Complex Momenta and Factorization

Complexifying external momenta turns factorization channels into isolated poles of a meromorphic deformation parameter while preserving momentum conservation and the on-shell conditions. At each pole an internal propagator goes on shell, and tree-level unitarity fixes the residue as a product of lower-point on-shell amplitudes summed over physical internal states. This is the analytic engine behind recursion; it does not by itself determine terms with no factorization pole.

Required background. Three-Point Amplitudes supplies the local on-shell seeds. Physical Poles and Tree-Level Factorization supplies the real-channel residue statement that is analytically continued here.

Helpful background. Laurent Series, Poles, and Residues supplies the one-complex-variable residue theorem.

An on-shell two-line deformation

For two massless legs $i$ and $j$, define the $[i,j\rangle$ shift

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

while $|i\rangle$ and $|j]$ are unchanged. In momentum form,

$$\widehat p_i(z)=p_i+zq, \qquad \widehat p_j(z)=p_j-zq, \qquad q_{\alpha\dot\alpha}=\lambda_{i\alpha}\widetilde\lambda_{j\dot\alpha}.$$

Because

$$q^2=q\!\cdot p_i=q\!\cdot p_j=0,$$

the shifted legs stay on shell and their sum is unchanged:

$$\widehat p_i^2=\widehat p_j^2=0, \qquad \widehat p_i+\widehat p_j=p_i+p_j.$$

For generic complex $z$, dotted and undotted spinors are independent. The deformation is not a path through real scattering events; it is an analytic family whose value at $z=0$ is the desired amplitude. The construction and Cauchy-contour setup appear in Elvang and Huang 2014, § 3.1, pp. 34–36, PDF.

Pole locations

Let a channel momentum $P_I$ be the sum of momenta in a proper nonempty subset $I$. If $I$ contains $i$ but not $j$, then

$$\widehat P_I(z)=P_I+zq,$$

and

$$\widehat P_I^2(z) =P_I^2+z\langle i|P_I|j].$$

The internal line becomes null at

$$\boxed{ z_I=-\frac{P_I^2}{\langle i|P_I|j]} }.$$

The displayed formulas take the exchanged state to be massless. For a state of mass $m_a$, replace $\widehat P_I^2(z)$ by $\widehat P_I^2(z)-m_a^2$ and hence $P_I^2$ by $P_I^2-m_a^2$ in the pole location and propagator. The internal sum must then use its $2s+1$ massive spin states rather than helicities.

Only channels that separate the two shifted legs depend on $z$. A channel containing both shifted legs, or neither, has no finite $z$ pole from this deformation. If $\langle i|P_I|j]=0$, the would-be pole is not exposed by this particular shift and a different deformation may be needed.

At $z=z_I$, choose spinors $|\widehat P_I\rangle,|\widehat P_I]$ for the null internal momentum. Their little-group scaling cancels between the two subamplitudes after the internal helicity sum. This cancellation is a valuable implementation check: a residue cannot depend on the arbitrary phase chosen for the internal spinors.

Tree factorization residue

Near a simple channel pole, the complete tree amplitude has the form

$$\mathcal A_n(z) \underset{z\to z_I}{=} \sum_{h,a} \mathcal A_L(z_I;-\widehat P_I^{-h,a}) \frac{1}{\widehat P_I^2(z)} \mathcal A_R(z_I;\widehat P_I^{h,\bar a}) +O(1).$$

Here $h$ labels physical helicity or spin states, while $a$ labels the internal species and representation and $\bar a$ its dual state on the other side of the cut line. For a self-conjugate state $\bar a=a$. For the on-shell functions in this chapter, the overall propagator $i$ has been absorbed into the convention for the stripped amplitude. When matching a Feynman-rule calculation, restore its declared $i\mathcal M$ and propagator conventions before comparing signs.

Since

$$\widehat P_I^2(z) =(z-z_I)\langle i|P_I|j],$$

the $z$-plane residue is

$$\operatorname*{Res}_{z=z_I}\mathcal A_n(z) =\sum_{h,a} \frac{\mathcal A_L(z_I)\mathcal A_R(z_I)} {\langle i|P_I|j]}.$$

The sign assigned to each internal leg matters. Because $P_I$ is the sum of the external momenta on the left, all-outgoing momentum conservation requires $-\widehat P_I$ on the left subamplitude and $+\widehat P_I$ on the right; their helicity labels are related accordingly.

From factorization to a contour identity

Apply Cauchy’s theorem to $\mathcal A_n(z)/z$:

$$\mathcal A_n(0) =-\sum_{z_I\ne0} \operatorname*{Res}_{z=z_I}\frac{\mathcal A_n(z)}{z} -\operatorname*{Res}_{z=\infty}\frac{\mathcal A_n(z)}{z}.$$

Using the pole location converts each finite residue to the familiar factorized term,

$$-\operatorname*{Res}_{z=z_I} \frac{\mathcal A_n(z)}{z} =\sum_{h,a} \mathcal A_L(z_I) \frac{1}{P_I^2} \mathcal A_R(z_I).$$

This identity is always the correct starting point. BCFW recursion is the special case in which the residue at infinity vanishes. If it does not, factorization still fixes every finite-pole residue but leaves a boundary contribution. The original recursion argument and its finite-pole sum are given in Britto et al. 2005, § 2, eqs. (2.3)–(2.7), pp. 3–5, PDF.

Physical and complex poles

A finite $z_I$ is generally complex even when the unshifted momenta are real. It marks a point where the analytically continued channel goes on shell. The physical pole is the singularity as $P_I^2\to0$ on the appropriate real boundary value with the Feynman prescription restored. These are related but not identical statements:

• the complex shift exposes a residue efficiently;
• the $+i0$ prescription and physical channel determine the boundary value;
• crossing and branch continuation determine how a formula reaches another physical region; and
• loop amplitudes also have cuts and non-rational dependence, so a tree-level meromorphic argument cannot be copied unchanged.

Spurious poles in a particular spinor representation may appear in individual recursive terms. They must cancel in the complete amplitude. A spurious-pole cancellation is therefore a representation check, while physical poles must remain with the correct residues.

A four-point channel check

For a four-point amplitude shifted on legs 1 and 4, the channel $P_{12}=p_1+p_2$ separates the shifted legs and acquires a pole. At that pole,

$$\widehat P_{12}^2(z_{12})=0,$$

and the coefficient of $1/\widehat P_{12}^2$ must be

$$\sum_{h,a} \mathcal A_3(\widehat1,2,-\widehat P_{12}^{-h,a}) \mathcal A_3(\widehat P_{12}^{h,\bar a},3,\widehat4).$$

The diagnostic sequence is: verify each three-point branch; check the internal little-group phase cancels; divide by the unshifted $P_{12}^2$ after the contour residue is taken; and compare the resulting physical residue with a direct tree calculation. This checks more than agreement of the final compact formula.

Common pitfalls

Shifting both halves of one momentum independently. A generic change spoils $p^2=0$. Use a rank-one null shift orthogonal to both shifted legs.

Including channels that do not separate the shifted legs. Their momenta are $z$ independent and they do not produce finite poles in this contour.

Equating correct residues with a complete amplitude. A polynomial contact term has no factorization pole. It appears in the residue at infinity or as independent local input.

Dropping the internal state sum. Factorization sums a complete physical basis, including species and internal labels. A single helicity product is not generally the residue.

Exercises

Derive $z_I$ for the $[i,j\rangle$ shift and then show algebraically that

$$-\operatorname*{Res}_{z=z_I} \frac{1}{z\widehat P_I^2(z)}=\frac{1}{P_I^2}.$$

Solution

Write $\widehat P_I^2=P_I^2+zD_I$ with $D_I=\langle i|P_I|j]$. Then $z_I=-P_I^2/D_I$ and

$$\operatorname*{Res}_{z=z_I} \frac{1}{z\widehat P_I^2(z)} =\frac{1}{z_ID_I} =-\frac1{P_I^2}.$$

The unshifted propagator appears only after both the $z$-Jacobian and the sign of $z_I$ are included.

Where to continue

• BCFW Recursion evaluates the contour when the large-$z$ term is controlled.
• Constructibility, Boundary Terms, and Failure Modes interprets a nonzero residue at infinity.
• Generalized Unitarity extends on-shell sewing to loop integrands with additional completeness qualifications.

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.