We study the higher-derivative extensions of the D=3 Abelian Chern--Simons
topological invariant that would appear in a perturbative effective action's
momentum expansion. The leading, third-derivative, extension I_ECS turns out to
be unique. It remains parity-odd but depends only on the field strength, hence
no longer carries large gauge information, nor is it topological because metric
dependence accompanies the additional covariant derivatives, whose positions
are seen to be fixed by gauge invariance. Viewed as an independent action,
I_ECS requires the field strength to obey the wave equation. The more
interesting model, adjoining I_ECS to the Maxwell action, describes a pair of
excitations. One is massless, the other a massive ghost, as we exhibit both via
the propagator and by performing the Hamiltonian decomposition. We also present
this model's total stress tensor and energy. Other actions involving I_ECS are
also noted.Comment: 3 typos fixed. 5 page

Barnich

Binegar

Deser

R. Jackiw

S. Deser

Physics Letters B

English

arXiv

We study the higher-derivative extensions of the D=3 Abelian Chern–Simons topological invariant that would appear in a perturbative effective action's momentum expansion. The leading, third-derivative, extension /_(ECS) turns out to be unique. It remains parity-odd but depends only on the field strength, hence no longer carries large gauge information, nor is it topological because metric dependence accompanies the additional covariant derivatives, whose positions are seen to be fixed by gauge invariance. Viewed as an independent action, /_(ECS) requires the field strength to obey the wave equation. The more interesting model, adjoining /_(ECS) to the Maxwell action, describes a pair of excitations. One is massless, the other a massive ghost, as we exhibit both via the propagator and by performing the Hamiltonian decomposition. We also present this model's total stress tensor and energy. Other actions involving /_(ECS) are noted, as is the corresponding extension of the D=4 θ-term

Deser, S.

Jackiw, R.

Caltech Authors - Main

Higher derivative Chern–Simons extensions

Crossref

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BRX TH–449
CTP#2818
Higher Derivative Chern–Simons Extensions
S. Deser
Department of Physics, Brandeis University
Waltham, MA 02454, USA
and
R. Jackiw
Department of Physics, Massachusetts Institute of Technology
Cambridge, MA 02139, USA
We study the higher-derivative extensions of the D=3 Abelian Chern–Simons topological invariant
that would appear in a perturbative effective action’s momentum expansion. The leading, third-
derivative, extension IECS turns out to be unique. It remains parity-odd but depends only on the
field strength, hence no longer carries large gauge information, nor is it topological because metric
dependence accompanies the additional covariant derivatives, whose positions are seen to be fixed
by gauge invariance. Viewed as an independent action, IECS requires the field strength to obey the
wave equation. The more interesting model, adjoining IECS to the Maxwell action, describes a pair
of excitations. One is massless, the other a massive ghost, as we exhibit both via the propagator
and by performing the Hamiltonian decomposition. We also present this model’s total stress tensor
and energy. Other actions involving IECS are also noted.
1. Introduction
The remarkable properties of the Chern–Simons (CS) topological invariant in D=3 gauge theories
[1, 2] are by now well-appreciated. For Abelian vector fields, ICS = m/2
∫
d3x ǫαβγAα∂βAγ is parity
violating, of first derivative order, metric-independent, and gauge invariant. Viewed as an action,
ICS leads to the locally “flat” field equation Fαβ ≡ ∂αAβ − ∂βAα = 0. Instead, when the Maxwell
action IMAX is adjoined, the resulting topologically massive electrodynamics (TME) describes a
helicity ±1 (depending on the sign of the mass m) mode [2]. The gravitational CS analog is of third
derivative order (ICS ∼ m−1
∫
ΓR), and its Euler–Lagrange equation Cµν ≡ ǫµαβDα(Rνβ− 14δνβR) = 0
states that space is locally conformally flat: the Cotton tensor Cµν is the 3-space conformal tensor.
Adjoining the Einstein action leads to a dynamical system, topologically massive gravity (TMG);
despite its higher order, the linearized limit of TMG describes a massive helicity ±2 excitation [2].
The first derivative ICS appears naturally in a perturbative effective action expansion of QED3,
because the electron’s mass term is also P-violating in D=3, and higher derivative extensions of it
(which we denote by IECS) should appear as well [3] in a (∂/m) power series,
IEF F [Aµ] = ICS + IMAX + IECS +O(m−2) .
It is therefore a natural question, both in connection with this expansion and for comparison with
the third derivative TMG, to consider such IECS. Additional derivative powers must be even (if the
1
parity violating ǫαβγ is retained), the lowest extensions being the most illuminating. In actuality,
there is only one such extension, as exhibited by the equalities
IECS = (2m)
−1
∫
d3x ǫαβγ2Aα∂βAγ = −(2m)−1
∫
d3xǫαβγ∂λAα∂
λ∂βAγ
= −(2m)−1
∫
d3x ǫαβγfα∂βfγ , f
α ≡ 1
2
ǫαµνFµν . (1)
Each term follows from its predecessor by an obvious integration by parts. Hence, unlike the
original ICS, IECS depends locally on the field strength and not on the potential, and so carries no
“large gauge” information. Our signature conventions are (+,–,–), ǫ012 = +1 = ǫ012.
We shall firstly exhibit the excitations described by actions containing IECS, including especially its
sum with the Maxwell action (ETME). There, we find a massless particle plus a massive ghost. We
then consider IECS in a gravitational background, where the higher derivatives necessarily engender
metric dependence, in contrast to the topological character of ICS, and give rise to a stress tensor
that contributes explicitly to the energy of ETME, as we shall display in terms of the two degrees
of freedom.
2. Maxwell–ECS Dynamics
We shall work in source-free flat space in this section, our aim being to characterize the excitations
described by actions that include IECS. It is clear from (1) that, taken alone, IECS yields the
unconstrained massless propagation of the field strength,
2fµ = 0 . (2)
Next, define the extended system by adjoining to IECS the Maxwell action
IETME = −12
∫
d3x[f2µ +m
−1ǫαβγfα∂βfγ ] (3a)
resulting in the field equations
m δIETME/δAµ = 2f
µ −mǫµαβ∂αfβ = −ǫµαβ∂α(mfβ + ǫβγδ ∂γfδ) = 0 . (3b)
Herem is seen to have dimensions of inverse length or mass. [We remark that (3a) is known [4] to be
precisely equivalent to TME, the Maxwell–CS action, if fµ is taken to be the fundamental variable
rather than, as for us, the curl of an underlying vector potential. Our equations are correspondingly
the curl of those of TME, as shown by the last equality in (3b).] Note that ETME can be formally
obtained from (the appropriate helicity branch of) TME by the replacement m → 2/m. Hence,
we can immediately write the form of our propagator, using that of TME. There [2],
GµνTME = (2 +m
2)−1(gµν − (m/2)ǫµαν∂α) (4)
when acting on conserved sources; it clearly described a massive excitation. Hence, the ETME
propagator becomes
GµνETME = (2 + 2
2/m2)−1(gµν −m−1ǫµαν∂α) , (5)
2
as of course follows directly from (3a). This time, the denominator describes two excitations,
m22−1(2 +m2)−1 = 2−1 − (2 +m2)−1 . (6)
One is massless, the other is massive, with a relative ghost sign. The limit m → ∞ of GETME
correctly reproduces the Maxwell propagator gµν/2. Its small m limit should correspond to pure
IECS and indeed we find GETME → −m/22 ǫµαν∂α, the propagator corresponding to (2). Note
that, irrespective of the signs in (3), there is never a tachyon: there cannot be a (2−m2) pole.
We next perform a detailed canonical analysis, decomposing fµ in terms of the vector potential
(Ai, A0), and writing
Ai ≡ ǫi j ∂ˆja+ ∂iΛ , ∂ˆi ≡ ∂i/
√−∇2 , (ǫi j ≡ −ǫij) , (7)
to yield
f0 = ǫij∂iAj = −
√−∇2 a , fi = ∂ˆia˙+ ǫij ∂ˆj E , E ≡ −
√−∇2(A0 − Λ˙) . (8)
The action IETME then reduces to
IETME =
1
2
∫
d3x(−a2a+ E2) +m−1 ∫ d3xE2a . (9)
The Maxwell contribution (m = ∞) is that of the transverse a-mode, together with a nonpropa-
gating longitudinal electric field term. Note the absence of dangerous explicit third time derivative
terms. This is unsurprising: they can only come from the
∫
ǫijfi∂0fj part of IECS. Due to the
explicit ǫij, all “diagonal” terms (aa) and (EE) vanish by antisymmetry (after spatial partial in-
tegration) since these scalars would have to carry ∂i, ∂j to saturate ǫ
ij . The aE cross-term has no
ǫij and so can and does contain the third derivative a ∂30 Λ, but (since Λ is a gauge parameter) one
∂0 is harmlessly buried as part of the gauge-invariant field variable E. The field equations from (9)
are obviously
2(a− E¯) = 0 , 2a+m2E¯ = 0 , E¯ ≡ m−1E . (10)
The appropriate diagonalization is also clear from (9):
IETME = −12
∫
d3x a¯2a¯+ 1
2
∫
d3x E¯(2 +m2)E¯ (11)
in terms of a¯ ≡ a − E¯. Here we have the normal transverse massless photon a¯, together with
a massive ghost E¯ (longitudinal electric field), as evidenced by the relative minus sign between
the two modes. We again recover the Maxwell theory in the m → ∞ limit, while as m → 0, we
find 2fµ = 0 in terms of the two invariant parts (a,E) of fµ. All this agrees with the analysis
above of the propagator (5,6). We have not studied the spin character of our excitations; any
massless particle must be spinless in D=3 [5], while the massive mode presumably has helicity
±1 according to the sign of m. We also omit details of coupling ETME to sources, which is a
straightforward exercise in the propagator’s properties for minimally coupled ∼ Aµjµ currents.
Non-minimal interactions ∼ fµkµ are also permitted here, and would presumably be equivalent to
minimal couplings in TME, in view of the equivalence of (3) to TME, in terms of fµ alone.
The most general gauge invariant action involving the terms discussed so far (apart from obvious
2
2 or higher insertions in ICS) would be the linear combination
ITOT = IECS + IMAX + ICS . (12)
3
The only unexplored 2-term action, IECS + ICS, obviously just corresponds to massive propagation,
(2 + m2)fµ = 0, of the field strength. If we keep all three terms, the propagator can once
again be read off from GµνTME by the replacement m → m + c2/m, where we have allowed for
different mass parameters m, m/c in the two CS variants. The GTOT denominator has the form
m−2[c222 +m22(1 + 2c) +m4] with roots m2[−(1 + 2c) ± (1 + 4c)1/2]/2c2. The degenerate root
at c = −1
4
corresponds to a double (massive) pole. Clearly there is a range of special cases, both
physical and not, that can be explored, but no massless excitation remains.
3. Curved Space Gauge Invariance
When the geometry is nonflat, even the Maxwell action must be written properly to preserve
gauge invariance. For example if instead of writing the manifestly invariant form FµνF
µν , we
had continued from its equally correct flat space expansion (∂µAν)
2 − (∂µAµ)2 to the covariant
expression (DµAν)
2 − (DµAµ)2, gauge invariance would be lost; the only correct order in the last
term is (DµAν)(D
νAµ), which differs [6] from (DµAµ)
2 by a gauge-variant, curvature-dependent
term RµνAµAν . It is similarly easy to see that only the last, manifestly gauge invariant derivative
ordering of (1) preserves curved space gauge invariance
IECS = −(2m)−1
∫
d3x ǫαβγfα∂βfγ , fα ≡ g−1/2gαβǫβµν∂µAν . (13)
Note that as defined here, fα is a covariant vector; f
a will denote the contravariant vector (not
the usual density) gαβfβ. The metric dependence IECS is now entirely contained in fα, so that the
stress tensor is
√
g T µνECS = 2δIECS/δgµν = −m−1
{
(ǫµαβf ν + ǫναβfµ)∂αfβ − gµνǫαβγfα∂βfγ
}
, (14)
in contrast to T µνCS ≡ 0. The (covariant) conservation of T µν on ECS shell is easily checked, using
the Bianchi identities ∂µ(
√
g fµ) ≡ 0, the (covariant) field equations
ǫµ
αλ ∂λ(g
−1/2ǫα
βγ∂βfγ) ≡ Dν(−∂µfν + ∂νfµ) ≡ [gµν D2 −DνDµ]f ν = D2fµ −Rµνfν = 0 (15)
and the identity ǫαβγ∂βfγ(∂αfµ − ∂µfα) ≡ 0.
The total ECSE stress tensor includes the usual Maxwell contribution,
T µνECSE = −12(fµf ν + f νfµ − gµνfλfσgλσ) + T µνECS (16a)
and is likewise conserved on combined shell, where is can be written very simply: By virtue of
the middle term in (3b), T µνECSE reduces to its Maxwell part but (in flat space) with the operator
(1 + 2m−22) rather than just unity, between the f ’s:
−2m2 T µνECSE = fµ(m2 + 22)f ν + f ν(m2 + 22)fµ − ηµνfα(m2 + 22)fα . (16b)
[In curved space, gαβ2 is replaced by the operator (D
2gαβ −Rαβ) of (15).] Inserting the canonical
decomposition (7,8) into T 00
ECSE
directly gives the total energy as the expected difference between
photon and ghost mode contributions; in our signature,
P0 = −
∫
d2r T 00 = 1
2
m−2
∫
d2r[fi(m
2 + 22)fi + f0(m
2 + 22)f0]
= 1
2
∫
d2r
{[
(∂0a¯)
2 + (∇a¯)2
]− [(∂0E¯)2 + (∇E¯)2 +m2E¯2]} . (17)
4
4. Summary
We have studied the first higher derivative analog of the CS topological invariant, which would
arise in the effective QED3 action’s expansion in powers of ∂/m. This IECS invariant turns out
to be unique, and while formally similar to ICS, differs profoundly from it in two respects: first,
IECS is a local function of the field strength, insensitive to the “large gauge” aspects captured
by ICS; second, it is no longer topological but depends explicitly on the background geometry.
When IECS is added to the Maxwell action, the resulting ETME system describes two degrees of
freedom, one massless, the other a massive ghost. This is in contrast with the otherwise similar
gravitational TMG model: while both are of overall third (but of second time) derivative order,
TMG represents a single massive excitation. The reasons for this difference can be traced to the
roles played by the respective component actions: First, the Maxwell term, unlike the Einstein one
in TMG, already describes a (massless) degree of freedom. Second, the triple derivative CS term in
gravity is conformally invariant and this higher symmetry, absent in IESC, eliminates one candidate
mode.
This work was supported by NSF grant PHY93–18511 and DOE grant DE–FC02–94ER40818.
References
[1] W. Siegel, Nucl. Phys. B156 (1979) 139; R. Jackiw and S. Templeton, Phys. Rev. D23 (1981)
2291; J. Schonfeld, Nucl. Phys. B185 (1981) 157.
[2] S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. 48 (1982) 475; Ann. Phys. 140 (1982)
372.
[3] S. Deser and A.N. Redlich, Phys. Rev. Lett. 61 (1988) 1541.
[4] S. Deser and R. Jackiw, Phys. Lett. 139B (1984) 371.
[5] B. Binegar, J. Math. Phys. 23 (1982) 1511.
[6] S. Deser, Nuovo Cim. Lett. 4 (1970) 1130.
5


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Higher Derivative Chern--Simons Extensions

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