On-shell recursion relations are a powerful method that enable us to build up higher-point amplitudes recursively from lower-point amplitudes. The starting point is to study the amplitude as a function of complex on-shell momenta by deforming external momenta into the complex plane while maintaining on-shell and momentum conservation. Since tree-level amplitudes are meromorphic functions, they are completely characterized by their poles, which correspond to propagators going on-shell. The recursion relations then arise as amplitudes factorize into lower-point subamplitudes on the poles.
Utilizing on-shell recursion relations, we may systematically reduce higher-point am-plitudes to basic building blocks – the 3-point amam-plitudes, which are fixed by little group scaling and locality. As such, we are able to construct all tree-level on-shell amplitudes of the theory without ever having to reference the Lagrangian or any off-shell information.
2.4.1. On-shell recursion relations: a general formulation
Consider a generic n-point amplitude An[phii] as a function of the momenta piand helicities hi, where i = 1 . . . n labels the particle number. In the following, the dependence on hi will be omitted unless needed to be stated explicitly. Since the amplitude is an analytic function of its momenta, we can reconstruct its functional form from the information of its analytical structure (for tree level, the poles and residues.) In practice, it is difficult to deal with a function of n momenta, instead a useful method is to introduce a single complex parameter z.
Let us introduce a deformation in the momenta
pi → ˆpi = pi+ zri (2.106)
depending on a complex parameter z for some ri. By considering the amplitude as a function of the parameter z and varying it, we care able probe the behavior of the ampli-tude in the resulting complex plane. The deformed momenta ˆpi must satisfy momentum
conservation ∑
ipi = 0 and remain on-shell ˆp2i = 0 in order for An[ˆpi(z)] to remain being an on-shell amplitude. We also impose the condition (∑
ri)2 = 0, in order for (∑
ipˆi)2 to be linear in z such that each propagator develops only one pole. In terms of the ri, these conditions become
Using the residue theorem, we can consider the original amplitude as a contour integral of the function An[ ˆpzi(z)] around the origin,
By Cauchy’s theorem, we can consider the same integral as surrounding the poles outside of the loop around the origin. Then the amplitude become a sum over residues,
An[pi] =−∑
I
Resz=zI An[ˆpi(z)]
z + Bn, (2.111)
with a possible boundary term Bn arising from z → ∞. The validity of the recursion relation derived in the following requires the boundary term to vanish. Assuming no contribution of the boundary term, let us specialize to the case of tree-level amplitudes, then the places z = zI where An[ ˆpzi(z)] develop singularities are where a certain internal propagator ˆpI goes on shell, ˆp2I(z = zI) = 0. On such a propagator pole, the deformed amplitude factorizes into two on-shell amplitudes on each side,
Atreen [ˆpi(z)] = AL[ˆpi, i∈ L]1 ˆ
p2IAR[ˆpi, i∈ R] (2.112) The locations where poles develop are at ˆp2I = 0, which we can solve for. Noting that the internal momentum is the sum of the momenta on the right hand side, pI =∑
i∈Rpi,
The residue is
Therefore we conclude the original amplitude is An[pi] =−∑ which is given in terms of two lower-point on-shell amplitudes AL and AR of deformed momenta evaluated at z = zI. We have to sum over all possible helicity configurations hI corresponding to the same propagator momentum pole. Writing AL/R[ˆpi(z)] = ˆAL/R(z), the on-shell recursion relation is with I summed over all factorization channels. In the following, we will choose certain schemes of deforming the momenta to obtain specific recursion relations.
Note that the boundary term Bn arising from the pole at infinity is not readily known in general, so recursion is preferably constructed using a shift under which the boundary term vanishes. In most applications the recursion relations are justified by proving that
Aˆn(z)→ 0 for z → ∞ (2.119)
Such a shift is called a valid (or good) shift, while shifts without this property are called bad shifts.
2.4.2. BCFW recursion relations
The most famous of recursion relations are the Britto, Cachazo, Feng, and Witten (BCFW) recursion relations, which we will use extensively in the later sections. In this
scheme, only two momenta, which we label as i and j, are selected to be shifted, In terms of the spinors, the shift reads as
|ˆi] = |i] + z|j], |ˆj⟩ = |j⟩ − z|i⟩. (2.122) Such a shift is called the [i, j⟩ shift. The brackets [ˆik] and ⟨ˆjk⟩ are linear in z while all other brackets are unshifted. The only propagators detectable by the BCFW shift are those which separate the shifted legs i and j onto opposite sides, since if the two shifted momenta are on the same side, the internal propagator will remain unshifted due to momentum conservation,
Using BCFW recursion, we can obtain the classic Parke-Taylor formula for MHV gluon amplitudes iteratively,
A[1−2−3+. . . n+] = ⟨12⟩4
⟨12⟩ . . . ⟨n1⟩. (2.124)
2.4.3. Super-BCFW recursion relations
In accordance with the on-shell superspace formalism, we can extend BCFW to include a shift in the Grassmann variable and preserve supermomentum conservation. The [i, j⟩ supershift is
|ˆi] = |i] + z|j], |ˆj⟩ = |j⟩ − z|i⟩, ˆηi = ηi+ zηj. (2.125) In evaluating the BCFW recursion relations, we have to sum over all possible states that can be exchanged over the internal propagator, as in the non-supersymmetric case. Due
to the on-shell superspace formalism, we can represent the sum as a Grassmann integral,
The development of the Grassmannian picture for N = 4 SYM is then related to the realization that the terms arising from super-BCFW recursion actually manifest the Yangian symmetry of the full amplitude.
2.4.4. Large z behavior under BCFW shifts
The validity of BCFW recursion relations requires the boundary term Bn in (2.111) be absent1. This is typical approach is to show that the shifted amplitude vanish as z → ∞,
Aˆn(z)→ 0 for z → ∞ (2.127)
The large z behavior of the amplitude, and hence the validity of BCFW recursion, depends on the helicities of the shifted legs. For pure Yang-Mills theory, an argument based on the background field method establishes the following large z behavior of color-ordered gluon tree amplitudes under shifts of adjacent gluons with indicated helicites:
[−, +⟩ [−, −⟩ [+, +⟩ [+, −⟩
For non-adjacent shifted legs, an extra power 1/z is gained in each case. Thus the [−, +⟩, [−, −⟩, [+, +⟩ shifts give valid recursion relations, while [+, −⟩ shifts do not and is termed the bad shift as such.
For amplitudes of pure gravity, the large z behavior were shown by [31][32][33] to be [−, +⟩ [−, −⟩ [+, +⟩ [+, −⟩
Since gravity amplitudes are not color ordered, there is no notion of adjacency. That the large z behavior of gravity amplitudes is the square of Yang-Mills can be antici-pated via the KLT relations which express graviton amplitudes as products of Yang-Mills amplitudes: Mn∼ An× An.
Naively, the large z behavior of YM and gravity amplitudes should have been much worse from the result of power counting looking at Feynman diagrams [34]. The better than expected large z behavior signals that YM and gravity amplitudes are actually tamed in the high energy limit. We can use BCFW shifts as a tool to probe the high energy behavior of amplitudes. The large z limit under opposite helicity shifts is especially noteworthy, since it has the physical interpretation of a hard light-like particle shooting
1Recently, a new algorithm known as multi-step BCFW recursion relations has been developed that can systematically deal with situations with the boundary term being present [29][30].
through a soft background [34]. This can be understood by noting that for an [i, j⟩ shift, as z→ ∞, ˆpi and ˆpj become opposite, while other moment become irrelevant,
ˆ
pi → z|j]⟨i|, ˆpj → −z|j]⟨i| for z → ∞ (2.130) Since we have chosen to represent all momenta as outgoing, for this to be interpreted as a single particle moving through a background, the helicites of i and j must be opposite (such that the helcities are actually the same when we reverse pj to be incoming.) This is drawn schematically below, where the dashed lines indicate other momenta, which are suppressed in the large z limit.
A(z) −→ˆ pˆi pˆj for z → ∞ (2.131)
For maximally supersymmetric theories, all particles are of the same superfield, so there is no distinction between different kinds of super-BCFW shifts. Specifically,N = 4 SYM amplitudes behave as 1/z and N = 8 supergravity amplitudes as 1/z2. The large z behavior of component amplitudes can be recovered from the maximally supersymmetric superamplitude by integrating away some Grassmann variables.