Stueckelberg Trick

Prerequisites: Gauge Symmetry, Proca Equation. Full path (23 pages)

Intuitive
Concrete
Abstract

Intuitive

A massless particle of spin 1, such as the photon, has two polarization states. A massive spin-1 particle, such as the W or Z boson, has three. The extra one is the longitudinal polarization, which oscillates along the direction of motion. The theory of a massive vector field written in the simplest way, by adding a mass term to Maxwell's theory, has no gauge symmetry, and many of the tools developed for gauge theories cannot be used.

The Stueckelberg trick restores the gauge symmetry by adding a new field and at the same time a new redundancy that exactly removes it again. The bookkeeping is the same as rewriting a function of one variable as a function of two variables that depends only on their difference. Nothing physical changes: the number of degrees of freedom stays the same, and one can always go back to the original description by choosing a gauge in which the new field is zero.

The point of the rewriting is that the new field makes one piece of physics explicit. At energies far above the mass, the longitudinal polarization of the massive vector behaves like a spin-0 particle, and in this regime the Stueckelberg field describes it. Questions about high-energy scattering of massive vector bosons can then be answered by studying a scalar, which is often much easier. The same idea is used for massive gravitons, where the broken symmetry is general coordinate invariance instead of an internal gauge symmetry.

Concrete

[W]e review the implementation of the Stückelberg trick in the theory of massive $SU(N)$ YM gauge bosons which, for simplicity, all have the same mass $m$ , $\begin{align} \mathcal{L}&=-\frac{1}{4g^{2}}{\rm tr}F_{\mu\nu}F^{\mu\nu}-\frac{m^{2}}{2g^{2}}{\rm tr} A_{\mu}A^{\mu}\ . \tag{ 3.22} \end{align}$

This lagrangian is not gauge invariant, but gauge invariance can be restored by coupling in new fields, $\pi^{a}(x)$ , with $a\in\{1,\ldots,N^2-1\}$ , i.e. , one field for each generator of $SU(N)$ . In order to insert the $\pi^{a}(x)$ 's appropriately, one first performs a gauge transformation with $\pi^{a}(x)$ as the gauge parameter, $\begin{align} A_{\mu}&\longmapsto U^\dagger(\pi)(A_{\mu}+\partial_{\mu})U(\pi)\equiv A'_{\mu} \\ F_{\mu\nu}&\longmapsto U^\dagger(\pi) F_{\mu\nu}U(\pi)\equiv F'_{\mu\nu} \end{align}$ where $U(\pi)=e^{\pi^{a}(x)T_{a}}$ is an element of $SU(N)$ . We then define a new lagrangian $\mathcal{L}'$ by taking (3.22) and replacing $A_{\mu}\mapsto A_{\mu}'$ and $F_{\mu\nu}\mapsto F_{\mu\nu}'$ . That is, $\begin{align} \mathcal{L}'&=-\frac{1}{4g^{2}}{\rm tr}F'_{\mu\nu}F'^{\mu\nu}-\frac{m^{2}}{2g^{2}}{\rm tr}A'_{\mu}A'^{\mu} \\ &=\!-\frac{1}{4g^{2}}\!{\rm tr}F_{\mu\nu}F^{\mu\nu}\!-\!\frac{m^{2}}{2g^{2}} {\rm tr}D_{\mu}U(\pi)D^{\mu}U^\dagger(\pi)~, \end{align}$ where $D_{\mu}U(\pi)= \partial_{\mu}U(\pi)+A_{\mu}U(\pi)$ is the gauge covariant derivative of $U(\pi)$ . The lagrangian $\mathcal{L}'$ then enjoys a gauge symmetry under which we simultaneously change $\begin{align} A_{\mu}&\longmapsto V^\dagger(x)(A_{\mu}+ {1}\partial_{\mu})V(x) \\ U(\pi)&\longmapsto V^\dagger(x)U(\pi)~, \end{align}$ where $V(x)\in SU(N)$ .

The physics of the $\mathcal{L}$ and $\mathcal{L}'$ lagrangians is the same, we have just made the degrees of freedom in Eq. (3.22) manifest. In $\mathcal{L}'$ , we introduced $N^{2}-1$ new fields, but also restored $N^{2}-1$ gauge symmetries and hence degree of freedom counting is the same for both cases. We can demonstrate the equivalence explicitly by using the gauge symmetry of $\mathcal{L}'$ to go ``unitary gauge" in which we set $U(\pi)\to {1}$ , where the two lagrangians coincide.

The above process and its generalizations are known collectively as the Stückelberg trick, which is often a useful tool for elucidating the physics in certain regimes of theories, especially at high energies.

https://arxiv.org/abs/1405.5532

Abstract

Abelian case. With metric signature $(+,-,-,-)$ , the Proca Lagrangian $\mathcal{L}=-\tfrac14F_{\mu\nu}F^{\mu\nu}+\tfrac12m^2A_\mu A^\mu$ is not invariant under $A_\mu\to A_\mu+\partial_\mu\lambda$ . Replacing $A_\mu\to A_\mu-\tfrac1m\partial_\mu\phi$ gives $\mathcal{L}'=-\tfrac14F_{\mu\nu}F^{\mu\nu}+\tfrac12m^2A_\mu A^\mu-m\,A^\mu\partial_\mu\phi+\tfrac12\partial_\mu\phi\,\partial^\mu\phi ,$ which is invariant under $A_\mu\to A_\mu+\partial_\mu\lambda$ , $\phi\to\phi+m\lambda$ . The gauge $\phi=0$ (unitary gauge) gives back the Proca theory. Alternatively, the gauge fixing term $-\frac{1}{2\xi}(\partial_\mu A^\mu+\xi m\phi)^2$ cancels the mixing term $-mA^\mu\partial_\mu\phi$ up to a total derivative. The vector then has the propagator of the $R_\xi$ gauges, which falls off like $1/k^2$ at large momentum, and $\phi$ becomes a field of mass squared $\xi m^2$ whose contributions cancel against the unphysical vector modes in physical amplitudes. For a vector coupled to a conserved current the abelian theory is renormalizable.

General structure. Let $G$ be a gauge group and $\mathcal{L}[A]$ a Lagrangian that is not $G$ -invariant. Introduce a field $U:M\to G$ and define $\mathcal{L}'[A,U]=\mathcal{L}[A^U]$ , where $A^U=U^{-1}AU+U^{-1}dU$ is the gauge transform of $A$ by $U$ . Under a gauge transformation $V$ , acting as $A\to A^V$ and $U\to V^{-1}U$ , we have $A^U\to (A^V)^{V^{-1}U}=A^U$ , so $\mathcal{L}'$ is invariant. The gauge $U=1$ recovers $\mathcal{L}$ , so the two theories are equivalent. For a mass term $\operatorname{tr}A_\mu A^\mu$ the result is a $G$ -valued nonlinear sigma model coupled to the gauge field, which is the limit of a Higgs model in which the radial Higgs modes are infinitely heavy. When only a subgroup $H\subset G$ is unbroken, the Stueckelberg fields take values in the coset $G/H$ . Stueckelberg fields are therefore Goldstone bosons in a nonlinear realization of the gauge symmetry.

Equivalence theorem. For momentum $k^\mu$ with energy $E\gg m$ the longitudinal polarization vector satisfies $\epsilon_L^\mu=k^\mu/m+O(m/E)$ . In the Stueckelberg description this means that amplitudes with external longitudinal vectors agree, at leading order in $m/E$ , with amplitudes in which they are replaced by the Stueckelberg scalars (the Goldstone boson equivalence theorem). In the non-abelian case the sigma model interactions are nonrenormalizable, with derivative couplings suppressed by $m/g$ , and tree-level unitarity of longitudinal vector scattering fails at energies of order $4\pi m/g$ unless new states, such as a Higgs boson, appear. In massive gravity the same construction applied to diffeomorphisms introduces Stueckelberg fields that restore general covariance, and their scalar part controls the strong coupling scale of the theory.

Stueckelberg Trick

Intuitive

Concrete

Abstract

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