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# **2 On-Shell Techniques for Tree-Level Amplitudes**

### **Abstract**

In this chapter we focus on the pole structure of tree-level amplitudes. We argue that amplitudes factorise on these poles into lower-point amplitudes. Moreover, universal factorisation structures emerge when two momenta become collinear as well as in the limit of low energy of a single particle—the soft limit. These factorisation properties are the basis of an efficient technique for computing treelevel scattering amplitudes in gauge theories and gravity recursively—without ever referring to Feynman rules or even a Lagrangian. These recursion relations use as input lower-point amplitudes, so that the gauge redundancy, which is partly responsible for the complexity of conventional Feynman graph calculations, is absent in this entirely on-shell based formalism. We then show the invariance of scattering amplitudes under Poincaré transformations, and introduce the conformal symmetry of gauge-theory tree-level amplitudes. Finally, we highlight a surprising double-copy relation between gluon and graviton amplitudes.

### **2.1 Factorisation Properties of Tree-Level Amplitudes**

Important insights and constraints on tree-level scattering amplitudes may be gained by thinking about them as analytic functions of the external momenta. In this section we will restrict ourselves to the case of massless particles. As we already argued in Chap. 1 with Fig. 1.2, tree-level amplitudes have simple poles in multi-particle channels. This can be seen from the Feynman diagrammatic expansion. Take all diagrams which have a propagator ∼ 1*/P*2 *ij* , where *Pij* = *pi* + *...* + *pj* is a sum of external momenta (which will be adjacent for colour-ordered amplitudes, or an arbitrary subset in gravity). We call *Pij* the *region momentum*, as it is the total momentum associated to the region of momenta {*pi,...,pj* }. As *Pij* goes on shell, *P*2 *ij* → 0, these singular diagrams will collect into a product of two

on-shell amplitudes, to the left and right hand sides of the divergent propagator, in a mechanism known as *factorisation*. This process is illustrated in Fig. 2.1. We can understand the details of the procedure by studying the Feynman rules for the propagator that goes on-shell directly, each of which has the generic form

$$\frac{\mathrm{i}\,N(P\_{lj})}{P\_{ij}^2} \,\mathrm{,}\tag{2.1}$$

where the numerator *N (Pij )* depends on the type of particle. For example, for the specific case of gluons in the axial gauge (*nμAμ* = 0) we would have *N (P )* → *Nμν g (P , n)* = −*ημν* + *(P μnν* + *Pνnμ)/(P* · *n)*. In the limit *P*2 *ij* → 0 the numerator *N* can be rewritten in terms of the spin sum over external polarisation vectors or wave-functions. Again for the case of gluons, following the results of Exercise 1.7, we have that

$$N\_{\mathcal{S}}^{\mu\upsilon}(P, n) \stackrel{P^2 \to 0}{\to} \sum\_{h=\pm} \epsilon\_h^{\mu}(P)\epsilon\_h^{\ast\upsilon}(P) = \sum\_{h=\pm} \epsilon\_h^{\mu}(P)\epsilon\_{-h}^{\upsilon}(-P) \,. \tag{2.2}$$

The polarisation vectors combine with the Feynman diagram components on either side of the divergent propagator to form on-shell amplitudes. A schematic form for a general particle type can be written as

$$A\_n^{\text{tree}}(1, \ldots, n) \stackrel{p\_{ij}^2 \to 0}{\longrightarrow} \sum\_{s \in s\text{p}} A\_L^{\text{tree}}(i, i+1, \ldots, j, -P\_{ij}^{\bar{s}}) \frac{p\_{\bar{P}}}{P\_{ij}^2} A\_R^{\text{tree}} \left(P\_{ij}^s, j+1, \ldots, i-1\right),\tag{2.3}$$

where P indicates the particle type of the propagator with momentum *Pij* , *s*P are its possible spin states and *n*P is a particle-dependent constant. As we see from the discussion above, for gluons *s*gluon = ± are the two helicities and *n*gluon = i, for spin- 1 2 fermions one can show that *s*fermion = ±1*/*2 and *n*fermion = 1, while spin 0 has *s*scalar = 0 and *n*scalar = i. Other particle types are simple to determine following the same argument. Factorisation will also occur in the case of massive propagators where *P*2 *ij* → *m*2 *ij* where *mij* is the mass of propagating particle.

### **2.1.1 Collinear Limits**

A special case of factorisation is the two-particle pole also known as *collinear*  singularity. Without loss of generality we take the two collinear particles to be 1 and 2. We then have *(p*1 + *p*2*)*2 = 0, which implies *p*1 · *p*2 = 0 or *collinearity p*1 | *p*2. We again concentrate on the massless case. In fact, since the factorisation now involves a three-particle amplitude, such a pole can only occur for collinear external momenta. We already know from the discussion in the previous chapter that three-point amplitudes are subtle. In the strict collinear limit *p*1 | *p*2 we may parametrise the collinear momenta *p*1 and *p*2 as

$$p\_1^{\mu} = \mathbf{x} \, P^{\mu}, \qquad \qquad p\_2^{\mu} = (1 - \mathbf{x}) \, P^{\mu}, \tag{2.4}$$

with the total collinear momentum *P* = *p*1 + *p*2, and *x* parametrising the amount of *P* distribution over *p*1 and *p*2. Tree-level amplitudes have a universal (singular) behaviour in the collinear limit governed by the *splitting functions*,

$$A\_n^{\text{tree}}(1^{h\_1}, 2^{h\_2}, \dots) \xrightarrow{\text{l} \wr \text{l}} \sum\_{h=\pm} \text{Split}\_h^{\text{tree}}(\ge, 1^{h\_1}, 2^{h\_2}) \, A\_{n-1}^{\text{tree}}(P^{-h}, \dots) \, , \tag{2.5}$$

as a special case of Eq. (2.3). In fact, the splitting functions are related to the threeparticle amplitudes as

$$\text{Split}\_{h}^{\text{tree}}\left(\mathbf{x}, \mathbf{1}^{h\_1}, \mathbf{2}^{h\_2}\right) = \lim\_{P^2 \to 0} A\_3^{\text{tree}}\left(\mathbf{1}^{h\_1}, \mathbf{2}^{h\_2}, -P^h\right) \frac{\mathbf{i}}{P^2} \,. \tag{2.6}$$

The splitting functions depend on the helicities of the collinear gluons but are independent of the helicities of the other legs {3*,...,n*} not participating in the collinear limit. This is known as the universality of the splitting functions.

### **Gluon Splitting Functions**

For collinear gluons the splitting functions are given by

$$\begin{aligned} \text{Split}\_{+}^{\text{tree}}(x, a^+, b^+) &= 0\\ \text{Split}\_{-}^{\text{tree}}(x, a^+, b^+) &= \frac{\text{g}}{\sqrt{x(1-x)}\langle ab \rangle} \\ \text{Split}\_{+}^{\text{tree}}(x, a^+, b^-) &= -\frac{(1-x)^2 \text{ g}}{\sqrt{x(1-x)}\langle ab \rangle} \\ \text{Split}\_{-}^{\text{tree}}(x, a^+, b^-) &= -\frac{x^2 \text{ g}}{\sqrt{x(1-x)}[ab]} \end{aligned} \tag{2.7}$$

(continued)

The remaining splitting functions may be obtained by parity

$$\text{Split}\_{h}^{\text{tree}}(\mathbf{x}, a^{-\lambda\_{a}}, b^{-\lambda\_{b}}) = \text{Split}\_{-h}^{\text{tree}}(\mathbf{x}, a^{\lambda\_{a}}, b^{\lambda\_{b}}) \Big|\_{\langle ab \rangle \leftrightarrow [ba]}.\tag{2.8}$$

We shall now derive these expressions from our knowledge of the three-point MHV amplitude (1.170 and (1.171). As the collinear kinematics is subtle, it is advantageous to systematically approach the collinear configuration as

$$\begin{aligned} \left| \left| \right> &= \cos\phi \left| P \right> - z \sin\phi \left| r \right> , & \quad \left| \mathbf{1} \right] &= \cos\phi \left| P \right> - z \sin\phi \left| r \right> , \\\left| \left| \mathbf{2} \right> &= \sin\phi \left| P \right> + z \cos\phi \left| r \right> , & \quad \left| \mathbf{2} \right] &= \sin\phi \left| P \right> + z \cos\phi \left| r \right> \end{aligned} \tag{2.9}$$

in the limit *z* → 0. Here *P μ* = *pμ* 1 +*pμ* 2 +O*(z*2*)* is the limiting collinear momentum vector, and *rμ* is a null reference momentum not parallel to *P μ*. Moreover, the collinear parameter above is *x* = cos2 *φ*. This parametrises the four-momenta of the collinear particles as

$$\begin{aligned} p\_1 &= \cos^2\phi \, P - z \cos\phi \sin\phi \left( |P\rangle[r| + |r\rangle][P|\right) + z^2 \sin^2\phi \, r, \\ p\_2 &= \sin^2\phi \, P + z \cos\phi \sin\phi \left( |P\rangle[r| + |r\rangle][P|\right) + z^2 \cos^2\phi \, r, \end{aligned} \tag{2.10}$$

implying that *p*1 + *p*2 = *P* + *z*2 *r*, as claimed. One then has

$$\langle 12 \rangle = z \langle Pr \rangle \,, \qquad \langle 1P \rangle = z \sin \phi \left\langle Pr \right\rangle \,, \qquad \langle 2P \rangle = -z \cos \phi \left\langle Pr \right\rangle \,, \tag{2.1}$$

$$\langle 12 \rangle = z \lbrack Pr \rceil \,, \qquad \langle 1P \rangle = z \sin \phi \left\lbrack Pr \right\rbrack \,, \qquad \left[ 2P \right] = -z \cos \phi \left\lbrack Pr \right\rbrack \,, \tag{2.11}$$

$$\left(\left(p\_1+p\_2\right)^2 = \mathrm{z}^2\langle Pr \rangle [r\,P]\right).$$

The splitting functions (2.7) follow immediately from Eq. (2.6) and the two MHV three-point amplitudes (1.170) and (1.171) upon using the above identities. The vanishing of Splittree + *(x,* 1+*,* 2+*)* follows from the vanishing all-plus amplitude. Using the MHV3 amplitude of Eq. (1.170) 1 we find

$$\text{Split}\_{-}^{\text{tree}}(\mathbf{x}, 1^{+}, 2^{-}) = \frac{-\text{ig}\,\langle 2(-P) \rangle^{3}}{\langle 21 \rangle \langle 1(-P) \rangle} \frac{\text{i}}{(p\_{1} + p\_{2})^{2}} = -\frac{\cos \phi^{3} g}{z \sin \phi [Pr]},\tag{2.12}$$

1 Recall our convention (1.113) that | − *P*> = i|*P*> and | − *P*] = i|*P*].

as well as using the MHV3 amplitude of Eq. (1.171),

$$\text{Split}\_{-}^{\text{tree}}(\mathbf{x}, 1^{+}, 2^{+}) = \frac{\text{ig}[12]^{3}}{[1(-P)][(-P)2]} \frac{\text{i}}{(p\_{1} + p\_{2})^{2}} = \frac{\text{g}}{z \cos \phi \sin \phi \langle Pr \rangle},$$

$$\text{Split}\_{+}^{\text{tree}}(\mathbf{x}, 1^{+}, 2^{-}) = \frac{\text{ig}[1(-P)]^{3}}{[12][2(-P)]} \frac{\text{i}}{(p\_{1} + p\_{2})^{2}} = -\frac{\sin \phi^{3} g}{z \cos \phi \langle Pr \rangle},$$

which prove the relations (2.7).

#### **Example: Collinear Limits of the Five-Point MHV Amplitude**

Let us use this result to test our conjectured five-point MHV amplitude (1.192) from Chap. 1 for consistency under collinear limits. We set *g* = 1 for convenience. We have

$$\begin{aligned} \mathrm{i}A\_{\mathfrak{F}}^{\mathrm{tree}}(\mathrm{l}^{-}, \mathrm{2}^{-}, \mathrm{3}^{+}, \mathrm{4}^{+}, \mathrm{5}^{+}) &= \frac{\langle 12 \rangle^{4}}{\langle 12 \rangle \langle 23 \rangle \langle 34 \rangle \langle 45 \rangle \langle 51 \rangle} \\\\ &\xrightarrow{4 \parallel \mathfrak{F}} \frac{1}{\sqrt{\chi(1-\chi)} \langle 45 \rangle} \times \frac{\langle 12 \rangle^{4}}{\langle 12 \rangle \langle 23 \rangle \langle 3P \rangle \langle P1 \rangle} \\\\ &= \mathrm{Split\\_{\mathrm{re}}^{\mathrm{tree}}}(\mathrm{x}, \mathrm{4}^{+}, \mathrm{5}^{+}) \times \mathrm{i}A\_{\mathfrak{A}}^{\mathrm{tree}}(\mathrm{l}^{-}, \mathrm{2}^{-}, \mathrm{3}^{+}, P^{+}), \end{aligned} \tag{2.13}$$

where we parametrised the collinear limit by |4> = √ √ *x*|*P*> and |5> = 1 − *x*|*P*>. Indeed we find agreement with the second function in Eq. (2.7). Next, we take the collinear limit in a *(*−+*)* channel. We have

$$\begin{split} \mathrm{i}A\_{\mathsf{S}}^{\mathrm{tree}}(\mathrm{l}^{-}, \mathrm{2}^{-}, \mathrm{3}^{+}, \mathrm{4}^{+}, \mathrm{5}^{+}) & \stackrel{\mathrm{2}\|3}{\longrightarrow} \frac{\mathrm{x}^{2}}{\sqrt{\mathrm{x}(1-\mathrm{x})}} \frac{1}{\langle 23 \rangle} \frac{\langle 1P \rangle^{4}}{\langle 1P \rangle \langle P4 \rangle \langle 45 \rangle \langle \mathsf{5} \,\mathrm{l} \rangle} \\ &= \mathrm{Spli}\_{+}^{\mathrm{tree}}(z, \mathrm{2}^{-}, \mathrm{3}^{+}) \times \mathrm{i}A\_{4}^{\mathrm{tree}}(\mathrm{l}^{-}, P^{-}, \mathrm{4}^{+}, \mathrm{5}^{+}) \, . \end{split} \tag{2.14}$$

from which we deduce

$$\text{Split}\_{+}^{\text{tree}}(x, a^{-}, b^{+}) = \frac{x^{2}}{\sqrt{x(1-x)}\langle ab \rangle},\tag{2.15}$$

in agreement with the third expression in Eq. (2.7) whilst swapping *a* and *b* (and *x* → 1 − *x*). In order to check the fourth function we consider the collinear limit

in a *(*+−*)*-channel,

$$A\_{\mathfrak{F}}^{\text{tree}}({\bf l}^{-},{\bf 2}^{-},{\bf 3}^{+},{\bf 4}^{+},{\bf 5}^{+}) \xrightarrow{\mathfrak{f} \parallel \mathbf{1}} \underbrace{\frac{({\bf 1}-{\bf x})^{2}}{\sqrt{{\bf x}({\bf 1}-{\bf x})} \left({\bf 5}\mathbf{1}\right)}}\_{\text{Spli}^{\text{tree}}\_{+}({\bf x},{\bf 5}^{+},{\bf 1}^{-})} A\_{\mathfrak{k}}^{\text{tree}}({\bf P}^{-},{\bf 2}^{-},{\bf 3}^{+},{\bf 4}^{+}), \tag{2.16}$$

yielding the desired result via parity (2.8). In order to check the vanishing of the uniform helicity splitting function in Eq. (2.7) we have study the collinear factorisation of the 6-point MHV amplitude with the helicity distributions *A*tree 6 *(*1−*,* 2−*,* 3+*,* 4+*,* 5+*,* 6+*)* along the legs 5 and 6.

**Exercise 2.1 (The Vanishing Splitting Function** Splittree + *(x, a*+*, b*+*)***)** Show that Splittree + *(x, a*+*, b*+*)* = 0 by studying the factorisation properties of the six-gluon MHV tree amplitude *A*tree 6 *(*1−*,* 2−*,* 3+*,* 4+*,* 5+*,* 6+*)* from Eq. (1.192) in the collinear limit 5 | 6. For the solution see Chap. 5.

### **Absence of Multi-Particle Poles in MHV Amplitudes**

General *n*-gluon scattering amplitudes have multi-particle poles when region momenta go on-shell. However, MHV tree-amplitudes are special, and in fact *only* have two-particle poles or collinear singularities. The reason is the following. Consider the general factorisation formula (2.3). In a factorisation of an MHV amplitude there are only three negative-helicity legs (corresponding to the two external negative helicities, and one for the internal on-shell propagator) that are distributed over two partial amplitudes. However, we saw in the previous chapter that *An(*1±*,* 2+*,...,n*+*)* = 0 (for *n >* 3). Therefore, either *AL* or *AR* in Eq. (2.3) is always zero unless one partial amplitude is a three-particle amplitude. The latter case corresponds to a two-particle pole or collinear singularity, as discussed above.

### **2.1.2 Soft Theorems**

We continue our quest in the factorisations of scattering amplitudes with a kinematical limit subject to just a *single* leg: the soft limit. Here one particle involved in the scattering process has a very low energy—it is soft. Specifically, it refers to the limit where the four-momentum of the particle goes to zero. Again the treeamplitude displays a universal factorisation property for photon, gluon and graviton amplitudes into a lower-point amplitude and a soft function. In order to take the limit, we parametrise the soft momentum of leg *s* as *pμ s* = *δ qμ* and take *δ* → 0 (do

**Fig. 2.2** The soft factorisation of a generic *(n* + 1*)*-particle amplitude. The soft function S[*δ q,*{*pi*}] only depends on the momenta (not the polarisations) of the hard legs

not confuse *δ qμ* with a variation). The soft theorems state that

$$A\_{n+1}^{\text{tree}}(\delta \neq q, 1, \ldots, n) \xrightarrow{\delta \to 0} \mathbb{S}[\delta \neq q, \{p\_1, \ldots, p\_n\}] \\ A\_n^{\text{tree}}(1, \ldots, n) \,,\tag{2.17}$$

which is illustrated in Fig. 2.2. The factorised soft function *S*[*δ q,*{*p*1*,...,pn*}] depends on the momentum *δ q* and helicity of the soft particle, as well as the momenta of the remaining hard legs {*pa*}. It is however independent of the helicities and even particle types of the remaining hard legs, which may be massless or massive. The soft function diverges as 1*/δ* at leading order, and also contains universal sub-leading pieces:

$$\mathcal{S}[\delta\,\,q,\,\{p\_a\}] = \frac{1}{\delta}\mathcal{S}^{[0]}(q) + \mathcal{S}^{[1]}(q) + \delta\,\mathcal{S}^{[2]}(q) + \mathcal{O}(\delta^2). \tag{2.18}$$

It takes a universal form for photons [1], gluons and gravity [2], remarkably not only to leading order, but also to sub-leading order S[1] *(q)* for gauge theories, and even to sub-sub-leading order S[2] *(q)* for gravity [3].

#### **Leading Soft Theorems**  The leading soft divergences take the universal form 1 *δ* S[0] *(δq)* = { ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 1 *δ* S[0] EM = 1 *δ* Σ*n a*=1 *ea Σ* · *pa pa* · *q ,* photon, 1 *δ* S[0] YM = *g δ* √2 (*Σ* · *p*1 *p*1 · *q* − *Σ* · *pn pn* · *q* ) *,* colour-ordered gluon, 1 *δ* S[0] GR = *κ δ* Σ*n Σμν pμ a pν a pa* · *q ,* graviton.

*a*=1

(2.19)

(continued)

Here *Σ* denotes the polarisation vector (tensor) of the soft particle, *ea* denotes the U*(*1*)* (electromagnetic) charge of the hard particle *a*.

These results can be made plausible through the following argument based on Feynman diagrams. For a tree-level amplitude the soft leg *δ q* is attached either via a three-point coupling to an outgoing hard leg *a*, or to the "bulk" of the remaining amplitude:

$$A\_{n}^{\text{tree}}(\delta \neq \emptyset, \{p\_{a}\}) = \sum\_{a=1}^{n} \vdots \underbrace{\vdots \cdot \sum\_{\substack{\uparrow \\ \downarrow \\ \downarrow \\ (p\_{a} + \delta \neq q)^{2} = m\_{a}^{2}}}\_{\delta \neq q} + \underbrace{\vdots \cdot \sum\_{\substack{\delta \\ \downarrow \\ (p\_{a} + \delta \neq q)^{2} = m\_{a}^{2}}}\_{\delta \neq q} a}\_{\delta \neq q} \tag{2.20}$$

We see that the divergence of the leading order soft function solely arises from the three-point coupling of the gauge or graviton field to the hard leg *a*. In the case of an amplitude with *n*-scalars and a soft photon or graviton these couplings take the form

$$\begin{aligned} \xleftarrow{\longleftrightarrow\limits\_{\mathbb{R}}} \quad \text{or} \quad \begin{cases} a \\ \kappa \end{cases} \quad \begin{cases} e\_a \,\epsilon \cdot p\_a \,, \qquad \text{photon}, \\ \kappa \,\epsilon\_{\mu\nu} \, p\_a^{\mu} \, p\_a^{\nu} \,, \quad \text{graviton}, \end{cases} \end{aligned} \tag{2.21}$$

which fixes the soft functions in Eq. (2.19) up to an overall factor. In the case of a colour-ordered pure gluon amplitude with one soft gluon leg, *A*tree *n (*1*, . . . , n, δ q)*, the soft leg can couple either to the hard gluon leg 1 or *n* due to colour ordering. A detailed look at the Feynman rules (1.149) then reveals the coupling ±*g pa* · *Σ* with *a* = 1*, n* at leading order in *δ* with a relative sign factor reproducing the form of Eq. (2.19).

It is instructive to study the gauge invariance of the leading soft functions (2.19). Gauge invariance requires the full amplitude *An*+1*(δ q, p*1*,...,pn)* to vanish under the transformation *Σμ* → *qμ* (in the graviton case we again take *Σμν* = *ΣμΣν* ). Hence, by consistency, the soft functions in Eq. (2.19) need to vanish:

$$\mathcal{S}^{[0]}(\delta \, q)\Big|\_{\epsilon \to \,\_q} = \begin{cases} \sum\_{a=1}^n e\_a = 0, & \text{photon}, \\ \frac{g}{\sqrt{2}}(1-1) = 0, & \text{gluon}, \\ \kappa \, q\_\mu \sum\_{a=1}^n p\_a^\mu = 0 \,, & \text{graviton}. \end{cases} \tag{2.22}$$

They indeed vanish due to total charge conservation in the electromagnetic or total momentum conservation in the gravitational case (provided the gravitational coupling is universal to all matter). Hence these soft theorems are intimately connected to fundamental symmetries of space-time-matter.

### **2.1.3 Spinor-Helicity Formulation of Soft Theorems**

It is instructive to translate the leading soft theorems to spinor-helicity language for the colour-ordered gluon and graviton cases. The leading order soft factorisation states that

$$A\_{n}^{\text{tree}}(\ldots,a,\delta q^{\pm},b,\ldots) \stackrel{\delta \to 0}{\longrightarrow} \mathcal{S}^{[0]}(a,q^{\pm},b) \, A\_{n-1}^{\text{tree}}(\ldots,a,b,\ldots)\,. \tag{2.23}$$

The factorised soft function depends on the momenta and helicities of the soft gluon and the momenta of the colour-ordered neighbours *a* and *b*. It is independent, however, of the helicities and particle types of the neighbouring legs. From considering the soft limit of an MHV amplitude one easily establishes that

$$\mathcal{S}\_{\rm YM}^{[0]}(a, q^{+}, b) = g \frac{\langle ab \rangle}{\langle aq \rangle \langle qb \rangle} \,. \tag{2.24}$$

Via parity, in analogy to Eq. (2.8), we find

$$\mathcal{S}\_{\rm YM}^{[0]}(a, q^{-}, b) = -g \frac{[ab]}{[aq][qb]} \,. \tag{2.25}$$

Both results directly follow from Eq. (2.19) as well. In the graviton case we deduce

$$\mathcal{S}\_{\text{GR}}^{[0]}(q^{++},1,\ldots,n) = \kappa \sum\_{a=1}^{n} \frac{\langle \mathbf{x}a \rangle \langle \mathbf{y}a \rangle [qa]}{\langle \mathbf{x}q \rangle \langle \mathbf{y}q \rangle \langle aq \rangle},\tag{2.26}$$

where *x* and *y* are arbitrary reference spinors associated to the polarisation vectors of the soft leg.

**Exercise 2.2 (Soft Functions in the Spinor-Helicity Formalism)** Starting from Eq. (2.19), derive the leading soft functions for a colour-ordered gluon and a graviton in the spinor-helicity formalism. For the solution see Chap. 5.

### **2.1.4 Subleading Soft Theorems**

Remarkably, the universal soft factorisation survives to subleading order (O ( *δ*0 ) ) in the gauge theory, and even sub-sub-leading order (O ( *δ* ) ) in the gravitational case, cf. Eq. (2.18). Here we just state the results. They may again be derived from a careful study of gauge invariance and consistency arguments [4, 5]. The novelty is that the sub-leading soft functions are (necessarily) no longer true functions but rather *differential operators* in the external kinematical data. These then act on the factorised hard scattering-amplitude. Explicitly, one has the following sub-leading soft operator for a soft photon *qμ* with polarisation *Σμ*,

$$\mathcal{S}\_{\rm EM}^{[1]}(q) = \sum\_{a=1}^{n} e\_a \frac{\epsilon\_{\mu} q\_{\nu} \, J\_a^{\mu \nu}}{p\_a \cdot q},\tag{2.27}$$

with the local angular momentum operator

$$J\_a^{\mu\upsilon} = 2 \, p\_a^{[\mu} \frac{\partial}{\partial p\_{a,\,\upsilon}} + 2 \, \epsilon\_a^{[\mu} \frac{\partial}{\partial \epsilon\_{a,\,\upsilon}} \,. \tag{2.28}$$

In the above [*μν*] denotes anti-symmetrisation with unit weight, see Appendix A. Similarly, for a colour-ordered soft gluon we have

$$\mathcal{S}\_{\rm YM}^{[1]}(n,q,1) = -\mathrm{i}\frac{\mathrm{g}}{\sqrt{2}} \left( \frac{\epsilon\_{\mu}q\_{\nu}\,\,J\_{1}^{\mu\nu}}{p\_{1}\cdot q} - \frac{\epsilon\_{\mu}q\_{\nu}\,\,J\_{n}^{\mu\nu}}{p\_{n}\cdot q} \right),\tag{2.29}$$

while for a soft gravitons we have a sub-leading and sub-sub-leading soft operators of the forms

$$\mathcal{S}\_{\text{GR}}^{[1]}(q) = -i\kappa \sum\_{a=1}^{n} \frac{\epsilon \cdot p\_a \epsilon\_\mu q\_\nu \, J\_a^{\mu\nu}}{p\_a \cdot q}, \qquad \mathcal{S}\_{\text{GR}}^{[2]}(q) = -\frac{\kappa}{2} \sum\_{a=1}^{n} \frac{\left(\epsilon\_\mu q\_\nu \, J\_a^{\mu\nu}\right)^2}{p\_a \cdot q}. \tag{2.30}$$

Again, one may show that these sub-leading soft operators are consistent with gauge invariance. While this is trivial for the gauge-theory operators by virtue of the antisymmetry of *J μν* , the gauge invariance of the gravitational S[1] EM*(q)* leads us to total

angular momentum conservation,

$$\mathcal{S}\_{\rm GR}^{\rm [1]}(q)\Big|\_{\epsilon \to q} = -\text{i}\kappa \,\epsilon\_{\mu} q\_{\upsilon} \sum\_{a=1}^{n} J\_{a}^{\mu \upsilon} = 0 \,,\tag{2.31}$$

nicely teaming up with the total momentum conservation in the leading case. Hence, the gravitational soft theorems are directly connected to the Poincaré invariance of scattering amplitudes.

Expressed in spinor-helicity variables the sub-leading soft operators take the form

$$\begin{aligned} \mathcal{S}\_{\text{YM}}^{\text{[1]}}(n, q^+, 1) &= g \left( \frac{[q \partial\_n]}{\langle qn \rangle} - \frac{[q \partial\_1]}{\langle q 1 \rangle} \right), \\\\ \mathcal{S}\_{\text{GR}}^{\text{[1]}}(q^+, 1, \dots, n) &= \frac{\kappa}{2} \sum\_{a=1}^n \frac{[aq]}{\langle aq \rangle} \left( \frac{\langle ax \rangle}{\langle qx \rangle} + \frac{\langle a \mathbf{y} \rangle}{\langle q \mathbf{y} \rangle} \right) \left[ q \partial\_a \right], \end{aligned} \tag{2.32}$$

where as before *x* and *y* are arbitrary reference spinors. The sub-sub-leading gravity operator of Eq. (2.30) in spinor-helicity variables simply reads

$$\mathcal{S}\_{\text{GR}}^{[2]}(q^+,1,\ldots,n) = \frac{\kappa}{2} \sum\_{a=1}^n \frac{[aq]}{\langle aq \rangle} [q \,\partial\_a]^2,\tag{2.33}$$

and was found in [3].

An important subtlety for the sub-leading soft theorems lies in balancing the total momentum conservation of the *(n* + 1*)*-leg amplitude with the soft leg and the factorised *n*-point hard amplitude. The soft factorisation (2.17) is really a distributional identity involving delta-functions:

$$\begin{aligned} \delta^{(D)}(\delta \, q + P) \, \mathcal{R}\_{n+1}^{\text{tree}}(\delta \, q, \{p\_a\}) \stackrel{\delta \to 0}{\longrightarrow} \\ \mathcal{S}[\delta \, q, \{p\_a\}] \, \delta^{(D)}(P) \, \mathcal{R}\_n^{\text{tree}}(1, \dots, n) + \mathcal{O}(\delta^{\text{J}}) \,, \end{aligned} \tag{2.34}$$

with *P* = *p*1 + *...* + *pn* and *j* = 1 or 2 for gauge theory or gravity, respectively. This in fact implies that

$$\mathcal{S}[\delta\,\!\!\/\,q,\,\{p\_d\}]\,\delta^{(D)}(P) = \delta^{(D)}(\delta\,\!\/\,q+P)\,\mathcal{S}[\delta\,\!\/\,q,\,\{p\_d\}],\tag{2.35}$$

from which one may strongly constrain the *δ* expansion of the soft function *S*[*δ q,*{*pa*}] using Eq. (2.18). It necessitates the sub-leading terms to be differential operators, and even the functional forms of Eq. (2.32) are fixed by the knowledge of the leading soft functions [4].

**Exercise 2.3 (A** *qqggg* ¯ **Amplitude from Collinear and Soft Limits)** In Chap. 1 we established the following colour-ordered *qqgg* ¯ amplitudes involving a massless quark and anti-quark using colour-ordered Feynman rules:

$$A^{\rm tree}\_{\bar{q}\,\otimes \bar{g}}(1^-\_{\bar{q}}, 2^+\_q, 3^+, 4^+) = 0 \,, \tag{2.36}$$

$$A^{\rm tree}\_{\bar{q}\,ggg}(1^-\_{\bar{q}}, 2^+\_q, 3^-, 4^+) = -ig^2 \frac{\langle 13 \rangle^3 \langle 23 \rangle}{\langle 12 \rangle \langle 23 \rangle \langle 34 \rangle \langle 41 \rangle} \,. \tag{2.37}$$

Use these and the discussed splitting and soft factorisation properties for gluonic legs to make a guess for the five-point single quark-line tree amplitude *A*tree *qqggg* ¯ *(*1− *q*¯ *,* 2+ *q ,* 3−*,* 4+*,* 5+*)*. Check your guess against all known factorisation properties. Can you generalise your guess to the *n*-particle partial amplitudes *A*tree *qqg...g* ¯ *(*1− *q*¯ *,* 2+ *q ,* 3+*,...,n*+*)* and *A*tree *qqg...g* ¯ *(*1− *q*¯ *,* 2+ *q ,* 3−*,* 4+*,...,n*+*)*? For the solution see Chap. 5.

### **2.2 BCFW Recursion for Gluon Amplitudes**

The Britto-Cachazo-Feng-Witten (BCFW) recursion relation [6, 7] is an efficient way to compute higher-point tree-level amplitudes from lower-point ones. It does not make use of Feynman rules but builds upon unitarity by artfully exploiting the factorisation property of scattering amplitudes (2.3) when region momenta go on-shell. As we will see, our knowledge of the gluon (and graviton) three-point amplitudes of Eqs. (1.170) and (1.171) allows for the construction of *all* higher-point tree-level amplitudes in a recursive fashion. To begin with, let us concentrate on the colour-ordered case and leave the discussion of gravitons for later. The central idea of the recursion is to consider a deformation in a *single* complex variable *z* of two adjacent momenta in a colour-ordered amplitude that maps the singularities of the amplitude into poles in *z* ∈ C. For the tree-level *n*-gluon amplitude *An(p*1*,...pn)* we introduce the following complex shift of the helicity spinors of two arbitrary adjacent particles, taken to be 1 and *n* without loss of generality:

$$\begin{array}{ll} \lambda\_1 \to \hat{\lambda}\_1(z) = \lambda\_1 - z\lambda\_n \,, & \qquad \tilde{\lambda}\_1 \to \tilde{\lambda}\_1 \,, \\\\ \lambda\_n \to \lambda\_n \,, & \qquad \tilde{\lambda}\_n \to \hat{\tilde{\lambda}}\_n(z) = \tilde{\lambda}\_n + z\tilde{\lambda}\_1 \, . \end{array} \tag{2.38}$$

We denote the shifted, *z*-dependent quantities by a hat. This shift is often termed an [*n* 1> shift. It results in a deformation of the momenta,

$$p\_1^{\dot{\alpha}\alpha} \rightarrow \hat{p}\_1^{\dot{\alpha}\alpha}(z) = \tilde{\lambda}\_1^{\dot{\alpha}} (\lambda\_1 - z\lambda\_n)^{\alpha}, \qquad p\_n^{\dot{\alpha}\alpha} \rightarrow \hat{p}\_n^{\dot{\alpha}\alpha}(z) = (\tilde{\lambda}\_n + z\tilde{\lambda}\_1)^{\dot{\alpha}} \lambda\_n^{\alpha}. \tag{2.39}$$

Importantly, the shift preserves both overall momentum conservation and the onshell conditions:

$$
\hat{p}\_1(z) + \hat{p}\_n(z) = p\_1 + p\_n \,, \qquad \hat{p}\_1^2(z) = 0 \,, \qquad \hat{p}\_n^2(z) = 0 \,. \tag{2.40}
$$

The [*n* 1> shift generates a one-parameter family of amplitudes,

$$A\_n(z) := A\_n\left(\hat{p}\_1(z), p\_2, \dots, p\_{n-1}, \hat{p}\_n(z)\right). \tag{2.41}$$

Note that *p*ˆ1 and *p*ˆ*n* in Eq. (2.39) are now complex, as the underlying helicity spinors *λ*1*,n* and *λ*˜ 1*,n* are no-longer complex conjugates of each other. This makes the three-point amplitudes involving these states of Eqs. (1.170) and (1.171) nonvanishing. They will become the seeds of the recursion relation. What are the analytic properties of the deformed amplitude *An(z)*? Factorisation implies that the deformed amplitude *An(z)* has precisely *n*−3 *simple* poles in *z*. Using the region momenta *Pi* := Σ*i*−1 *j*=1 *pj* , these *n*−3 poles take the form

$$\frac{\mathrm{i}}{\hat{P}\_l^2(z)} := \frac{\mathrm{i}}{P\_l^2 - z \,\langle n \,|P\_l|1\rangle} = -\frac{1}{\langle n \,|P\_l|1\rangle} \frac{\mathrm{i}}{z - z\_{P\_l}} \,, \tag{2.42}$$

where *P*ˆ *i(z)* = ˆ*p*1*(z)* + *p*2 +··· *pi*−1, and

$$z\_{P\_l} = \frac{P\_l^2}{\langle n \vert P\_l \vert 1 \rangle}, \qquad \forall i \in \{3, \dots, n-1\}. \tag{2.43}$$

Note that any region momentum containing *both p*ˆ1*(z)* and *p*ˆ*n(z)* is independent of *z* by virtue of Eq. (2.40), and hence cannot contribute to a *z*-pole. This is why we find *n*−3 poles. It follows that, as *z* → *zPi* , the amplitude *An(z)* factorises as

$$\begin{aligned} A\_{\boldsymbol{\mu}}(\boldsymbol{z}) & \stackrel{\boldsymbol{z}\to\boldsymbol{z}\_{P\_{l}}}{\longrightarrow} \\ \frac{\mathbf{i}}{\hat{P}\_{l}^{2}(\boldsymbol{z})} & \sum\_{h} A\_{L} \Big[ \hat{\mathbf{l}}(\boldsymbol{z}\_{P\_{l}}), \, 2, \ldots, \, \boldsymbol{i} - 1, \, -\hat{P}^{-h}(\boldsymbol{z}\_{P\_{l}}) \Big] A\_{R} \Big[ \hat{P}^{h}(\boldsymbol{z}\_{P\_{l}}), \, \boldsymbol{i}, \, \ldots, \, \boldsymbol{n} - 1, \, \hat{n}(\boldsymbol{z}\_{P\_{l}}) \Big], \end{aligned} \tag{2.44}$$

**Fig. 2.3** Factorisation of the *z*-deformed amplitude *An(z)*

as depicted in Fig. 2.3. The sum on *h* runs over all possible helicity states *h* propagating between *AL* and *AR*, and depends on the field content of the theory considered. For gluons it is a sum over *h* = {+1*,* −1}.

In the end we are only interested in the undeformed amplitude, i.e. *An(z*=0*)*, and we can use complex analysis to construct it from the knowledge of the residues of *An(z)*:

$$\begin{split} A\_{\boldsymbol{n}}(\boldsymbol{z}=\boldsymbol{0}) &= \frac{1}{2\pi\mathrm{i}} \oint\_{C\_{0}} \frac{\mathrm{d}\boldsymbol{z}}{\boldsymbol{z}} \, A\_{\boldsymbol{n}}(\boldsymbol{z}) \\ &= \sum\_{l=2}^{n-1} \sum\_{h=\pm} A\_{L}^{-h}(\boldsymbol{z}\_{\boldsymbol{P}l}) \frac{\mathrm{i}}{\boldsymbol{P}\_{l}^{2}} A\_{R}^{h}(\boldsymbol{z}\_{\boldsymbol{P}l}) + \mathrm{Res}(\boldsymbol{z}=\boldsymbol{\infty}) \,. \end{split} \tag{2.45}$$

Here *C*0 is a small circle around *z*=0 that only contains the pole around the origin. To obtain Eq. (2.45) we have deformed this into a large circle at infinity, now encircling all the poles *zPi* in the complex plane but with an opposite orientation. See Fig. 2.4. If *An(z)* → 0 as *z* → ∞ we can drop the boundary term Res*(z* = ∞*)*. As we shall argue in a moment, this is the case for gauge theories under certain conditions.

### **BCFW Recursion Relation for Gluon Amplitudes**

With this assumption, we arrive at the BCFW recursion relation [7]:

$$\begin{aligned} A\_n(1, \ldots, n) &= \sum\_{l=3}^{n-1} \sum\_{h=\pm} A\_l \left( \hat{1}(z\_{P\_l}), 2, \ldots, -\hat{P}\_l^{-h}(z\_{P\_l}) \right) \\ &\quad \frac{\mathbf{i}}{P\_l^2} A\_{n+2-l} \left( \hat{P}\_l^h(z\_{P\_l}), i, \ldots, n-1, \hat{n}(z\_{P\_l}) \right), \end{aligned} \tag{2.46}$$

(continued)

**Fig. 2.4** Using Cauchy's theorem to obtain Eq. (2.45) we may pull the initial circle *C*0 off to infinity thereby encircling the other poles clock-wise

with *zPi* defined in (2.43), and *Pi* = *p*1 + *p*2 + *...* + *pi*−1. This relation is constructive: the amplitudes appearing on the right-hand side have lower multiplicity than the initial *An*. Hence, with the seed three-gluon amplitudes (1.170) and (1.171) we can bootstrap this relation to construct *all n*-gluon trees without using Feynman diagrams at all! Even more, our nuclei—the three-point amplitudes—were obtained from helicity scaling arguments alone, as discussed in Sect. 1.11.3. In this derivation we chose to shift two neighbouring legs 1ˆ and *n*ˆ. In fact, one can also shift nonneighbouring legs or even more than two legs to obtain alternative recursion relations, see e.g. [8, 9].

An open issue is the vanishing of the boundary term in (2.45). For this we need to have that

$$\frac{1}{2\pi \mathrm{i}} \oint\_{\infty} \frac{dz}{z} \, A\_{\mathrm{n}}(z) = 0 \,, \tag{2.47}$$

which in turns requires a large-*z* falloff of the amplitude as *An(z)*∼*z*−1. In fact, the large-*z* behaviour depends on the helicities of the shifted legs, and one can show that

$$A(\hat{\mathbf{l}}^{+}, \hat{n}^{-}) \stackrel{z \to \infty}{\sim} \frac{1}{z}, \qquad A(\hat{\mathbf{l}}^{+}, \hat{n}^{+}) \stackrel{z \to \infty}{\sim} \frac{1}{z}, \qquad A(\hat{\mathbf{l}}^{-}, \hat{n}^{-}) \stackrel{z \to \infty}{\sim} \frac{1}{z}, \qquad (2.48)$$

yet *A* ( 1ˆ−*, n*ˆ+) *z*→∞ ∼ *z*3, which is then a forbidden [*n*1> shift. In the following we show the first relation; the other scalings are more technical to derive, and may be found in [10].

### **2.2.1 Large** *z* **Falloff**

In order to estimate the large *z* behaviour of generic tree-level amplitudes we perform an analysis based on Feynman graphs. There are three sources for *z* dependence in a generic colour-ordered amplitude: the propagators, the interaction vertices, and the polarisation vectors. Consider a generic graph contributing to the tree-level *n*-gluon amplitude (1ˆ and *n*ˆ are assumed to be neighbours). The *z* dependence occurs only along the path from 1ˆ to *n*ˆ, see Fig. 2.5. Along this path, each three-gluon vertex, being linear in the momenta, maximally contributes a factor of *z*, while four-gluon vertices do not contribute, and all propagators along the path contribute a factor of 1*/z*. We may derive an upper bound for the *z*-scaling by considering the diagrams with maximal powers of *z*. This happens when the path from 1ˆ to *n*ˆ contains *only* three-valent vertices. In that case it is easy to see that the graph scales as *z*:

$$\begin{array}{cccccccccc}\downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ \end{array} \tag{2.49}$$

Finally, there is an additional *z* dependence arising from the polarisation vectors at legs 1 and *n*:

$$\epsilon\_{1}^{+\alpha\dot{\alpha}} = -\sqrt{2} \frac{\bar{\lambda}\_{1}^{\dot{\alpha}} \mu\_{1}^{\alpha}}{\langle \hat{\lambda}\_{1}(z) \mu\_{1} \rangle} \sim \frac{1}{z}, \qquad \epsilon\_{1}^{-\alpha\dot{\alpha}} = \sqrt{2} \frac{\hat{\lambda}\_{1}^{\alpha}(z) \tilde{\mu}\_{1}^{\dot{\alpha}}}{[\lambda\_{1} \mu\_{1}]} \sim z, \tag{2.50}$$

$$\epsilon\_n^{+\alpha \dot{\mu}} = -\sqrt{2} \frac{\hat{\tilde{\lambda}}\_n^{\dot{\tilde{\alpha}}}(z) \mu\_n^{\alpha}}{\langle \lambda\_n \mu\_n \rangle} \sim z, \qquad \epsilon\_n^{-\alpha \dot{\mu}} = \sqrt{2} \frac{\lambda\_n^{\alpha} \tilde{\mu}\_n^{\dot{\alpha}}}{[\hat{\lambda}\_n(z) \mu\_n]} \sim \frac{1}{z}. \tag{2.51}$$

**Fig. 2.5** The *z* scaling of a generic graph: along the path from 1ˆ to *n*ˆ the propagators scale as 1*/z*, the three-point vertices as *z*, while four-point vertices do not scale. This sample graph scales as 1. However, if we would replace the four-point vertex by a three-point one, it would scale as *z*. On top of that we have to consider the *z* scaling of the polarisation vectors

Therefore, taking all sources of *z* dependence into account, we conclude that individual graphs scale at worst as

$$\begin{array}{ll} A(\hat{\mathbf{l}}^{+}, \hat{n}^{-}) \stackrel{z \to \infty}{\sim} \frac{1}{z}, & \qquad A(\hat{\mathbf{l}}^{+}, \hat{n}^{+}) \stackrel{z \to \infty}{\sim} z, \\\\ A(\hat{\mathbf{l}}^{-}, \hat{n}^{-}) \stackrel{z \to \infty}{\sim} z, & \qquad A(\hat{\mathbf{l}}^{-}, \hat{n}^{+}) \stackrel{z \to \infty}{\sim} z^{3}. \end{array} \tag{2.52}$$

This shows that the [−+>-shift has the desired falloff properties that allow us to drop the boundary term at infinity in the BCFW formula (2.46). By cyclicity, it is always possible to find a {1ˆ+*, n*ˆ−} pair. In fact, the above bound is too conservative. It was shown in [10] that the [++> and [−−>-shifts also lead to an overall 1*/z* scaling once the sum over all Feynman graphs is performed, as the terms scaling as *z* or 1 cancel out. Only the [+−>-shift gives a non-vanishing Res*(z* = ∞*)* in general, and may not be used as basis for a BCFW recursion.

### **2.3 BCFW Recursion for Gravity and Other Theories**

Can we generalise the BCFW recursion to other *massless* quantum field theories? If we analyse its derivation in Sect. 2.2, we see that only two ingredients were needed to establish it:

Therefore, in order to construct tree-level amplitudes recursively without colour ordering from their factorisation properties, we need to consider all multi-particle channels that may occur. We thus generalise the region momenta to include any subset *I* of the momenta {*p*1*,...,pn*},

$$P\_I^{\mu} := \sum\_{i \in I} p\_i^{\mu} \,. \tag{2.53}$$

Whenever *P*2 *I* = 0 we have a pole, and, if a two-particle BCFW shift is used, the set *I* must contain only one of the shifted momenta so that *P*2 *I* becomes *z*-dependent. Concretely, the BCFW recursion for a shift of legs 1 and *n* as in Eq. (2.38) in gravity

takes the form [11, 12]

$$M\_n = \sum\_{\mathcal{Q}} \sum\_{h=\pm\pm} M\_L \left( \hat{1}(z\_{P\_{\mathcal{Q}}}), \mathcal{Q}, -\hat{P}\_{\mathcal{Q}}^{-h}(z\_{P\_{\mathcal{Q}}}) \right) \frac{\mathbf{i}}{P\_{\mathcal{Q}}^2} \, M\_R \left( \hat{P}\_{\mathcal{Q}}^h(z\_{P\_{\mathcal{Q}}}), \bar{\mathcal{Q}}, \hat{n}(z\_{P\_{\mathcal{Q}}}) \right), \tag{2.54}$$

where the first sum runs over *all* subsets *Q* of momenta in {*p*2*,...,pn*−1}, *Q*¯ is the complement of *Q*, and *PQ* = *p*1 + Σ *i*∈*Q Pi*. Again, the recursion is only valid for the [*n*1> shifts

$$\left|\left|\dot{1}\right> = \left|1\right> - z\left|n\right>\right, \qquad \left|\hat{n}\right] = \left|n\right] + z\left|1\right], \tag{2.55}$$

of the types [−+>, [++>, and [−−>. For a derivation see [10].

Finally, we note that the BCFW recursion can be generalised to massive theories [13, 14] to be discussed in Sect. 2.5, to the rational parts of one-loop amplitudes in QCD and gravity [15–19] and form factors [20, 21]. In supersymmetric Yang-Mills theory a supersymmetric version of the BCFW recursion may be formulated [22, 23]. In fact this recursion could be solved analytically [24].

### **2.4 MHV Amplitudes from the BCFW Recursion Relation**

### **2.4.1 Proof of the Parke-Taylor Formula**

As an application of the colour-ordered BCFW recursion (2.46), we now derive the Parke-Taylor formula (1.192) for MHV amplitudes. We already know from Sect. 1.11 that it is true for *n* = 3 and *n* = 4 through an explicit computation. Therefore we shall prove by induction that the formula is correct. We focus here on the case where particles *n* and 1 have negative helicity, and perform the [*n*1> shifts of Eq. (2.38). The MHV amplitude has no multi-particle factorisation, as was discussed in Sect. 2.1.1. Hence, only the two BCFW diagrams of Fig. 2.6 contribute to the BCFW recursion of Eq. (2.46). We recall the [*n*1> shift,

$$\left|\hat{1}\right\rangle = \left|1\right\rangle - z\left|n\right\rangle \,, \quad \left|\hat{n}\right\rangle = \left|n\right\rangle + z\left|1\right\rangle \,, \quad \hat{P} = P - z\left|n\right\rangle \lbrack1\rangle \, , \tag{2.56}$$

whereas |1ˆ]=|1] and | ˆ*n*>=|*n*> are left inert.

In fact, diagram *(*II*)* does not contribute. Here, the right diagram is of MHV3 type. Its numerator reads [*P*ˆ|*n* − 1] 3, cf. Eq. (1.171), which vanishes:

$$[n-1|\hat{P}] = \frac{[n-1|\hat{P}|n)}{\langle \hat{P}|n\rangle} = \frac{[n-1|(P-z|1]\langle n|)|n\rangle}{\langle \hat{P}|n\rangle}$$

$$=\frac{[n-1|P|n)}{\langle \hat{P}|n\rangle} \stackrel{P=-\underleftarrow{\frac{p\_{n-1}}{\langle n\rangle}-p\_n}}{\langle \hat{P}|n\rangle} \stackrel{}{\phantom{\frac{p\_n}{\langle n\rangle}-p\_n}}{\langle \hat{P}|n\rangle} \tag{2.57}$$

**Fig. 2.6** The two BCFW diagrams contributing to the MHV*n* amplitude. In fact, diagram *(*II*)* does not contribute

In fact, this vanishing is consistent with the observation that the MHV3 kinematical assumption of collinear left-handed spinors, i.e. <*(n*−1*)*| ˆ*n*>=<*(n*−1*)*|*n*> = 0, forces the two momenta *pn*−1 and *pn* to be collinear, *pn*−1 | *pn*. This is an inconsistent assumption on the *n*-particle kinematics. This is not a problem for diagram *(*I*)*, where the analogue condition reads <2|1ˆ> = 0, which does not imply collinearity of *p*1 and *p*2 as |1ˆ> /= |1>.

Therefore only the BCFW diagram *(*I*)* contributes, where *AL* is a three-point MHV amplitude, and *AR* is an *(n* − 1*)*-point MHV amplitude. From Eq. (2.43), the position of the pole is

$$z\_P = \frac{\left(p\_1 + p\_2\right)^2}{\langle n \vert P \vert 1 \rangle} = \frac{\langle 12 \rangle \lbrack 21 \rbrack}{\langle n \vert 2 \rbrack \lbrack 21 \rbrack} = \frac{\langle 12 \rangle}{\langle n \vert 2 \rangle} \,. \tag{2.58}$$

The amplitudes *AL* and *AR* are then given by

$$\begin{aligned} A\_L &= A\_3^{\overline{\text{MHV}}} (\hat{1}^-, 2^+, -\hat{P}^+) = \text{ig} \frac{[2(-\hat{P})]^3}{[\hat{1}2][(-\hat{P})\hat{1}]}, \\\\ A\_R &= A\_{n-1}^{\text{MHV}} (\hat{P}^-, 3^+, 4^+, \dots (n-1)^+, \hat{n}^-) = -\text{ig}^{n-3} \frac{\langle \hat{n}\hat{P} \rangle^3}{\langle \hat{P}3 \rangle \langle 34 \rangle \cdots \langle (n-1)\hat{n} \rangle}. \end{aligned} \tag{2.59}$$

Using (1.113), the fact that *λn* and *λ*˜ 1 are not shifted in our [*n*1> shift of Eq. (2.55), as well as

$$
\langle \hat{\mathbf{n}} | \hat{P} \rangle [\hat{P} 2] = \langle \mathbf{n} | \hat{\mathbf{l}} \rangle \begin{bmatrix} 12 \\ \end{bmatrix} = \langle \mathbf{n} | \mathbf{l} \rangle \begin{bmatrix} 12 \\ \end{bmatrix}, \qquad \langle \mathbf{3} \hat{P} \rangle \begin{bmatrix} \hat{P} 1 \\ \end{bmatrix} = \langle \mathbf{3} 2 \rangle \begin{bmatrix} 2 \hat{1} \\ \end{bmatrix} = \langle \mathbf{3} 2 \rangle \begin{bmatrix} 2 \mathbf{1} \\ \end{bmatrix}, \tag{2.60}
$$

we find

$$\begin{split} A\_L \frac{\text{i}}{(p\_1 + p\_2)^2} A\_R &= \text{i} \text{g}^{n-2} \frac{\langle n \text{1} \rangle^3 \text{[12]}^3}{\text{[12]} \text{[21]} \langle 32 \rangle \text{[21]} \text{\{12\}} \text{\{12\}} \text{\{34\}} \cdots \text{\{(n-1)n\}}} \\ &= -\text{i} \text{g}^{n-2} \frac{\langle n \text{1} \rangle^4}{\text{\{12\}} \cdots \text{\{n1\}}} \,, \end{split} \tag{2.61}$$

in agreement with the conjecture (1.192) for the chosen helicities. This proves the Parke-Taylor formula for adjacent negative-helicity states.

### **2.4.2 The Four-Graviton MHV Amplitude**

As a second example using the BCFW recursion for non-colour-ordered amplitudes (2.54) we compute the four-graviton amplitude *M*4*(*1−−*,* 2−−*,* 3++*,* 4++*)*. For this we perform a [2−−1−−> shift,

$$\left|\hat{1}\right\rangle = \left|1\right\rangle - z\left|2\right\rangle, \quad \left|\hat{2}\right\rangle = \left|2\right\rangle + z\left|1\right\rangle. \tag{2.62}$$

As *Mn(*−−*,* ++*,...,* ++*)* = 0 for *n >* 3, we again only find two channels in the BCFW recursion, as shown in Fig. 2.6 with the hatted leg *n*ˆ now replaced by 2ˆ, while the positive-helicity legs are summed over. Still, the same argument for the vanishing of the type-*(*II*)* diagrams applies. For the special case of the four-point graviton amplitude we therefore have

$$M\_4(1^{--},2^{--},3^{++},4^{++}) = \underset{3^{++}\stackrel{\triangle}{\mathbb{R}}}{\underset{3^{++}\stackrel{\triangle}{\mathbb{R}}}{\stackrel{\triangle}{\mathbb{R}}}} \underbrace{\underset{\stackrel{\triangle}{\mathbb{R}}}{\stackrel{\triangle}{\mathbb{R}}}}\_{\stackrel{\triangle}{\mathbb{R}}\_{\stackrel{\triangle}{\mathbb{R}}}} \underbrace{\underset{\stackrel{\triangle}{\mathbb{R}}}{\stackrel{\triangle}{\mathbb{R}}}}\_{\stackrel{\triangle}{\mathbb{R}}\_{\stackrel{\triangle}{\mathbb{R}}}} + \left(3 \leftrightarrow 4\right),\tag{2.63}$$

with *P* = −*p*1 − *p*3. Inserting the three-graviton amplitudes (1.178) this becomes (setting *κ* = 1)

$$M\_4(1^{--}, 2^{--}, 3^{++}, 4^{++}) = \left(\frac{[\hat{P}\mathfrak{J}]^3}{[31][1\hat{P}]}\right)^2 \frac{\mathrm{i}}{(p\_1 + p\_3)^2} \left(\frac{\langle\hat{P}\mathfrak{J}\rangle^3}{\langle 24\rangle\langle 4\hat{P}\rangle}\right)^2 + (3 \leftrightarrow 4)$$

$$= \frac{\mathrm{i}}{s\_{13}} \frac{\langle 2|\hat{P}|\mathfrak{J}\rangle^6}{\langle 24\rangle^2 [31]^2 \langle 4|\hat{P}|\mathfrak{J}\rangle^2} + (3 \leftrightarrow 4)\,\mathrm{i}$$

We now use *P*ˆ = *p*2 + *p*4 + *z*|2>[1| to find <2|*P*ˆ|3]=<24>[43] and <4|*P*ˆ|1] = <42>[21]. Hence, the *z* dependance drops out! Inserting these two relations one finds

$$M\_4(1^{--}, 2^{--}, 3^{++}, 4^{++}) = \mathrm{i} \frac{[34]^6}{[12]^2} \left( \frac{\langle 24 \rangle^2}{\langle 13 \rangle [31]^3} + \frac{\langle 23 \rangle^2}{\langle 14 \rangle [41]^3} \right),\tag{2.64}$$

where *sij* = *(pi* + *pj )*2. While this is the final result, one may write it using the Mandelstam variables

$$s = \left(p\_1 + p\_2\right)^2, \quad t = \left(p\_2 + p\_3\right)^2, \quad u = \left(p\_1 + p\_3\right)^2. \tag{2.65}$$

Doing this one finally arrives at the compact result (reinstating the coupling)

$$M\_4(1^{--}, 2^{--}, 3^{++}, 4^{++}) = \dot{\kappa}^2 \frac{\langle 12 \rangle^4 [34]^4}{stu},\tag{2.66}$$

with a peculiar pole structure. The correct helicity scaling is easily checked. Similarly to the gluon case, a closed expression for the MHV *n*-graviton tree-level amplitude may also be conjectured and proven via BCFW recursion. Yet, it is more involved and may be found in [25].

**Exercise 2.4 (The Six-Gluon Split-Helicity NMHV Amplitude)** Determine the first non-trivial next-to-maximally-helicity-violating (NMHV) amplitude

$$A\_6^{\text{tree}}({\bf l}^+, {\bf 2}^+, {\bf 3}^+, {\bf 4}^-, {\bf 5}^-, {\bf 6}^-)$$

from a BCFW recursion relation and our knowledge of the MHV amplitudes. Consider a shift of the two helicity states 1+ and 6−, and show that

$$A\_6^{\text{tree}}(1^+,2^+,3^+,4^-,5^-,6^-)=\text{ig}^4\left(\frac{\langle 6|P\_{12}|3\rangle^3}{\langle 61\rangle\langle 12\rangle\langle 13\rangle\langle 44\rangle\|\mathfrak{E}\rangle\langle \mathfrak{E}|P\_{16}|2\rangle}\right)$$

$$\frac{1}{(p\_6+p\_1+p\_2)^2}+\frac{\langle 4|P\_{56}|1\rangle^3}{\langle 23\rangle\langle 34\rangle\langle 16\rangle\|\mathfrak{E}\mathfrak{E}\rangle\langle \mathfrak{E}|P\_{16}|2\rangle}$$

$$\frac{1}{(p\_5+p\_6+p\_1)^2}\right),\tag{2.67}$$

where *Pij* = *pi* + *pj* . For the solution see Chap. 5.

**Exercise 2.5 (Soft Limit of the Six-Gluon Split-Helicity Amplitude)**  Check the consistency of the six-gluon split-helicity amplitude of Eq. (2.67) with the soft limit of leg 5. For the solution see Chap. 5.

### **2.5 BCFW Recursion with Massive Particles**

So far we restricted our attention to amplitudes involving massless particles, i.e. gluons, gravitons, and massless fermions and scalars. Yet, scattering amplitudes involving massive particles are of great relevance in physics, and consequently on-shell recursions have been devised for this case as well. Let us focus here on colour-ordered gauge-theory amplitudes involving also massive coloured matter fields (for the colour-ordered amplitudes the representation of the matter field is irrelevant). Concretely, we consider amplitudes involving at least two massless gluons, which we take to be neighbours for simplicity of the discussion, say at positions 1 and *n*, together with *n* − 2 massive states:

$$A\_{\hbar}(p\_1, p\_2, \dots, p\_n) \,, \qquad p\_1^2 = 0 = p\_n^2 \,, \qquad p\_l^2 = m\_l^2 \,. \tag{2.68}$$

Let us now see how the BCFW on-shell recursion derived in Sect. 2.2 may be generalised to gauge theory amplitudes with massive particles. We closely follow reference [13] in our exposition.

As before in Eq. (2.38), we consider a complex shift of the null gluon momenta at positions 1 and *n* by a parameter *z* ∈ C,

$$
\hat{p}\_1(z) = p\_1 - z|n\rangle\langle 1|\,, \qquad \hat{p}\_n(z) = p\_n + z|n\rangle\langle 1|\,. \tag{2.69}
$$

This entails a *z*-shift of the region momentum *Pi* := *p*1 + *...pi*−1,

$$\vec{P}\_l(z) = P\_l - z|n\rangle\langle 1|\,, \qquad i \in \{3, \ldots, n-1\}\,,\tag{2.70}$$

which also featured in the BCFW recursion relation of Eq. (2.42). Importantly, the on-shellness of the deformed legs 1ˆ and *n*ˆ as well as total momentum conservation is preserved under the shift. As the arguments leading to the BCFW recursion relation are purely based on factorisation, they are applicable to a generic quantum field theory involving massive particles as well. The BCFW recursion was obtained by thinking about the deformed amplitude *An(z)* as an analytic function in *z*. Its poles arise whenever an internal propagator associated to the *z*-shifted region momentum *P*ˆ *i(z)* goes on-shell. This reasoning does not change in the massive case, i.e. we will have a pole whenever a *z*-shifted region momentum goes on-shell, i.e. *P*ˆ 2 *i (z)* = *m*2

with *i* ∈ {3*,...,n* − 1} . The pole then reads in generalisation of Eq. (2.42) as

$$\frac{1}{\hat{P}\_l(z)^2 - m\_{P\_l}^2} = \frac{1}{(\hat{P}\_l(z) + p\_2 + \dots \ p\_{l-1})^2 - m\_{P\_l}^2}$$

$$= \frac{1}{P\_l^2 - m\_{P\_l}^2 - z \langle n | P\_l | 1 \rangle} \,, \tag{2.71}$$

where *mPi* is the mass of the associated intermediate particle going on-shell. Generalising Eq. (2.43), the location of the pole is shifted to

$$z\_{P\_l} = \frac{P\_l^2 - m\_{P\_l}^2}{\langle n | P\_l | 1 \rangle}, \qquad \forall i \in \{3, \dots, n - 1\}. \tag{2.72}$$

### **BCFW Recursion Relation with Massive Particles**

Using again the complex analysis arguments of Fig. 2.4, one immediately arrives at the on-shell recursion relation for amplitudes including massive particles:

$$A\_{n}(1,\ldots,n) = \sum\_{l=3}^{n-1} \sum\_{s \in \text{sp}} A\_{L}(\hat{1}(z\_{P\_{l}}), 2, \ldots, i-1, -\hat{P}^{\bar{s}}(z\_{P\_{l}})) \frac{\text{i} \, n\_{\text{P}}}{P\_{l}^{2} - m\_{P\_{l}}^{2}}$$

$$\times A\_{R}(\hat{P}^{s}(z\_{P\_{l}}), i, \ldots, n-1, \hat{n}(z\_{P\_{l}})) + \text{Res}(z = \infty),\tag{2.73}$$

where the sum now is over the spins *s*P of the intermediate particle P and *n*P is the particle-dependent constant appearing in the factorisation as described below Eq. (2.3). We recall that the legs 1 and *n* are assumed to be massless.

Again, this formula is only of use if the residue at infinity, Res*(z* = ∞*)*, vanishes. This turns out to be the case if the gluon helicities of the shifted legs are not of the [*n*+1−> type, just as in Eq. (2.52). Hence, the statement is

$$\operatorname{Res}(z = \infty) = 0 \quad \text{iff} \quad (h\_1, h\_n) = (+, -), \ (+, +), \ (-, -). \tag{2.74}$$

See [13] for a derivation. This renders the massive BCFW recursion relation (2.73) very useful.

### **2.5.1 Four-Point Amplitudes with Gluons and Massive Scalars**

Let us construct an explicit example. We consider a theory of a massive complex scalar field coupled to gauge theory. Concretely, we want to evaluate the four-point amplitude involving two neighbouring gluons of positive helicity and two scalars,

$$A\_4(1^+, \mathcal{Z}\_\phi, \mathcal{Z}\_{\bar{\phi}}, 4^+) \,. \tag{2.75}$$

The scalars have mass *m*2. This amplitude vanishes in the massless limit *m* = 0, similarly to the vanishing of the above amplitude when the scalars are replaced by massless fermions, as was shown in Eq. (1.164). In fact, amplitudes of the above type are of interest even in massless theories at the one-loop level. There, the need to regulate divergences leads one to consider internal particles propagating in *D* = 4 − 2*Σ* dimensions which may be modelled using masses, as we shall discuss in detail in the next chapter.

Returning to our concrete example we employ the massive on-shell recursion of Eq. (2.73). Only the scalar channel contributes,

$$A\_4(\mathbf{l}^+, \mathbf{2}\_\phi, \mathbf{3}\_{\bar{\phi}}, 4^+) = A\_3(\hat{\mathbf{l}}^+, \mathbf{2}\_\phi, -\hat{P}\_{\bar{\phi}}) \frac{\mathbf{i}}{P^2 - m^2} A\_3(\hat{P}\_\phi, \mathbf{3}\_{\bar{\phi}}, \hat{4}^+),\qquad(2.76)$$

as an amplitude with a single scalar vanishes, *A*3*(*1ˆ+*,* 2*φ,* 3±*)* = 0. All that is needed are the *(φgφ)*¯ -amplitudes. These follow from the colour-ordered Feynman vertices of two charged scalars and a gluon of Eq. (1.151),

$$V\_3(l\_1, p^\mu, l\_2) = \underset{p \stackrel{\oplus}{\sim} \mu}{l\_1^\mu} \underbrace{\stackrel{\bullet}{\sim}}\_{\mu} - l\_2 \tag{2.77} = \underset{p \stackrel{\oplus}{\sim} \mu}{\text{arg}} \left(l\_2^\mu - l\_1^\mu\right),\tag{2.77}$$

where 1 (2) denotes a *φ*¯ (*φ*) leg, respectively. Contracting this with the positivehelicity gluon polarisation of Eq. (1.124) one obtains the on-shell three-point amplitudes (setting *g* = 1)

$$A\_{\Im}(l\_{1\cdot\bar{\phi}},p^+,l\_{2\cdot\phi}) = -\mathbf{i}\frac{\langle r|l\_1|p\rangle}{\langle rp\rangle} = A\_{\Im}(l\_{1\cdot\phi},p^+,l\_{2\cdot\bar{\phi}})\,,\tag{2.78}$$

where the last relation follows by reflection. Note that here *r* is the arbitrary null reference momentum of the gluon leg related to the local gauge invariance of the theory. By similar arguments one establishes the three-point amplitudes involving a negative helicity gluon:

$$A\_{\Im}(l\_{1\cdot\bar{\phi}},p^-,l\_{2\cdot\phi}) = -\mathbf{i}\frac{\langle p|l\_1|r\rangle}{\lbrack pr\rbrack} = A\_{\Im}(l\_{1\cdot\phi},p^-,l\_{2\cdot\bar{\phi}})\,. \tag{2.79}$$

Before moving on with the recursion, let us address a seemingly dramatic problem: the amplitudes of Eqs. (2.78) and (2.79) apparently depend on the reference momentum *r*—how can that be? It turns out that, despite their representation, the amplitudes in Eqs. (2.78) and (2.79) are actually *independent* of the choice of *r*. Taking the initial reference spinor |*r*> and |*p*> as a basis in Weyl spinor space, we may parametrise an arbitrary reference spinor different from |*r*> as |*r*' > = *α*|*r*> + *β*|*p*>. Clearly, Eq. (2.78) is invariant under rescaling of the reference spinor |*r*> → *Λ* |*r*>, thus without loss of generality we may parameterise |*r*' >=|*r*>+*γ* |*p*>, or infinitesimally write *δr*|*r*>∝|*p*>. This entails that the amplitude Eq. (2.78) changes under a variation of the reference spinor |*r*> as

$$
\delta\_r A\_3(l\_1^+, p^+, l\_2^-) \propto \frac{\langle p | l\_1 | p \rangle}{\langle rp \rangle} = 0 \,, \tag{2.80}
$$

where the vanishing follows from <*p*|*l*1|*p*] = 2*p* · *l*1 = 0, which is a consequence of the three-point kinematics:

$$(l\_1 + p)^2 = l\_2^2 \quad \rightarrow \quad l\_1 \cdot p = 0 \quad \text{as} \quad l\_1^2 = l\_2^2 = m^2 \,, \ p^2 = 0 \,. \tag{2.81}$$

A similar argument applies to Eq. (2.79). Again we see the subtleties in three-point amplitudes: the expressions in Eqs. (2.78) and (2.79) are actually independent of *r*.

Coming back to the recursive construction of the amplitude *A*4*(*1+*,* 2*φ,* 3*φ*¯*,* 4+*)*, we have (cf. Fig. 2.7)

$$\begin{split} A\_{4}(\mathbf{1}^{+},2\_{\phi},3\_{\tilde{\phi}},4^{+}) &= A\_{3}(-\hat{P}\_{\tilde{\phi}},\hat{\mathbf{1}}^{+},2\_{\phi}) \frac{\mathbf{i}}{P^{2}-m^{2}} A\_{3}(3\_{\tilde{\phi}},\hat{\mathbf{4}}^{+},\hat{P}\_{\phi}) \\ &= \left(\mathbf{i}\frac{\langle r\_{1}|\hat{P}|\hat{\mathbf{1}}\rangle}{\langle r\_{1}\hat{\mathbf{1}}\rangle}\right) \frac{\mathbf{i}}{P^{2}-m^{2}} \left(-\mathbf{i}\frac{\langle r\_{4}|p\_{3}|\hat{\mathbf{4}}\rangle}{\langle r\_{4}\hat{\mathbf{4}}\rangle}\right), \end{split} \tag{2.82}$$

with *P* = *p*1 + *p*2. Here *r*1*/*4 denote the reference momenta of the gluon legs 1 and 4. Things are simplified considerably with the gauge choice

$$r\_1 = \hat{p}\_4\,, \qquad r\_4 = \hat{p}\_1\,\,. \tag{2.83}$$

**Fig. 2.7** On-shell recursion for the massive *A*4*(*1+*,* 2*φ,* 3*φ*¯*,* 4+*)* amplitude. All external momenta are outgoing, *P* runs from right to left

Noting that |1ˆ]=|1] and |4ˆ>=|4>, we then have

$$\begin{aligned} \langle r\_1 | \hat{P} | \hat{1} \rangle &= \langle 4 | \hat{P} | 1 \rangle = \langle 4 | P | 1 \rangle = -\langle 4 | p\_3 | 1 \rangle, & \langle r\_1 \hat{1} \rangle = \langle 4 \hat{1} \rangle = \langle 4 \hat{1} \rangle \,. \\\\ \langle r\_4 | p\_3 | \hat{4} \rangle &= \langle \hat{1} | p\_3 | \hat{4} \rangle \,. \end{aligned} \tag{2.84}$$

Plugging these into the above we find

$$A\_4 = \mathrm{i} \frac{\langle 4 | p\_3 | 1 \rangle \langle \hat{1} | p\_3 | \hat{4} \rangle}{\langle 14 \rangle^2 \left[ (p\_1 + p\_2)^2 - m^2 \right]}. \tag{2.85}$$

The numerator may be simplified with a trace identity (see Eq. (1.29)) to

$$
\begin{split}
\langle 4|p\_3|1\rangle \langle \hat{1}|p\_3|\hat{4}\rangle &= \frac{1}{2} \operatorname{Tr} \left(\hat{p}\_4 \,\hat{p}\_3 \,\hat{p}\_1 \,\hat{p}\_3\right) \\ &= 2 \left( 2(p\_3 \cdot \hat{p}\_4) \,(p\_3 \cdot \hat{p}\_1) - p\_3^2 \left(\hat{p}\_1 \cdot \hat{p}\_4\right) \right). \end{split} \tag{2.86}$$

In fact *p*3 · ˆ*p*4 = 0, which follows from momentum conservation:

$$
\hat{P}^2 = (p\_3 + \hat{p}\_4)^2 \quad \Rightarrow \quad m^2 = 2p\_3 \cdot \hat{p}\_4 + p\_3^2 \quad \Rightarrow \quad p\_3 \cdot \hat{p}\_4 = 0. \tag{2.87}
$$

Moreover, we find 2*p*ˆ1 · ˆ*p*4 = <14ˆ >[41ˆ ]=<14>[41]. Putting everything together we arrive at the compact expression for the four-point amplitude

$$A\_4(1^+,2\_\phi,3\_{\tilde{\phi}},4^+) = \operatorname{i}\frac{\left[14\right]m^2}{\left(14\right)\left[(p\_1+p\_2)^2-m^2\right]},\tag{2.88}$$

which indeed vanishes in the massless limit, as claimed.

**Exercise 2.6 (Mixed-Helicity Four-Point Scalar-Gluon Amplitude)** Compute the four-point massive-scalar-gluon amplitude with one positive and one negative gluon,

$$A\_4(1^+,2\_\phi, \mathfrak{z}\_{\tilde{\phi}}, 4^-) = \mathrm{i} \frac{\langle 4 | p\_3 | 1 \rangle^2}{(p\_1 + p\_4)^2 \left[ (p\_1 + p\_2)^2 - m^2 \right]},\tag{2.89}$$

using the above recursive techniques. For the solution see Chap. 5.

### **2.6 Symmetries of Scattering Amplitudes**

We now turn to a more conceptual yet very important subject: the question of how the space-time symmetries of the Poincaré group (and beyond) that we discussed in Sect. 1.1 manifest themselves at the level of scattering amplitudes. This has proven to be a very rich subject in particular at tree level. The space-time symmetries of scattering amplitudes may be grouped into obvious and less obvious symmetries.

The obvious symmetries are the Poincaré transformations of Sect. 1.1, under which scattering amplitudes should be invariant. Sticking to massless amplitudes, working in the spinor-helicity formulation of momentum space is highly advantageous. Here the momentum generator *pαα*˙ is represented by a multiplicative operator,

$$p^{\alpha \dot{\alpha}} = \sum\_{l=1}^{n} \lambda\_l^{\alpha} \,\tilde{\lambda}\_l^{\dot{\alpha}}\,,\tag{2.90}$$

and any amplitude A*n* must obey

$$p^{\alpha \dot{\alpha}} \, \mathcal{R}\_{\pi}(\{\lambda\_i, \tilde{\lambda}\_i\}) = 0 \,. \tag{2.91}$$

This is in fact true in the distributional sense of the relation *p δ(p)* = 0 *,* thanks to the total momentum conserving delta-function present in each amplitude (that we often drop):

$$\mathcal{A}\_{\hbar}(\lambda\_{i}, \tilde{\lambda}\_{i}) = \delta^{(4)}\left(\sum\_{l} p\_{l}\right) A\_{\hbar}\left(\lambda\_{i}, \tilde{\lambda}\_{l}\right) \,. \tag{2.92}$$

In our notation we distinguish the full amplitude A*n* and the delta-function "stripped" amplitude *An*.

The Lorentz generators in the helicity spinor basis come in two pairs of symmetric rank-two tensors *mαβ* and *mα*˙*β*˙, originating from the projections *Mμν (σμν )αβ* = *mαβ* and *Mμν (σ*¯*μν )α*˙*β*˙ = *mα*˙*β*˙. They are first-order differential operators in helicity spinor space,

$$m\_{\alpha\beta} = \sum\_{l=1}^{n} \lambda\_{l\ (\alpha\ \partial\_l \beta)} \,, \quad \overline{m}\_{\dot{\alpha}\dot{\beta}} = \sum\_{l=1}^{n} \tilde{\lambda}\_{l\ (\dot{\alpha}\ \partial\_l \dot{\beta})} \,, \tag{2.93}$$

where *∂iα* := *∂ ∂λα i* , *∂iα*˙ := *∂ ∂λ*˜ *α*˙ *i* and *r(αβ)* := *(rαβ* + *rβα)/*2 denotes symmetrisation with unit weight, cf. Exercise 1.6. The invariance of *An(λi, λ*˜*i)* under Lorentztransformations,

$$m\_{\alpha\beta} \, A\_n(\lambda\_i, \bar{\lambda}\_i) = 0 = \overline{m}\_{\dot{\alpha}\dot{\beta}} \, A\_n(\lambda\_i, \bar{\lambda}\_i) \,, \tag{2.94}$$

is manifest, as it is an immediate consequence of the proper contraction of all Weyl indices within *An*, i.e. the fact that the spinor brackets <*ij* > and [*ij* ] are invariant under *mαβ* and *mα*˙*β*˙. See the solution of Exercise 1.6 for an explicit calculation.

Let us now discuss a set of less obvious symmetries of A*n(λi, λ*˜*i)* in the case of pure colour-ordered gluon amplitudes. Classical Yang-Mills theory is in fact invariant under a larger symmetry group than Poincaré: due to the absence of dimensionful parameters in the theory (the coupling *g* is dimensionless) pure Yang-Mills theory (as well as massless QCD or scalar QCD) is invariant under a scale transformation,

$$\mathbf{x}^{\mu} \to \Lambda^{-1} \mathbf{x}^{\mu}, \quad \text{respectively} \quad p^{\mu} \to \Lambda \, p^{\mu} \,. \tag{2.95}$$

The scale transformations of the momenta are generated by the dilatation operator *d*, whose representation in spinor-helicity variables acting on amplitudes is

$$d = \sum\_{l=1}^{n} \left(\frac{1}{2}\lambda\_l^{\underline{\alpha}}\partial\_{l\;\alpha} + \frac{1}{2}\tilde{\lambda}\_l^{\dot{\underline{\alpha}}}\partial\_{l\;\dot{\alpha}} + d\_0\right), \qquad d\_0 \in \mathbb{R} \,, \tag{2.96}$$

reflecting the mass dimension 1*/*2 of the *λi* and *λ*˜*i* helicity spinors, i.e. [*d,λi*] = *λi/*2 and [*d, λ*˜*i*] = *λ*˜*i/*2. The constant *d*0 is undetermined at this point. It may be fixed by requiring invariance of the MHV gluon amplitudes,

$$\mathcal{R}\_n^{\rm MHV} = \delta^{(4)}(p) \frac{\langle \text{st} \rangle^4}{\langle 12 \rangle \dots \langle n1 \rangle} \,, \tag{2.97}$$

where *p* = Σ*n i*=1 *pi*. The dilatation operator *d* of Eq. (2.96) simply measures the weight in units of mass of the amplitude it acts on adding a factor of *n d*0, namely

$$d\,\mathcal{H}\_{\hbar} = \left( [\mathcal{H}\_{\hbar}] + n \, d\_{0} \right) \mathcal{H}\_{\hbar} \,, \tag{2.98}$$

where [O] indicates the weight of O in dimensions of mass. We note the weights [*δ(*4*) (p)*]=−4, [<*st*>4] = 4 and [<12>*...*<*n*1>] = *n*, hence

$$d\,\mathcal{R}\_n^{\text{MHV}} = (-4 + 4 - n + n \,d\_0) \,\mathcal{R}\_n^{\text{MHV}},\tag{2.99}$$

which vanishes for the choice *d*0 = 1. One easily checks the invariance under dilatations of the *qqgg* ¯ -amplitude of Eq. (1.194) and of the MHV*n* amplitudes as well.

The scaling symmetry comes with a further less obvious symmetry of vectorial nature known as special conformal transformations. Their generators, denoted by *kαα*˙ , are realised in terms of a second-order differential operator in the spinor

variables,

$$k\_{a\dot{\alpha}} = \sum\_{l=1}^{n} \partial\_{l\;\alpha} \,\partial\_{l\;\dot{\alpha}}\,. \tag{2.100}$$

Together with the Poincaré and dilatation generators, the set {*pαα*˙ *, kαα*˙ *, mαβ, mα*˙*β*˙*, d*} generates the *conformal group* in four dimensions, SO*(*2*,* 4*)*.

Let us now prove the invariance of the MHV gluon amplitudes under special conformal transformations. As the only dependence of AMHV *n* on the conjugate spinors *λ*˜*i* resides in the momentum conserving delta-function, we have

$$k\_{a\dot{\alpha}} \cdot \mathcal{J}\_n^{\text{MWV}} = \sum\_{i=1}^n \frac{\partial}{\partial \dot{\lambda}\_i^{\alpha}} \frac{\partial}{\partial \dot{\bar{\lambda}}\_i^{\dot{\alpha}}} \left(\delta^{(4)}(p) \, A\_n^{\text{MWV}}\right)$$

$$= \sum\_{i=1}^n \frac{\partial}{\partial \dot{\lambda}\_i^{\alpha}} \left[\frac{\partial p^{\beta\dot{\beta}}}{\partial \dot{\lambda}\_i^{\dot{\alpha}}} \left(\frac{\partial}{\partial p^{\beta\dot{\beta}}} \,\delta^{(4)}(p)\right) \, A\_n^{\text{MWV}}\right]$$

$$= \left[\left(n \, \frac{\partial}{\partial p^{\alpha\dot{\alpha}}} + p^{\beta\dot{\beta}} \,\frac{\partial}{\partial p^{\beta\dot{\alpha}}} \,\frac{\partial}{\partial p^{\alpha\dot{\beta}}}\right) \delta^{(4)}(p)\right] A\_n^{\text{MWV}}$$

$$+ \left(\frac{\partial \,\delta^{(4)}(p)}{\partial p^{\beta\dot{\alpha}}}\right) \sum\_{i=1}^n \lambda\_i^{\beta} \, \frac{\partial}{\partial \lambda\_i^{\alpha}} A\_n^{\text{MWV}} \,. \tag{2.101}$$

The last term may be rewritten as follows. First, we note the relation

$$\sum\_{l=1}^{n} \lambda\_{l\,\alpha} \,\partial\_{l\,\beta} = \sum\_{l=1}^{n} \lambda\_{l\,\,(\alpha} \,\partial\_{l\,\,\beta)} + \frac{1}{2} \epsilon\_{\alpha\beta} \sum\_{l} \lambda\_{l}^{\prime} \,\partial\_{l\,\,\,\gamma} \,, \tag{2.102}$$

which follows from decomposing the left-hand side in a symmetric and antisymmetric piece. The first term on the right-hand side is the Lorentz generator *mαβ*, which we already know annihilates *A*MHV *n* . The remaining term yields

$$\sum\_{l=1}^{n} \lambda\_l^{\beta} \frac{\partial}{\partial \lambda\_l^{\alpha}} A\_n^{\text{MHV}} = \frac{1}{2} \delta\_\alpha^\beta \sum\_l \lambda\_l^\delta \,\partial\_{l\;\delta} A\_n^{\text{MHV}} = (4-n) A\_n^{\text{MHV}}.\tag{2.103}$$

Hence Eq. (2.101) turns into

$$k\_{\alpha \dot{\alpha}} \, \mathcal{R}\_n^{\text{MHV}} = \left[ \left( 4 \frac{\partial}{\partial p^{\alpha \dot{\alpha}}} + p^{\beta \dot{\beta}} \frac{\partial}{\partial p^{\beta \dot{\alpha}}} \frac{\partial}{\partial p^{\alpha \dot{\beta}}} \right) \delta^{(4)}(p) \right] A\_n^{\text{MHV}} \,. \tag{2.104}$$

Indeed in a distributional sense we have

$$p^{\beta \dot{\beta}} \frac{\partial}{\partial p^{\beta \dot{\alpha}}} \frac{\partial}{\partial p^{\alpha \dot{\beta}}} \delta^{(4)}(p) = -4 \frac{\partial}{\partial p^{a \dot{\alpha}}} \delta^{(4)}(p) \,, \tag{2.105}$$

which one may see by integrating the second derivative expression against a test function *F (p)*:

$$\begin{split} \int \mathrm{d}^{4}p \, F(p) \, p^{\beta \dot{\beta}} \, \frac{\partial}{\partial p^{\beta \dot{\alpha}}} \, \frac{\partial}{\partial p^{\alpha \dot{\beta}}} \, \delta^{(4)}(p) &= \\ &= \int \mathrm{d}^{4}p \left[ \left( \frac{\partial}{\partial p^{\beta \dot{\alpha}}} F(p) \right) \, 2 \, \delta^{\beta}\_{\dot{\alpha}} + \left( \frac{\partial}{\partial p^{\alpha \dot{\beta}}} F(p) \right) 2 \, \delta^{\dot{\beta}}\_{\dot{\alpha}} \right] \\ &= 4 \int \mathrm{d}^{4}p \left( \frac{\partial}{\partial p^{\alpha \dot{\alpha}}} F(p) \right) \, \delta^{(4)}(p) \, . \end{split} \tag{2.106}$$

This proves the vanishing of *kαα*˙ AMHV *n* , as claimed.

Summarising, we have constructed a representation of the conformal group whose generators are represented by differential operators of degree one (*mαβ*, *mα*˙*β*˙, *d*), of degree two (*kαα*˙ ), and as a multiplicative operator (*pαα*˙ ) in an *n*particle helicity spinor space. This representation is natural, as amplitudes are functions in this space. All these generators annihilate the scattering amplitudes. We have verified this explicitly for the MHV amplitudes. The representation obeys the commutation relations of the conformal algebra so*(*2*,* 4*)*,

$$\begin{aligned} [d, p^{a\dot{\alpha}}] &= p^{a\dot{\alpha}}, \quad [d, k\_{a\dot{\alpha}}] = -k\_{a\dot{\alpha}}, \quad [d, m\_{a\beta}] = 0 = [d, \overline{m}\_{\dot{\alpha}\dot{\beta}}], \\\ [k\_{a\dot{\alpha}}, p^{\beta}{}^{\dot{\beta}}] &= \delta\_{\alpha}{}^{\beta}\delta\_{\dot{\alpha}}{}^{\dot{\beta}}d + m\_{\alpha}{}^{\beta}\delta\_{\dot{\alpha}}{}^{\dot{\beta}} + \overline{m}\_{\dot{\alpha}}{}^{\dot{\beta}}\delta\_{\alpha}{}^{\beta}, \end{aligned} \tag{2.107}$$

plus the Poincaré commutators discussed in Sect. 1.1.

The origin of this helicity spinor space representation becomes clear if one looks at the more familiar representation of the conformal group in configuration space *xμ*. For scalar fields, it reads

$$\mathcal{M}\_{\mu\nu} = \mathbf{i} \left( \mathbf{x}\_{\mu} \,\partial\_{\nu} - \mathbf{x}\_{\nu} \,\partial\_{\mu} \right), \qquad \qquad P\_{\mu} = -\mathbf{i} \,\partial\_{\mu} \,, \tag{2.108}$$

$$\mathcal{D} = -\mathbf{i} \left( \mathbf{x}^{\mu} \,\partial\_{\mu} + \Delta \right) \,, \qquad \qquad K\_{\mu} = \mathbf{i} \left[ \mathbf{x}^{2} \,\partial\_{\mu} - 2 \,\mathbf{x}\_{\mu} \left( \mathbf{x}^{\nu} \,\partial\_{\nu} + \Delta \right) \right] \,, \tag{2.108}$$

where *Δ* is the scaling dimension and *∂μ* := *∂/∂xμ*. In quantum field theory the conformal symmetry is distinguished by the Haag-Lopuszanski-Sohnius theorem [26] as the maximal bosonic extension of the space-time symmetry of the *S*-matrix generalising the Poincaré algebra.

A Fourier transform f d4*x* ei*p*·*x*O*(x, ∂x )* brings this representation into momentum space. From this point of view, it is clear why *pαα*˙ becomes a multiplication operator and *kαα*˙ a second-order derivative operator in momentum space as seen above. The momentum-space representation of the conformal symmetry as it applies to scattering amplitudes then takes the form

$$\begin{aligned} m^{\mu\upsilon} &= p^{\mu} \, \partial\_p^{\upsilon} - p^{\upsilon} \, \partial\_p^{\mu} \,, \qquad &p^{\mu} = p^{\mu} \,, \\ d &= p\_{\mu} \, \partial\_p^{\mu} + \tilde{\Delta} \,, \qquad &k^{\mu} = p^{\mu} \, \partial\_p^2 - 2(p\_{\upsilon} \partial\_p^{\upsilon} + \tilde{\Delta}) \partial\_p^{\mu} \,, \end{aligned} \tag{2.109}$$

where *Δ*¯ = 4 − *Δ*. This representation may be mapped to the helicity-spinor one discussed above.

> **The Conformal Generators in Spinor Helicity Space** Here we collect the generators of the conformal algebra in their single-particle action (with *Δ*¯ = 1, which is relevant for gauge bosons and scalar fields):

$$\begin{array}{ll} p^{\alpha \dot{\mu}} = \lambda^{\alpha} \tilde{\lambda}^{\dot{\mu}}, & k\_{\alpha \dot{\alpha}} & = \partial\_{\alpha} \partial\_{\dot{\alpha}} \end{array},$$

$$\begin{array}{ll} m\_{\alpha \dot{\beta}} = \lambda\_{(\alpha} \partial\_{\beta)} \end{array} , \qquad \qquad \overline{m}\_{\dot{\alpha} \dot{\beta}} = \tilde{\lambda}\_{(\dot{\alpha}} \partial\_{\dot{\beta})} \,, \tag{2.110}$$

$$d = \frac{1}{2} \lambda^{\alpha} \partial\_{\alpha} + \frac{1}{2} \tilde{\lambda}^{\dot{\alpha}} \partial\_{\dot{\alpha}} + 1 \; ,$$

where *r(αβ)* := *(rαβ* + *rβα)/*2 denotes symmetrisation of the indices. The helicity generator is given by *h* = −1 2*λα∂α* + 1 2*λ*˜ *α*˙ *∂α*˙ . It commutes with all generators of the conformal algebra.

Scaleless quantum field theories—such as pure Yang-Mills or massless QCD enjoy conformal symmetry at the tree-level. Yet, this symmetry is usually broken at the loop level, as the need to regularise divergencies introduces a scale into the quantum theory. It is manifested by a non-vanishing *β*-function of the coupling *g*. In fact, understanding the implications of broken conformal symmetry for loop amplitudes is an area of ongoing research. In the maximally supersymmetric generalisations of Yang-Mills theory, N = 4 super Yang-Mills theory, this property is prominently absent. It is a quantum conformal field theory, with far-reaching consequences, including a hidden infinite dimensional symmetry known as the Yangian symmetry. In fact, tree-level amplitudes are invariant under this extension of the super-conformal group [27] and the hidden integrability of the leading colour limit of the theory allows for exact non-perturbative results, see [28] for a review.

**Exercise 2.7 (Conformal Algebra)** Show that the representation constructed in the above Eq. (2.110) indeed obeys the commutation relations of the conformal algebra given in Eq. (2.107). For the solution see Chap. 5.

**Exercise 2.8 (Inversion and Special Conformal Transformations)** The generator *Kμ* of Eq. (2.108) generates infinitesimal special conformal transformations. A finite special conformal transformation is given by

$$\mathbf{x}^{\mu} \rightarrow \mathbf{x}^{\prime \mu} = \frac{\mathbf{x}^{\mu} - a^{\mu}\mathbf{x}^{2}}{1 - 2a \cdot \mathbf{x} + a^{2}\mathbf{x}^{2}},\tag{2.111}$$

where *aμ* is the transformation parameter.

$$\Phi(\mathbf{x}) \longrightarrow \Phi'(\mathbf{x}') = \left| \frac{\partial \mathbf{x}'}{\partial \mathbf{x}} \right|^{-\Delta/4} \Phi(\mathbf{x}) \,, \tag{2.112}$$

where |*∂x*' */∂x*| is the Jacobian of the transformation, and *Δ* is the scaling dimension. Show that the generator *Kμ* of Eq. (2.108) indeed generates the transformation of Eq. (2.111) for a scalar field. In other words, prove that

$$\Phi'(\mathbf{x}) = \left[1 - \mathrm{i} \, a^{\mu} \, K\_{\mu} + \mathcal{O}(a^{2})\right] \, \Phi(\mathbf{x}) \,. \tag{2.113}$$

Hint: in order to compute the Jacobian factor, treat the finite special conformal transformation as the composition of inversion, translation and inversion.

For the solution see Chap. 5.

### **2.7 Double-Copy Relations for Gluon and Graviton Amplitudes**

So far we have discussed gauge theories and gravity rather in parallel. While their Lagrangians and Feynman rules look very different, there exist intriguing relationships between gluon amplitudes and graviton amplitudes that suggest a deeper relationship of these two theories—at least in their perturbative domain. In a nutshell, they express gravity as the square of Yang-Mills theory, a property we already saw at the level of the polarisations and of the three-point amplitudes.

### **2.7.1 Lower-Point Examples**

The squaring relation between gravity and gluon amplitudes is manifest at the level of three-point amplitudes:

$$\begin{aligned} M\_3^{\text{tree}}(1^-, 2^{--}, \mathfrak{Z}^{++}) &= \frac{\langle 12 \rangle^6}{\langle 23 \rangle^2 \langle 31 \rangle^2} = \left[ A\_3^{\text{tree}}(1^-, 2^-, \mathfrak{Z}^+) \right]^2, \\\\ M\_3^{\text{tree}}(1^-, 2^{++}, \mathfrak{Z}^{++}) &= \frac{[23]^6}{[1!2]^2 [31]^2} = \left[ A\_3^{\text{tree}}(1^-, 2^+, \mathfrak{Z}^+) \right]^2. \end{aligned} \tag{2.114}$$

For simplicity we set all couplings to unity here and absorbed a factor of i in the amplitudes. Hence, for any choice of polarisations we find

$$M\_3^{\text{tree}}(1,2,3) = A\_3^{\text{tree}}(1,2,3)^2 \,. \tag{2.115}$$

Turning to the four-point case, we need to compare the MHV4 gluon amplitude and the MHV4 four-graviton amplitude of Eq. (2.66). We first expose the *s*, *t*, *u* poles in the MHV colour-ordered gluon amplitude explicitly. For *A*4*(*1−*,* 2−*,* 3+*,* 4+*)* we have

$$\frac{\langle 12 \rangle^3}{\langle 23 \rangle \langle 34 \rangle \langle 41 \rangle} = -\frac{1}{s\_{12}} \frac{\langle 12 \rangle^3 [34]}{\langle 23 \rangle \langle 41 \rangle} = -\frac{1}{s\_{12}s\_{23}} \frac{\langle 12 \rangle^2 \langle 12 \rangle [32][34]}{\langle 41 \rangle} = -\frac{\langle 12 \rangle^2 [34]^2}{s\_{12}s\_{23}}.$$

This gets close to the four-graviton amplitude (2.66) but is not just a simple square. Looking at *A*4*(*1−*,* 2−*,* 4+*,* 3+*)*, that is obtained by swapping 3 ↔ 4 in the above, we arrive at

$$M\_4^{\text{tree}}(1^-,2^-,\mathfrak{Z}^{++},4^{++}) = s\_{12}A\_4^{\text{tree}}(1^-,2^-,\mathfrak{Z}^+,4^+)A\_4^{\text{tree}}(1^-,2^-,4^+,\mathfrak{Z}^+) \,. \tag{2.116}$$

Again resulting in a squaring relation between the two. In fact, such squaring relations turn out to be generally true for all multiplicities.

### **2.7.2 Colour-Kinematics Duality: A Four-Point Example**

The relations (2.115) and (2.116) suggest a general squaring relation of the structure *Mn* ∼ *A*2 *n*. This can be made precise in the context of the colour-kinematic duality, which we now want to discuss in a four-point example.

For this, let us look at the *coloured* tree-level four-gluon amplitude in *D* dimensions using the polarisation *Σ μ i* and momentum *pμ i* vectors with *i* = 1*,...,* 4 to describe the kinematics. It may be written as

$$\mathbf{A}\_4^{\text{tree}} = -\text{ig}^2 \left( \frac{n\_s c\_s}{s} + \frac{n\_l c\_l}{t} + \frac{n\_u c\_s}{u} \right), \tag{2.117}$$

split into the *s*, *t* and *u* channels, with

$$\begin{aligned} c\_{\mathcal{S}} &= -2f^{d\_1 a\_2 \epsilon} \int \epsilon^{\epsilon a\_3 a\_4}, \\\\ n\_{\mathcal{S}} &= -\frac{1}{2} \Big[ \left[ \left( \epsilon\_1 \cdot \epsilon\_2 \right) p\_1^{\mu} + 2 \left( \epsilon\_1 \cdot p\_2 \right) \epsilon\_2^{\mu} - (1 \leftrightarrow 2) \right] \\\\ &\times \left[ \left( \epsilon\_3 \cdot \epsilon\_4 \right) p\_3 \, \_{,\mu} + 2 \left( \epsilon\_3 \cdot p\_4 \right) \epsilon\_{4,\mu} - (3 \leftrightarrow 4) \right] \\\\ &\quad + s \left[ \left( \epsilon\_1 \cdot \epsilon\_3 \right) \left( \epsilon\_2 \cdot \epsilon\_4 \right) - \left( \epsilon\_1 \cdot \epsilon\_4 \right) \left( \epsilon\_2 \cdot \epsilon\_3 \right) \right] \right], \end{aligned} \tag{2.119}$$

and

$$c\_{l}n\_{l} = c\_{s}n\_{s}\Big|\_{\substack{1 \rightarrow 2 \rightarrow 3 \rightarrow 1}}, \qquad c\_{u}n\_{u} = c\_{s}n\_{s}\Big|\_{\substack{1 \rightarrow 3 \rightarrow 2 \rightarrow 1}}.\tag{2.120}$$

In writing the amplitude in this fashion we have split up contact terms emerging from the four-gluon vertex (1.66) into the *s*, *t* and *u* channels by multiplying the corresponding colour factors *ns* by *s s* , and so on. These are the terms proportional to *s* in the last line of Eq. (2.119). It is instructive to study the gauge invariance of leg 4. Replacing *Σ*4 → *p*4 yields

$$m\_s \Big|\_{\epsilon\_4 \to p\_4} = \frac{s}{2} \left[ (\epsilon\_1 \cdot \epsilon\_2)(\epsilon\_3 \cdot p\_{12}) + \text{cyclic}(1, 2, 3) \right] =: s \, a(\{\epsilon\_l, p\_l\})\,,\qquad(2.121)$$

where *p*12 = *p*1 − *p*2. Crucially, the function *α* is cyclically invariant. Therefore the gauge transformations of the other kinematical numerators read

$$n\_{l}\Big|\_{\epsilon\_{4}\to p\_{4}} = t\,\alpha(\{\epsilon\_{l},p\_{l}\})\,,\qquad n\_{\mu}\Big|\_{\epsilon\_{4}\to p\_{4}} = \mu\,\alpha(\{\epsilon\_{l},p\_{l}\})\,.\tag{2.122}$$

Hence, the numerators are not gauge invariant individually. This is to be expected, as only the full amplitude and not individual graphs (or parts thereof) are gauge

invariant. How does **A**tree 4 then become gauge invariant? We have

$$\mathbf{A}\_4^{\text{tree}}\Big|\_{\epsilon\_4 \to p\_4} = \left(c\_s + c\_l + c\_u\right) \alpha(\{\epsilon\_l, p\_l\})\,,\tag{2.123}$$

which is zero by virtue of Jacobi's identity (1.135),

$$c\_s + c\_l + c\_u = -2 \left( f^{a\_1 a\_2 e} f^{\epsilon a\_3 a\_4} + f^{a\_2 a\_3 e} f^{\epsilon a\_1 a\_4} + f^{a\_3 a\_1 e} f^{\epsilon a\_2 a\_4} \right) = 0. \tag{2.124}$$

Remarkably, one also sees that the kinematical numerators *ni* obey a Jacobi-like identity,

$$n\_s + n\_l + n\_\mu = 0\,,\tag{2.125}$$

upon using the on-shell identities. This is known as the *kinematical Jacobi identity*. This property allows us to construct a gauge invariant object that has all the required properties to be the four-graviton amplitude: we simply replace the colour numerators *ci* by the kinematical ones *ni* in Eq. (2.117), obtaining

$$M\_4^{\text{tree}} = \mathbf{A}\_4^{\text{tree}} \Big|\_{\substack{c\_i \to n\_i \\ g \to \kappa/2}} = -\mathbf{i} \left(\frac{\kappa}{2}\right)^2 \left(\frac{n\_s^2}{s} + \frac{n\_l^2}{t} + \frac{n\_u^2}{u}\right) \,. \tag{2.126}$$

Clearly, it is bi-linear in the polarisation vectors *Σ μ i* and displays a consistent pole structure, which are necessary ingredients for it to be a graviton amplitude. Also the gauge invariance may be tested straightforwardly using Eqs. (2.121) and (2.122),

$$\left.M\_4^{\text{tree}}\right|\_{\epsilon\_4 \to p\_4} = 2\left(n\_s + n\_l + n\_u\right)\alpha\left(\{\epsilon\_l, p\_l\}\right),\tag{2.127}$$

which now vanishes by virtue of the kinematical Jacobi identity (2.125). We shall show momentarily that the result (2.126) is equivalent to Eq. (2.116). In order to do so, let us express the gluon amplitude (2.117) in a minimal colour and kinematical basis. Going to the DDM basis of Sect. 1.10 amounts to eliminating *ct* via *ct* = −*cs* − *cu* as

$$c\_s = f^{a\_1 a\_2 e} f^{e a\_3 a\_4} = \begin{array}{c} 2 \\ \stackrel{\triangle}{\mathbb{R}} \\ 1 \cdot \mathbb{R} \otimes \mathbb{R} \otimes \mathbb{R} \otimes \mathbb{R} \\ \end{array},$$

$$-c\_{\iota t} = f^{a\_1 a\_2 e} f^{e a\_2 a\_4} = \begin{array}{c} \mathbb{R} & 2 \\ \stackrel{\triangle}{\mathbb{R}} & \stackrel{\triangle}{\mathbb{R}} \\ 1 \cdot \mathbb{R} \otimes \mathbb{R} \otimes \mathbb{R} \otimes \mathbb{R} \otimes \mathbb{R} & 4 \end{array}.\tag{2.128}$$

The amplitude then takes the form

$$\begin{split} \frac{\mathbf{i}}{\mathcal{S}^2} \mathbf{A}\_4^{\text{tree}} &= c\_s \left( \frac{n\_s}{s} - \frac{n\_l}{t} \right) - c\_u \left( \frac{n\_l}{t} - \frac{n\_u}{u} \right) \\ &= c\_s \left| A\_4^{\text{tree}}(1, 2, 3, 4) - c\_u \left| A\_4^{\text{tree}}(1, 3, 2, 4) \right. \end{split} \tag{2.129}$$

hence the two colour-ordered amplitudes are to be identified as

$$A\_4^{\text{tree}}(1,2,3,4) = \frac{n\_s}{s} - \frac{n\_l}{t}, \qquad A\_4^{\text{tree}}(1,3,2,4) = \frac{n\_l}{t} - \frac{n\_u}{u} \,. \tag{2.130}$$

Their gauge invariance follows from Eqs. (2.121) and (2.122), e.g.

$$A\_4^{\text{tree}}(1,2,3,4)\Big|\_{\epsilon\_4 \to p\_4} = \left(\frac{s}{s} - \frac{t}{t}\right)\alpha = 0\dots$$

This must be the case, as we argued before for the gauge invariance of the colourordered amplitudes. Due to the kinematical Jacobi identity, the above representation of the amplitude is not yet minimal. By eliminating (in analogy to *ct*) now *nt* as well via *nt* = −*ns* − *nu* in Eq. (2.129) we find the relation

$$
\begin{pmatrix} A\_4^{\text{tree}}(1,2,3,4) \\ A\_4^{\text{tree}}(1,3,2,4) \end{pmatrix} = \begin{pmatrix} s^{-1} + t^{-1} & t^{-1} \\ -t^{-1} & -u^{-1} - t^{-1} \end{pmatrix} \begin{pmatrix} n\_s \\ n\_u \end{pmatrix} \,. \tag{2.131}
$$

From this expression we learn that the two colour-ordered amplitudes *A*tree 4 *(*1*,* 2*,* 3*,* 4*)* and *A*tree 4 *(*1*,* 3*,* 2*,* 4*) cannot* be independent of each other: while they are gauge invariant, the kinematical numerators *ns* and *nu* are not, and hence the 2 × 2 matrix relating them is not invertible. The linear dependance reads

$$\text{is } A\_4^{\text{tree}}(1,2,3,4) = \mu \, A\_4^{\text{tree}}(1,3,2,4) \,. \tag{2.132}$$

This is the Bern-Carrasco-Johannson (BCJ) relation for a four-point amplitude. The general relation reads, cf. (1.158),

$$\sum\_{l=2}^{n-1} p\_l \cdot (p\_2 + \dots + p\_l) \, A\_n^{\text{tree}}(2, \dots, i, 1, i+1, \dots, n) = 0 \,, \tag{2.133}$$

which for *n* = 4 reduces to Eq. (2.132). As a matter of fact, using this kinematical 2 × 2 matrix the coloured amplitude may be written as

$$\mathbf{A}\_{4}^{\text{tree}} = -\mathrm{ig}^{2} \begin{pmatrix} c\_{s} \ -c\_{u} \end{pmatrix} \begin{pmatrix} s^{-1} + t^{-1} & t^{-1} \\ -t^{-1} & -u^{-1} - t^{-1} \end{pmatrix} \begin{pmatrix} n\_{s} \\ n\_{u} \end{pmatrix} . \tag{2.134}$$

Returning to the conjectured form of the four-graviton amplitude (2.126), we can use there the relations *nt* = −*nu* −*ns* and *nu* = *tA*tree 4 *(*1*,* 2*,* 3*,* 4*)*+*nsu/s*. One finds

$$M\_4^{\text{tree}}(1,2,3,4) = -\text{i}\frac{\text{st}}{\mu} \Big[A\_4^{\text{tree}}(1,2,3,4)\Big]^2. \tag{2.135}$$

This very compact result may be transformed into the one we derived in Eq. (2.116) upon using the BCJ relation (2.132). Cyclically permute 1 → 2 → 3 → 4 → 1 in (2.132) (under which *s* ↔ *t* but *u* is inert) to reach

$$\iota \, A\_4^{\text{tree}}(2,3,4,1) = \iota \, A\_4^{\text{tree}}(2,4,3,1) \,. \tag{2.136}$$

With the cyclicity of the colour-ordered amplitudes, one has *A*tree 4 *(*1*,* 2*,* 4*,* 3*)* = *t uA*tree 4 *(*1*,* 2*,* 3*,* 4*)*, which inserted in Eq. (2.116) yields Eq. (2.135).

**Exercise 2.9 (Kinematical Jacobi Identity)** Prove the kinematical Jacobi identity (2.125) for the coloured tree-level four-gluon amplitude in *D* dimensions given in Eq. (2.117). For the solution see Chap. 5.

### **2.7.3 The Double-Copy Relation**

The general statement of the duality between colour and kinematics is as follows. A general coloured *n*-gluon amplitude may be written as

$$\mathbf{A}\_{n}^{\text{tree}} = -\text{ig}^{n-2} \sum\_{l} \frac{c\_{l} n\_{l}}{\prod\_{\alpha\_{l}} D\_{\alpha\_{l}}} \,, \tag{2.137}$$

where the sum is over all *n*-point diagrams with a trivalent vertex structure. Here *ci* denote the colour factors (made up of the structure constants and possibly of generators), *ni* the numerators (made up of the momenta and polarisation vectors), and the *Dαi* = *p*2 *αi* − *m*2 *αi* are the inverse propagators. We note that any graph may be made tri-valent upon inserting the identity 1 = *Dαi /Dαi* in order to lift the four-gluon vertices to a sum of three *s*-*t*-*u* channel diagrams as dictated by the colour structure. Due to the Jacobi identity, the colour factors obey algebraic relations of the form *ci* − *cj* = *ck*. Colour-kinematic duality now asserts that it is always possible to find a representation of the amplitude (2.137) in which also the kinematical numerators obey an analogous identity *ni* − *nj* = *nk*. This may be reached by possibly adding overall zeros to the amplitude.

The *n*-graviton scattering amplitude is then obtained upon replacing colour by kinematics *ci* → *ni*,

$$M\_n^{\text{tree}} = -\text{i}\left(\frac{k}{2}\right)^{n-2} \sum\_{l} \frac{n\_l^2}{\prod\_{\alpha\_l} D\_{\alpha\_l}} \,. \tag{2.138}$$

In general it is non-trivial to find numerators which satisfy the colour-kinematics duality. A possible route is to start out from an ansatz, which one then constrains to match the amplitude and to obey the duality. At tree level the duality has been proven [29–31], while at the Lagrangian level a full understanding is still lacking [30, 32, 33]. A comprehensive review of the double-copy relation is given in [34].

The *n*-point generalisation of the squaring relation (2.135) is known as the Kawai-Lewellen-Tye (KLT) relation and takes the form [35]

$$M\_n^{\text{tree}} = \sum\_{\sigma, \rho \in S\_{n-3}} A\_n^{\text{tree}}(1, \sigma, n-1, n) \, S[\sigma \, | \rho \, | \, A\_n^{\text{tree}}(1, \rho, n, n-1) \,. \tag{2.139}$$

Here *σ* and *ρ* range over the *(n* − 3*)*! permutations of the elements {2*,...,n* − 2}. The KLT kernel *S*[*σ*|*ρ*] are the entries of an *(n* − 3*)*! × *(n* − 3*)*! matrix of kinematic polynomials. A closed form expression reads [29, 36]

$$S[\sigma|\rho] = \prod\_{l=2}^{n-2} \left[ 2 \ p\_l \cdot p\_{\sigma l} + \sum\_{j=2}^{l} 2 \ p\_{\sigma l} \cdot p\_{\sigma j} \,\theta(\sigma\_j, \sigma\_l)\_{\rho} \right],\tag{2.140}$$

where *θ (σj , σi)ρ* = 1 when *σj* is before *σi* in the permutation *ρ*, and zero otherwise. For *n* = 4 the matrix degenerates to a scalar and reproduces Eq. (2.116).

**Exercise 2.10 (Five-Point KLT Relation)** Prove the five-point KLT relation

$$M\_{\mathfrak{F}}^{\text{tree}}(1,2,3,4,5) = s\_{12}s\_{34}A\_{\mathfrak{F}}^{\text{tree}}(1,2,3,4,5)A\_{\mathfrak{F}}^{\text{tree}}(1,2,5,3,4) + (2\leftrightarrow 3)\dots \tag{2.141}$$

For the solution see Chap. 5.

### **References**

1. F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions. Phys. Rev. **110**, 974–977 (1958). https://doi.org/10.1103/PhysRev.110.974

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# **3 Loop Integrands and Amplitudes**

### **Abstract**

In this chapter we study the structure of loop-level scattering amplitudes. The appearance of integrals over internal loop momenta gives rise to a new set of functions that go beyond the rational functions of spinor products seen at tree-level. We will use the unitarity of scattering amplitudes to show that discontinuities in loop amplitudes can be determined from tree-level information as a result of factorisation when loop momentum dependent propagators go on-shell. We then show that generalised discontinuities can be used to break loop amplitudes further into small tree-level building blocks. We then turn our attention to a general method for one-loop dimensionally regulated amplitudes in which a basis of functions is determined as well as a technique to determine their coefficients from on-shell data.

### **3.1 Introduction to Loop Amplitudes**

Perturbative predictions for scattering amplitudes allow us to explore the quantum nature of fundamental interactions. Explicit computations within quantum field theory, in particular using the method of Feynman diagrams, quickly lead to an explosion of both analytic and algebraic complexity.

Loop-level amplitudes involve integration of internal—virtual—momenta, which takes us beyond the simple rational functions we have encountered at tree level. In Chap. 2 we have seen that the analysis of the poles of tree-level amplitudes led to factorisation when the poles vanish. Equivalently we may say amplitudes factorise when the internal momenta go on-shell. This factorisation was observed when considering the soft and collinear limits of amplitudes and also, after analytic continuation to complex momenta, on the residues in the BCFW construction. As we will see, the integrals over the virtual momenta give rise to functions with branch

cuts, such as logarithms. These branch cuts lead to discontinuities, which are a new feature of loop-level amplitudes. At the level of the integrand, which is a rational function of the internal and external momenta, we may associate discontinuities with poles dependent on the virtual (or loop) momentum. Analysing the integrand at points where these poles vanish will again lead to factorisation into simpler objects. Our main aim for this chapter is to turn these words into a concrete computational method in which we may re-use on-shell tree-level amplitudes to directly obtain information about loop amplitudes.

There is, however, another major new feature of loop amplitudes. Integrals over virtual momenta can lead to divergences, which have to be regulated. Divergences at large values of the loop momentum are known as ultraviolet (UV), while divergences at small values of the loop momentum are known as infrared (IR). In these lecture notes we will follow the procedure of dimensional regularisation, which regulates both IR and UV regions using an analytical continuation of the space-time dimension to *D* = 4 − 2*ε*, where *ε* is a small parameter. We postpone a more detailed discussion of the dimensional regularisation until Chap. 4 (see Sect. 4.2.2), where we consider the evaluation of the loop integrals. UV and IR divergences must cancel for physical predictions. UV divergences are removed through the procedure of renormalisation, which is covered in standard field theory textbooks (e.g. [1–3]). The cancellation of IR singularities is more complicated and in general beyond the scope of these lecture notes. We have seen that IR singularities also appear in tree-level amplitudes in soft and collinear limits, and it is these divergences that must cancel the IR divergences in the virtual amplitudes. This cancellation only happens at the level of cross-sections, where the squared amplitude is integrated over an inclusive phase space. The topic is worthy of study in its own right, and the interested reader may like to explore the review [4].

We begin this discussion with some general observations on the structure of loop amplitudes. We will consider loop amplitudes in Yang-Mills theory (YM) in which the colour structure has been stripped off as discussed in Sect. 1.11. An amplitude with *n* external legs at *L* loops may be written in terms of a set of Feynman integrals, *F*, together with rational coefficients *c*. The amplitude will depend on the external momenta of each leg as well as their helicity, and mass (we may consider YM coupled to matter). We will write the arguments of the amplitudes as a list of integers which represent these properties. The external momenta will be denoted *pi* with *i* = 1*,...,n*. As in the previous chapters, we take them to be all outgoing, and hence they satisfy momentum conservation in the form Σ*n i*=1 *pi* = 0. When analytically continuing the dimension we must rescale the coupling to make sure that we are still expanding in a dimensionless quantity. This scale is arbitrary and we will represent it with the symbol *μ*R. To write down a general expression for the amplitude we must introduce a number of conventions. Let us first write the expression and then proceed with the explanation of the normalisations and symbols:

$$A\_n^{(L),[D]}(1,\ldots,n) = \left(g\_{\rm YM}\mu\_{\rm R}^{(4-D)/2}\right)^{n-2} \left(\frac{\mathrm{i}\,\alpha\_{\rm YM}\,\mu\_{\rm R}^{(4-D)}}{(4\pi)^{(D-2)/2}}\right)^L$$

$$\sum\_T c\_T^{[D]}(1,\cdots,n) \, F\_T^{(L),[D]}(p\_1,\cdots,p\_{n-1})\,.$$

Here the expansion is in the coupling *α*YM = *g*2 YM*/(*4*π )*. The linear combination sums over a set of loop *topologies*, *T* , which are defined by the set of propagators and loop-momentum dependent numerators. They may also potentially contain propagators with higher powers. The coefficients *c* [*D*] *T* depend on the momenta, helicities and masses of the external legs, and on the masses of the internal particles. On the other hand, the integrals *F(L),*[*D*] *T* only depend on the *n* − 1 independent external momenta and on the masses of the internal and external particles (which are suppressed in the notation above for conciseness). The factors of i and *π* are due to the normalisation of the integrals, which is given below. Example graphs of possible loop topologies are shown in Fig. 3.1. The structure of the amplitude is not specific to YM theory apart from the couplings. For a useful separation of coefficients and integrals, we need to identify a (linearly) independent basis of integrals, which defines the sum over topologies *T* . A precise definition of this basis at one loop is one of the main aims of this chapter. In addition, we will show how on-shell techniques can be used to directly extract the coefficients of the basis integrals. The couplings and dependence on *μ*R can be easily restored at the end of a calculation through dimensional analysis, and so we set *g*YM = 1 and *μ*R = 1 for the remainder of this chapter. Furthermore, the factors of *α*YM*μ(*4−*D)* R */(*4*π )(D*−2*)/*2

will also be suppressed, since they may be restored at the end of the computation as well.

At one loop all topologies take the form of an *n*-gon with a potential numerator function *N*,

$$F\_n^{(1),[D]}[N] = \int\_k \frac{N}{\prod\_{a=1}^n \left[ -(k - q\_a)^2 + m\_a^2 - \mathrm{i}0 \right]} \,, \tag{3.2}$$

where *qa* = Σ*a*−1 *b*=1 *pb* (with *q*1 = 0) are the momenta flowing in each propagator, which also have a mass *ma*. We have also introduced a short hand for the integration measure:

$$\int\_{k} := \int \frac{\mathbf{d}^{D}k}{\mathbf{i}\pi^{D/2}} \,. \tag{3.3}$$

Note the different loop integration measure with respect to the d*Dk/(*2*π )D* in the Feynman rules. This difference is responsible for the factors of i and *π* in Eq. (3.1), and will be motivated in Sect. 4.2. The configuration of momenta and propagators of Eq. (3.2) is shown graphically in Fig. 3.2. Cases in which the numerator is one, *F(*1*),*[*D*] *n* [1], are referred to as *scalar integrals*. When using an integer *n* to represent the topology we are already indicating that it is a one-loop integral, and so the loop-order superscript will be dropped for the remainder of this chapter. If no numerator is specified it should considered to be a scalar integral and so *F*[*D*] *n* ≡ *F*[*D*] *n* [1] ≡ *F(*1*),*[*D*] *n* [1]. A small imaginary part i0 follows from the Feynman prescription for the propagators that was introduced in the Feynman rules. This i0 prescription will mainly play a role only when evaluating the integrals, and so we will drop it from the propagator expressions in cases where it is not necessary. We will follow the standard convention to refer to the simple one-loop topologies according to the polygon that represents the number of propagators, e.g. bubble for two propagators, triangle for three propagators, box for four propagators, and so

**Fig. 3.2** The generic

on (see Fig. 3.1 again). An integral with one propagator is referred to as a tadpole integral.

The coefficients of the integrals in Eq. (3.1) can be expanded around the physical space-time dimension *D* = 4, as

$$c\_T^{[D]} = c\_T^{(0)} + \epsilon \left. c\_T^{(1)} + \epsilon^2 \left. c\_T^{(2)} + \dots \right. \tag{3.4}$$

The order at which we must expand the coefficients to ensure the correct result for the amplitudes as *ε* → 0 will depend on the overall divergences present in the loop integrals.

We may also consider the *integrand* of an amplitude *A(L),*[*D*] *n (*1*, . . . , n)*, which we denote *I (L),*[*D*] *n (*1*, . . . , n)*, and is a rational function satisfying

$$A\_n^{(L),[D]}(1,\ldots,n) = \int \prod\_{l=1}^L \frac{\mathbf{d}^D k\_l}{\mathbf{i}\pi^{D/2}} \, I\_n^{(L),[D]}(k,1,\ldots,n)\,. \tag{3.5}$$

The amplitude integrands are not uniquely defined, as they may differ by terms which integrate to zero, thus giving rise to the same amplitude. The simplest choice is to use the Feynman-diagram expansion to define the integrand. If we make the assumption that a colour-ordered one-loop amplitude has a single ordering of the external legs for the topology with *n* propagators (this is the case for the leadingcolour approximation in Yang-Mills theory), we can be more explicit and write

$$I\_n^{(1),[D]}(k,1,\ldots,n) = \frac{N(k,1,\ldots,n)}{\prod\_{a=1}^n \left[ -(k-q\_a)^2 + m\_a^2 - \mathrm{i}0 \right]} \,. \tag{3.6}$$

In the same way that we can try to find a basis of Feynman integrals for the amplitude, we may also ask if there exists a basis of loop-momentum dependent numerator functions {*fx* } such that the integrand numerator in Eq. (3.6) can be written as

$$N(k,1,\ldots,n) = \sum\_{\chi} c\_{\chi}(1,\ldots,n) \, f\_{\chi}(k,p\_1,\ldots,p\_{n-1})\,,\tag{3.7}$$

where the coefficients *cx* are rational functions of the external kinematics, and *fx* are independent scalar products dependent on the loop momentum. Again, the sum over *x* and the definition of "independent" here are not yet defined but we can motivate the construction with a simple example. If we consider a one-loop integrand *I (*1*),*[*D*] *n (k,* 1*,* 2*,* 3*)* in which *N (k,* 1*,* 2*,* 3*)* = *k* · *q*2, then we can express

**Fig. 3.3** Sample diagrams contributing to the four-gluon scattering amplitude at one-loop order

the numerator in terms of the difference of two propagators in order to rewrite everything in terms of scalar integral functions:

$$\begin{split} N(k,1,2,3) &= \frac{1}{2} \left[ \left( -(k-q\_2)^2 + m\_2^2 \right) - \left( -k^2 + m\_1^2 \right) + q\_2^2 - m\_2^2 + m\_1^2 \right] \\ &= \frac{1}{2} \left\{ 1, -1, q\_2^2 - m\_2^2 + m\_1^2 \right\} \cdot \left\{ -(k-q\_2)^2 + m\_2^2, -k^2 + m\_1^2, 1 \right\}^\top. \end{split} \tag{3.8}$$

So in this case the functions *fx* are the inverse propagators and 1.

Throughout this chapter we will use the example of four-gluon scattering to illustrate general methods for loop integrands and amplitudes. Sample diagrams for this process are shown in Fig. 3.3. In this case the most complicated topology is the box graph which, following the structure of the three-gluon vertex, has a maximum of four powers of the loop momentum in the numerator function. Graphs containing vertex corrections or those with bubble insertions will contain triangle and bubble tensor integrals respectively. As we will see, writing the loop-momentum dependence of the numerator in terms of the propagators in each graph will allow us to find a basis of scalar Feynman integrals, so that the particular form of Eq. (3.1) relevant for four-gluon scattering becomes

$$\begin{split} A\_4^{(1),[D]}(1,2,3,4) &= \mathrm{i}\,c\_{\mathrm{box}}^{[D]}F\_4^{[D]}[1](p\_1, p\_2, p\_3) \\ &+ \mathrm{i}\,c\_{\mathrm{mi},1}^{[D]}F\_3^{[D]}[1](p\_1, p\_2) + \mathrm{i}\,c\_{\mathrm{mi},2}^{[D]}F\_3^{[D]}[1](p\_1, p\_2) \\ &+ \mathrm{i}\,c\_{\mathrm{mi},3}^{[D]}F\_3^{[D]}[1](p\_{12}, p\_3) + \mathrm{i}\,c\_{\mathrm{mi},4}^{[D]}F\_3^{[D]}[1](p\_2, p\_3) \\ &+ \mathrm{i}\,c\_{\mathrm{bub},1}^{[D]}F\_2^{[D]}[1](p\_{12}) + \mathrm{i}\,c\_{\mathrm{bub},2}^{[D]}F\_2^{[D]}[1](p\_{23}) \,. \end{split} \tag{3.9}$$

where we have used the notation,

$$p\_{l\_1\cdots l\_n} = p\_{l\_1} + p\_{l\_2} + \cdots + p\_{l\_n},\tag{3.10}$$

which will also be used for the invariants,

$$s\_{i\_1\cdots i\_n} = p\_{i\_1\cdots i\_n}^2.\tag{3.11}$$

The fact that only scalar integrals appear remains to be proven of course. We will also set about the task of extracting the coefficients of these scalar integrals and the task of generalising to the *n*-point case.

The chapter is organised as follows. We start in Sect. 3.2 by demonstrating that the unitarity of the *S*-matrix leads to deep insights into the structure of loop amplitudes. This will lead us to consider the discontinuities of loop amplitudes, and show that they may be computed from the product of tree-level amplitudes. Using the example of four-gluon scattering, we will demonstrate that this results in an extremely efficient technique to identify simple integral structure in the amplitude. We will then explore generalised discontinuities of loop amplitudes in Sect. 3.3, and use the "Cutkosky rules" to make a direct connection with the pole structure of the integrand. The factorisation on these poles leads to the generalised unitarity method, which allows for the computation of the coefficients of the scalar box integrals. Section 3.4 lays the ground work for the general treatment of any loop amplitude, as we use tensor reduction and integrand-level analysis of transverse spaces to identify relations between Feynman integrals. Section 3.5 is dedicated to the derivation of the complete decomposition of a general one-loop amplitude into a basis of scalar integrals, and how their coefficients may be extracted from products of tree-level amplitudes via generalised unitarity. In Sect. 3.6 we put all of this technology to work to complete the computation of the one-loop four-gluon scattering amplitude in dimensional regularisation. Finally we give some outlook and extensions of the ideas presented here, and consider efficient computations using rational parametrisations of the external kinematics in Sect. 3.7 and extensions to two-loop integrands in Sect. 3.8.

Further information on the topics presented in this chapter can be found in a number of comprehensive reviews, for example see [5–7].

**Ultraviolet Power Counting** Before we get started with the main topics of this chapter it is useful to recall how we can quantify UV divergences. The divergences at large values of the loop momentum can be estimated at the integrand level by considering the scaling behaviour. For example, using general polar coordinates, one-loop scalar integrals become

$$F\_n^{[D]}[1] \stackrel{|k| \to \infty}{\to} \int \frac{(-1)^n |k|^{D-1} \mathbf{d} |k| \, \mathbf{d} \mathcal{Q}}{\text{i} \pi^{D/2}} \frac{1}{k^{2n}} \,, \tag{3.12}$$

from which we see a divergence if *n* ≤ *D/*2. If *n* = 2 in *D* = 4−2*ε*, we have that

$$\left[F\_2^{\left[4-2\epsilon\right]}\right]\left[1\right] \stackrel{|k|\to\infty}{\to} \int \frac{\mathbf{d}|k|\,\mathbf{d}\mathcal{Q}}{\mathrm{i}\pi^{2-\epsilon}}\frac{1}{|k|}\,,\tag{3.13}$$

(continued)

and we see a logarithmic divergence. Infrared divergences, such as the soft and collinear configurations discussed in Chap. 2, are more difficult to classify but can also be regulated by the analytic continuation of the dimension. We will discuss this further in Chap. 4, where we focus on the integration over loop momenta.

### **3.2 Unitarity and Cut Construction**

The unitarity of the *S*-matrix provides some of the most fundamental constraints on the analytic form of on-shell amplitudes. The initial steps follow the discussion of the optical theorem and Cutkosky analysis of the discontinuities of Feynman integrals [8] that may be familiar to many readers.

The scattering amplitudes associated with an *S*-matrix *S* = 1+i*T* are determined through the transition matrix *T* . Transition matrix elements <*F*|*T* |*I* > are a measure of the probability of a initial state *I* evolving into a final state *F*.

The unitarity condition of the *S*-matrix

$$S^{\dagger}S = \mathbb{1}\tag{3.14}$$

implies a non-linear constraint on the transition matrix:

$$-\left(T - T^{\dagger}\right) = T^{\dagger}T\,. \tag{3.15}$$

The asymptotic states obey the completeness relations

$$\sum\_{n} \int \prod\_{j=1}^{n} \frac{\mathbf{d}^{\mathbf{j}} \mathbf{k}\_{j}}{(2\pi)^{3} 2E\_{j}} \left| \langle k \rangle\_{n} \right\rangle \langle \{k\}\_{n} \right| = \mathbf{1} \,, \tag{3.16}$$

where |{*k*}*n*> indicate multi-particle states |{*k*}*j* > := |*k*1*, k*2*,...,kn*>, *k μ i* = *(Ei,* **k***i)*, and *Ei* = |**k***i*| 2 + *m*2 *i* , *mi* being the mass of the *i*th particle. Therefore, when we contract Eq. (3.15) with the initial and final states and insert a complete set of states in the product *T T* †, we determine that

$$-\operatorname{i}\langle F|(T-T^{\dagger})|I\rangle = \sum\_{n} \int \prod\_{j=1}^{n} \frac{\mathbf{d}^{3}\mathbf{k}\_{j}}{(2\pi)^{3} 2E\_{j}} \left\langle F|T^{\dagger}|\{k\}\_{n}\right\rangle \left\langle \{k\}\_{n}|T|I\right\rangle. \tag{3.17}$$

The scattering amplitude *A(I* → *F )* is extracted from the transition matrix element stripped of the overall momentum-conserving delta function. If we represent this amplitude by a picture,

$$A(I \to F) = \bigwedge\brace{\bigvee\cdots\bigvee^{F}}{\times}^{F},\tag{3.18}$$

we can show the relation (3.17) graphically, as

$$-2i\left(\widehat{\rho\left\{\sum\_{j}\hat{\beta}\_{j} - I\left(\sum\_{i}\hat{\beta}\_{j}\right)\right\}}\right) = \sum\_{n=2}^{\infty} \int d\Phi\_{n}(P, \{k\}\_{n}) / \left\langle \underbrace{\sum\_{j}\hat{\beta}\_{j}}\_{} \right\rangle / \sum\_{n=2}^{\infty} \delta\_{n}.\tag{3.19}$$

where the integration measure now includes the on-shell delta function that ensures an on-shell final state phase-space integral,

$$\mathrm{d}\Phi\_n(P,\{k\}\_n) = (2\pi)^4 \delta^{(4)}\left(P - \sum\_n k\_n\right) \prod\_{l=1}^n \frac{\mathrm{d}^3 \mathbf{k}\_l}{(2\pi)^3 \, E\_l}.\tag{3.20}$$

The momentum *P* above represents the total incoming momentum *P* = Σ *i pi*.

The LHS of Eq. (3.19) may be shown to be proportional to the discontinuity of the amplitude across the real *P*2 axis. While we will not see any specific examples until Chap. 4, Feynman integrals will in general contain branch cuts. The simplest function of this type is the logarithm log*(x)*, 1 which has a branch cut across the (negative) real *x* axis. For a generic function *f (x)*, the discontinuity across the real *x* axis is defined as Disc*xf (x)* := *f (x* + i0*)* − *f (x* − i0*)*. For the logarithm this gives Disc*x* log*(x)* = 2*π*i *Θ(*−*x)*, where *Θ(*−*x)* is the Heaviside step function (for further details see Exercise 4.7). Rational functions, such as the tree-level amplitudes we have encountered up until now, do not contain branch cuts. We can take the unitarity constraint to imply that scattering amplitudes do contain branch cuts. Following this argument, it is possible to show that

$$-2i\left(\brace{I\bigvee\cdots I}\_{I}F-I\bigvee\cdots\right)\_{F}^{F}\big|\_{F}=\text{Disc}\_{P^{2}}\left(\brace{I\bigvee\cdots\bigvee}\_{I}F\right)\tag{3.21}$$

where the discontinuity is across the branch cut in the invariant *P*2, Disc*P*2 *A(*··· *, P*2*,* ···*)* = *A(. . . , P*2 + i0*, . . .)* − *A(. . . , P*2 − i0*, . . .)*. This is referred to as the discontinuity in the *P*2 channel.

1 We saw the logarithm appear in the UV limit of of the one-loop bubble function *F*[4−2*ε*] 2 [1] in Eq. (3.13).

We may now expanding the relation (3.19) perturbatively, in a coupling *g*, which results in a set of extremely useful equations for on-shell amplitudes. We can represent the perturbative expansion of the amplitudes this pictorially by,

$$\begin{aligned} \bullet \quad \bigvee\_{I} \quad \bigvee\_{F} &= g^{1,0} \left( \bigvee\_{I} \quad \bigvee\_{F} + g^{2\_{I}} \bigvee \bigvee\_{I} \bigvee\_{F} + g^{4\_{I}} \bigvee \bigvee\_{\widehat{Q}} \bigvee\_{F} + \dots \right), \end{aligned} \tag{3.22}$$

where *g*LO is the leading-order coupling (e.g. *g*LO = *gn*−2 for a *n*-gluon amplitude in YM theory). By substituting this expansion into the unitarity relation (3.19) we find equations order by order in the coupling *g*. Explicitly, up to third order we find

$$\text{Disc}\_{P^2} \left( \bigwedge \stackrel{\text{def}}{\bigvee} \stackrel{\text{def}}{\bigvee} \right) = \mathbf{0},$$

$$\mathsf{Disc}\_{P^2}\left(\bigwedge\limits^k\middle|\bigvee^k\right) = \int d\Phi\_{2,P}\left(\bigvee\hbox{
ormalfont
}\bigg|\bigvee^{k\_1}\_{k\_2}\right) \wedge \dots \wedge \bigvee^{k\_{|P|}}\_{|P|},\tag{3.24}$$

$$\operatorname{Disc}\_{P^2}\left(I\bigvee\bigvee\bigvee F\right) = \int d\Phi\_2\left(I\bigvee\bigvee\_{k\_2}^{k\_1}\bigvee\_{k\_2}^{k\_1}\bigvee\_{i^F}^{k\_1}\bigvee\_{j^F}^{k\_2}\bigvee\_{k\_2}^{k\_1}\bigvee\_{i^F}^{k\_1}\right)$$

$$+\int d\Phi\_3\left(\bigvee\bigvee\_{k\_3}^{k\_1}\bigvee\_{k\_3}^{k\_1}\bigvee\_{i^F}^{k\_2}\right).\tag{3.25}$$

The first of these equations confirms that rational tree-level amplitudes do not contain branch cuts. At one-loop order we find that the product of two *on-shell*  tree-level amplitudes is directly related to the discontinuity in the channel *P*2 of the one-loop amplitude.

Re-using the on-shell tree-level amplitudes we have found in Chaps. 1 and 2 inside the factorised loop integrand is an extremely efficient way of computing the discontinuity of a loop amplitude. It avoids some of the large intermediate algebraic steps that would be found following the expansion of the loop amplitude into Feynman diagrams. The act of putting an internal propagator on-shell, as we have done through the insertion of a complete set of states, is referred to a the *cut* of a loop amplitude.

However, we must still find a way to upgrade the discontinuity of the amplitude to the full amplitude. One method to do this is to perform a dispersion integral. To express this concretely, let us specify that the amplitude depends on *r* invariants *s*1 = *P*2 1 *,...,sr* = *P*2 *r* . Then, we have that

$$A^{(1)}(\mathbf{s}\_1', \dots, \mathbf{s}\_r') = \sum\_{l=1}^r \int \frac{\mathbf{ds}\_l'}{\mathbf{s}\_l' - \mathbf{s}\_l} \operatorname{Disc}\_{\mathbf{s}\_l} A^{(1)}.\tag{3.26}$$

This story has been known for a long time and traces back to the work of Cutkosky [8] and the days of the analytic *S*-matrix [9]. The modern unitarity method (Bern et al. [10]) transformed the approach into a powerful computational tool by the cut constraints with knowledge of a *basis* of loop integrals, and the spinor-helicity method for the compact representation of on-shell tree amplitudes. Rather than computing the dispersion relation, one uses the unitarity cut constraints to project out information about the rational integral coefficients from a representation of the amplitudes such as the one shown in Eq. (3.1). It is this procedure that we refer to as *cut construction* of a scattering amplitude.

The on-shell phase space d*Φn* contains Dirac *δ* functions which ensure the intermediate particles are on-shell. We recall in fact that Eq. (3.20) can be rewritten in a manifestly Lorentz-invariant form as

$$\mathrm{d}\Phi\_{\hbar}(P,\{k\}\_{n}) = (2\pi)^{4}\delta^{(4)}\left(P - \sum\_{n} k\_{n}\right)\prod\_{l=1}^{n} \frac{\mathrm{d}^{4}k\_{l}}{(2\pi)^{4}}(2\pi)\delta^{(+)}\left(k\_{l}^{2} - m\_{l}^{2}\right),\tag{3.27}$$

where

$$\delta^{(+)} \left( k\_l^2 - m\_l^2 \right) \coloneqq \delta \left( k\_l^2 - m\_l^2 \right) \Theta \left( k\_l^0 \right), \tag{3.28}$$

with the Heaviside step function *Θ* ensuring the positivity of the energy.

Since it will become a fundamental part of our amplitude analysis, it is useful to introduce some notation for the action of imposing these on-shell constraints on internal particles of a loop diagram. The operation of computing the discontinuity across a two-particle factorisation or *cut* will be represented as C*L*|*R*, where the indices *L* and *R* will be the list of external particles entering the left and right side of the cut respectively,

$$\begin{split} \mathcal{C}\_{l\_1 \ldots l\_m | l\_{m+1} \ldots l\_n} \left( A\_n^{(1), [D]} \right) &:= \text{Disc}\_{s\_{l\_1 \ldots l\_m}} \left( A\_n^{(1), [D]} \right) \\ = \int \mathbf{d} \Phi\_2 \sum\_{h\_l = \pm} \mathbf{i} A\_{m+1}^{(0)} \left( -l\_1^{-h\_1}, i\_1, \ldots, i\_m, l\_2^{h\_2} \right) \mathbf{i} A\_{n-m+2}^{(0)} \left( -l\_2^{-h\_2}, i\_{m+1}, \ldots, i\_n, l\_1^{h\_1} \right), \end{split} \tag{3.29}$$

where

$$\begin{split} \mathrm{d}\Phi\_{2} &= \frac{\mathrm{d}^{4}l\_{1}}{(2\pi)^{4}} \frac{\mathrm{d}^{4}l\_{2}}{(2\pi)^{4}} (2\pi)^{4} \delta^{(4)}(l\_{1} - l\_{2} - p\_{i\_{1}...i\_{n}}) \\ &\times (2\pi)\delta^{(+)}(l\_{1}^{2} - m\_{1}^{2}) \,(2\pi)\delta^{(+)}(l\_{2}^{2} - m\_{2}^{2}) \,. \end{split} \tag{3.30}$$

The *δ* functions ensure momentum conservation, and that the internal momenta *l*1 and *l*2 are at their on-shell values *m*1 and *m*2. Notice that two factors of i appear in the factorisation into tree amplitudes. This appears for exactly the same reason as it did in BCFW recursion, where each factorised on-shell gluon propagator contributed a factor of i (see below Eq. (2.3)). For fermion propagators this factor would change as discussed in Sect. 2.1 and Exercise 3.1.

We will also use the operation C*L*|*R* to act on the integrand of the amplitude, and therefore only represent the product of tree-level amplitudes,

$$\begin{aligned} \mathbf{C}\_{l\_1 \ldots l\_m | l\_{m+1} \ldots l\_n} \left( I\_n^{(1), [D]} \right) &= \\ \sum\_{h\_l = \pm} \mathrm{i}A\_{m+2}^{(0)} \left( -l\_1^{-h\_1}, i\_1, \ldots, i\_m, l\_2^{h\_2} \right) \mathrm{i}A\_{n-m+2}^{(0)} \left( -l\_2^{-h\_2}, i\_{m+1}, \ldots, i\_n, l\_1^{h\_1} \right), \end{aligned} \tag{3.31}$$

where the on-shell conditions for *l*1, *l*2 are understood to be imposed.

### **Example: The** *s*12**-Channel Cut of the** *gg* → *gg* **MHV Scattering Amplitude**

Let us consider the leading-colour2 four-gluon MHV amplitude in pure Yang-Mills theory, *A(*1*) (*1−*,* 2−*,* 3+*,* 4+*)*. We begin with the familiar Parke-Taylor formula (1.192) for the tree amplitudes (this time setting the coupling to 1),

$$A^{(0)}(1^-,2^-,3^+,4^+) = \frac{\mathrm{i}\langle 12 \rangle^3}{\langle 23 \rangle \langle 34 \rangle \langle 41 \rangle} \,. \tag{3.32}$$

Note that by using the four-dimensional tree-level amplitudes inside the cut we are only resolving the first term in the expansion of the integral coefficient expressed in Eq. (3.4). The *s*12-channel is associated with the invariant *s*12 = *(p*1 + *p*2*)*2, and the discontinuity is obtained from the following product of two tree amplitudes summed over all possible helicity states,

$$\mathcal{C}\_{12|34}\left(I\_4^{(1)}(1^-,2^-,3^+,4^+)\right) = \sum\_{h\_l=\pm} \text{i}A^{(0)}(-l\_1^{-h\_1},1^-,2^-,l\_2^{h\_2})\,\text{i}A^{(0)}$$

$$\times \left(-l\_2^{-h\_2},3^+,4^+,l\_1^{h\_1}\right),\tag{3.33}$$

where, as imposed by the Lorentz invariant phase-space measure d*Φ*2, *l*2 = *l*1 − *p*12 and *l* 2 *i* = 0. Since all tree amplitudes with all-like helicities or those with a single positive (or negative) helicity vanish, only a single term contributes to the cut. In order to keep the notation as compact as possible, we will use C12|34 to

2 This is the coefficient of the single-trace term in the colour ordered one-loop amplitude given in Eq. (1.141), denoted there as *A(*1*) n*;1.

refer to C12|34( *I (*1*)* 4 *(*1−*,* 2−*,* 3+*,* 4+*)* ) for the remainder of this section. Replacing the tree-level amplitudes with their spinor-bracket forms leads to

$$\mathcal{C}\_{12|34} = \mathrm{i}\,A^{(0)}(-l\_1^+, 1^-, 2^-, l\_2^+) \,\mathrm{i}\,A^{(0)}(-l\_2^-, 3^+, 4^+, l\_1^-)$$

$$= \frac{\langle 12 \rangle^3}{\langle 2l\_2 \rangle \langle l\_2(-l\_1) \rangle \langle (-l\_1) \rangle \mathbf{1}} \frac{\langle l\_1(-l\_2) \rangle^3}{\langle (-l\_2) \rangle \langle 34 \rangle \langle 4l\_1 \rangle}$$

$$= \frac{\langle 12 \rangle^3}{\langle 2l\_2 \rangle \langle l\_2 l\_1 \rangle \langle l\_1 \mathbf{1} \rangle} \frac{\langle l\_1 l\_2 \rangle^3}{\langle l\_2 \mathbf{3} \rangle \langle 34 \rangle \langle 4l\_1 \rangle} \,. \tag{3.34}$$

Above we have used the phase convention (1.113) for the spinors of the loop momenta, namely |*(*−*li)*> = i|*li*>. We can now apply some spinor identities to recast the integrand into a form that can be identified with one-loop integral topologies. For example,

$$\langle l\_{\rm l}1\rangle = \frac{2\,l\_{\rm l}\cdot p\_{\rm l}}{\left[1l\_{\rm l}\right]} = -\frac{\left(l\_{\rm l} - p\_{\rm l}\right)^{2}}{\left[1l\_{\rm l}\right]}\,. \tag{3.35}$$

Hence we find

$$C\_{12|34} = \frac{\langle 12 \rangle^3}{\langle 34 \rangle} \frac{\langle l\_1 l\_2 \rangle^2 [2l\_2][l\_1 1][l\_2 3][4l\_1]}{(l\_2 + p\_2)^2 (l\_1 - p\_1)^2 (l\_2 - p\_3)^2 (l\_1 + p\_4)^2}$$

$$= \frac{\langle 12 \rangle^3}{\langle 34 \rangle} \frac{[1|l\_1 l\_2|2][3|l\_2 l\_1 |4]}{(l\_2 + p\_2)^2 (l\_1 - p\_1)^2 (l\_2 - p\_3)^2 (l\_1 + p\_4)^2} \,. \tag{3.36}$$

The numerator can also be reduced using the on-shell kinematics,

$$\left[1\middle|l\_{\mathbf{l}}l\_{\mathbf{2}}\middle|2\right] = \left[1\middle|l\_{\mathbf{l}}(l\_{\mathbf{l}}-p\_{\mathbf{l}2})\middle|2\right] = -\left(1\middle|l\_{\mathbf{l}}\middle|1\middle|\right]\left[1\middle|2\right] = \left(l\_{\mathbf{l}}-p\_{\mathbf{l}}\right)^{2}\left[1\middle|2\right].\tag{3.37}$$

This leads us to the simple result

$$\begin{split} \mathcal{C}\_{12|34} &= \frac{\langle 12 \rangle^3}{\langle 34 \rangle} \frac{[12][34]}{(l\_1 - p\_1)^2 (l\_1 + p\_4)^2} \\ &= -\frac{\langle 12 \rangle^3}{\langle 23 \rangle \langle 34 \rangle \langle 41 \rangle} \frac{s\_{12} s\_{23}}{(l\_1 - p\_1)^2 (l\_1 + p\_4)^2} . \end{split} \tag{3.38}$$

We can now identify the integrand of the double cut of the one-loop scalar box integral,

$$F\_4^{[D]}(p\_1, p\_2, p\_3) \equiv F\_4^{[D]}(s\_{12}, s\_{23}) = \int\_k \frac{1}{k^2 (k - p\_1)^2 (k - p\_{12})^2 (k + p\_4)^2} \,. \tag{3.39}$$

<

The discontinuity of this function in the *s*12 channel is given by

$$\operatorname{Disc}\_{\mathfrak{s}\_{12}} F\_4^{[D]}(\mathfrak{s}\_{12}, \mathfrak{s}\_{23}) = \int \operatorname{d}\Phi\_2 \, \frac{1}{(l\_1 - p\_1)^2 (l\_1 + p\_4)^2} \,, \tag{3.40}$$

where the cut propagators have been replaced by on-shell delta functions. We will justify this step in the next section. Our final result for the discontinuity for the one-loop amplitude is then

$$\begin{split} \text{Disc}\_{\mathfrak{s}\_{12}} A^{(1)}(1^-, 2^-, 3^+, 4^+) &= -\text{s}\_{12} \text{s}\_{23} A^{(0)} \\ &\quad \times (1^-, 2^-, 3^+, 4^+) \,\text{Disc}\_{\mathfrak{s}\_{12}} F\_4^{[D]}(\mathfrak{s}\_{12}, \mathfrak{s}\_{23}) . \end{split} \tag{3.41}$$

This result deserves a few remarks. The on-shell approach has uncovered some dramatic simplifications and the final cut contains only one scalar integral function. As we will see, this is not the most general structure we can encounter, and we will need to work harder to find a complete function basis. It should also be noted that we have only identified the leading term in the *ε* expansion (3.4) of the integral coefficient, since the tree amplitudes were evaluated in four dimensions. These contributions to loop amplitudes are usually referred to as *cut-constructible*.

**Which Feynman Diagrams Have We Calculated?** We can try to put this in the context of the Feynman diagram expansion. In an axial gauge without ghosts, one can check that there are 39 diagrams contributing to the full colour four-gluon one-loop amplitude, of which 17 contribute at leading colour. In the *s*12-channel cut only 9 of the 17 ordered diagrams contribute. What we should take from this is that a direct Feynman diagram computation is not at all prohibitive here with only a modest number of diagrams contributing. However, one of these diagrams is the most complicated tensor integral we can find for massless four-particle kinematics, and will contain four powers of loop momentum in the numerator. The number of three- and four-point vertices also means each diagram will expand to a large number of terms. The use of compact on-shell trees has allowed us to avoid a lot of this complexity. As we have highlighted, some contributions have been dropped but we have obtained a lot of information about the amplitude.

We may now try to complete the cut-constructible part of the four-gluon scattering amplitude by considering the cut in the other independent invariant.

### **Example: The** *s*23**-Channel Cut of** *gg* → *gg* **MHV Scattering Amplitude**

The *s*23-channel cut associated with the invariant *s*23 = *(p*2+*p*3*)*2 of our example *A(*1*) (*1−*,* 2−*,* 3+*,* 4+*)* is slightly more complicated, since the sum over internal helicities contains more non-zero elements. The cut integrand can be written as