudes [11, 15, 16]. These amplitudes turn out to be also intimately connected to those of self-dual Yang Mills [143, 53, 93, 92, 4, 30, 33] and gravity [108, 52, 109]. The method has also been used to explicitly compute two-loop supergravity amplitudes [19] in dimension D = 11, that were then used to check M-theory dualities [68].

Although the KLT relations have been exploited to obtain non-trivial results in quantum gravity theories, a derivation of these relations from the Einstein-Hilbert Lagrangian is lacking. There has, however, been some progress in this regard. It turns out that with an appropriate choice of field variables one can separate the space-time indices appearing in the Lagrangian into ‘left’ and ‘right’ classes [124, 123, 125, 26], mimicking the similar separation that occurs in string theory. Moreover, with further field redefinitions and a non-linear gauge choice, it is possible to arrange the off-shell three-graviton vertex so that it is expressible in terms of a sum of squares of Yang-Mills three-gluon vertices [26]. It might be possible to extend this more generally starting from the formalism of Siegel [124, 123, 125], which contains a complete gravity Lagrangian with the required factorization of space-time indices.

This review is organized as follows. In Section 2 the Feynman diagram approach to perturbative quantum gravity is outlined. The Kawai, Lewellen, and Tye relations between open and closed string tree amplitudes and their field theory limit are described in Section 3. Applications to understanding and constructing tree-level gravity amplitudes are also described in this section. In Section 4 the implications for the Einstein-Hilbert Lagrangian are presented. The procedure for obtaining quantum loop amplitudes from gravity tree amplitudes is then given in Section 5. The application of this method to obtain quantum gravity loop amplitudes is described in Section 6. In Section 7 the quantum divergence properties of maximally supersymmetric supergravity obtained from this method are described. The conclusions are found in Section 8.

There are a number of excellent sources for various subtopics described in this review. For a recent review of the status of quantum gravity the reader may consult the article by Carlip [31]. The conventional Feynman diagram approach to quantum gravity can be found in the Les Houches lectures of Veltman [138]. A review article containing an early version of the method described here of using unitarity to construct complete loop amplitudes is ref. [20]. Excellent reviews containing the quantum chromodynamics amplitudes used to obtain corresponding gravity amplitudes are the ones by Mangano and Parke [99], and by Lance Dixon [48]. These reviews also provide a good description of helicity techniques which are extremely useful for explicitly constructing scattering amplitude in gravity and gauge theories. Broader textbooks describing quantum chromodynamics are Refs. [107, 141, 58]. Chapter 7 of Superstring Theory by Green, Schwarz, and Witten [70] contains an illuminating discussion of the relationship of closed and open string tree amplitudes, especially at the four-point level. A somewhat more modern description of string theory may be found in the book by Polchinski [110, 111]. Applications of string methods to quantum field theory are described in a recent review by Schubert [120].

Our starting point for constructing perturbative quantum gravity is the Kawai, Lewellen, and Tye (KLT) relations [85] between closed and open string tree-level amplitudes. Since closed string theories are theories of gravity and open string theories include gauge bosons, in the low energy limit, where string theory reduces to field theory, these relations then necessarily imply relations between gravity and gauge theories. The realization that ordinary gauge and gravity field theories emerge from the low energy limit of string theories has been appreciated for nearly three decades. (See, for example, Refs. [144, 118, 117, 69, 70, 110, 111].)

3.2 The KLT relations in field theory

The fact that the KLT relations hold for the extensive variety of compactified string models [102, 103, 49, 50, 86, 3] implies that they should also be generally true in field theories of gravity. For the cases of four- and five-particle scattering amplitudes, in the field theory limit the KLT relations [85] reduce to:

$$M_4^{tree}(1,2,3,4) = - i{s_{12}}A_4^{tree}(1,2,3,4)\;A_4^{tree}(1,2,3,4),$$

(10)

$$\begin{array}{*{20}{c}} {M_5^{tree}(1,2,3,4,5) = - i{s_{12}}{s_{34}}A_5^{tree}(1,2,3,4,5)\;A_5^{tree}(2,1,4,3,5)\;\;} \\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + i{s_{13}}{s_{24}}A_5^{tree}(1,2,3,4,5)\;A_5^{tree}(3,1,4,2,5)\,,} \end{array}$$

(11)

where the M _n ’s are tree-level amplitudes in a gravity theory, the A _n ’s are color-stripped tree-level amplitudes in a gauge theory, and s _ij ≡ (k _i + k _j )². In these equations the polarization and momentum labels are suppressed, but the label “j = 1, ..., n” is kept to distinguish the external legs. The coupling constants have been removed from the amplitudes, but are reinserted below in Eqs. (12) and (13). An explicit generalization to n-point field theory gravity amplitudes may be found in Appendix A of Ref. [23]. The KLT relations before the field theory limit is taken may, of course, be found in the original paper [85].

The KLT equations generically hold for any closed string states, using their Fock space factorization into pairs of open string states. Although not obvious, the gravity amplitudes (10) and (11) have all the required symmetry under interchanges of identical particles. (This is easiest to demonstrate in string theory by making use of an SL(2, Z) symmetry on the string world sheet.)

In the field theory limit the KLT equations must hold in any dimension, because the gauge theory amplitudes appearing on the right-hand side have no explicit dependence on the space-time dimension; the only dependence is implicit in the number of components of momenta or polarizations. Moreover, if the equations hold in, say, ten dimensions, they must also hold in all lower dimensions since one can truncate the theory to a lower-dimensional subspace.

The amplitudes on the left-hand side of Eqs. (10) and (11) are exactly the scattering amplitudes that one obtains via standard gravity Feynman rules [44, 45, 138]. The gauge theory amplitudes on the right-hand side may be computed via standard Feynman rules available in any modern textbook on quantum field theory [107, 141]. After computing the full gauge theory amplitude, the color-stripped partial amplitudes A _n appearing in the KLT relations (10) and (11), may then be obtained by expressing the full amplitudes in a color trace basis [8, 90, 98, 99, 48]:

$$\mathcal{A}_n^{tree}(1,2,...\;n) = {g^{(n - 2)}}\sum\limits_\sigma {Tr\left( {{T^{{a_{\sigma (1)}}}}...\;{T^{{a_{\sigma (n)}}}}} \right)A_n^{tree}(\sigma (1),\;...,\;\sigma (n)),} $$

(12)

where the sum runs over the set of all permutations, but with cyclic rotations removed and g as the gauge theory coupling constant. The A _n partial amplitudes that appear in the KLT relations are defined as the coefficients of each of the independent color traces. In this formula, the T ^ai are fundamental representation matrices for the Yang-Mills gauge group SU(N _c ), normalized so that Tr (T ^a T ^b ) - δ ^ab . Note that the A _n are completely independent of the color and are the same for any value of N _c . Eq. (12) is quite similar to the way full open string amplitudes are expressed in terms of the string partial amplitudes by dressing them with Chan-Paton color factors [106].

Instead, it is somewhat more convenient to use color-ordered Feynman rules [99] since they directly give the A _n color-stripped gauge theory amplitudes appearing in the KLT equations. These Feynman rules are depicted in Fig. 6. When obtaining the partial amplitudes from these Feynman rules it is essential to order the external legs following the order appearing in the corresponding color trace. One may view the color-ordered gauge theory rules as a new set of Feynman rules for gravity theories at tree-level, since the KLT relations allow one to convert the obtained diagrams to tree-level gravity amplitudes [14] as shown in Fig. 6.

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To obtain the full amplitudes from the KLT relations in Eqs. (10), (11) and their n-point generalization, the couplings need to be reinserted. In particular, when all states couple gravitationally, the full gravity amplitudes including the gravitational coupling constant are:

$$\mathcal{M}_n^{tree}(1,\;...\;n) = {\left( {\frac{\kappa }{2}} \right)^{(n - 2)}}M_n^{tree}(1,\;...\;n),$$

(13)

where κ² = 32πG_N expresses the coupling κ in terms of Newton’s constant G_N. In general, the precise combination of coupling constants depends on how many of the interactions are gauge or other interactions and how many are gravitational.

For the case of four space-time dimensions, it is very convenient to use helicity representation for the physical states [36, 87, 142]. With helicity amplitudes the scattering amplitudes in either gauge or gravity theories are, in general, remarkably compact, when compared with expressions where formal polarization vectors or tensors are used. For each helicity, the graviton polarization tensors satisfy a simple relation to gluon polarization vectors:

$$\varepsilon _{\mu \nu }^ + = \varepsilon _\mu ^ + \varepsilon _\nu ^ + ,\;\;\;\;\;\;\;\;\;\;\;\varepsilon _{\mu \nu }^ - = \varepsilon _\mu ^ - \varepsilon _\nu ^ - .$$

(14)

The ε _μ ^± are essentially ordinary circular polarization vectors associated with, for example, circularly polarized light. The graviton polarization tensors defined in this way automatically are traceless, ε _μ ^±μ = 0, because the gluon helicity polarization vectors satisfy ε^± · ε^± = 0. They are also transverse, ε _μν ^± k ^ν = ε _μν ^± k ^μ = 0, because the gluon polarization vectors satisfy ε^± · k = 0, where k ^μ is the four momentum of either the graviton or gluon.

3.3 Tree-level applications

Using a helicity representation [36, 87, 142], Berends, Giele, and Kuijf (BGK) [10] were the first to exploit the KLT relations to obtain amplitudes in Einstein gravity. In quantum chromodynamics (QCD) an infinite set of helicity amplitudes known as the Parke-Taylor amplitudes [105, 9, 89] were already known. These maximally helicity violating (MHV) amplitudes describe the tree-level scattering of n gluons when all gluons but two have the same helicity, treating all particles as outgoing. (The tree amplitudes in which all or all but one of the helicities are identical vanish.) BGK used the KLT relations to directly obtain graviton amplitudes in pure Einstein gravity, using the known QCD results as input. Remarkably, they also were able to obtain a compact formula for n-graviton scattering with the special helicity configuration in which two legs are of opposite helicity from the remaining ones.

As a particularly simple example, the color-stripped four-gluon tree amplitude with two minus helicities and two positive helicities in QCD is given by

$$A_4^{tree}(1_g^ - ,2_g^ - ,3_g^ + ,4_g^ + ) = \tfrac{{{S_{12}}}}{{{S_{23}}}},$$

(15)

$$A_4^{tree}(1_g^ - ,2_g^ - ,4_g^ + ,3_g^ + ) = \tfrac{{{S_{12}}}}{{S24}},$$

(16)

where the g subscripts signify that the legs are gluons and the ± superscripts signify the helicities. With the conventions used here, helicities are assigned by treating all particles as outgoing. (This differs from another common choice which is to keep track of which particles are incoming and which are outgoing.) In these amplitudes, for simplicity, overall phases have been removed. The gauge theory partial amplitude in Eq. (15) may be computed using the color-ordered Feynman diagrams depicted in Fig. 7. The diagrams for the partial amplitude in Eq. (16) are similar except that the labels for legs 3 and 4 are interchanged. Although QCD contains fermion quarks, they do not contribute to tree amplitudes with only external gluon legs because of fermion number conservation; for these amplitudes QCD is entirely equivalent to pure Yang-Mills theory.

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The corresponding four-graviton amplitude follows from the KLT equation (10). After including the coupling from Eq. (13), the four-graviton amplitude is:

$$\mathcal{M}_4^{tree}(1_h^ - ,2_h^ - ,3_h^ + ,4_h^ + ) = {\left( {\frac{\kappa }{2}} \right)^2}{s_{12}}\; \times \;\frac{{{S_{12}}}}{{{S_{23}}}}\; \times \;\frac{{{S_{12}}}}{{{S_{24}}}},$$

(17)

where the subscript h signifies that the particles are gravitons and, as with the gluon amplitudes, overall phases are removed. As for the case of gluons, the ± superscripts signify the helicity of the graviton. This amplitude necessarily must be identical to the result for pure Einstein gravity with no other fields present, because any other states, such as an anti-symmetric tensor, dilaton, or fermion, do not contribute to n-graviton tree amplitudes. The reason is similar to the reason why the quarks do not contribute to pure glue tree amplitudes in QCD. These other physical states contribute only when they appear as an external state, because they couple only in pairs to the graviton. Indeed, the amplitude (17) is in complete agreement with the result for this helicity amplitude obtained by direct diagrammatic calculation using the pure gravity Einstein-Hilbert action as the starting Point [7] (taking into account the different Hilbert action as the starting point [ ] (and conventions for helicity).

The KLT relations are not limited to pure gravity amplitudes. Cases of gauge theory coupled to gravity have also been discussed in Ref. [14]. For example, by applying the Feynman rules in Fig. 6, one can obtain amplitudes for gluon amplitudes dressed with gravitons. A sampling of these, to leading order in the graviton coupling, is

$$\begin{array}{*{20}{c}} {{M_4}(1_g^ + ,2_g^ + ,3_h^ - ,4_h^ - ) = 0,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {{M_4}(1_g^ - ,2_g^ + ,3_h^ - ,4_h^ + ) = {{\left( {\frac{\kappa }{2}} \right)}^2}{{\left| {\frac{{s_{23}^5}}{{{s_{13}}s_{12}^2}}} \right|}^{1/2}},} \\ {{M_5}(1_g^ + ,2_{g,}^ + 3_g^ + ,4_g^ - ,5_h^ - ) = {g^2}\left( {\frac{\kappa }{2}} \right){{\left| {\frac{{s_{45}^4}}{{{s_{12}}{s_{23}}{s_{34}}{s_{41}}}}} \right|}^{1/2}},} \\ {{M_5}(1_g^ + ,2_g^ + ,3_h^ + ,4_h^ - ,5_h^ - ) = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}$$

(18)

for the coefficients of the color traces Tr [T^a1 ... T ^am ] following the ordering of the gluon legs. Again, for simplicity, overall phases are eliminated from the amplitudes. (In Ref. [14] mixed graviton matter amplitudes including the phases may be found.)

These formulae have been generalized to infinite sequences of maximally helicity-violating tree amplitudes for gluon amplitudes dressed by external gravitons. The first of these were obtained by Selivanov using a generating function technique [121]. Another set was obtained using the KLT relations to find the pattern for an arbitrary number of legs [14]. In doing this, it is extremely helpful to make use of the analytic properties of amplitudes as the momenta of various external legs become soft (i. e. k _i → 0) or collinear (i. e. k _i parallel to k _j ), as discussed in the next subsection.

Cases involving fermions have not been systematically studied, but at least for the case with a single fermion pair the KLT equations can be directly applied using the Feynman rules in Fig. 6, without any modifications. For example, in a supergravity theory, the scattering of a gravitino by a graviton is

$$\begin{array}{*{20}{c}} {\mathcal{M}_4^{tree}(1_{\tilde h}^ - ,2_h^ - ,3_h^ + ,4_{\tilde h}^ + ) = {{\left( {\frac{\kappa }{2}} \right)}^2}{s_{12}}{A_4}(1_g^ - ,2_g^ - ,3_g^ + ,4_g^ + ){A_4}(1_{\tilde g}^ - ,2_g^ - ,4_{\tilde g}^ + ,3_g^ + )} \\ { = {{\left( {\frac{\kappa }{2}} \right)}^2}{s_{12}}\; \times \;\frac{{{s_{12}}}}{{{s_{23}}}}\; \times \;\sqrt {\frac{{{s_{12}}}}{{{s_{24}}}},} } \end{array}$$

(19)

where the subscript \(\tilde h\) signifies a spin 3/2 gravitino and \(\tilde g\) signifies a spin 1/2 gluino. As a more subtle example, the scattering of fundamental representation quarks by gluons via graviton exchange also has a KLT factorization:

$$\begin{array}{*{20}{c}} {\mathcal{M}_4^{\operatorname{ex} }(1_q^{ - {i_1}},2_{\bar q}^{ + {{\bar i}_2}},3_g^{ - {a_3}},4_g^{ + {a_4}})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ { = {{\left( {\frac{\kappa }{2}} \right)}^2}{s_{12}}{A_4}(1_s^{{i_1}},2_s^{{{\bar i}_2}},3_Q^{ - {a_3}},4_{\bar Q}^{ + {a_4}}){A_4}(1_q^ - ,2_{\bar q}^ + ,4_{\bar Q}^ + ,3_Q^ - )} \\ { = {{\left( {\frac{\kappa }{2}} \right)}^2}\frac{{\sqrt {\left| {s_{13}^3{s_{23}}} \right|} }}{{{s_{12}}}}{\delta _{{i_1}}}^{{{\bar i}_2}}{\delta ^{{a_3}{a_4}}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}$$

(20)

where q and Q are distinct massless fermions. In this equation, the gluons are factorized into products of fermions. On the right-hand side the group theory indices (i₁, ī₂, a₃, a₄) are interpreted as global flavor indices but on the left-hand side they should be interpreted as color indices of local gauge symmetry. As a check, in Ref. [14], for both amplitudes (19) and (20), ordinary gravity Feynman rules were used to explicitly verify that the expressions for the amplitudes are correct.

Cases with multiple fermion pairs are more involved. In particular, for the KLT factorization to work in general, auxiliary rules for assigning global charges in the color-ordered amplitudes appear to be necessary. This is presumably related to the intricacies associated with fermions in string theory [62].

When an underlying string theory does exist, such as for the case of maximal supergravity discussed in Section 7, then the KLT equations necessarily must hold for all amplitudes in the field theory limit. The above examples, however, demonstrate that the KLT factorization of amplitudes is not restricted only to the cases where an underlying string theory exists.

3.4 Soft and collinear properties of gravity amplitudes from gauge theory

The analytic properties of gravity amplitudes as momenta become either soft (k _i → 0) or collinear (k _j parallel to k _j ) are especially interesting because they supply a simple demonstration of the tight link between the two theories. Moreover, these analytic properties are crucial for constructing and checking gravity amplitudes with an arbitrary number of external legs. The properties as gravitons become soft have been known for a long time [139, 10] but the collinear properties were first obtained using the known collinear properties of gauge theories together with the KLT relations.

Helicity amplitudes in quantum chromodynamics have a well-known behavior as momenta of external legs become collinear or soft [92, 20]. For the collinear case, at tree-level in quantum chromodynamics when two nearest neighboring legs in the color-stripped amplitudes become collinear, e.g., k₁ → zP, k₂ → (1 - z)P, and P = k₁ + k₂, the amplitude behaves as [99]:

$$A_n^{tree}(1,2,...,n)\;\mathop \to \limits^{{k_1}||{k_2}} \sum\limits_{\lambda = \pm } {Split_{ - \lambda }^{QCD}{\;^{tree}}(1,2)A_{n - 1}^{tree}({P^\lambda },3,...,\;n)} .$$

(21)

The function Split _-λ ^{QCD tree} (1, 2) is a splitting amplitude, and λ is the helicity of the intermediate state P. (The other helicity labels are implicit.) The contribution given in Eq. (21) is singular for k₁ parallel to k₂; other terms in the amplitude are suppressed by a power of \(\sqrt {{s_{12}}} \), which vanishes in the collinear limit, compared to the ones in Eq. (21). For the pure glue case, one such splitting amplitude is

$$Split_ - ^{QCD}{\;^{tree}}({1^ + },{2^ + }) = \frac{1}{{\sqrt {z(1 - z)} }}\frac{1}{{\left\langle {12} \right\rangle }},$$

(22)

where

$$\left\langle {jl} \right\rangle = \sqrt {{s_{ij}}} {e^{i\phi jl}},\;\;\;\;\;\;\;\;\;\;\;\left| {jl} \right| = - \sqrt {{s_{ij}}} {e^{ - i\phi jl}},$$

(23)

are spinor inner products, and φ _jl is a momentum-dependent phase that may be found in, for example, Ref. [99]. In general, it is convenient to express splitting amplitudes in terms of these spinor inner products. The ‘+’ and ‘−’ labels refer to the helicity of the outgoing gluons. Since the spinor inner products behave as \(\sqrt {{s_{ij}}} \), the splitting amplitudes develop square-root singularities in the collinear limits. If the two collinear legs are not next to each other in the color ordering, then there is no singular contribution, e.g. no singularity develops in A _n ^tree (1,2,3, ..., n) for k₁ collinear to k₃.

From the structure of the KLT relations it is clear that a universal collinear behavior similar to Eq. (21) must hold for gravity since gravity amplitudes can be obtained from gauge theory ones. The KLT relations give a simple way to determine the gravity splitting amplitudes, Split^{gravity tree}. The value of the splitting amplitude may be obtained by taking the collinear limit of two of the legs in, for example, the five-point amplitude. Taking k₁ parallel to k₂ in the five-point relation (11) and using Eq. (13) yields:

$$\mathcal{M}_5^{tree}(1,2,3,4,5)\;\mathop \to \limits^{{k_1}||{k_2}} \;\frac{\kappa }{2}\sum\limits_{\lambda = \pm } {Split_{ - \lambda }^{gravity\;tree}(1,2)\mathcal{M}_4^{tree}({P^\lambda },3,4,5),} $$

(24)

where

$$Spli{t^{gravity\;tree}}(1,2)\; = \; - {s_{12}}\; \times \;Spli{t^{QCD}}{\;^{tree}}(1,2)\; \times \;Spli{t^{QCD}}{\;^{tree}}(2,1).$$

(25)

More explicitly, using Eq. (22) then gives:

$$Split_ - ^{gravity}{\;^{tree}}({1^ + },{2^ + }) = \frac{{ - 1}}{{z(1 - z)}}\frac{{[12]}}{{\left\langle {12} \right\rangle }}.$$

(26)

Using the KLT relations at n-points, it is not difficult to verify that the splitting behavior is universal for an arbitrary number of external legs, i. e.:

$$\mathcal{M}_n^{tree}(1,2,...,\;n)\;\mathop \to \limits^{{k_1}||{k_2}} \;\frac{\kappa }{2}\sum\limits_{\lambda = \pm } {Split_{ - \lambda }^{gravity}{\;^{tree}}(1,2)} \;\mathcal{M}_{n - 1}^{tree}({P^\lambda },3,...,\;n).$$

(27)

(Since the KLT relations are not manifestly crossing-symmetric, it is simpler to check this formula for some legs being collinear rather than others; at the end all possible combinations of legs must give the same results, though.) The general structure holds for any particle content of the theory because of the general applicability of the KLT relations.

In contrast to the gauge theory splitting amplitude (22), the gravity splitting amplitude (26) is not singular in the collinear limit. The s₁₂ factor in Eq. (25) has canceled the pole. However, a phase singularity remains from the form of the spinor inner products given in Eq. (23), which distinguishes terms with the splitting amplitude from any others. In Eq. (23), the phase factor φ₁₂ rotates by 2π as \({\vec k_1}\) and \({\vec k_2}\) rotate once around their sum \({\vec P}\) as shown in Fig. 8. The ratio of spinors in Eq. (26) then undergoes a 4π rotation accounting for the angular-momentum mismatch of 2ħ between the graviton P⁺ and the pair of gravitons 1⁺ and 2⁺. In the gauge theory case, the terms proportional to the splitting amplitudes (21) dominate the collinear limit. In the gravitational formula (27), there are other terms of the same magnitude as [12]/〈12〉 as s₁₂ → 0. However, these non-universal terms do not acquire any additional phase as the collinear vectors \({\vec k_1}\) and \({\vec k_2}\) are rotated around each other. Thus, they can be separated from the universal terms. The collinear limit of any gravity tree amplitude must contain the universal terms given in Eq. (27) thereby putting a severe restriction on the analytic structure of the amplitudes.

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Even for the well-studied case of momenta becoming soft one may again use the KLT relation to extract the behavior and to rewrite it in terms of the soft behavior of gauge theory amplitudes. Gravity tree amplitudes have the well known behavior [139]

$$\mathcal{M}_n^{tree}(1,2,...,\;{n^ + })\;\mathop \to \limits^{{k_n} \to 0} \;\frac{\kappa }{2}\;{\mathcal{S}^{gravity}}({n^ + })\; \times \;\mathcal{M}_{n - 1}^{tree}(1,2,...,n - 1)$$

(28)

as the momentum of graviton n becomes soft. In Eq. (28) the soft graviton is taken to carry positive helicity; parity can be used to obtain the other helicity case.

One can obtain the explicit form of the soft factors directly from the KLT relations, but a more symmetric looking soft factor can be obtained by first expressing the three-graviton vertex in terms of a Yang-Mills three-vertex [26] (see Eq. (40)). This three-vertex can then be used to directly construct the soft factor. The result is a simple formula expressing the universal function describing soft gravitons in terms of the universal functions describing soft gluons [26]:

$${\mathcal{S}^{gravity}}({n^ + }) = \sum\limits_{i = 1}^{n - 1} {{s_{ni}}{\mathcal{S}^{QCD}}({q_l},{n^ + },i)\; \times \;{\mathcal{S}^{QCD}}({q_r},{n^ + },i)} ,$$

(29)

where

$${\mathcal{S}^{QCD}}(a,{n^ + },b) = \frac{{\left\langle {ab} \right\rangle }}{{\left\langle {an} \right\rangle \left\langle {nb} \right\rangle }}$$

(30)

is the eikonal factor for a positive helicity soft gluon in QCD labeled by n, and a and b are labels for legs neighboring the soft gluon. In Eq. (29) the momenta q _l and q _r are arbitrary null “reference” momenta. Although not manifest, the soft factor (29) is independent of the choices of these reference momenta. By choosing q _l = k₁ and q _r = k_n-1 the form of the soft graviton factor for k _n → 0 used in, for example, Refs. [10, 22, 23] is recovered. The important point is that in the form (29), the graviton soft factor is expressed directly in terms of the QCD gluon soft factor. Since the soft amplitudes for gravity are expressed in terms of gauge theory ones, the probability of emitting a soft graviton can also be expressed in terms of the probability of emitting a soft gluon.

One interesting feature of the gravitational soft and collinear functions is that, unlike the gauge theory case, they do not suffer any quantum corrections [23]. This is due to the dimensionful nature of the gravity coupling κ, which causes any quantum corrections to be suppressed by powers of a vanishing kinematic invariant. The dimensions of the coupling constant must be absorbed by additional powers of the kinematic invariants appearing in the problem, which all vanish in the collinear or soft limits. This observation is helpful because it can be used to put severe constraints on the analytic structure of gravity amplitudes at any loop order.

One of the key properties exhibited by the KLT relations (10) and (11) is the separation of graviton space-time indices into ‘left’ and ‘right’ sets. This is a direct consequence of the factorization properties of closed strings into open strings. Consider the graviton field, h _μν . We define the μ index to be a “left” index and the v index to be a “right” one. In string theory, the “left” space-time indices would arise from the world-sheet left-mover oscillator and the “right” ones from the right-mover oscillators. Of course, since h _μν is a symmetric tensor it does not matter which index is assigned to the left or to the right. In the KLT relations each of the two indices of a graviton are associated with two distinct gauge theories. For convenience, we similarly call one of the gauge theories the “left” one and the other the “right” one. Since the indices from each gauge theory can never contract with the indices of the other gauge theory, it must be possible to separate all the indices appearing in a gravity amplitude into left and right classes such that the ones in the left class only contract with left ones and the ones in the right class only with right ones.

This was first noted by Siegel, who observed that it should be possible to construct a complete field theory formalism that naturally reflects the left-right string theory factorization of space-time indices. In a set of remarkable papers [124, 123, 125], he constructed exactly such a formalism. With appropriate gauge choices, indices separate exactly into “right” and “left” categories, which do not contract with each other. This does not provide a complete explanation of the KLT relations, since one would still need to demonstrate that the gravity amplitudes can be expressed directly in terms of gauge theory ones. Nevertheless, this formalism is clearly a sensible starting point for trying to derive the KLT relations directly from Einstein gravity. Hopefully, this will be the subject of future studies, since it may lead to a deeper understanding of the relationship of gravity to gauge theory. A Lagrangian with the desired properties could, for example, lead to more general relations between gravity and gauge theory classical solutions.

Here we outline a more straightforward order-by-order rearrangement of the Einstein-Hilbert Lagrangian, making it compatible with the KLT relations [26]. A useful side-benefit is that this provides a direct verification of the KLT relations up to five points starting from the Einstein-Hilbert Lagrangian in its usual form. This is a rather non-trivial direct verification of the KLT relations, given the algebraic complexity of the gravity Feynman rules.

The above ideas represent some initial steps in reorganizing the Einstein-Hilbert Lagrangian so that it respects the KLT relations. An important missing ingredient is a derivation of the KLT equations starting from the Einstein-Hilbert Lagrangian (and also when matter fields are present).

In this section, the above discussion is extended to quantum loops through use of D-dimensional unitarity [15, 16, 28, 20, 115]. The KLT relations provide gravity amplitudes only at tree-level; D-dimensional unitarity then provides a means of obtaining quantum loop amplitudes. In perturbation theory this is tantamount to quantizing the theory since the complete scattering matrix can, at least in principle, be systematically constructed this way. Amusingly, this bypasses the usual formal apparatus [60, 61, 5, 79] associated with quantizing constrained systems. More generally, the unitarity method provides a way to systematically obtain the complete set of quantum loop corrections order-by-order in the perturbative expansion whenever the full analytic behavior of tree amplitudes as a function of D is known. It always works when the particles in the theory are all massless. The method is well tested in explicit calculations and has, for example, recently been applied to state-of-the-art perturbative QCD loop computations [21, 12].

In quantum field theory the S-matrix links initial and final states. A basic physical property is that the S matrix must be unitary [95, 91, 96, 35]: S†S = 1. In perturbation theory the Feynman diagrams describe a transition matrix T defined by S - 1 + iT, so that the unitarity condition reads

$$2\operatorname{Im} {T_{if}} = \sum\limits_j {T_{ij}^*{T_{jf}},} $$

(42)

where i and f are initial and final states, and the “sum” is over intermediate states j (and includes an integral over intermediate on-mass-shell momenta). Perturbative unitarity consists of expanding both sides of Eq. (42) in terms of coupling constants, g for gauge theory and κ for gravity, and collecting terms of the same order. For example, the imaginary (or absorptive) parts of one-loop four-point amplitudes, which is order κ⁴ in gravity, are given in terms of the product of two four-point tree amplitudes, each carrying a power of κ². This is then summed over all two-particle states that can appear and integrated over the intermediate phase space (see Fig. 9).

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This provides a means of obtaining loop amplitudes from tree amplitudes. However, if one were to directly apply Eq. (42) in integer dimensions one would encounter a difficulty with fully reconstructing the loop scattering amplitudes. Since Eq. (42) gives only the imaginary part one then needs to reconstruct the real part. This is traditionally done via dispersion relations, which are based on the analytic properties of the S matrix [95, 91, 96, 35]. However, the dispersion integrals do not generally converge. This leads to a set of subtraction ambiguities in the real part. These ambiguities are related to the appearance of rational functions with vanishing imaginary parts R(s _i ), where the s _i are the kinematic variables for the amplitude.

A convenient way to deal with this problem [15, 16, 28, 20, 115] is to consider unitarity in the context of dimensional regularization [131, 137]. By considering the amplitudes as an analytic function of dimension, at least for a massless theory, these ambiguities are not present, and the only remaining ambiguities are the usual ones associated with renormalization in quantum field theory. The reason there can be no ambiguity relative to Feynman diagrams follows from simple dimensional analysis for amplitudes in dimension D = 4 - 2ε. With dimensional regularization, amplitudes for massless particles necessarily acquire a factor of (−s _i )^−ε for each loop, from the measure \(\int {{d^D}L} \). For small ε, \({( - {s_i})^{ - \epsilon}}R({s_i}) = R({s_i}) -\epsilon \ln ( - {s_i})R({s_i}) + \mathcal{O}({\epsilon^2})\), so every term has an imaginary part (for some s _i > 0), though not necessarily in terms which survive as ε → 0. Thus, the unitarity cuts evaluated to \(\mathcal{O}({\epsilon})\) provide sufficient information for the complete reconstruction of an amplitude. Furthermore, by adjusting the specific rules for the analytic continuation of the tree amplitudes to D-dimensions one can obtain results in the different varieties of dimensional regularization, such as the conventional one [34], the t’ Hooft-Veltman scheme [131], dimensional reduction [122], and the four-dimensional helicity scheme [27, 13].

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.

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

$$\begin{array}{*{20}{c}} {\sum\limits_{states} {\mathcal{A}_4^{tree}(1,2, - {L_2}, - {L_1})\; \times \;\mathcal{A}_4^{tree}({L_1},{L_2}, - {L_3}, - {L_4})} } \\ {{{\left. { \times \;\mathcal{A}_4^{tree}({L_4},{L_3},3,4)} \right|}_{2 \times 2 - cut}},} \end{array}$$

(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.

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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^−2ε 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 one- and two-loop gauge theory amplitudes [15, 16, 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.

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, 84, 100], 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 s₁₂ channel unitarity cut depicted in Fig. 9 is

$$\mathcal{M}_4^{tree}( - L_1^s,1_h^ + ,2_h^ + ,L_3^s)\; \times \;\mathcal{M}_4^{tree}( - L_3^s,3_h^ + ,4_h^ + ,L_1^s),$$

(45)

where the superscript s indicates that the cut lines are scalars. The h subscripts on legs 1...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

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

(46)

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

(47)

The external gluon momenta are four-dimensional, but the scalar momenta L₁ and L₃ are D-dimensional since they will form the loop momenta. In general, loop momenta will have a non-vanishing (−2ε)-dimensional component \(\vec \mu \), with \(\vec \mu \cdot \vec \mu = {\mu ^2} > 0\). The factors of μ² 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

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

(48)

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

$$\begin{array}{*{20}{c}} {\int {\frac{{{d^D}{L_1}}}{{{{(2\pi )}^D}}}{\mu ^8}\frac{i}{{L_1^2}}\left[ {\frac{i}{{{{({L_1} - {k_1})}^2}}} + \frac{i}{{{{({L_1} - {k_2})}^2}}}} \right]} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ { \times \;\frac{i}{{{{({L_1} - {k_1} - {k_2})}^2}}}\left[ {\frac{i}{{{{({L_1} - {k_3})}^2}}} + \frac{i}{{{{({L_1} - {k_4})}^2}}}} \right].} \end{array}$$

(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

$$\begin{array}{*{20}{c}} {M_4^{1\;loop}(1_h^ + ,2_h^ + ,3_h^ + ,4_h^ + ) = 2\left( {\mathcal{I}_4^{1\;loop}[{\mu ^8}]({s_{12}},{s_{23}}) + \mathcal{I}_4^{1\;loop}[{\mu ^8}]({s_{12}},{s_{13}})} \right.} \\ {\left. { + \mathcal{I}_4^{1\;loop}[{\mu ^8}]({s_{23}},{s_{13}})} \right),} \end{array}$$

(50)

and where

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

(51)

is the box integral depicted in Fig. 11 with the external legs arranged in the order 1234. In Eq. (50) \(\mathcal{P}\) is μ⁸. 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.

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Since the factor of μ⁸ is of \(\mathcal{O}(\epsilon)\), the only non-vanishing contributions come where the e from the μ⁸ 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

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

(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.

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:

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

(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 [71, 104, 20], which provide a set of simple linear relations between the various contributions showing that they must be proportional to each other.

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 [129, 97, 80], 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.

7.2 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:

$$\begin{array}{*{20}{c}} {\mathcal{M}_4^{2\;loop}(1,2,3,4) = {{\left( {\frac{\kappa }{2}} \right)}^6}{s_{12}}{s_{23}}{s_{13}}\;M_4^{tree}(1,2,3,4)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ { \times \left( {s_{12}^2\mathcal{I}_4^{2\;loop,P}\left( {{s_{12}},{s_{23}}} \right) + s_{12}^2\mathcal{I}_4^{2\;loop,P}\left( {{s_{12}},{s_{13}}} \right)} \right.} \\ {\;\;\;\;\;\; + s_{12}^2\mathcal{I}_4^{2loop,NP}({s_{12}},{s_{23}}) + s_{12}^2\mathcal{I}_4^{2\;loop,NP}({s_{12}},{s_{13}})} \\ {\left. { + \;cyclic} \right),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}$$

(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, \(\mathcal{I}_4^{2\;\text{loop},\;{\text{P}}}({s_{12}},{s_{23}})\) and \(\mathcal{I}_4^{2\;\text{loop},\;{\text{NP}}}({s_{12}},{s_{23}})\), 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].

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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 in sufficient 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.

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This review described how the notion that gravity ∼ (gauge theory) × (gauge theory) can be exploited to develop a better understanding of perturbative quantum gravity. The Kawai-Lewellen-Tye (KLT) string theory relations [85] 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 [15, 16, 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, 128]. 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.