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**Unformatted text preview: **he gauge
symmetry which kills the extra degrees of freedom appears
only when the mass is strictly zero. The extra degrees
of freedom are a massless vector and a massless scalar
which couples to the trace of the energy momentum tensor.
This extra scalar coupling is responsible for the vDVZ
discontinuity.
Taking m ! 0 straight away in the Lagrangian (3.1) does
not yield a smooth limit, because degrees of freedom are lost.
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 To ﬁnd the correct limit, the trick is to introduce new ﬁelds
and gauge symmetries into the massive theory in a way that
¨
does not alter the theory. This is the Stuckelberg trick. Once
this is done, a limit can be found in which no degrees of
freedom are gained or lost. To introduce the idea, we consider a simpler case, the
theory of a massive photon A coupled to a (not necessarily
conserved) source J ,
Z
1
1
S ¼ dD x À F F À m2 A A þ A J ;
(4.1)
4
2
where F @ A À @ A . The mass term breaks the
would-be gauge invariance A ¼ @ Ã, and for D ¼ 4 this
theory describes the 3 degrees of freedom of a massive spin 1
particle. Recall that the propagator for a massive vector is
½Ài=ðp2 þ m2 Þð þ p p =m2 Þ, which is similar to
$1=m2 for large momenta, invalidating the usual power
counting arguments.
As it stands, the limit m ! 0 of the Lagrangian (4.1) is not
a smooth limit because we lose a degree of freedom; for
m ¼ 0 we have Maxwell electromagnetism which in D ¼ 4
propagates only 2 degrees of freedom, the two polarizations
of a massless helicity 1 particle. Also, the limit does not exist
unless the source is conserved, as this is a consistency
requirement in the massless case.
¨
The Stuckelberg trick consists of introducing a new scalar
ﬁeld , in such a way that the new action has gauge symmetry but is still dynamically equivalent to the original action. It
will expose a different m ! 0 limit which is smooth, in that
no degrees of freedom are gained or lost. We introduce a ﬁeld
by making the replacement
A ! A þ @ ; (4.2) following the pattern of the gauge symmetry we want to
¨
introduce (Stuckelberg, 1957). This is emphatically not a
change of ﬁeld variables. It is not a decomposition of A
into transverse and longitudinal parts (A is not meant to
identically satisfy @ A ¼ 0 after the replacement), and it is
not a gauge transformation [the Lagrangian (4.1) is not gauge
invariant]. Rather, this is creating a new Lagrangian from the
old one, by the addition of a new ﬁeld . F is invariant
under this replacement, since the replacement looks similar to
a gauge transformation and F is gauge invariant. The only
thing that changes is the mass term and the coupling to the
source,
Z
1
1
S ¼ dD x À F F À m2 ðA þ @ Þ2
4
2
À @ J :
þ A J
(4.3)
We have integrated by parts in the coupling to the source. The
new action now has the gauge symmetry
A ¼ @ Ã; ¼ ÀÃ: (4.4) By ﬁxing the gauge ¼ 0, called...

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