Four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory

We present an expression for the leading-color (planar) four-loop four-point amplitude of $\mathcal{N}=4$ supersymmetric Yang-Mills theory in $4-2ϵ$ dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around $ϵ=0$, and obtain analytic expressions for the poles from $1/{ϵ}^8$ through $1/{ϵ}^4$. We give numerical results for the coefficients of the $1/{ϵ}^3$ and $1/{ϵ}^2$ poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the $1/{ϵ}^2$ coefficient allows us to test a conjecture of Eden and Staudacher for the four-loop cusp (soft) anomalous dimension. We find that the conjecture is incorrect, although our numerical results suggest that a simple modification of the expression, flipping the sign of the term containing ${\zeta }_3^2$, may yield the correct answer. Our numerical value can be used, in a scheme proposed by Kotikov, Lipatov, and Velizhanin, to estimate the two constants in the strong-coupling expansion of the cusp anomalous dimension that are known from string theory. The estimate works to 2.6% and 5% accuracy, providing nontrivial evidence in support of the AdS/CFT correspondence. We also use the known constants in the strong-coupling expansion as additional input to provide approximations to the cusp anomalous dimension which should be accurate to under 1% for all values of the coupling. When the evaluations of the integrals are completed through the finite terms, it will be possible to test the iterative, exponentiated structure of the finite terms in the four-loop four-point amplitude, which was uncovered earlier at two and three loops.

Figure 1

Integrals required for $gg\to gg$ scattering in planar MSYM at one loop (1), two loops (2), and three loops [(3)a and (3)b]. The box (1), planar double-box (2), and three-loop ladder (3)a integrals are scalar integrals, with no loop-momentum dependent factors in the numerator. The tennis-court integral (3)b contains a factor of $(l_1+l_2{)}^2$, where $l_1$ and $l_2$ are marked with arrows in the figure.

Figure 2

“Rung-rule” contributions to the leading-color four-loop amplitude, in terms of integral functions given in Eqs. (a9, a10, a11, a12, a13, a14). An overall factor of $st$ has been suppressed in each figure, compared with the definitions in Eqs. (a9, a10, a11, a12, a13, a14).

Figure 3

Generalized cuts that provide information about the planar four-loop amplitude. (i) A two-particle cut separating a tree amplitude from a three-loop amplitude. (ii) A two-particle cut separating a one-loop amplitude from a two-loop amplitude. (iii) A “3–3” cut separating the amplitude into a product of three tree amplitudes. (iv) An “upper-2–3–lower-2” cut separating the amplitude into a product of four tree amplitudes. (v) A “lower-2–3–lower-2” cut. (vi) A “3–lower-3” cut.

Figure 4

Non-rung-rule contributions to the leading-color four-loop amplitude, in terms of integral functions defined in Eqs. (a15, a16). Integral $({\mathrm{d}}_2)$ follows the labeling of integral (d) in Fig. 2, and integral $({\mathrm{f}}_2)$ follows the labeling of integral (f). An overall factor of $st$ has been suppressed in each figure, compared with Eqs. (a15, a16).

Figure 5

The only one-particle-irreducible, purely cubic, four-loop, four-point graph with no triangle or bubble subgraphs, besides the rung-rule graphs in Fig. 2.

Figure 6

No-triangle planar four-loop graphs, in addition to those given in Figs. 2, 4. The notation indicates how a given graph here can be derived from a rung-rule graph in Fig. 2, or else from graph (g) in Fig. 5, by canceling propagators.

Figure 7

The two-loop planar double box and its dual diagram. The double box is represented by light colored lines and the dual diagram by dark (blue) lines. A dark line connecting $x_i$ with $x_j$ represents the factor $1/x_{ij}^2$. A dotted line signifies a numerator factor of $x_{ij}^2$. The momentum corresponding to any $x_{ij}$ is given by the sum of momenta of the light lines crossing the dark line joining $x_i$ and $x_j$. An overall factor of $st$ has been removed for clarity.

Figure 8

The rung-rule dual diagrams. A factor of $st$ has been removed.

Figure 9

The non-rung-rule dual diagrams. A factor of $st$ has been removed.

Figure 10

The two four-loop dual integrals, in addition to those given in Figs. 2, 4, that survive the requirement of conformal invariance. Both integrals are ruled out by two-particle cuts. In this case, no factor of $st$ has been removed.

Figure 11

Two examples of dual diagrams which do not lead to new conformal integrals in the on-shell limit, for reasons discussed in the text.

Figure 12

Approximations to the cusp anomalous dimension in planar MSYM based on the formula (7.1) and perturbative information through two loops (dotted line), three loops (dot-dash line), and four loops (solid line). Also shown is the strong-coupling prediction from string theory (dashed line).

Figure 13

Approximations to the cusp anomalous dimension in planar MSYM based on the formula (7.1), four-loop perturbative information, and zero, one, or two constants from the strong-coupling expansion.

Figure 14

Value estimated for the four-loop cusp anomalous dimension $f_0^{(4)}$ by the $[3/2]$ Padé approximant (7.35), as a function of the parameter $\xi$ controlling the point at which the cut is assumed to terminate, $\widehat{a}=-{\xi }^2/8$.

Figure 15

Ratio of the $[3/2]$ Padé approximant (7.35), as a function of the parameter $\xi$, to the KLV approximate formula (7.21), as a function of the coupling $\widehat{a}$.