1 Introduction
$$\begin{array}{c}\text{gravity}\sim \text{(gauge theory)}\times \text{(gauge theory)}.\end{array}$$ | (1) |
2 Outline of the Traditional Approach to Perturbative Quantum Gravity
2.1 Overview of gravity Feynman rules
$$\begin{array}{c}{P}_{{\mu}_{1}{\nu}_{1};{\mu}_{2}{\nu}_{2}}=\frac{1}{2}\left[{\eta}_{{\mu}_{1}{\mu}_{2}}{\eta}_{{\mu}_{2}{\nu}_{2}}+{\eta}_{{\mu}_{1}{\nu}_{2}}{\eta}_{{\nu}_{1}{\mu}_{2}}-\frac{2}{D-2}{\eta}_{{\mu}_{1}{\nu}_{1}}{\eta}_{{\mu}_{2}{\nu}_{2}}\right]\frac{i}{{k}^{2}+i\epsilon}.\end{array}$$ | (2) |
$$\begin{array}{ccc}{G}_{deDonder}^{{\mu}_{1}{\nu}_{1},{\mu}_{2}{\nu}_{2},{\mu}_{3}{\nu}_{3}}({k}_{1},{k}_{2},{k}_{3})& \sim & {k}_{1}\cdot {k}_{2}{\eta}^{{\mu}_{1}{\nu}_{1}}{\eta}^{{\mu}_{2}{\nu}_{2}}{\eta}^{{\mu}_{3}{\nu}_{3}}+{k}_{1}^{{\mu}_{3}}{k}_{2}^{{\nu}_{3}}{\eta}^{{\mu}_{1}{\mu}_{2}}{\eta}^{{\nu}_{1}{\nu}_{2}}\end{array}$$ |
$$\begin{array}{ccc}& & +\text{many other terms},\end{array}$$ | (3) |
2.2 Divergences in quantum gravity
^{ $\text{1}$ } This happens because field redefinitions exist that can be used to remove the potential divergences.
2.3 Gravity and gauge theory Feynman rules
$$\begin{array}{c}{G}_{factorizing}^{{\mu}_{1}{\nu}_{1},{\mu}_{2}{\nu}_{2},{\mu}_{3}{\nu}_{3}}({k}_{1},{k}_{2},{k}_{3})\sim {V}_{YM}^{{\mu}_{1}{\mu}_{2}{\mu}_{3}}({k}_{1},{k}_{2},{k}_{3}){V}_{YM}^{{\nu}_{1}{\nu}_{2}{\nu}_{3}}({k}_{1},{k}_{2},{k}_{3}),\end{array}$$ | (4) |
3 The Kawai–Lewellen–Tye Relations
3.1 The KLT relations in string theory
$$\begin{array}{c}{V}^{closed}={V}_{left}^{open}\times {\overline{V}}_{right}^{open}.\end{array}$$ | (5) |
$$\begin{array}{c}{A}_{4}^{open}=-\frac{1}{2}{g}^{2}\frac{\Gamma \left({\alpha}^{\prime}s\right)\Gamma \left({\alpha}^{\prime}t\right)}{\Gamma (1+{\alpha}^{\prime}s+{\alpha}^{\prime}t)}{\xi}^{A}{\xi}^{B}{\xi}^{C}{\xi}^{D}{K}_{ABCD}({k}_{1},{k}_{2},{k}_{3},{k}_{4}),\end{array}$$ | (6) |
$$\begin{array}{ccc}{M}_{4}^{closed}& =& {\kappa}^{2}sin\left(\frac{{\alpha}^{\prime}\pi t}{4}\right)\times \frac{\Gamma ({\alpha}^{\prime}s/4)\Gamma ({\alpha}^{\prime}t/4)}{\Gamma (1+{\alpha}^{\prime}s/4+{\alpha}^{\prime}t/4)}\times \frac{\Gamma ({\alpha}^{\prime}t/4)\Gamma ({\alpha}^{\prime}u/4)}{\Gamma (1+{\alpha}^{\prime}t/4+{\alpha}^{\prime}u/4)}\end{array}$$ |
$$\begin{array}{ccc}& & \times {\xi}^{A{A}^{\prime}}{\xi}^{B{B}^{\prime}}{\xi}^{C{C}^{\prime}}{\xi}^{D{D}^{\prime}}{K}_{ABCD}({k}_{1}/2,{k}_{2}/2,{k}_{3}/2,{k}_{4}/2)\end{array}$$ |
$$\begin{array}{ccc}& & \times {K}_{{A}^{\prime}{B}^{\prime}{C}^{\prime}{D}^{\prime}}({k}_{1}/2,{k}_{2}/2,{k}_{3}/2,{k}_{4}/2),\end{array}$$ | (7) |
$$\begin{array}{c}|\text{closed string state}\rangle =|\text{open string state}\rangle \otimes |\text{open string state}\rangle .\end{array}$$ | (8) |
$$\begin{array}{c}|\text{gravity theory state}\rangle =|\text{gauge theory state}\rangle \otimes |\text{gauge theory state}\rangle .\end{array}$$ | (9) |
3.2 The KLT relations in field theory
$$\begin{array}{c}{M}_{4}^{tree}(1,2,3,4)=-i{s}_{12}{A}_{4}^{tree}(1,2,3,4){A}_{4}^{tree}(1,2,4,3),\end{array}$$ | (10) |
$$\begin{array}{ccc}{M}_{5}^{tree}(1,2,3,4,5)& =& i{s}_{12}{s}_{34}{A}_{5}^{tree}(1,2,3,4,5){A}_{5}^{tree}(2,1,4,3,5)\end{array}$$ |
$$\begin{array}{ccc}& & +i{s}_{13}{s}_{24}{A}_{5}^{tree}(1,3,2,4,5){A}_{5}^{tree}(3,1,4,2,5),\end{array}$$ | (11) |
$$\begin{array}{c}{\mathcal{A}}_{n}^{tree}(1,2,\dots n)={g}^{(n-2)}{\sum}_{\sigma}Tr({T}^{{a}_{\sigma \left(1\right)}}\cdots {T}^{{a}_{\sigma \left(n\right)}}){A}_{n}^{tree}\left(\sigma \right(1),\dots ,\sigma (n\left)\right),\end{array}$$ | (12) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{n}^{tree}(1,\dots n)={\left(\frac{\kappa}{2}\right)}^{(n-2)}{M}_{n}^{tree}(1,\dots n),\end{array}$$ | (13) |
$$\begin{array}{c}{\varepsilon}_{\mu \nu}^{+}={\varepsilon}_{\mu}^{+}{\varepsilon}_{\nu}^{+},{\varepsilon}_{\mu \nu}^{-}={\varepsilon}_{\mu}^{-}{\varepsilon}_{\nu}^{-}.\end{array}$$ | (14) |
3.3 Tree-level applications
$$\begin{array}{c}{A}_{4}^{tree}({1}_{g}^{-},{2}_{g}^{-},{3}_{g}^{+},{4}_{g}^{+})=\frac{{s}_{12}}{{s}_{23}},\end{array}$$ | (15) |
$$\begin{array}{c}{A}_{4}^{tree}({1}_{g}^{-},{2}_{g}^{-},{4}_{g}^{+},{3}_{g}^{+})=\frac{{s}_{12}}{{s}_{24}},\end{array}$$ | (16) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{4}^{tree}({1}_{h}^{-},{2}_{h}^{-},{3}_{h}^{+},{4}_{h}^{+})={\left(\frac{\kappa}{2}\right)}^{2}{s}_{12}\times \frac{{s}_{12}}{{s}_{23}}\times \frac{{s}_{12}}{{s}_{24}},\end{array}$$ | (17) |
$$\begin{array}{c}\begin{array}{ccc}{M}_{4}({1}_{g}^{+},{2}_{g}^{+},{3}_{h}^{-},{4}_{h}^{-})& =& 0,\\ {M}_{4}({1}_{g}^{-},{2}_{g}^{+},{3}_{h}^{-},{4}_{h}^{+})& =& {\left(\frac{\kappa}{2}\right)}^{2}{\left|\frac{{s}_{23}^{5}}{{s}_{13}{s}_{12}^{2}}\right|}^{1/2},\\ {M}_{5}({1}_{g}^{+},{2}_{g}^{+},{3}_{g}^{+},{4}_{g}^{-},{5}_{h}^{-})& =& {g}^{2}\left(\frac{\kappa}{2}\right){\left|\frac{{s}_{45}^{4}}{{s}_{12}{s}_{23}{s}_{34}{s}_{41}}\right|}^{1/2},\\ {M}_{5}({1}_{g}^{+},{2}_{g}^{+},{3}_{h}^{+},{4}_{h}^{-},{5}_{h}^{-})& =& 0\end{array}\end{array}$$ | (18) |
$$\begin{array}{ccc}{\mathcal{\mathcal{M}}}_{4}^{tree}({1}_{\stackrel{~}{h}}^{-},{2}_{h}^{-},{3}_{h}^{+},{4}_{\stackrel{~}{h}}^{+})& =& {\left(\frac{\kappa}{2}\right)}^{2}{s}_{12}{A}_{4}({1}_{g}^{-},{2}_{g}^{-},{3}_{g}^{+},{4}_{g}^{+}){A}_{4}({1}_{\stackrel{~}{g}}^{-},{2}_{g}^{-},{4}_{\stackrel{~}{g}}^{+},{3}_{g}^{+})\end{array}$$ |
$$\begin{array}{ccc}& =& {\left(\frac{\kappa}{2}\right)}^{2}{s}_{12}\times \frac{{s}_{12}}{{s}_{23}}\times \sqrt{\frac{{s}_{12}}{{s}_{24}}},\end{array}$$ | (19) |
$$\begin{array}{ccc}& & {\mathcal{\mathcal{M}}}_{4}^{ex}({1}_{q}^{-{i}_{1}},{2}_{\overline{q}}^{+{\overline{\u0131}}_{2}},{3}_{g}^{-{a}_{3}},{4}_{g}^{+{a}_{4}})\end{array}$$ |
$$\begin{array}{ccc}& & ={\left(\frac{\kappa}{2}\right)}^{2}{s}_{12}{A}_{4}({1}_{s}^{{i}_{1}},{2}_{s}^{{\overline{\u0131}}_{2}},{3}_{Q}^{-{a}_{3}},{4}_{\overline{Q}}^{+{a}_{4}}){A}_{4}({1}_{q}^{-},{2}_{\overline{q}}^{+},{4}_{\overline{Q}}^{+},{3}_{Q}^{-})\end{array}$$ |
$$\begin{array}{ccc}& & ={\left(\frac{\kappa}{2}\right)}^{2}\frac{\sqrt{\left|{s}_{13}^{3}{s}_{23}\right|}}{{s}_{12}}{{\delta}_{{i}_{1}}}^{{\overline{\u0131}}_{2}}{\delta}^{{a}_{3}{a}_{4}},\end{array}$$ | (20) |
3.4 Soft and collinear properties of gravity amplitudes from gauge theory
$$\begin{array}{c}{A}_{n}^{tree}(1,2,\dots ,n{)}^{{k}_{1}\parallel {k}_{2}}\u27f6{\sum}_{\lambda =\pm}{\text{Split}}_{-\lambda}^{\text{QCD tree}}(1,2\left){A}_{n-1}^{\text{tree}}\right({P}^{\lambda},3,\dots ,n).\end{array}$$ | (21) |
$$\begin{array}{c}Spli{t}_{-}^{QCDtree}({1}^{+},{2}^{+})=\frac{1}{\sqrt{z(1-z)}}\frac{1}{\langle 12\rangle},\end{array}$$ | (22) |
$$\begin{array}{c}\langle jl\rangle =\sqrt{{s}_{ij}}{e}^{i{\phi}_{jl}},\left[jl\right]=-\sqrt{{s}_{ij}}{e}^{-i{\phi}_{jl}},\end{array}$$ | (23) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{5}^{tree}(1,2,3,4,5{)}^{{k}_{1}\parallel {k}_{2}}\u27f6\frac{\kappa}{2}{\sum}_{\lambda =\pm}Spli{t}_{-\lambda}^{gravitytree}(1,2\left){\mathcal{\mathcal{M}}}_{4}^{tree}\right({P}^{\lambda},3,4,5),\end{array}$$ | (24) |
$$\begin{array}{c}Spli{t}^{gravitytree}(1,2)=-{s}_{12}\times Spli{t}^{QCDtree}(1,2)\times Spli{t}^{QCDtree}(2,1).\end{array}$$ | (25) |
$$\begin{array}{c}Spli{t}_{-}^{gravitytree}({1}^{+},{2}^{+})=\frac{-1}{z(1-z)}\frac{\left[12\right]}{\langle 12\rangle}.\end{array}$$ | (26) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{n}^{tree}(1,2,\dots ,n{)}^{{k}_{1}\parallel {k}_{2}}\u27f6\frac{\kappa}{2}{\sum}_{\lambda =\pm}Spli{t}_{-\lambda}^{gravitytree}(1,2\left){\mathcal{\mathcal{M}}}_{n-1}^{tree}\right({P}^{\lambda},3,\dots ,n).\end{array}$$ | (27) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{n}^{tree}(1,2,\dots ,{n}^{+}{)}^{{k}_{n}\to 0}\u27f6\frac{\kappa}{2}{\mathcal{S}}^{gravity}({n}^{+})\times {\mathcal{\mathcal{M}}}_{n-1}^{tree}(1,2,\dots ,n-1)\end{array}$$ | (28) |
$$\begin{array}{c}{\mathcal{S}}^{gravity}\left({n}^{+}\right)={\sum}_{i=1}^{n-1}{s}_{ni}{\mathcal{S}}^{QCD}({q}_{l},{n}^{+},i)\times {\mathcal{S}}^{QCD}({q}_{r},{n}^{+},i),\end{array}$$ | (29) |
$$\begin{array}{c}{\mathcal{S}}^{QCD}(a,{n}^{+},b)=\frac{\langle ab\rangle}{\langle an\rangle \langle nb\rangle}\end{array}$$ | (30) |
4 The Einstein–Hilbert Lagrangian and Gauge Theory
$$\begin{array}{c}{L}_{EH}=\frac{2}{{\kappa}^{2}}\sqrt{-g}R,{L}_{YM}=-\frac{1}{4}{F}_{\mu \nu}^{a}{F}^{a\mu \nu},\end{array}$$ | (31) |
$$\begin{array}{c}{L}_{2}=-\frac{1}{2}{h}_{\mu \nu}{\partial}^{2}{h}^{\mu \nu}+\frac{1}{4}{{h}_{\mu}}^{\mu}{\partial}^{2}{{h}_{\nu}}^{\nu},\end{array}$$ | (32) |
$$\begin{array}{c}{L}_{EH}=\frac{2}{{\kappa}^{2}}\sqrt{-g}R+\sqrt{-g}{\partial}^{\mu}\phi {\partial}_{\mu}\phi .\end{array}$$ | (33) |
$$\begin{array}{c}{L}_{2}=-\frac{1}{2}{h}_{\mu \nu}{\partial}^{2}{h}^{\mu \nu}+\frac{1}{4}{{h}_{\mu}}^{\mu}{\partial}^{2}{{h}_{\nu}}^{\nu}-\phi {\partial}^{2}\phi .\end{array}$$ | (34) |
$$\begin{array}{c}{h}_{\mu \nu}\to {h}_{\mu \nu}+{\eta}_{\mu \nu}\sqrt{\frac{2}{D-2}}\phi \end{array}$$ | (35) |
$$\begin{array}{c}\phi \to \frac{1}{2}{{h}_{\mu}}^{\mu}+\sqrt{\frac{D-2}{2}}\phi ,\end{array}$$ | (36) |
$$\begin{array}{c}{L}_{2}\to -\frac{1}{2}{h}_{\nu}^{\mu}{\partial}^{2}{{h}_{\mu}}^{\nu}+\phi {\partial}^{2}\phi .\end{array}$$ | (37) |
$$\begin{array}{c}{{h}_{\mu}}^{\mu},{{h}_{\mu}}^{\nu}{{h}_{\nu}}^{\lambda}{{h}_{\lambda}}^{\mu},\cdots ,\end{array}$$ | (38) |
$$\begin{array}{c}{g}_{\mu \nu}={e}^{\sqrt{\frac{2}{D-2}}\kappa \phi}{e}^{\kappa {h}_{\mu \nu}}\equiv {e}^{\sqrt{\frac{2}{D-2}}\kappa \phi}\left({\eta}_{\mu \nu}+\kappa {h}_{\mu \nu}+\frac{{\kappa}^{2}}{2}{{h}_{\mu}}^{\rho}{h}_{\rho \nu}+\cdots \right).\end{array}$$ | (39) |
$$\begin{array}{ccc}i{G}^{{\mu}_{1}{\nu}_{1},{\mu}_{2}{\nu}_{2},{\mu}_{3}{\nu}_{3}}& & ({k}_{1},{k}_{2},{k}_{3})=\end{array}$$ |
$$\begin{array}{ccc}-\frac{i}{2}\left(\frac{\kappa}{2}\right)& & [{V}_{GN}^{{\mu}_{1}{\mu}_{2}{\mu}_{3}}({k}_{1},{k}_{2},{k}_{3})\times {V}_{GN}^{{\nu}_{1}{\nu}_{2}{\nu}_{3}}({k}_{1},{k}_{2},{k}_{3})\end{array}$$ |
$$\begin{array}{ccc}& & +{V}_{GN}^{{\mu}_{2}{\mu}_{1}{\mu}_{3}}({k}_{2},{k}_{1},{k}_{3})\times {V}_{GN}^{{\mu}_{2}{\mu}_{1}{\mu}_{3}}({k}_{2},{k}_{1},{k}_{3})],\end{array}$$ | (40) |
$$\begin{array}{c}{V}_{GN}^{\mu \nu \rho}({k}_{1},{k}_{2},{k}_{3})=i\sqrt{2}({k}_{1}^{\rho}{\eta}^{\mu \nu}+{k}_{2}^{\mu}{\eta}^{\nu \rho}+{k}_{3}^{\nu}{\eta}^{\rho \mu}),\end{array}$$ | (41) |
5 From Trees to Loops
$$\begin{array}{c}2Im{T}_{if}={\sum}_{j}{T}_{ij}^{*}{T}_{jf},\end{array}$$ | (42) |
$$\begin{array}{c}{{\sum}_{states}\int \frac{{d}^{D}{L}_{1}}{(4\pi {)}^{D}}\frac{i}{{L}_{1}^{2}}{\mathcal{A}}_{4}^{tree}(-{L}_{1},1,2,{L}_{2})\frac{i}{{L}_{2}^{2}}{\mathcal{A}}_{4}^{tree}(-{L}_{2},3,4,{L}_{1})|}_{2-cut},\end{array}$$ | (43) |
$$\begin{array}{ccc}{\sum}_{states}{\mathcal{A}}_{4}^{tree}(1,2,-{L}_{2},-{L}_{1})& \times & {\mathcal{A}}_{4}^{tree}({L}_{1},{L}_{2},-{L}_{3},-{L}_{4})\end{array}$$ |
$$\begin{array}{ccc}& \times & {{\mathcal{A}}_{4}^{tree}({L}_{4},{L}_{3},3,4)|}_{2\times 2-cut},\end{array}$$ | (44) |
6 Gravity Loop Amplitudes from Gauge Theory
6.1 One-loop four-point example
$$\begin{array}{c}{M}_{4}^{tree}(-{L}_{1}^{s},{1}_{h}^{+},{2}_{h}^{+},{L}_{3}^{s})\times {M}_{4}^{tree}(-{L}_{3}^{s},{3}_{h}^{+},{4}_{h}^{+},{L}_{1}^{s}),\end{array}$$ | (45) |
$$\begin{array}{ccc}{A}_{4}^{tree}(-{L}_{1}^{s},{1}_{g}^{+},{2}_{g}^{+},{L}_{3}^{s})& =& i\frac{{\mu}^{2}}{({L}_{1}-{k}_{1}{)}^{2}},\end{array}$$ | (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) |
$$\begin{array}{c}{M}_{4}^{tree}(-{L}_{1}^{s},{1}_{h}^{+},{2}_{h}^{+},{L}_{3}^{s})=-i{\mu}^{4}\left[\frac{1}{({L}_{1}-{k}_{1}{)}^{2}}+\frac{1}{({L}_{1}-{k}_{2}{)}^{2}}\right].\end{array}$$ | (48) |
$$\begin{array}{ccc}& \int \frac{{d}^{D}{L}_{1}}{(2\pi {)}^{D}}{\mu}^{8}& \frac{i}{{L}_{1}^{2}}\left[\frac{i}{({L}_{1}-{k}_{1}{)}^{2}}+\frac{i}{({L}_{1}-{k}_{2}{)}^{2}}\right]\end{array}$$ |
$$\begin{array}{ccc}& & \times \frac{i}{({L}_{1}-{k}_{1}-{k}_{2}{)}^{2}}\left[\frac{i}{({L}_{1}+{k}_{3}{)}^{2}}+\frac{i}{({L}_{1}+{k}_{4}{)}^{2}}\right].\end{array}$$ | (49) |
$$\begin{array}{ccc}{M}_{4}^{1loop}({1}_{h}^{+},{2}_{h}^{+},{3}_{h}^{+},{4}_{h}^{+})& =& 2({\mathcal{\mathcal{I}}}_{4}^{1loop}\left[{\mu}^{8}\right]({s}_{12},{s}_{23})+{\mathcal{\mathcal{I}}}_{4}^{1loop}\left[{\mu}^{8}\right]({s}_{12},{s}_{13})\end{array}$$ |
$$\begin{array}{ccc}& & +{\mathcal{\mathcal{I}}}_{4}^{1loop}\left[{\mu}^{8}\right]({s}_{23},{s}_{13})),\end{array}$$ | (50) |
$$\begin{array}{c}{\mathcal{\mathcal{I}}}_{4}^{1loop}[\mathcal{P}]({s}_{12},{s}_{23})=\int \frac{{d}^{D}L}{(2\pi {)}^{D}}\frac{\mathcal{P}}{{L}^{2}(L-{k}_{1}{)}^{2}(L-{k}_{1}-{k}_{2}{)}^{2}(L+{k}_{4}{)}^{2}}\end{array}$$ | (51) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{4}^{1loop}({1}_{h}^{+},{2}_{h}^{+},{3}_{h}^{+},{4}_{h}^{+})=\frac{1}{(4\pi {)}^{2}}{\left(\frac{\kappa}{2}\right)}^{4}\frac{{s}_{12}^{2}+{s}_{23}^{2}+{s}_{13}^{2}}{120},\end{array}$$ | (52) |
$$\begin{array}{c}{\mathcal{\mathcal{M}}}_{4}^{1loop}({1}_{h}^{+},{2}_{h}^{+},{3}_{h}^{+},{4}_{h}^{+})=\frac{{N}_{s}}{2}\frac{1}{(4\pi {)}^{2}}{\left(\frac{\kappa}{2}\right)}^{4}\frac{{s}_{12}^{2}+{s}_{23}^{2}+{s}_{13}^{2}}{120},\end{array}$$ | (53) |
6.2 Arbitrary numbers of legs at one loop
7 Divergence Properties of Maximal Supergravity
7.1 One-loop cut construction
$$\begin{array}{ccc}{\sum}_{\genfrac{}{}{0ex}{}{\text{}N=8\text{}}{\text{states}}}& & {M}_{4}^{tree}(-{L}_{1},1,2,{L}_{3})\times {M}_{4}^{tree}(-{L}_{3},3,4,{L}_{1})\end{array}$$ |
$$\begin{array}{ccc}& =& -{s}_{12}^{2}{\sum}_{\genfrac{}{}{0ex}{}{\text{}N=4\text{}}{\text{states}}}{A}_{4}^{tree}(-{L}_{1},1,2,{L}_{3})\times {A}_{4}^{tree}(-{L}_{3},3,4,{L}_{1})\end{array}$$ |
$$\begin{array}{ccc}& & \times {\sum}_{\genfrac{}{}{0ex}{}{\text{}N=4\text{}}{\text{states}}}{A}_{4}^{tree}({L}_{3},1,2,-{L}_{1})\times {A}_{4}^{tree}({L}_{1},3,4,-{L}_{3}),\end{array}$$ | (54) |
$$\begin{array}{ccc}& & {\sum}_{\genfrac{}{}{0ex}{}{\text{}N=4\text{}}{\text{states}}}{A}_{4}^{tree}(-{L}_{1},1,2,{L}_{3})\times {A}_{4}^{tree}(-{L}_{3},3,4,{L}_{1})\end{array}$$ |
$$\begin{array}{ccc}& & =-i{s}_{12}{s}_{23}{A}_{4}^{tree}(1,2,3,4)\frac{1}{({L}_{1}-{k}_{1}{)}^{2}}\frac{1}{({L}_{3}-{k}_{3}{)}^{2}},\end{array}$$ | (55) |
$$\begin{array}{ccc}{\sum}_{\genfrac{}{}{0ex}{}{\text{}N=8\text{}}{\text{states}}}& & {M}_{4}^{tree}(-{L}_{1},1,2,{L}_{3})\times {M}_{4}^{tree}(-{L}_{3},3,4,{L}_{1})\end{array}$$ |
$$\begin{array}{ccc}& =& i{s}_{12}{s}_{23}{s}_{13}{M}_{4}^{tree}(1,2,3,4)\left[\frac{1}{({L}_{1}-{k}_{1}{)}^{2}}+\frac{1}{({L}_{1}-{k}_{2}{)}^{2}}\right]\end{array}$$ |
$$\begin{array}{ccc}& & \times \left[\frac{1}{({L}_{3}-{k}_{3}{)}^{2}}+\frac{1}{({L}_{3}-{k}_{4}{)}^{2}}\right].\end{array}$$ | (56) |
$$\begin{array}{ccc}{\mathcal{\mathcal{M}}}_{4}^{1loop}(1,2,3,4)& =& -i{\left(\frac{\kappa}{2}\right)}^{4}{s}_{12}{s}_{23}{s}_{13}{M}_{4}^{tree}(1,2,3,4)\end{array}$$ |
$$\begin{array}{ccc}& & \times \left({\mathcal{\mathcal{I}}}_{4}^{1loop}({s}_{12},{s}_{23})+{\mathcal{\mathcal{I}}}_{4}^{1loop}({s}_{12},{s}_{13})+{\mathcal{\mathcal{I}}}_{4}^{1loop}({s}_{23},{s}_{13})\right),\end{array}$$ |
$$\begin{array}{ccc}& & \end{array}$$ | (57) |
7.2 Higher loops
$$\begin{array}{ccc}{\mathcal{\mathcal{M}}}_{4}^{2loop}(1,2,3,4)& =& {\left(\frac{\kappa}{2}\right)}^{6}{s}_{12}{s}_{23}{s}_{13}{M}_{4}^{tree}(1,2,3,4)\end{array}$$ |
$$\begin{array}{ccc}& & \times ({s}_{12}^{2}{\mathcal{\mathcal{I}}}_{4}^{2loop,P}({s}_{12},{s}_{23})+{s}_{12}^{2}{\mathcal{\mathcal{I}}}_{4}^{2loop,P}({s}_{12},{s}_{13})\end{array}$$ |
$$\begin{array}{ccc}& & +{s}_{12}^{2}{\mathcal{\mathcal{I}}}_{4}^{2loop,NP}({s}_{12},{s}_{23})+{s}_{12}^{2}{\mathcal{\mathcal{I}}}_{4}^{2loop,NP}({s}_{12},{s}_{13})\end{array}$$ |
$$\begin{array}{ccc}& & +\text{cyclic}),\end{array}$$ | (58) |
7.3 Divergence properties of $\mathit{N}\mathbf{=}8$ supergravity
$$\begin{array}{c}{{\mathcal{\mathcal{M}}}_{4}^{two-loop,D=7-2\epsilon}|}_{div.}=\frac{1}{2\epsilon (4\pi {)}^{7}}\frac{\pi}{3}({s}_{12}^{2}+{s}_{23}^{2}+{s}_{13}^{2}){\left(\frac{\kappa}{2}\right)}^{6}\times {s}_{12}{s}_{23}{s}_{13}{M}_{4}^{tree},\end{array}$$ | (59) |
$$\begin{array}{c}l<\frac{10}{D-2}\end{array}$$ | (60) |
8 Conclusions
9 Acknowledgments
Note: The reference version of this article is published by Living Reviews in Relativity