Massive On-Shell Variables and Little-Group Covariance

A four-dimensional massive momentum is a rank-two bispinor, so it requires two spinors rather than the single outer product used for a null momentum. The extra label is not arbitrary bookkeeping: it transforms under the massive little group $SU(2)$. A spin-$s$ amplitude is therefore a symmetric rank-$2s$ tensor in that label, which packages all $2s+1$ physical spin states without choosing a preferred spin axis.

Required background. On-Shell States and Little-Group Scaling distinguishes particle-state covariance from field covariance. The shared Spinor-Helicity Variables dictionary supplies the four-dimensional bispinor and bracket conventions.

Helpful background. Lorentz Field Representations and Poincaré Particle Representations supplies the Wigner little-group construction. LSZ for Spinor and Vector External States fixes the external-state interpretation, and Three-Point Amplitudes supplies the massless comparison.

Rank-two factorization and SU(2) covariance

For $p^2=m^2>0$ in the mostly-minus convention,

$$\det p_{\alpha\dot\alpha}=m^2,$$

so $p_{\alpha\dot\alpha}$ has rank two. Introduce a pair of spinors labeled by $I=1,2$:

$$\boxed{ p_{\alpha\dot\alpha} =\lambda_\alpha^I\widetilde\lambda_{\dot\alpha I} }.$$

We choose the determinant normalization

$$\det(\lambda_\alpha^I)=m, \qquad \det(\widetilde\lambda_{\dot\alpha I})=m.$$

The factorization is invariant under

$$\lambda^I\longmapsto W^I{}_J\lambda^J, \qquad \widetilde\lambda_I\longmapsto \widetilde\lambda_J(W^{-1})^J{}_I.$$

For real Lorentzian momenta and the fixed mass normalization, $W\in SU(2)$ is the massive little group. For complexified kinematics it is natural to use its complexification $SL(2,\mathbb C)$ on the $I$ index. This $SL(2)$ is a little-group action and must not be confused with the Lorentz $SL(2,\mathbb C)$ acting on $\alpha$ or $\dot\alpha$.

Arkani-Hamed, Huang, and Huang construct the variables and the little-group action in Arkani-Hamed, Huang, and Huang 2021, §§ 2–2.1, pp. 3–8, PDF.

Indices are raised and lowered with $\epsilon^{IJ}$ and $\epsilon_{IJ}$. With a consistent phase convention the spinors satisfy massive Weyl equations of the form

$$p_{\alpha\dot\alpha}\widetilde\lambda^{\dot\alpha I} =m\lambda_\alpha^I, \qquad p^{\dot\alpha\alpha}\lambda_\alpha^I =m\widetilde\lambda^{\dot\alpha I},$$

where a sign can move between the second equation and the raising/lowering convention. The invariant checks are $p^2=m^2$ and the determinant normalization; they should be tested before comparing phases across sources.

Spin states as symmetric tensors

The spin-$s$ irreducible representation of $SU(2)$ is the completely symmetric tensor with $2s$ fundamental indices. An amplitude with one massive spin-$s$ leg therefore has the form

$$\mathcal M^{(I_1\cdots I_{2s})} (\lambda^I,\widetilde\lambda^I;\ldots).$$

Its number of independent components is

$$\dim\operatorname{Sym}^{2s}(\mathbf2)=2s+1,$$

exactly the physical spin multiplicity. Examples are

Spin	Little-group tensor	Components	Covariant wavefunction interface
$0$	scalar	1	no $I$ index
$\tfrac12$	$\mathcal M^I$	2	$\lambda_\alpha^I$ or $\widetilde\lambda_{\dot\alpha}^I$
$1$	$\mathcal M^{(IJ)}$	3	$\varepsilon_{\alpha\dot\alpha}^{IJ}=\sqrt2\,\lambda_\alpha^{(I}\widetilde\lambda_{\dot\alpha}^{J)}/m$
$s$	$\mathcal M^{(I_1\cdots I_{2s})}$	$2s+1$	symmetric product of fundamental spinors

Here parentheses include the factor $1/2$. For spin one, the symmetric tensor is transverse and contains the two transverse modes plus the longitudinal mode. Contracting it with an orthonormal basis of symmetric $SU(2)$ tensors gives polarization vectors normalized by $\varepsilon_{m_s}^*\!\cdot\varepsilon_{m_s'}=-\delta_{m_sm_s'}$ on the real physical slice. No gauge redundancy is required for a genuinely massive vector. A choice of spin quantization axis amounts to choosing a basis in the $I$ space; the covariant tensor exists before that choice.

Mixed massive and massless brackets

Brackets carrying a massive leg retain its little-group index:

$$\langle i^I j\rangle =\epsilon^{\alpha\beta}\lambda_{i\alpha}^I\lambda_{j\beta}, \qquad [i^I j] =\epsilon_{\dot\alpha\dot\beta} \widetilde\lambda_i^{I\dot\alpha} \widetilde\lambda_j^{\dot\beta}.$$

A legal amplitude must contract Lorentz spinor indices while leaving, for each massive spin-$s$ external leg, exactly a symmetric rank-$2s$ little-group tensor. There is no single homogeneous $U(1)$ weight for that leg. Expressions that would appear to mix massless helicities can be different components of one perfectly covariant massive tensor.

Useful invariant sandwiches include

$$\langle i^I|P|j] =\lambda_i^{I\alpha}P_{\alpha\dot\alpha} \widetilde\lambda_j^{\dot\alpha},$$

and massive–massive brackets $\langle i^I j^J\rangle$ and $[i^I j^J]$. Before simplifying, keep every $I,J$ index visible; suppressing them is safe only after the symmetrization and contraction pattern is unambiguous.

Controlled high-energy limit

Choose a little-group basis $\zeta_\pm^I$ and decompose

$$\lambda_\alpha^I =\lambda_\alpha\zeta_-^I +\eta_\alpha\zeta_+^I, \qquad \widetilde\lambda_{\dot\alpha}^I =\widetilde\lambda_{\dot\alpha}\zeta_+^I +\widetilde\eta_{\dot\alpha}\zeta_-^I,$$

with

$$\langle\lambda\eta\rangle=m, \qquad [\widetilde\lambda\widetilde\eta]=m.$$

At energy $E\gg m$, the large spinors scale as $\lambda,\widetilde\lambda\sim\sqrt E$, while $\eta,\widetilde\eta\sim m/\sqrt E$. Projecting a symmetric rank-$2s$ tensor onto powers of $\zeta_+$ and $\zeta_-$ isolates helicities $h=-s,-s+1,\ldots,s$. The expansion and its normalization are given in Arkani-Hamed, Huang, and Huang 2021, § 2.2, pp. 9–10, PDF.

This limit is more informative than setting $m=0$ blindly. It records which helicity components survive, which are suppressed by $m/E$, and whether explicit $1/m$ factors cancel. At degenerate three-point kinematics, setting the small spinors to zero can produce $0/0$ expressions; expand at finite $m$ first and only then take the limit.

For a massive vector, the three projections reproduce transverse helicities $-1$ and $+1$ and a longitudinal component. Whether the longitudinal amplitude stays finite, grows as $E/m$, or cancels depends on the interactions and accompanying states. A divergent high-energy limit is a dynamical consistency warning, not a failure of the variables.

An exact spin frame and polarization check

Take a future-directed momentum $p^\mu=(E,0,0,k)$ with $E>k>0$ and $m^2=E^2-k^2$. The bispinor is

$$p_{\alpha\dot\alpha} =\begin{pmatrix}E-k&0\\0&E+k\end{pmatrix}.$$

Writing $a=\sqrt{E-k}$ and $b=\sqrt{E+k}$, so $ab=m$, one exact frame is

$$\lambda^1=\widetilde\lambda_1= \begin{pmatrix}a\\0\end{pmatrix}, \qquad \lambda^2=\widetilde\lambda_2= \begin{pmatrix}0\\b\end{pmatrix}.$$

Column multiplication reconstructs $p_{\alpha\dot\alpha}$ and $\det\lambda=\det\widetilde\lambda=m$. Projecting the symmetric spin-one tensor onto three orthonormal $SU(2)$ components gives, up to harmless state phases,

$$\varepsilon_+^\mu=\frac1{\sqrt2}(0,-1,-i,0), \quad \varepsilon_-^\mu=\frac1{\sqrt2}(0,+1,-i,0), \quad \varepsilon_0^\mu=\frac1m(k,0,0,E).$$

They obey

$$p\!\cdot\varepsilon_a=0, \qquad \varepsilon_a\!\cdot\varepsilon_b^*=-\delta_{ab}, \qquad \sum_{a=1}^3\varepsilon_a^\mu\varepsilon_a^{*\nu} =-\eta^{\mu\nu}+\frac{p^\mu p^\nu}{m^2}.$$

For $(E,k,m)=(5,3,4)$ the longitudinal state is $(3/4,0,0,5/4)$ and has norm exactly $-1$. This checks the spinor factorization against the usual massive-vector completeness relation rather than only counting its three components Schwartz 2014, § 8.2.2, pp. 117–118.

One massive and two massless legs

If $k_1^2=k_2^2=0$, $P_3^2=m^2$, and $k_1+k_2+P_3=0$, then

$$m^2=(k_1+k_2)^2=\langle12\rangle[21].$$

Unlike all-massless three-point kinematics, neither bracket family vanishes for $m\ne0$. As a bounded example, a massive vector coupled to two scalar species through an antisymmetric derivative current has the seed

$$\mathcal M_3^{IJ} =g\,\varepsilon_3^{IJ}\!\cdot(k_1-k_2) =\frac{g}{\sqrt2m} \left( \langle1,3^{(I}\rangle[3^{J)}1] -\langle2,3^{(I}\rangle[3^{J)}2] \right).$$

The open pair $(IJ)$ is symmetric, so the expression is a spin-one triplet. It has dimension one for dimensionless $g$, and current conservation follows directly:

$$P_3\!\cdot(k_1-k_2) =-(k_1+k_2)\!\cdot(k_1-k_2)=0.$$

The tensor therefore passes covariance, dimension, and transversality checks. These do not fix $g$ or prove that a complete higher-point theory exists; if the vector is unstable, the same tensor describes its pole residue but not an exact asymptotic state Arkani-Hamed, Huang, and Huang 2021, § 4.1, pp. 18–19, PDF.

A state-sum covariance check

Consider a massive spin-$\tfrac12$ leg represented by $\mathcal M^I$. Under $W\in SU(2)$,

$$\mathcal M^I\longmapsto W^I{}_J\mathcal M^J.$$

On the real physical slice, the state-summed square contracts the fundamental representation with its conjugate using the invariant Hermitian form,

$$\delta_{I\bar J}\,\mathcal M^I (\mathcal M^J)^* =\sum_{I=1}^2|\mathcal M^I|^2,$$

and is basis independent. For spin one, the analogous contraction uses the symmetric two-index representation. This gives a direct check that an apparent spin-axis dependence is only a basis choice; a leftover dependence after the complete state sum signals an inconsistent projection or normalization.

Common pitfalls

Treating the index $I$ as flavor. It transforms with momentum-dependent little-group data and labels spin states. Flavor is an additional internal index with different transformation rules.

Assigning a massless helicity weight to a massive leg. The correct requirement is $SU(2)$ tensor covariance. Helicity emerges only after a basis projection, especially in the high-energy limit.

Taking $m\to0$ term by term in a singular basis. First decompose into large and small spinors, project the desired spin component, and check cancellation of explicit $1/m$ factors.

Assuming high-energy agreement proves a complete massive amplitude. Terms suppressed by powers of $m/E$ remain invisible in that limit. Factorization, locality, permutation symmetry, and low-energy checks are still needed.

Exercises

Count the independent components of a symmetric tensor $\mathcal M^{I_1\cdots I_{2s}}$ and project it onto a basis of $\zeta_\pm$.

Solution

A symmetric tensor with $2s$ indices, each taking two values, is labeled only by the number $r=0,\ldots,2s$ of $\zeta_+$ factors. It therefore has $2s+1$ components. The spin projection is $h=r-s$, so $h=-s,-s+1,\ldots,s$. At finite mass an $SU(2)$ rotation mixes these components; they are components of one covariant amplitude, not independent helicity amplitudes.

Where to continue

• Three-Point Amplitudes supplies the massless seeds used as high-energy checks.
• Complex Momenta and Factorization explains how massive or massless internal states enter a residue.
• LSZ for Spinor and Vector External States connects the covariant on-shell tensors to normalized external states.

References

• Arkani-Hamed, Nima, Tzu-Chen Huang, and Yu-tin Huang. “Scattering Amplitudes for All Masses and Spins.” Journal of High Energy Physics 11 (2021): 070. DOI. Open PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.