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--- advanced_tools:gauge_symmetry:stueckelberg_trick [2017/11/08 14:33]
jakobadmin created
+++ advanced_tools:gauge_symmetry:stueckelberg_trick [2022/06/07 10:04] (current)
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 <tabbox Student>
-<blockquote>we review the implementation of the St\"uckelberg trick in the theory of massive $SU(N)$ YM gauge bosons which, for simplicity, all have the same mass $m$,
+<blockquote>[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}\ .\label{MassiveYMforStuckelbergSection}
+\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}\ .\label{MassiveYMforStuckelbergSection} \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 $a\in\{1,\ldots,N^2-1\}$, {\it 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,
+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}\nn
+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}\label{StuckelbergTransformationExample}
 \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 \eqref{MassiveYMforStuckelbergSection} 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'^{\nu}\nn
+\mathcal{L}'&=-\frac{1}{4g^{2}}{\rm tr}F'_{\mu\nu}F'^{\mu\nu}-\frac{m^{2}}{2g^{2}}{\rm tr}A'_{\mu}A'^{\nu}\\
 &=\!-\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)~, \label{StuckelbergedLagrangianEx}
 \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}+\mathds{1}\partial_{\mu})V(x)\nn
+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~\eqref{MassiveYMforStuckelbergSection} 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 $U(\pi)\to \mathds{1}$, where the two lagrangians coincide.
+**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 St\"uckelberg trick, which is often a useful tool for elucidating the physics in certain regimes of theories, especially at high energies. See \cite{Burgess:1992gx,Hinterbichler:2011tt,Preskill:1990fr,ArkaniHamed:2002sp} for good discussions of St\"uckelberg fields in various contexts.
+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.
 <cite>https://arxiv.org/abs/1405.5532</cite></blockquote>

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