^{1}

^{1,2}

^{1,3,4}

^{3}.

We consider finite superamplitudes of

Conformal symmetry has played a central role in quantum field theory for many decades. In recent years, its consequences are being actively explored within the AdS/CFT correspondence, the bootstrap program, and high-energy QCD.

In particle physics, scattering amplitudes are fundamental objects, relevant for collider physics. In the high-energy regime the masses can often be neglected, and the Lagrangian becomes conformal. Implications of conformal symmetry were successfully investigated in maximally supersymmetric Yang-Mills theory, based on the remarkable duality between scattering amplitudes and Wilson loops. The conformal symmetry used there is that of the dual Wilson loop. However, the native (super)conformal symmetry of the amplitudes is largely unexplored. The purpose of this Letter is to study its consequences in a broader framework.

One reason why this is a difficult problem is that putting the external particles on shell can render the symmetry anomalous. Tree-level amplitudes have a holomorphic anomaly

The conformal symmetry of finite loop integrals and the associated anomalous Ward identities were studied in Ref.

In the present Letter, we show how to obtain powerful first-order differential equations in a model of

As a first application, we focus on the five-particle case, which is interesting for several reasons. First, it turns out that for four particles (super)conformal symmetry does not give any restrictions. So, from this point of view, five particles is the first nontrivial case. On the other hand, five-particle scattering at higher loops involves intricate transcendental functions

We show how the differential equations can be used to fully determine the answer. The fermionic generator is reduced to the so-called twistor collinearity operator of Ref.

This model is superconformal at the classical level. At the quantum level, the symmetry is broken, but only by propagator corrections, and the beta function is proportional to the anomalous dimension of the superfield

One-loop

Nonplanar two-loop

Such graphs are naively superconformal. We will see that the symmetry is in fact broken by on-shell collinear effects. The latter can be controlled and give rise to powerful anomalous Ward identities.

In order to discuss superamplitudes, we introduce the on-shell superstates

In our

The anomalous Ward identity Eq.

Let us now apply this anomaly equation to computing

The five-leg amplitudes we consider have the structure

Let us now turn to the superconformal Ward identities. When

In order to determine the right-hand side of the Ward identity we need to evaluate explicitly the

It is convenient to define the dimensionless, helicity-neutral function

Let us comment on the solutions of the homogeneous equation

Let us now derive the result for

The system of differential equations, Eq.

We find it amusing to note that, although the superconformal Ward identities are trivial for four particles, one can nonetheless obtain the result for the four-particle amplitude by taking the (finite) collinear limit

The anomalous superconformal Ward identity for this integral involves four anomaly terms

Using known expressions for these one-loop functions, we find

Integrating over

One method follows the one-loop example above. Using a Feynman parameter representation of the integral Eq.

This remarkable fact is not a coincidence and leads us to a second method of fixing the boundary data. Imagine that we have already found the correct solution for

We can write the solution for

In principle, one could rewrite the answer in terms of a minimal function basis, see Ref.

Let us discuss checks of our result. The symbol of the result agrees with Ref.

Let us study the symbol of our answer in light of the structure suggested by the differential equation, Eq.

We obtained a first-order differential equation for a given

It is important to note that our method applies equally to planar and nonplanar amplitudes. We illustrated this by evaluating a nonplanar two-loop five-particle Feynman integral.

The differential equations shed light on the class of special functions needed. This information can be valuable input for “bootstrap” approaches that have been successfully used in several cases, such as amplitudes in

The class of integrals considered here is generated by

We are grateful to Simon Caron-Huot for stimulating discussions. The authors were supported in part by the PRISMA Cluster of Excellence at Mainz university. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 725110), “Novel structures in scattering amplitudes.”