Complete four-loop four-point amplitude in $\mathcal{N}=4$ super-Yang-Mills theory

We present the complete four-loop four-point amplitude in $\mathcal{N}=4$ super-Yang-Mills theory, for a general gauge group and general $D$-dimensional covariant kinematics, and including all nonplanar contributions. We use the method of maximal cuts—an efficient application of the unitarity method—to construct the result in terms of 50 four-loop integrals. We give graphical rules, valid in $D$ dimensions, for obtaining various nonplanar contributions from previously-determined terms. We examine the ultraviolet behavior of the amplitude near $D=11/2$. The nonplanar terms are as well-behaved in the ultraviolet as the planar terms. However, in the color decomposition of the three- and four-loop amplitude for an $SU(N_c)$ gauge group, the coefficients of the double-trace terms are better behaved in the ultraviolet than are the single-trace terms. The results from this paper were an important step toward obtaining the corresponding amplitude in $\mathcal{N}=8$ supergravity, which confirmed the existence of cancellations beyond those needed for ultraviolet finiteness at four loops in four dimensions. Evaluation of the loop integrals near $D=4$ would permit tests of recent conjectures and results concerning the infrared behavior of four-dimensional massless gauge theory.

Figure 1

Rules for obtaining the color factors associated with cubic parent graphs. The roman letters $a$, $b$, $c$ are color indices. The sign of each structure constant ${\tilde{f}}^{abc}$ is fixed by the clockwise ordering of each vertex, as drawn.

Figure 2

The box diagram is the only parent graph at one loop (up to permutations of the external legs).

Figure 3

The planar and nonplanar parent graphs contributing to the two-loop four-point amplitude. The prefactors $s_{12}$ are the integral numerators $N^{(\mathrm{P})}$ and $N^{(\mathrm{NP})}$, respectively.

Figure 4

The nine parent graphs for the three-loop four-point $\mathcal{N}=4$ sYM amplitude. The prefactor of each diagram is the numerator polynomial $N^{(x)}$. The kinematic invariants are defined in Eq. (2.15).

Figure 5

Planar contributions to the four-loop four-point amplitude.

Figure 6

Contact terms in the planar four-point amplitude, which can be absorbed into the numerator of graph (f) of Fig. 5. [Note that here (${\mathrm{f}}_2$) and (${\mathrm{d}}_2$) refers to the labeling used in Ref. 13; unlike Ref. 13, we choose to write out explicitly the relative factors of these integrals with respect to diagram (f).]

Figure 7

Examples of maximal and near-maximal cuts used to construct a complete ansatz for the four-loop $\mathcal{N}=4$ sYM amplitude. Cuts (a) and (b) have the maximal 13 cut conditions. Near-maximal cuts (c) have 12 conditions, and (d) and (e) have 11 conditions. No cuts with fewer than 11 on-shell propagators are needed to build the complete ansatz.

Figure 8

These 11 cuts, along with the two-particle cuts in Fig. 9, suffice to determine any massless four-loop four-point amplitude. As displayed they determine any planar amplitude; to determine any nonplanar contributions one must also include cuts with the legs of each tree subamplitude permuted arbitrarily. The cuts (h) and (k) have been evaluated in $D$ dimensions, as they are included in the box cuts Fig. 10c and (e), respectively.

Figure 9

The two-particle cuts. These cuts have been evaluated in $D$ dimensions. Together with the cuts of Fig. 8 these form a spanning set of cuts for any massless four-loop four-point amplitude.

Figure 10

The three box cuts. These cuts have been evaluated in $D$ dimensions.

Figure 11

A two-particle cut that may be used to construct contributions to the $(L+L^{\prime }+1)$-loop amplitude from those at $L$ and $L^{\prime }$ loop orders.

Figure 12

Sample contributions to the full color-dressed two-particle cut for $L=2$ and $L^{\prime }=1$. The diagrams on the right-hand side show some of the terms in $U^{(4)}(1,2,3,4)$ that are constructed from these cuts, using Fig. 13. The explicit color factors, as well as factors of $i$, have been omitted.

Figure 13

The lower-loop integral functions entering the cuts on the left-hand side of Fig. 12. This figure displays in detail how the prefactor of the planar double-box integral appears in $U^{(2)}(-l_1,1,2,l_2)$ as either $s_{12}$ or $(k_1-l_1{)}^2$, depending on the permutation.

Figure 14

The box-substitution rule [64] for generating a higher-loop contribution by inserting a one-loop four-point box subintegral into a four-point vertex. In this example, we substitute a box into the central four-point vertex in the four-loop “window” diagram. The result is a five-loop integral that cannot be obtained from two-particle cuts. (Note that an overall normalization factor of $s_{12}{s}_{23}$ has been absorbed into $\mathcal{K}$, relative to Ref. 64.)

Figure 15

Two examples of multiparticle multiloop “box cuts.” They reduce to lower-loop cuts by replacing the darker (red) four-point subamplitudes by four-point color-ordered trees, multiplied by known numerator and denominator factors. This property allows these cuts to be computed easily in $D$ dimensions, once the amplitudes with fewer loops are known. White holes represent loops and the darker subamplitudes mark four-point amplitudes amenable to reduction.

Figure 16

Application of the box cut to determine the numerator $N$ for the four-loop parent graph in Fig. 5f. The (red) dashed cut conditions around the upper left box in this diagram allow us to replace it by the product of a reduced cut diagram, some kinematical factors and a box integral. The reduced cut diagram is then expanded into two three-loop “tennis-court” diagrams, corresponding to the two allowed channels of the marked four-point vertex. The relevant kinematical pieces of the tennis-court diagrams, i.e. the numerators and the spurious propagators, are extracted from the known three-loop contribution in Fig. 4e. Assembling all the kinematical factors gives the result for $N$, which is free of spurious propagators. The result is consistent with Fig. 5f; here the numerator is symmetrized with respect to the ($1\leftrightarrow 3$) symmetry of this parent graph.

Figure 17

Graphs for the four-point tree amplitude. Contact terms are absorbed into the diagrams as inverse propagators. Each diagram is associated with a color factor obtained by dressing the vertices with an ${\tilde{f}}^{abc}$, as in Sec. 2a.

Figure 18

Two nontrivial relations between numerators of planar and nonplanar contributions at four loops, which follow from the tree-level numerator relations in Fig. 17. For the three graphs in each relation, the configuration of lines outside the region marked by a dashed circle is identical; the only difference is for the lines fully inside this region. The relation on the second line involves the same graph numerator $N_{28}$ with two different labelings of momenta.

Figure 19

Four-loop examples of “MHV/$\overline{\mathrm{MHV}}$-amplitude cuts”, which are composed entirely of four- and five-point tree amplitudes.

Figure 20

Integrals (1)–(11) appearing in the four-loop amplitude. The graphs encode denominator factors as the Lorentz squares of the momenta flowing along every internal line, and color factors following the rules defined in Fig. 1. The momentum-dependent factor in front of each graph represents the numerator factor $N_i$ that resides inside the integral $I_i$, where $(i)$ is the label below each graph. Numbers 1, 2, 3, 4 label the external (outgoing) momenta. The internal lines carry momenta as signified by the arrows (here only line 5 is labeled). The kinematic variables are defined as $s_{ij}=s_{i,j}=(l_i+l_j{)}^2$ and in the following figures $s_{i,\overline{j}}=(l_i-l_j{)}^2$ and ${\tau }_{ij}=2l_i·l_j$, where $l_i$ is the momentum of line $i$. A specific (clockwise) orientation of each cubic vertex (in the plane of the figure) is implied here. Because of the antisymmetry of the structure constants, any noncyclic reordering of a vertex should be accompanied by a sign flip of the numerator factor.

Figure 21

Integrals (12)–(23) appearing in the four-loop amplitude. The notation follows that of Fig. 20.

Figure 22

Integrals (24)–(31) appearing in the four-loop amplitude. The notation follows that of Fig. 20.

Figure 23

Integrals (32)–(39) appearing in the four-loop amplitude. The notation follows that of Fig. 20.

Figure 24

Integrals (40)–(47) appearing in the four-loop amplitude. The notation follows that of Fig. 20.

Figure 25

Integrals (48)–(49) appearing in the four-loop amplitude. Integral (50) is required for the ansatz to match the color-stripped cut at the level of the integrand. However, it does not contribute to the color-dressed amplitude. The notation follows that of Fig. 20.

Figure 26

The vacuumlike graphs describing the three-loop UV divergences in $D=6$. The large (blue) dots indicate that a propagator appears squared, or doubled.

Figure 27

The leading contributions with numerators quadratic in loop momentum in Fig. 4 can be absorbed into the two contact diagrams displayed here, up to relabeling of external legs. Further rearrangements push the leading terms into the diagrams of Fig. 28.

Figure 28

At three loops the leading divergence can be rearranged so that it comes from parent graphs which are one-particle reducible, depicted here in the $s_{12}$ channel. The color factors of these graphs have no double-trace contributions. The complete contribution comes from summing over all 24 permutations of external legs and multiplication by a symmetry factor of $1/4$ to remove double counts in both graphs.

Figure 29

Four-loop vacuum diagrams describing the UV divergences of individual graphs in the four-loop $\mathcal{N}=4$ sYM amplitude. A dot indicates that a given propagator appears squared in the integral. For $V_{10}$ and $V_{11}$, the factor of $s_{56}$ indicates the insertion of $s_{56}=(l_5+l_6{)}^2$ into the numerator of the integral, where lines 5 and 6 are marked in the figure by arrows.

Figure 30

At four loops the leading divergence can be rearranged so that it comes from graphs with two spurious external propagators. The graphs with $1/s_{12}$ propagators are displayed here. This rearrangement makes manifest that the color factors of these graphs have no double-trace contributions. The complete contribution comes from summing over all 24 permutations of external legs and multiplication by a symmetry factor of $1/16$, $1/8$ and $1/16$, respectively, to remove double counts in all three graphs.

Figure 31

A two-loop integral with central propagator raised to the power $n$. If $n$ is not an integer, $G^{(2)}(n,D)$ cannot be reduced to one-loop integrals.

Figure 32

The UV poles in the vacuum diagram $V_1$ can be determined from the product of a finite three-loop two-point integral with a UV-divergent one-loop two-point integral. There are four inequivalent ways of doing the reduction, corresponding to different propagators connecting the two points at which the momentum $k^{\mu }$ is injected.

Figure 33

Four inequivalent ways of reducing vacuum diagram $V_2$ to a product of a finite three-loop two-point integral with a UV-divergent one-loop two-point integral.

Figure 34

Four inequivalent ways of reducing vacuum diagram $V_8$ to a product of a finite nonplanar three-loop two-point integral with a UV-divergent one-loop two-point integral.

Figure 35

The three-loop vacuum diagram appearing in the UV divergence of the three-loop four-point $\mathcal{N}=4$ sYM amplitude in $D=20/3-2ϵ$. Dots indicate the appearance of squared propagator factors in the integral.

Figure 36

A nontrivial nonplanar cut at four loops. The cuts (a) and (b) represent the two distinct internal helicity configurations contributing to this seven-particle cut. The blobs representing MHV trees are labeled by a “$+$” and the blobs representing $\overline{\mathrm{MHV}}$ trees are labeled by a “$-$.” For a four-point amplitude, either assignment can be made; the assignment in the figure makes the supersum calculation most transparent.

Figure 37

Tree structures building up the cut in Fig. 36.