BCFW Recursion

BCFW recursion reconstructs a tree amplitude from lower-point on-shell amplitudes by applying Cauchy’s theorem to a complex two-line shift. The recursion closes only when the shifted amplitude has no residue at infinity. That large-$z$ condition depends on the theory, shifted species and helicities, ordering, dimension, and deformation; it is an input to verify, not a consequence of factorization alone.

Required background. Complex Momenta and Factorization supplies the on-shell shift, finite pole locations, and factorized residues.

The recursive formula

Apply the $[i,j\rangle$ shift

$$|\widehat i]=|i]+z|j], \qquad |\widehat j\rangle=|j\rangle-z|i\rangle.$$

Assume the tree amplitude is rational in $z$, its finite poles arise from physical propagators, and the boundary coefficient

$$B_\infty \equiv-\operatorname*{Res}_{z=\infty} \frac{\mathcal A_n(z)}{z} =0.$$

Then Cauchy’s theorem gives

$$\boxed{ \mathcal A_n(0) =\sum_{I,h,a} \mathcal A_L(z_I;\widehat i,\ldots,-\widehat P_I^{-h,a}) \frac{1}{P_I^2} \mathcal A_R(z_I;\widehat P_I^{h,\bar a},\ldots,\widehat j) }.$$

The sum runs over partitions that place $i$ and $j$ on opposite sides and leave at least three legs on each subamplitude after the internal leg is included. It also runs over a complete set of internal helicities, species, and internal quantum numbers; $\bar a$ denotes the dual representation on the other subamplitude and equals $a$ for a self-conjugate state. For a color-ordered amplitude, only partitions compatible with the cyclic ordering appear. The displayed propagator is for massless exchange; a massive pole uses $1/(P_I^2-m_a^2)$ and a complete massive spin sum.

The familiar condition $\mathcal A_n(z)\to0$ is sufficient for $B_\infty=0$, but it is not the exact logical criterion. For example,

$$F(z)=z+\frac{r}{z-z_*}$$

grows linearly, yet $F(z)/z=1+O(z^{-2})$ has no $1/z$ term at infinity, so $B_\infty=0$ and the finite residue reconstructs $F(0)=-r/z_*$. Power-law falloff is the robust theorem normally proved in gauge theory or gravity; if it is unavailable, compute or constrain the boundary coefficient rather than inferring it from growth alone.

The direct tree-level proof for Yang–Mills theory is the original result of Britto, Cachazo, Feng, and Witten 2005, §§ 2–3, pp. 2–7, PDF. A detailed pedagogical derivation, including the contour and examples, is Elvang and Huang 2014, §§ 3.1–3.2, pp. 34–44, PDF.

A reliable recursion workflow

Step	Calculation	Required check
1. Declare the object	full or color-ordered tree amplitude, external species, helicities, and normalization	no color or momentum-conserving delta function is silently changed
2. Choose a shift	select legs $i,j$ and which spinor half moves	on-shell conditions and total momentum are exact for all $z$
3. Establish the boundary condition	analytic argument, background-field power counting, or an independently known theorem in the stated theory	$B_\infty=0$; decay of $\mathcal A_n(z)$ is a sufficient test, not the definition
4. List partitions	channels separating $i$ and $j$	each physical channel appears once and only ordering-compatible channels are kept
5. Solve each pole	$z_I=-P_I^2/\langle i	P_I
6. Sew subamplitudes	evaluate $\mathcal A_L\mathcal A_R/P_I^2$	internal little-group phase cancels and the physical state sum is complete
7. Simplify	use Schouten and momentum conservation	spurious poles cancel without removing physical poles
8. Validate	compare residues, weights, dimensions, soft limits, and a direct low-point result	final answer is independent of a valid shift

This workflow is a static algorithm: it can be implemented symbolically or followed by hand. Numerical spot checks are useful after the analytic identities have been established, but they do not replace a proof of large-$z$ behavior or spurious-pole cancellation.

Four-gluon MHV example

As on the color-decomposition page, $A_n$ denotes the coupling-stripped partial amplitude; the reconstructed Yang–Mills tree carries the overall factor $g^{n-2}$. Consider

$$A_4(1^-,2^-,3^+,4^+)$$

Apply the $[1,4\rangle$ shift. In two-derivative Yang–Mills theory this helicity choice has adequate large-$z$ falloff. Cyclic ordering leaves one partition, with $P_{12}=p_1+p_2$. At $z=z_{12}$ the two subamplitudes are the three-point seeds

$$z_{12} =-\frac{s_{12}}{\langle1|P_{12}|4]} =\frac{[12]}{[24]}, \qquad [\widehat1,2]=0, \qquad \langle3\widehat4\rangle=0.$$

The last two equalities put the left seed on its holomorphic branch and the right seed on its antiholomorphic branch. The two factors are

$$A_3(\widehat1^-,2^-,-\widehat P_{12}^{+}) \quad\text{and}\quad A_3(\widehat P_{12}^{-},3^+,\widehat4^+).$$

Their internal little-group weights cancel. Substitution into the recursion, including the factor $1/s_{12}$, and use of momentum conservation gives

$$\boxed{ A_4(1^-,2^-,3^+,4^+) =\frac{\langle12\rangle^4} {\langle12\rangle\langle23\rangle \langle34\rangle\langle41\rangle} }.$$

The answer has weight $t_1^2t_2^2t_3^{-2}t_4^{-2}$, mass dimension zero, and the allowed adjacent color-ordered poles. Its compact denominator contains collinear factors, but no arbitrary reference spinor. A complete check also takes the $P_{12}^2\to0$ limit and reproduces the same product of three-point amplitudes used in the recursion.

The example illustrates an induction: three-point amplitudes are the base data, and every term on the right has fewer external legs. It does not prove the falloff for other helicities, matter content, higher-derivative interactions, or gravity.

Large-z behavior and shift selection

At large $z$, individual Feynman graphs often grow even when the gauge-invariant sum falls. The improvement can follow from gauge cancellations, helicity, color ordering, or enhanced symmetries. In four-dimensional two-derivative pure Yang–Mills theory, useful adjacent shifts can give $1/z$ behavior, while the opposite helicity orientation for the same shift convention can grow. Gravity frequently falls even faster for suitable shifts. These statements change when higher-derivative interactions or nonstandard external states are introduced.

The meaningful record is not merely “a BCFW shift was used,” but

$$(\text{theory},\text{ordered/full object},i^{h_i},j^{h_j}, \text{shift convention},\mathcal A(z)\text{ as }z\to\infty).$$

Elvang and Huang summarize Yang–Mills shift behavior and show explicit successes and failures in Elvang and Huang 2014, §§ 3.2–3.3, pp. 36–46, PDF. If no proof or trusted theorem applies to the declared object, retain a boundary term rather than assuming it vanishes.

What recursion does and does not determine

When the boundary term vanishes, factorization residues determine the amplitude uniquely within the assumed rational class. This simultaneously makes locality and tree-level unitarity visible: every recursive term is built from on-shell lower-point objects and a physical propagator.

When the boundary term does not vanish, the finite residues are still correct. What remains is information invisible to those residues, commonly a local contact interaction or a shift-dependent combination of such terms. A different good shift may expose more information, but agreement between shifts must be demonstrated.

At loop level, amplitudes have logarithms, branch cuts, rational pieces, and regulator dependence. Tree BCFW recursion is not a loop formula. Later chapters use generalized cuts and loop-specific reconstruction with explicit information-completeness checks.

Common pitfalls

Choosing a shift from a memorized helicity table without matching conventions. Tables depend on which angle and square spinors are shifted and on ordering. Write the deformation explicitly and test the declared amplitude.

Summing every factorization channel in a color-ordered amplitude. Only cyclically compatible partitions contribute. The full color-dressed amplitude has a different channel organization.

Using unphysical polarizations in the internal sum. BCFW sewing uses physical on-shell states. Gauge-fixed completeness relations require the corresponding ghost or Ward-identity treatment and are not interchangeable without proof.

Treating two different recursive representations as different amplitudes. Valid shifts can produce different-looking sums. Compare them using Schouten, momentum conservation, and their complete pole and boundary data.

Exercises

Boundary coefficient. For $F(z)=c+z+r/(z-z_*)$, determine the finite-pole contribution and $B_\infty$, and verify the contour identity at $z=0$.

Solution

The finite pole has $\operatorname{Res}_{z=z_*}F(z)/z=r/z_*$, so its contribution is $-r/z_*$. At large $z$,

$$\frac{F(z)}z=1+\frac{c}{z}+O(z^{-2}),$$

hence $\operatorname{Res}_{z=\infty}F(z)/z=-c$ and $B_\infty=c$. Therefore $F(0)=-r/z_*+c$, exactly as the original function gives. The growing $z$ term does not contribute because it vanishes at the physical point.

Four-gluon recursion. Reproduce the MHV result above. Identify the single color-ordered partition, fix the internal helicity from the two three-point branches, obtain the Parke–Taylor expression, and check all four little-group weights plus the $P_{12}^2$ residue.

Where to continue

• Constructibility, Boundary Terms, and Failure Modes diagnoses a nonzero contour at infinity.
• Color Decomposition and Partial Amplitudes explains which ordered object the recursion reconstructs.
• Generalized Unitarity uses products of trees to constrain loop data.

References

• Britto, Ruth, Freddy Cachazo, Bo Feng, and Edward Witten. “Direct Proof of Tree-Level Recursion Relation in Yang–Mills Theory.” Physical Review Letters 94 (2005): 181602. DOI. Open PDF.
• Elvang, Henriette, and Yu-tin Huang. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press, 2015. Open prepublication version. Open PDF.