Constructibility, Boundary Terms, and Failure Modes

An amplitude is constructible under a specified complex deformation when its finite factorization residues and lower-point seed data determine the value at the undeformed point. The obstruction is the contour at infinity. A nonzero boundary term is not an inconsistency: it records information that the selected residues cannot see, commonly a local contact interaction or a contribution exposed only by another shift.

Required background. BCFW Recursion supplies the boundary-free recursive formula and its factorization sum.

Helpful background. Derivative Interactions and Contact Terms supplies diagrammatic examples of local momentum-dependent vertices.

The complete contour equation

For any rational tree amplitude $\mathcal A_n(z)$ under a declared on-shell shift,

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

where

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

The finite residues are fixed by physical factorization. The exact closing condition is $B_\infty=0$. Decay of $\mathcal A_n(z)$ is sufficient, a nonzero constant gives a nonzero boundary, but growth alone is inconclusive: only the coefficient of $z^0$ in the Laurent expansion of $\mathcal A_n(z)$ contributes to $B_\infty$. Thus $\mathcal A_n(z)=z$ grows but has $B_\infty=0$, whereas $\mathcal A_n(z)=c+z$ has $B_\infty=c$.

Constructibility is therefore a relation among five declared items:

$$(\text{theory and particle content}, \text{amplitude class}, \text{seed data}, \text{deformation}, \text{large-}z\text{ behavior}).$$

It is not an intrinsic yes/no label attached to a Lagrangian without qualification. Benincasa and Cachazo define four-point constructibility by vanishing at infinity under the chosen BCFW deformation, so that three-point data determine the result Benincasa and Cachazo 2008, § IV.A, p. 10, PDF. The boundary examples below follow Elvang and Huang 2014, § 3.3, pp. 44–46, PDF.

Contact interactions are invisible to factorization

The cleanest example is massless $\lambda\phi^4$ theory. Its four-point tree amplitude is a constant,

$$\mathcal A_4=-\lambda$$

in a convention where the stripped vertex is $-\lambda$. Under any ordinary two-line on-shell shift,

$$\mathcal A_4(z)=-\lambda.$$

There are no three-point seeds and no finite factorization poles, so

$$B_\infty=-\lambda.$$

The recursion has not failed to reproduce known residues; there are none. It has correctly isolated the independent four-point local datum.

At six points, exchange graphs built from two quartic vertices do have physical poles. Their residues are determined by two four-point amplitudes, but a six-point contact operator would remain additional boundary data. Constructibility can therefore begin at a seed multiplicity higher than three when the theory contains irreducible contact interactions.

Derivative contacts make the diagnosis more pronounced. A local operator with $k$ derivatives produces a polynomial in momenta and can scale as $z^r$ under a two-line shift. Its lack of propagator poles means factorization cannot determine its coefficient unless other assumptions—symmetry, soft behavior, power counting, or a different recursion—supply the missing information.

Shift dependence

The same physical amplitude can fall for one deformation and grow for another. A poor shift does not prove nonconstructibility under all shifts; a good shift does not imply every related amplitude has the same falloff.

For a useful diagnosis, record a table like this:

Item	Question	Evidence needed
Shifted legs	Which species and helicities move?	explicit spinor deformation
Object	Full, color-ordered, flavor-ordered, or fixed component?	basis and normalization declaration
Power at infinity	Is $\mathcal A(z)=O(z^r)$?	theorem with matching hypotheses or a direct analytic derivation
Finite poles	Which channels separate the shifted legs?	complete partition list and internal state sums
Boundary meaning	Which local structures share all finite residues?	contact-term basis, dimension, symmetry, and soft checks
Cross-check	Does another valid shift give the same result?	analytic equality after including all boundary data

In two-derivative Yang–Mills theory, gauge cancellations improve large-$z$ behavior for suitable helicity shifts even though individual graphs grow. Higher-derivative operators generally weaken the falloff. Gravity can display stronger cancellations for suitable shifts, but adding higher-curvature interactions changes the result. The large-$z$ claim must match the exact amplitude under study.

The boundary coefficient is the regular part at the physical point

Suppose all finite residues of two rational functions agree. Their difference has no finite poles. At tree level, under locality and a fixed mass dimension, that difference is a polynomial or another allowed pole-free local structure. Thus

$$\mathcal A_n^{(1)}-\mathcal A_n^{(2)} =\mathcal C_n,$$

where $\mathcal C_n$ is contact data constrained by little-group covariance, dimensional analysis, permutation symmetry, internal symmetry, and any imposed soft behavior.

Equivalently, separate the shifted amplitude into its principal parts and a polynomial,

$$\mathcal A_n(z) =\sum_I\frac{r_I}{z-z_I}+P(z).$$

At the physical point, finite residues give $-\sum_I r_I/z_I$, while $B_\infty=P(0)$. Positive powers of $z$ affect the large-shift growth but vanish at $z=0$; the constant part is precisely the pole-free information missed by the chosen residues. This is why large-$z$ power counting is a useful sufficient test but not, by itself, a classification of boundary data.

This observation provides a constructive workflow even when ordinary BCFW does not close:

1. determine every factorization residue;
2. form one rational function with those residues;
3. enumerate the local structures allowed by weights, dimension, and symmetries;
4. fix their coefficients using independent input; and
5. verify the completed amplitude under a second deformation and in physical limits.

In an effective field theory, the independent coefficients are Wilson coefficients whose construction and matching belong to Renormalization and Effective Field Theory. On-shell methods organize the corresponding contact amplitudes but do not predict their numerical values without a UV or matching input.

Failure modes beyond a nonzero boundary

Incomplete pole set. A chosen shift exposes only channels that separate its two legs. If a recursive expression lacks an unexposed physical channel after simplification, the proposed result is incomplete even if its computed residues are correct.

Wrong internal spectrum. Omitting a particle or helicity from the state sum changes residues. This is a theory-definition error, not a boundary term.

Spurious poles that do not cancel. Individual recursive terms may contain reference or spinor denominators with no physical channel. A surviving spurious pole signals an incorrect sum or missing contribution.

Higher-order poles or non-rational dependence. Standard tree recursion assumes simple propagator poles in a rational function. Loop branch cuts, regulator-dependent rational terms, unstable-particle widths, and nonlocal form factors require different analytic machinery.

Unjustified complex continuation. Real-unitarity and crossing statements do not automatically provide a globally meromorphic function of the chosen $z$. The tree-level local QFT setting supplies that structure; more general objects require proof.

A four-point contact diagnostic

Let a crossing-symmetric amplitude for four identical massless scalars, with a scalar of mass $M$ exchanged in all three channels, be

$$\mathcal A_4(s,t,u) =g^2\left(\frac{1}{s-M^2}+\frac{1}{t-M^2} +\frac{1}{u-M^2}\right) +c_0+c_2(s^2+t^2+u^2).$$

The three pole residues determine the exchange coefficient $g^2$. They do not determine $c_0$ or $c_2$. Under a generic shift for which the Mandelstam invariants vary linearly with $z$, the $c_2$ term grows as $z^2$ and contributes at infinity; $c_0$ remains constant and also contributes. Crossing fixes the symmetric polynomial basis used here, but only an additional dynamical input fixes its coefficients.

This example also shows why matching only pole locations is insufficient. Residues, allowed contact basis, symmetry, and asymptotic behavior are independent pieces of the amplitude.

Common pitfalls

Calling a boundary term “unphysical.” It can be the entire physical effect of a local interaction. What is unphysical is to drop it without a valid falloff argument.

Using power counting graph by graph. Gauge and diffeomorphism cancellations can improve the complete amplitude. Establish large-$z$ behavior for the gauge-invariant object actually being recursed.

Claiming uniqueness from three-point data in an EFT. Higher-dimension local operators can share the same lower-point poles or begin at higher multiplicity. Power counting and matching data remain necessary.

Changing shifts without tracking boundary data. Different shifts reorganize the same amplitude. Agreement is expected only after every finite and infinite residue is included.

Exercises

Apply a two-line shift to the constant $\lambda\phi^4$ amplitude. Find its finite poles and $B_\infty$, and explain why the answer is an independent four-point seed rather than an inconsistency of on-shell factorization.

Solution

The amplitude $\mathcal A_4(z)=-\lambda$ has no finite pole. Since $\mathcal A_4(z)/z=-\lambda/z$, its residue at infinity is $+\lambda$, so $B_\infty=-\lambda$ and the complete contour equation returns $\mathcal A_4(0)=-\lambda$. Factorization is consistent but silent because the theory has no cubic seed and hence no exchange residue at four points.

Where to continue

• Derivative Interactions and Contact Terms derives the same missing local data from an action.
• Integration-by-Parts Identities and Master Integrals and Generalized Unitarity introduce loop-specific reconstruction ambiguities.
• Operator Bases supplies the EFT interpretation of independent contact structures.

References

• Benincasa, Paolo, and Freddy Cachazo. “Consistency Conditions on the S-Matrix of Massless Particles.” arXiv:0705.4305v2 [hep-th] (2008). 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.