Toward the Feynman rule for $n$-point gluon Mellin amplitudes in $\mathrm{AdS}/\mathrm{CFT}$

We investigate the embedding formalism in conjunction with the Mellin transform to determine tree-level gluon amplitudes in $\mathrm{AdS}/\mathrm{CFT}$. Detailed computations of three to five-point correlators are conducted, ultimately distilling what were previously complex results for five-point correlators into a more succinct and comprehensible form. We then proceed to derive a recursion relation applicable to a specific class of $n$-point gluon amplitudes. This relation is instrumental in systematically constructing amplitudes for a range of topologies. We illustrate its efficacy by specifically computing six to eight-point functions. Despite the complexity encountered in the intermediate steps of the recursion, the higher-point correlator is succinctly expressed as a polynomial in boundary coordinates, upon which a specific differential operator acts. Remarkably, we observe that these amplitudes strikingly mirror their counterparts in flat space, traditionally computed using standard Feynman rules. This intriguing similarity has led us to propose a novel dictionary: comprehensive rules that bridge AdS Mellin amplitudes with flat-space gluon amplitudes.

Figure 1

From left to right, (a) the three gluon amplitude, (b) the contact diagram of the four gluon amplitude, and (c) the $s$-channel representation of the four gluon amplitude.

Figure 2

Five-point channel consisting of contact and three-point vertex interactions.

Figure 3

The channel of five-point gluon amplitude for (3.30a).

Figure 4

A ($n+1$)-current amplitude for involving a three-vertex.

Figure 5

The “snowflake” channel of six point gluon amplitude for (4.8).

Figure 6

The channel of six point gluon amplitude for (4.16).

Figure 7

The “scarecrow” channel of seven-point gluon amplitude for (4.20).

Figure 8

The “drone” channel of eight-point gluon amplitude for (4.24).