Soft Limits as On-Shell Constraints

When an external massless gauge boson or graviton becomes soft, the leading pole comes from attaching it to the external hard legs. On shell, this factorization fixes a universal soft factor multiplying the lower-point amplitude. Gauge invariance then turns the soft theorem into a conservation-law check. This page treats the leading amplitude constraint; infrared divergences, inclusive cancellation, and dressed states are separate questions.

Required background. Three-Point Amplitudes supplies the emission seeds. Physical Poles and Tree-Level Factorization supplies the residue argument.

Helpful background. Soft Theorems develops the infrared, loop, and symmetry interfaces after the leading on-shell constraint is established.

Uniform soft scaling

Let an outgoing null momentum $q$ become soft as

$$q^\mu=\tau q_0^\mu, \qquad \tau\to0^+.$$

In spinor variables a uniform scaling is

$$|q\rangle\mapsto\sqrt{\tau}\,|q_0\rangle, \qquad |q]\mapsto\sqrt{\tau}\,|q_0].$$

This differs from a holomorphic soft limit, where only one spinor is scaled. State which limit is used before quoting powers of $\tau$. With the uniform limit, the leading gauge and gravity soft factors scale as $\tau^{-1}$.

For a hard external leg of mass $m_i$, the adjacent propagator supplies

$$\frac{1}{(p_i+q)^2-m_i^2} =\frac{1}{2p_i\!\cdot q} \sim\tau^{-1},$$

so only emissions from external hard legs contribute to the leading pole. Internal emissions and contact terms are less singular but are needed for subleading relations and gauge invariance away from the leading limit.

Photon and gluon soft factors

For a soft photon of polarization $\varepsilon(q)$ and all hard legs treated as outgoing, the leading theorem is

$$\boxed{ \mathcal A_{n+1}(q;1,\ldots,n) =e\sum_{i=1}^n Q_i \frac{p_i\!\cdot\varepsilon(q)}{p_i\!\cdot q} \mathcal A_n(1,\ldots,n) +O(\tau^0) }.$$

Here $Q_i$ is the charge appropriate to the outgoing state; an incoming physical particle is represented after crossing, so its sign must be translated consistently. Replacing $\varepsilon^\mu$ by $q^\mu$ gives

$$e\sum_iQ_i\,\mathcal A_n,$$

which vanishes by charge conservation. The leading soft theorem and its infrared interpretation originate in Weinberg 1965, pp. B516–B524.

For a soft gluon of adjoint color $a$, the full color-dressed statement is

$$\boxed{ \mathcal A_{n+1}^{a}(q) =g\sum_{i=1}^n \mathbf t_i^{a} \frac{p_i\!\cdot\varepsilon(q)}{p_i\!\cdot q} \mathcal A_n +O(\tau^0) }.$$

$\mathbf t_i^a$ acts in the representation of hard leg $i$ and uses the conventional normalization $\operatorname{tr}(t^at^b)=\delta^{ab}/2$ for a fundamental hard leg. The color-decomposition page instead writes $T^a=\sqrt2\,t^a$ in trace tensors; keeping this translation explicit prevents a spurious $\sqrt2$ in the soft current. Gauge invariance requires color conservation on the amplitude,

$$\sum_i\mathbf t_i^a\mathcal A_n=0.$$

For a color-ordered tree amplitude with the soft gluon $s$ inserted between adjacent legs $a$ and $b$, the color operators reduce to adjacent factors. In the spinor convention of this volume,

$$A_{n+1}(\ldots,a,s^+,b,\ldots) \longrightarrow \frac{\langle ab\rangle} {\langle as\rangle\langle sb\rangle} A_n(\ldots,a,b,\ldots),$$

and

$$A_{n+1}(\ldots,a,s^-,b,\ldots) \longrightarrow \frac{[ab]} {[as][sb]} A_n(\ldots,a,b,\ldots).$$

Here $A_n$ is the coupling-stripped partial amplitude defined on the color-decomposition page, so the extra power of $g$ resides in the color-dressed prefactor and does not appear again in these two equations. Each expression has the correct soft-leg little-group weight and scales as $\tau^{-1}$ under the uniform limit. Only the two neighbors appear because the partial amplitude has a fixed cyclic ordering; the full color-dressed sum restores emission from every charged leg.

Graviton soft factor

For Einstein gravity normalized by $S=2\kappa^{-2}\int\!\sqrt{-g}\,R+\cdots$, the leading soft-graviton theorem may be written

$$\boxed{ \mathcal M_{n+1}(q) =\frac{\kappa}{2} \sum_{i=1}^n \frac{p_i^\mu p_i^\nu\varepsilon_{\mu\nu}(q)} {p_i\!\cdot q} \mathcal M_n +O(\tau^0) }.$$

For a helicity graviton, $\varepsilon_{\mu\nu}=\varepsilon_\mu\varepsilon_\nu$. Under the linearized gauge shift

$$\varepsilon_{\mu\nu} \longmapsto \varepsilon_{\mu\nu}+q_\mu\xi_\nu+q_\nu\xi_\mu,$$

the leading factor changes by a term proportional to

$$\xi_\nu\sum_i p_i^\nu,$$

which vanishes by momentum conservation. In contrast with the photon theorem, no particle-dependent charge appears: every hard momentum enters with the universal gravitational coupling. Weinberg’s derivation and the universality argument are given in Weinberg 1965, pp. B516–B524.

What the leading theorem constrains

The soft pole checks several pieces of an amplitude at once:

Check	Gauge theory	Gravity
Factorization origin	external charged-leg pole	external hard-leg pole
Leading numerator	$p_i\cdot\varepsilon$ times charge/color action	$p_i^\mu p_i^\nu\varepsilon_{\mu\nu}$
Gauge-invariance condition	charge or color conservation	momentum conservation
Uniform soft degree	$\tau^{-1}$	$\tau^{-1}$
Information not fixed	finite terms, most contact data, loop corrections	finite terms, higher-derivative corrections beyond the leading pole

A candidate amplitude that has the correct physical poles but the wrong soft residue has an incorrect coupling, state assignment, color action, or normalization. Conversely, adding a local contact term can leave the leading soft pole unchanged. Soft consistency is powerful but does not determine every interaction.

Soft scalars are not generically universal. A Goldstone boson can exhibit an Adler zero or a symmetry-controlled soft theorem, but the symmetry representation and breaking pattern are essential inputs. They cannot be inferred by replacing a polarization in the gauge-boson formula.

A five-gluon check

Take leg $5^+$ soft in the color-ordered Parke–Taylor amplitude

$$A_5(1^-,2^-,3^+,4^+,5^+) =\frac{\langle12\rangle^4} {\langle12\rangle\langle23\rangle\langle34\rangle \langle45\rangle\langle51\rangle}.$$

With $|5\rangle=\sqrt\tau|q\rangle$ and $|5]=\sqrt\tau|q]$, while the hard momenta follow any momentum-conserving recoil family smooth at $\tau=0$, the ratio to the four-point hard amplitude is

$$\frac{A_5(\tau)}{A_4} =\frac1\tau \frac{\langle41\rangle} {\langle4q\rangle\langle q1\rangle} +O(\tau^0).$$

The two brackets containing the soft leg supply exactly one power of $\tau$, and only its cyclic neighbors 4 and 1 occur. Recoil changes the finite term, not the leading coefficient. This calculation checks the soft degree, helicity weight, adjacency, and normalization at once; it also explains why a statement of the on-shell soft family is needed when comparing subleading terms.

Amplitude limit versus infrared finiteness

The theorem describes a singular amplitude with an additional low-energy quantum. Integrating $|\mathcal A_{n+1}|^2$ over unresolved phase space can produce an infrared divergence. Whether it cancels depends on the observable, virtual corrections, inclusivity, masses, and state prescription.

Therefore:

• a correct soft factor does not make an exclusive Fock-space S-matrix element finite;
• Bloch–Nordsieck or KLN cancellation applies to specified inclusive sums or measurement functions;
• loop-level subleading soft behavior can receive corrections and regulator-ordering subtleties; and
• asymptotic-symmetry and memory interpretations require additional boundary and state hypotheses.

Those issues are developed in Infrared Structure and Factorization, not assumed here.

Common pitfalls

Not declaring the soft scaling. Uniform and holomorphic spinor limits assign different powers to intermediate expressions. State the scaling before comparing results.

Using the adjacent color-ordered factor as the full non-Abelian theorem. The full amplitude contains color-charge operators acting on every hard leg. Adjacency is a property of one ordered coefficient.

Treating the leading theorem as an all-orders subleading theorem. The leading pole is especially robust. Subleading terms depend more strongly on spin, loops, regulators, and higher-derivative interactions.

Confusing a soft constraint with cancellation of an infrared divergence. The former is an amplitude factorization statement; the latter is a statement about a specified measured sum over real and virtual contributions.

Exercises

Replace $\varepsilon^\mu$ by $q^\mu$ in the photon soft factor and $\varepsilon_{\mu\nu}$ by $q_\mu\xi_\nu+q_\nu\xi_\mu$ in the graviton factor.

Solution

The photon factor varies by $e\sum_iQ_i$, which annihilates the hard amplitude by charge conservation. The graviton numerator varies by $2(p_i\!\cdot q)(p_i\!\cdot\xi)$, so the denominators cancel and the result is proportional to $2\xi\!\cdot\sum_i p_i=0$. Both conclusions require one consistent all-outgoing convention.

Where to continue

• Soft Theorems develops subleading, loop, and symmetry qualifications.
• Soft and Collinear Singularities connects the amplitude limit to singular momentum regions.
• Bloch–Nordsieck and KLN Cancellation states the inclusive conditions needed for finite observables.

References

• Weinberg, Steven. “Infrared Photons and Gravitons.” Physical Review 140, no. 2B (1965): B516–B524. DOI.