Yang-Mills as massive Chern-Simons theory: a third way to three-dimensional gauge theories

0.12 MB·English

by  Alex S. Arvanitakis

DAMTP-2015-8
Yang-Mills as massive Chern-Simons theory:
a third way to three-dimensional gauge theories
Alex S. Arvanitakis,1,∗ Alexander Sevrin,2,† and Paul K. Townsend1,‡
1Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.
2Theoretische Natuurkunde, Vrije Universiteit Brussel,
and The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels,
Belgium, and Physics Department, Universiteit Antwerpen,
Campus Groenenborger, 2020 Antwerpen, Belgium
5
1 The Yang-Mills (YM) equation in three spacetime dimensions (3D) can be modified to include
0 a novel parity-preserving interaction term, with inverse mass parameter, in addition to a possible
2 topological mass term. The novelty is that the modified YM equation is not the Euler-Lagrange

r equationofanygauge-invariantlocalaction fortheYMgaugepotentialalone. Instead,consistency
p isachievedin the“third way”exploitedby3D “minimal massivegravity”. Werelateourresultsto
A the“novel Higgs mechanism” for Chern-Simons gauge theories.

6
PACSnumbers: 11.15.-q,11.15.Yc,11.30.Er
2

h] Inthreespacetimedimensions(3D)thegeneralgauge- Because of the Bianchi identity DµF˜µ ≡ 0, consistency
t invariant second-order action for a Yang-Mills (YM) requires the source current to be covariantly conserved:
-
p gauge potential A is
e DµJµ =0. (7)
h I [A]= 1 d3xF˜µ·F˜ + µ I [A], (1)
[ TMYM 2g2 Z µ g2 CS There are two standard ways to construct a source cur-

rent with this property:
2 where F˜ is the dual Yang-Mills field strength,
v 1. J =j(φ),theNoethercurrentinaYMbackground
48 F˜µ =εµνρ ∂νAρ+ 1Aν ×Aρ , (2) for lower-spinfields φ. In this case Dµjµ(φ)=0 as
(cid:18) 2 (cid:19) a consequence of the φ equations of motion.
5
7
0 and ICS[A] is the Chern-Simons action 2. J =δI[A]/δA, where I[A] is some gauge-invariant,
. andLorentzinvariant,functionalofA. Inthis case
01 12Z d3xεµνρ(cid:20)Aµ·∂νAρ+ 31Aµ·Aν ×Aρ(cid:21) . (3) DµJµ ≡ 0. This will lead to higher-derivative ad-
ditions to the action.
5
1 Here we suppose, for simplicity, that the gauge group is
There is, however, a third possibility, at least in 3D. In
v: SU(2)andweusevectoralgebranotationforproductsof the spin-2 context, this “third way”is realisedby “mini-
i SU(2) triplets; the generalisation to other gauge groups
X mal massive gravity”(MMG) [3–5], which is a modifica-
is straightforward.
tion of the much-studied “topologicallymassive gravity”
r Forµ=0,theaction(1)isthe3DYMactionwithcou-
a [6]. What we show here is that there is a spin-1 analog
pling constantg. Notice that g2 has dimensions ofmass,
of the constructionof [3], realisedas a particularmodifi-
so that µ/g2 is dimensionless. For non-zero µ the action
cation of either YM theory (if µ = 0) or TMYM theory
is that of “topologically-massive Yang-Mills” (TMYM)
(if µ6=0).
theory [1, 2], which propagates an SU(2) triplet of spin-
Consider the current
1 modes of mass µ. The field equation is
Jµ ∝ εµνρF˜ ×F˜ . (8)
εµνρD F˜ +µF˜µ =0, (4) ν ρ
ν ρ
This current involves only the gauge field A, through its
where D is the covariant derivative, defined such that
µ field strength. It is not identically conserved,
D V =∂ V +A ×V (5)
µ µ µ D Jµ ∝ εµνρD F˜ ×F˜ 6≡0, (9)
µ µ ν ρ
for any SU(2)-triplet V. (cid:16) (cid:17)
Let us now add a source current J to the right hand but using the source-free TMYM equation (4) we find
side of (4), so that that
εµνρD F˜ +µF˜µ =Jµ. (6) D Jµ ∝ µF˜µ×F˜ ≡0. (10)
ν ρ µ µ
2
Inotherwords,thethirdpossibilityisthatJ isconserved Thesamecannotbesaidofminimalcouplingtolower-
as a consequence of the YM or TMYM equation itself. spin “matter”. Consider the equation
The obvious difficulty with this idea is that we change
1
the YM or TMYM equation as soon as we include J as εµνρ D F˜ + F˜ ×F˜ +µF˜µ =Jµ, (17)
ν ρ ν ρ
a source, but in this case (cid:18) 2m (cid:19)
where J is a matter source current. Taking the diver-
D Jµ ∝Jµ×F˜ ∝εµνρ F˜ ×F˜ ×F˜ ≡0. (11)
µ µ µ ν ρ gence of this equation and then using it to simplify the
(cid:16) (cid:17)
result, we deduce that
The finalidentity is a consequence ofthe Lie algebraJa-
cobi identity, so the current J of (8) is conserved as a D Jµ+m−1F˜ ×Jµ =0. (18)
µ µ
consequence of the YM or TMYM equation even after
this equation is modified to include J. We have now ver- Only when m−1 = 0 can we take J to be a Noether
ified the consistency of the modified equation currentj(φ),soitisnotimmediatelyclearwhetherthere
is a consistent coupling to lower-spin matter. However,
1
εµνρ D F˜ + F˜ ×F˜ +µF˜µ =0, (12) given a covariantly conserved matter current j(φ), and
ν ρ ν ρ
(cid:18) 2m (cid:19) assuming that m 6= µ, the consistency condition (18) is
satisfied by a source current of the form
where m is a further mass parameter.
ItwouldappearthatthenewadditiontotheYMequa- 1 1
Jµ = jµ− εµνρD j − εµνρF˜ ×j
tionbreaksparity,evenwhenµ=0,becauseifthe1-form (m−µ) ν ρ m(m−µ) ν ρ
A is parity even, as it apparently must be for its field
1
strength 2-form F to have definite parity, then the dual + 2m(m−µ)2εµνρjν ×jρ. (19)
1-form F˜ is parity odd, implying that its covariantexte-
riorderivativeDF˜ is parityoddbutalsothatthe 2-form Noticethatthisisquadraticinthecovariantlyconserved
F˜ ×F˜ is parity even. Nevertheless, the equation (12) currentj,incloseanalogytothesourcetensorforMMG,
does not break parity when µ=0. To see this one must which is quadratic in the matter stress tensor [4]. To
assign the following parity transformation to the 1-form verify that (19) solves (18) one needs to use the Lie al-
A: gebra Jacobi identity and equation (17), which includes
the source.
P : A→A+m−1F˜. (13)
We shall now consider the particular case of coupling
to an adjoint Brout-Englert-Higgs (BEH) field; i.e. a
The parity transformation of F is then
triplet scalar φ for gauge group SU(2). Assuming that
1 1 the φ field equation is
P : F →F + DF˜+ F˜×F˜ , (14)
m(cid:18) 2m (cid:19)
[DµD +2V′]φ=0, (20)
µ
sothatF˜isstillparityodd,andhenceF˜×F˜isstillparity
for potential V(φ·φ), the covariantly conserved current
even, when one uses the µ = 0 equation of motion! The
is
clash with the apparent odd parity of the DF˜ term is
resolved by the shift of A, which flips the sign of the j =φ×D φ. (21)
µ µ
F˜×F˜ term in DF˜+ 1 F˜×F˜, so the µ=0 equationof
2m
motion preserves parity. We shall see later that the new In this case
YM theory can be formulated in a way that makes this
Jµ = φ×Dµφ−(m−µ)−1φ×(F˜µ×φ)+... (22)
feature manifest.
The 3D YM stress tensor is = φ× Aµ−(m−µ)−1εµνρ∂νAρ ×φ +...
(cid:8)(cid:2) (cid:3) (cid:9)
T =F˜ ·F˜ − 1η F˜ ·F˜ρ. (15) where omitted terms are non-linear, even when φ has a
µν µ ν µν ρ
2 non-zero vacuum value. Let us now suppose that
This tensor has the property that ∂µT =0 as a conse-
µν φ=v+ϕ, (23)
quence of either the YM equation or the TMYM equa-
tion. This remains true even if the equation used is the where v is a constant SU(2) triplet and ϕ has zero vac-
modified one of (12): uum value. Then,
∂µTµν = 2F˜µ·D[µF˜ν] =−2m−1F˜µ·F˜µ×F˜ν v·Jµ = 0+... , (24)
= −2m−1F˜µ×F˜ ·F˜ ≡0. (16) v×Jµ = v2 v×Aµ−(m−µ)−1εµνρ∂ (v×A ) +...
µ ν ν ρ
(cid:2) (cid:3)
This suggests that the coupling to 3D gravity will be where omitted terms are non-linear. We see that the
straightforward. vectorpotential gaugingthe unbrokenU(1)gaugegroup
3
is unaffected by the BEH field; it continues to propa- G = F˜ on shell, it follows that the stress tensor is the
gate a single spin-1 mode of mass µ. The other two vec- usual one, i.e. T of (15), times a factor of (m−µ)/m.
µν
tor potentials each acquire an explicit mass term (with We have already verified that ∂µT = 0 remains true
µν
mass-squaredv2)sothey eachpropagateapairofspin-1 for finite m. We now see that the energy will be positive
modes, but these vector potentials also have a topologi- or negative according to the sign of m(m−µ), and that
calmassterm,nowwithmassparameterµ+v2/(m−µ); positive energy requires
notice that this is non-zero even when µ=0.
In the special case that µ = 0 we have, in addition m(m−µ)>0. (27)
to one massive scalar mode, one massless mode prop-
agated by the U(1) vector potential and four massive We may also use (25) to recoverour earlier result (19)
modes propagated by the other two vector potentials; forthesourcecurrentJ. WejustaddtoLtheinteraction
each propagates two spin-1 modes of opposite (3D) he- term −Aµ · jµ. This is gauge invariant provided that
licities but with different masses because of the topolog- Dµjµ = 0, but (as will become clear shortly) it breaks
ical mass (v2/m) induced by the symmetry breaking. In parity when m is finite. With this term included, the A
a parity-preserving theory, massive spin-1 modes must equation becomes
appearin parity-doubletsofopposite helicities, soparity
1
is broken by the coupling to matter for finite m, even εµνρ D G + G ×G +µF˜µ =j. (28)
ν ρ ν ρ
though the source-free theory with µ = 0 preserves par- (cid:18) 2m (cid:19)
ity. ThisisaconsequenceofthefactthatAisnotparity
Recall now that this equation is needed, in addition to
inert for finite m. We shall see later how to modify the
the G equation, to determine G, so G will acquire a j-
construction so as to preserve parity when µ=0.
dependence. In fact,
AlthoughthereisnolocalgaugeinvariantactionforA
alone that yields the source-free equation (12), there is
Gµ =F˜µ+(m−µ)−1j. (29)
anactioninvolvingauxiliaryfields, provided thatm6=µ.
The Lagrangiandensity L is given by
If this is now substituted into the A equation and all j-
1 dependent terms are taken to the right hand side, the
g2L = Gµ·F˜µ− (m−µ)Gµ·Gµ+µLCS (25) result of (19) is recovered. This construction parallels
2m
the construction in [4] of the source tensor for MMG.
1 1
+ εµνρ Gµ·DνGρ+ Gµ·Gν ×Gρ . We now show how the parity invariance of the action
2m (cid:18) 3m (cid:19)
for µ=0 may be made manifest. First we introduce the
Inthem→∞limit,theauxiliaryvectorfieldG(whichis new gauge potential
alsoanSU(2)triplet)canbetriviallyeliminated,andwe
A¯=A+m−1G, (30)
arethen backto the standardTMYMaction. Moregen-
erally,a variationof both A and G induces the following
andthenrewritetheactionintermsofAandA¯byusing
variation of L:
G=m(A¯−A). The result is
(m−µ)
g2δL = δG · F˜µ−Gµ + δA +m−1δG
µ µ µ m (m−µ)
m (cid:16) 1 (cid:17) (cid:0) (cid:1) I = g2ICS[A¯]− g2 ICS[A] (31)
· εµνρ D G + G ×G +µF˜µ . (26)
(cid:20) (cid:18) ν ρ 2m ν ρ(cid:19) (cid:21) − 1 m(m−µ) d3x A¯−A · A¯−A µ .
2g2 Z µ
From this result we see that the field equations imply (cid:0) (cid:1) (cid:0) (cid:1)
bothGµ =F˜µandafurtherequationthatbecomesequa- When m = µ we may eliminate A, trivially, to obtain a
tion (12) upon substitution for G. Chern-Simons (CS) action for A¯, here for gauge group
Wealsoseefrom(26)thatitisnottheGfieldequation SU(2), but the equations of motion of this CS action
alonethatallowsustosolveforG; thatequationalsoin- are not equivalent to the m = µ case of (12), for which
volves DG and a term quadratic in G. It is a linear special case no action is known.
combination of the A and G field equations that allows When µ = 0 the action (31) preserves parity if par-
us to eliminate G, but for this reason back-substitution ity is assumed to exchange A with A¯. Then, although
in the action is illegitimate. This accords nicely with parity flips the sign of the two CS actions, it also ex-
ourearlierconclusionthat,asaconsequenceofits“third changes them, so their difference is parity even provided
way” construction, the modified YM equation is not the their coefficients sum to zero, which is the case when
Euler-Lagrangeequationforanylocalgauge-invariantac- µ=0. AftereliminationofGtorecoverthenewYMfield
tion constructed from A alone. equationof(12),theparitytransformationofAbecomes
Observe that only the G ·Gµ term in (25) involves A → A¯ = A+m−1F˜. It should now be clear how we
µ
the 3D Minkowski metric. From this, and the fact that must proceed if we wish minimal coupling to lower-spin
4
matter to preserve parity when µ = 0: we must choose valuesofH˜;weshouldnoteherethataspecialcase(m=
the gauge potential to be the parity-inert combination 2µ) has appeared previously in a related context [10].
The Lagrangian density (35) is a convenient starting
1 1
C = A+A¯ =A+ G, (32) pointfortheconstructionoftheHamiltonianformulation
2 2m
(cid:0) (cid:1) forthe µ=0 case. Performingatime-space splitwe find
and then add the parity-preserving interaction term that
Lint =−Cµ·jµ, ∂µjµ+Cµ×jµ =0. (33) g2L = 21G0·G0+G0·(cid:18)B+ 8m12εijEi×Ej(cid:19) (38)
Since parity flips the signofµ in (31), we may assume 1
that µ ≥ 0. In addition, the field redefinition A ↔ A¯ + Ei·C˙i+C0· ∂iEi+Ci×Ei − 2Ei·Ei,
(cid:0) (cid:1)
yields the same actionbut with m replacedby µ−m, so
where a sum over i=1,2 is implicit, and
we may further assume m ≥ µ/2. Thus m ≥ µ/2 ≥ 0
may be assumed without loss of generality, but we also
1
need m>µ for positive energy,in which case(excluding Ei =εijG , B =εij ∂ C + C ×C . (39)
j i j i j
(cid:18) 2 (cid:19)
m=µ) we have
The auxiliary field G may now be trivially eliminated;
m>µ≥0, (34) 0
this yields
which excludes m=0.
g2L=Ei·C˙ +C ·D Ei−H, (40)
Remarkably, the µ = 0 case of the action (31) has i 0 i
appeared previously [7], as a model designed to illus-
where the covariantderivativeis now defined with gauge
trate a “novel Higgs mechanism” [8]. In this context it
potential C and the Hamiltonian is
arises from a CS theory for gauge groupSU(n)×SU(n)
coupled to a bi-fundamental Higgs field that breaks 2
1 1 1
SU(n)×SU(n) to the diagonal SU(n) subgroup, here H = 2Ei·Ei+ 2(cid:12)B+ 8m2εijEi×Ej(cid:12) . (41)
SU(2). This construction yields an additional singlet (cid:12) (cid:12)
(cid:12) (cid:12)
massive scalar field, so once this is included the model (cid:12) (cid:12)
Here, |..| is the SU(2)-triplet norm. We see that the
becomes a CS gauge theory minimally coupled to scalar canonical variables {C ,Ei} are subject to the Gauss-
i
fields,whichisrenormalizableasa3Dquantumfieldthe- lawconstraintD Ei =0,asinthestandardHamiltonian
i
ory.
formulation of 3D YM theory. The only difference is in
If the source-free Lagrangian density (25) is rewritten
the Hamiltonian, which includes additional terms. No-
in terms of C rather than A, the result for µ = 0 is
tice, however, that these are such that the Hamiltonian
particularly simple:
remains manifestly positive.
For the generic case of non-zero µ it is simpler to per-
1 1
g2L=Gµ·H˜µ−2Gµ·Gµ+24m2 εµνρGµ·Gν×Gρ, (35) forma time-spacesplit inthe action(31). Providedthat
m(m−µ)6=0 we can then trivially eliminate (A −A¯ )
0 0
where to get
1
H˜µ =εµνρ(cid:18)∂µCν + 2Cµ×Cν(cid:19) . (36) L = (m2−g2µ)εijAi·A˙j − 2mg2εijA¯i·A¯˙j
1
Parityis manifestly preservedsince C is parity even and + C · mB¯+(m−µ)B −H, (42)
G is parity odd. This action was also given in [7, 9], g2 0
(cid:2) (cid:3)
where it was observed that the field equation for G can
where
besolvedrecursively,yieldinganinfiniteseriesexpansion
in powers of 1/m2: 1
H = m(m−µ) A −A¯ · A −A¯
2g2 i i i i
1 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)
Gµ =H˜µ+ 8m2εµνρH˜ν ×H˜ρ+O m−4 . (37) + 1 mB¯−(m−µ)B 2 . (43)
(cid:0) (cid:1) m(m−µ) (cid:21)
(cid:12) (cid:12)
Wemaythenback-substitutetogetanactionforCalone. (cid:12) (cid:12)
We may also substitute for G in the C equation to get Here,B isdefinedasin(39),andB¯ isthesamebutwith
(for µ = 0) an equation for C in the form of an infinite A¯insteadofA. ThefieldC isagainthetimecomponent
0
series,butthisseriesisnotexplicitlydefinedandwillnot of the parity-even gauge potential C, and it is again a
converge for all values of the dual field-strength H˜. In Lagrange multiplier for an SU(2)-triplet of “first-class”
contrast, our simple equation (12) for A is, in addition constraints,whichgenerateSU(2)gaugetransformations
to being more general, both explicit and defined for all of the canonical variables. Notice that the Hamiltonian
5
is positive only if m(m−µ) > 0, as expected from our ‡ Electronic address: [email protected]
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