On-Shell States and Little-Group Scaling

A massless amplitude is not a Lorentz scalar in each external wavefunction separately: it is a covariant multilinear function of one-particle helicity states. The subgroup of Lorentz transformations that preserves a null momentum acts by a phase on an ordinary helicity state. In four-dimensional spinor-helicity variables that phase becomes a homogeneous rescaling, so every external helicity fixes one exact weight of the amplitude.

Required background. One-Particle States: Mass, Spin, and Relativistic Normalization supplies the induced-representation construction and state normalization. Massive and Massless Spin-One Polarizations supplies the physical polarization states and their gauge redundancy.

Helpful background. Lorentz Field Representations and Poincaré Particle Representations develops the representation-theoretic distinction between fields and particles.

The massless little group

Choose the standard future-directed null momentum $k^\mu=(E,0,0,E)$ in the site’s mostly-minus metric. Its proper Lorentz stabilizer is isomorphic to $\operatorname{ISO}(2)$. For the finite-helicity particle representations used in perturbative QFT, the two translation-like generators act trivially and the remaining rotation acts as

$$U(R(\theta))\,|k,h\rangle=e^{-ih\theta}|k,h\rangle.$$

The nontrivial translation representations describe continuous-spin particles and are outside this chapter. The induced-representation derivation and this distinction are given in Weinberg 1995, § 2.5, pp. 62–74.

For a general null $p$, a choice of standard boost defines $|p,h\rangle$. Changing that choice by a little-group transformation changes the state by its helicity phase but leaves $p$ fixed. Consequently an $n$-particle amplitude must transform covariantly, once for every external leg. This is a constraint on the complete on-shell amplitude, not on an individual gauge-dependent graph.

Homogeneous weights in spinor variables

In four dimensions a complex null momentum may be factorized as

$$p_{\alpha\dot\alpha} =\lambda_\alpha\widetilde\lambda_{\dot\alpha}.$$

The factorization is unchanged under

$$\lambda_i\longmapsto t_i\lambda_i, \qquad \widetilde\lambda_i\longmapsto t_i^{-1}\widetilde\lambda_i, \qquad t_i\in\mathbb C^\times.$$

For real, positive-energy Lorentzian momenta, the reality condition restricts $t_i$ to a phase. Complexified kinematics permits any nonzero complex $t_i$. If a rotation by $\theta$ is represented by $t_i=e^{-i\theta/2}$, the all-outgoing external bra carries the phase conjugate to the ket in the preceding section. With that external-state convention,

$$\boxed{ \mathcal A_n(\ldots,t_i\lambda_i,t_i^{-1}\widetilde\lambda_i,h_i,\ldots) =t_i^{-2h_i}\mathcal A_n(\ldots,\lambda_i,\widetilde\lambda_i,h_i,\ldots) }.$$

Equivalently,

$$\left( \lambda_i^\alpha\frac{\partial}{\partial\lambda_i^\alpha} -\widetilde\lambda_i^{\dot\alpha} \frac{\partial}{\partial\widetilde\lambda_i^{\dot\alpha}} \right)\mathcal A_n=-2h_i\mathcal A_n.$$

Elvang and Huang derive the homogeneous weight and use it to determine massless three-point structures in Elvang and Huang 2014, § 2.6, pp. 27–30, PDF.

A convention-fixed dictionary

The following compact table records the weight checks used on this page. Angle and square brackets, polarization conventions, and the massive interface are collected in the shared spinor-helicity dictionary; the last row below prevents the massless $U(1)$ rule from being applied to a massive state.

Object	Spinor-helicity representation	Little-group behavior	Check
Null momentum $p_i$	$p_{i\alpha\dot\alpha}=\lambda_{i\alpha}\widetilde\lambda_{i\dot\alpha}$	invariant under $(\lambda_i,\widetilde\lambda_i)\mapsto(t_i\lambda_i,t_i^{-1}\widetilde\lambda_i)$	$\det p_i=0$
Angle bracket	$\langle ij\rangle=\epsilon^{\alpha\beta}\lambda_{i\alpha}\lambda_{j\beta}$	$t_it_j$	antisymmetric; mass dimension one
Square bracket	$[ij]=\epsilon_{\dot\alpha\dot\beta}\widetilde\lambda_i^{\dot\alpha}\widetilde\lambda_j^{\dot\beta}$	$t_i^{-1}t_j^{-1}$	antisymmetric; mass dimension one
Helicity-$h_i$ external state	label $h_i$ attached to leg $i$	amplitude weight $t_i^{-2h_i}$	differentiate with the weight operator above
Vector polarization	$\varepsilon_+^\mu(i;q)$ or $\varepsilon_-^\mu(i;q)$	$t_i^{-2}$ or $t_i^{+2}$	$p_i\!\cdot\varepsilon_i=0$ and reference-$q$ independence of the amplitude
Massive momentum	$p_{\alpha\dot\alpha}=\lambda_\alpha^I\widetilde\lambda_{\dot\alpha I}$	transforms under $SU(2)$ on $I$, not a single $U(1)$ weight	$\det p=m^2$ and $2s+1$ spin components

For a bracket monomial, the weight on leg $i$ is the signed number of angle spinors $\lambda_i$ minus the signed number of square spinors $\widetilde\lambda_i$, with denominator powers counted negatively. Matching that integer to $-2h_i$ is usually the fastest rejection test for a proposed expression.

Polarization redundancy and gauge invariance

A polarization vector for a massless spin-one leg also depends on a null reference momentum $q$. Changing $q$ shifts the polarization by a multiple of $p^\mu$,

$$\varepsilon^\mu_h(p;q')-\varepsilon^\mu_h(p;q) =c_h(p;q,q')\,p^\mu.$$

Thus the little-group weight and reference-vector independence test different properties. Homogeneous scaling confirms the helicity representation. Reference independence follows only when the complete amplitude obeys the relevant Ward identity. A single diagram can pass the weight check while retaining gauge-dependent $q$ terms that cancel only in the sum.

For an amplitude written as $\varepsilon_\mu(p;q)\mathcal M^\mu$, the decisive check is

$$p_\mu\mathcal M^\mu=0.$$

The spinor-helicity expressions for vector polarizations and their gauge shifts are developed in Elvang and Huang 2014, § 2.4, pp. 16–20, PDF.

Worked weight check

Consider the color-ordered three-gluon candidate

$$A_3(1^-,2^-,3^+) =g\frac{\langle12\rangle^3} {\langle23\rangle\langle31\rangle}.$$

Leg 1 appears three times in the numerator and once in the denominator, so its weight is $t_1^2=t_1^{-2(-1)}$. Leg 2 behaves identically. Leg 3 appears twice in the denominator, giving $t_3^{-2}=t_3^{-2(+1)}$. The bracket ratio has mass dimension one, so a dimensionless Yang–Mills coupling gives the required three-point amplitude dimension. These checks do not yet prove the coupling, color factor, or branch; they show that the kinematic structure is compatible with the declared helicities.

Common pitfalls

Calling every element of $\operatorname{ISO}(2)$ a helicity phase. The translation-like part acts trivially only for the ordinary finite-helicity representations assumed here. Continuous-spin representations are a distinct possibility, not additional helicities of the same particle.

Using a little-group check as a gauge-invariance proof. Correct homogeneous weight is necessary but does not remove reference-spinor dependence. Apply the Ward check to the complete color-dressed or color-ordered physical amplitude, as appropriate.

Applying a $U(1)$ weight to a massive leg. A four-dimensional massive particle carries an $SU(2)$ little-group index. The massive-variable page makes the replacement explicit.

Exercises

Determine the weight of

$$\frac{\langle12\rangle^4} {\langle12\rangle\langle23\rangle\langle34\rangle\langle41\rangle}$$

on every leg.

Solution

Legs 1 and 2 each have net angle-bracket power $+2$, so $h_1=h_2=-1$. Legs 3 and 4 have net power $-2$, so $h_3=h_4=+1$. The weights are therefore $(--++)$; they do not determine the coupling, color tensor, factorization residues, or boundary behavior.

Where to continue

• Spinor-Helicity Variables derives the bracket dictionary and invariant identities.
• Three-Point Amplitudes turns the homogeneous weights into local interaction seeds.
• Massive On-Shell Variables and Little-Group Covariance replaces the massless $U(1)$ weight by explicit $SU(2)$ covariance.

References

• Elvang, Henriette, and Yu-tin Huang. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press, 2015. Open prepublication version. Open PDF.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.