Unitarity analysis of a non-Abelian gauge invariant action with a mass
ABSTRACT In previous work done by us and coworkers, we have been able to construct a local, non-Abelian gauge invariant action with a mass parameter, based on the nonlocal gauge invariant mass dimension two operator Fμν(D2)-1Fμν. The renormalizability of the resulting action was proven to all orders of perturbation theory, in the class of linear covariant gauges. We also discussed the perturbative equivalence of the model with ordinary massless Yang-Mills gauge theories when the mass is identically zero. Furthermore, we pointed out the existence of a Becchi-Rouet-Stora-Tyutin (BRST) symmetry with corresponding nilpotent charge. In this paper, we study the issue of unitarity of this massive gauge model. First, we provide a short review how to discuss the unitarity making use of the BRST charge. Afterwards we make a detailed study of the most general version of our action, and we come to the conclusion that the model is not unitary, as we are unable to remove all the negative norm states from the physical spectrum in a consistent way.
- [Show abstract] [Hide abstract]
ABSTRACT: Our goal will be the description of a theory of Gribov's type as a physical process of phase transition in the context of a spontaneous symmetry breaking. We mainly focus at the quantum stability of the whole process.Physical review D: Particles and fields 01/2011; 84. - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: An equivalent formulation of the Gribov-Zwanziger theory accounting for the gauge fixing ambiguity in the Landau gauge is presented. The resulting action is constrained by a Slavnov-Taylor identity stemming from a nilpotent exact BRST invariance which is spontaneously broken due to the presence of the Gribov horizon. This spontaneous symmetry breaking can be described in a purely algebraic way through the introduction of a pair of auxiliary fields which give rise to a set of linearly broken Ward identities. The Goldstone sector turns out to be decoupled. The underlying exact nilpotent BRST invariance allows to employ BRST cohomology tools within the Gribov horizon to identify renormalizable extensions of gauge invariant operators. Using a simple toy model and appropriate Dirac bracket quantization, we discuss the time-evolution invariance of the operator cohomology. We further comment on the unitarity issue in a confining theory, and stress that BRST cohomology alone is not sufficient to ensure unitarity, a fact, although well known, frequently ignored.Physical review D: Particles and fields 05/2012; 86(4). - SourceAvailable from: R. F. Sobreiro[Show abstract] [Hide abstract]
ABSTRACT: We prove that the nonlocal gauge invariant mass dimension 2 operator Fμν(D2)-1Fμν can be consistently added to the Gribov–Zwanziger action, which implements the restriction of the path integral’s domain of integration to the first Gribov region when the Landau gauge is considered. We identify a local polynomial action and prove the renormalizability to all orders of perturbation theory by employing the algebraic renormalization formalism. Furthermore, we also pay attention to the breaking of the BRST invariance, and to the consequences that this has for the Slavnov–Taylor identity.European Physical Journal C 01/2007; 52(2):459-476. · 5.44 Impact Factor
Page 1
arXiv:0705.0871v1 [hep-th] 7 May 2007
Unitarity analysis of a non-Abelian gauge invariant
action with a mass
D. Dudal∗, N. Vandersickel†, H. Verschelde‡
Ghent University
Department of Mathematical Physics and Astronomy
Krijgslaan 281-S9, B-9000 Gent, Belgium
Abstract
In previous work done by us and coworkers, we have been able to construct a local,
non-Abelian gauge invariant action with a mass parameter, based on the nonlocal gauge
invariant mass dimension two operator Fµν(D2)−1Fµν. The renormalizability of the re-
sulting action was proven to all orders of perturbation theory, in the class of linear co-
variant gauges. We also discussed the perturbative equivalence of the model with ordi-
nary massless Yang-Mills gauge theories when the mass is identically zero. Furthermore,
we pointed out the existence of a BRST symmetry with corresponding nilpotent charge.
In this paper, we study the issue of unitarity of this massive gauge model. Firstly, we
provide a short review how to discuss the unitarity making use of the BRST charge. Af-
terwards we make a detailed study of the most general version of our action, and we
come to the conclusion that the model is not unitary, as we are unable to remove all the
negative norm states from the physical spectrum in a consistent way.
1Introduction
In the two previous papers [1, 2], the following action was constructed
Sphys
=
Scl+ Sgf,
?
3
8m2λ1
λabcd
16
?
(1.1)
?
Scl
=
d4x
?
−1
?Ba
Ba
?ξ
4Fa
µνFa
µν+im
4(B − B)a
?+ m2λ3
µνGb
µν
µνFa
µν+1
4
?
Ba
µνDab
σDbc
σBc
µν− Ga
µνDab
σDbc
σGc
µν
−
µνBa
µν− Ga
µνGa
µν
??
µ+ ca∂µDab
32
?Ba
µν− Ba
µν
??
?2
+
?
µνBb
µν− Ga
Bc
ρσBd
ρσ− Gc
?
ρσGd
ρσ
,
(1.2)
Sgf
=
d4x
2baba+ ba∂µAa
µcb
.
(1.3)
∗david.dudal@ugent.be
†nele.vandersickel@ugent.be
‡henri.verschelde@ugent.be
1
Page 2
The bosonic fields Ba
antisymmetric in their Lorentz indices and belong to the adjoint representation. λabcdis a
gauge invariant quartic tensor coupling, subject to a generalized Jacobi identity [3]
µν, its conjugate Ba
µνand the fermionic (ghost) fields Ga
µνand Ga
µνare
fmanλmbcd+ fmbnλamcd+ fmcnλabmd+ fmdnλabcm= 0,
(1.4)
and to the symmetry constraints
λabcd= λcdab,
λabcd= λbacd,
(1.5)
while λ1and λ3are mass couplings1.
To avoid confusion, let us mention here that we shall work in Minkowski space throughout
this paper, since we plan to come to the canonical quantization. In [1, 2], the action was
treated in Euclidean space.
The classical part of the action, Scl, enjoys a non-Abelian gauge invariance generated by
δAa
µ
=
−Dab
gfabcωbBc
gfabcωbBc
µωb,
δBa
δBa
µν
=
µν,
µν
=
µν,
δGa
δGa
µν
=
gfabcωbGc
gfabcωbGc
µν,
µν
=
µν,
(1.6)
with ωaparametrizing an arbitrary infinitesimal SU(N) gauge transformation.
Quite obviously, the gauge model (1.1) did not come out of thin air. Our original motivation
was based on the quest for a dynamical mass generation mechanism in gauge theories. We
do not plan to give a complete overview of this issue, but let us mention that this has been a
research topic since long, see e.g. [4] for a seminal work on this.
Morerecently,workappearedinwhichadynamicalgluonmasswasintroducedphenomeno-
logically based on the QCD sum rules [5]. Such a mass can account for
in certain physical correlators [5, 6, 7]. A natural question arising is where this mass scale
would originate from? The authors of [6, 7] invoked the condensation of the operator
1
Q2power corrections
A2
min= (V T)−1
min
U∈SU(N)
?
d4x?AU
µ
?2,
(1.7)
since it is gauge invariant due to the minimization along the gauge orbit2. As it is well
known, a local gauge invariant dimension two operator does not exist in Yang-Mills gauge
theories. The nonlocality of (1.7) is best seen when it is expressed as a series in Euclidean
space [10]
?
1In comparison with [1, 2], we changed the sign of m, λ3, λ1and λabcdto avoid a number of minus signs.
2One should however be aware of the problem of gauge (Gribov) ambiguities [8, 9] for determiningthe global
minimum.
A2
min=
1
2V T
d4x
?
Aa
µ
?
δµν−∂µ∂ν
∂2
?
Aa
ν− gfabc
?∂ν
∂2∂Aa
??1
∂2∂Ab
?
Ac
ν
?
+ ... (1.8)
2
Page 3
which contains the inverse Laplacian
be immediately inferred from its formal expression in d dimensions through
1
∂2several times. This is a nonlocal operator, as it can
1
∂2
x
f(x) = −
Γ?d
2
?
2π
d
2(d − 2)
?
ddy
f(y)
|x − y|d−2.
(1.9)
All efforts so far were concentrated on the Landau gauge ∂µAµ = 0. The preference for
this particular gauge is obvious since the nonlocal expression(1.8) reduces to an (integrated)
local operator, more precisely
∂µAµ= 0 ⇒ A2
min= (V T)−1
?
d4xA2
µ.
(1.10)
In the case of a local operator like A2
tance expansion, becomes applicable, and consequently a measurement of the soft (infrared)
part ?A2
appearance of
Q2power corrections in (gauge variant) quantities like the gluon propagator
or the strong coupling constant, defined in a particular way, from lattice simulations. Let
us mention that already two decades ago attention was paid to ?A2
applied to the propagators [12]. This condensate ?A2
gluon mass, see e.g. [13].
µ, the Operator Product Expansion (OPE), viz. short dis-
µ?OPEbecomes possible. Such an approach was followed in e.g. [11] by analyzing the
1
µ?OPEwhen the OPE was
µ?OPEcan also be related to an effective
A more direct approach to a determination of ?A2
[14, 15]. In [14], a meaningful effective potential for the condensation of the local composite
operator A2
ishinggluonmass ofa fewhundredMeV was found. Therenormalizability ofthistechnique
was proven to all orders of perturbation theory in [16].
µ? in the Landau gauge was presented in
µwas constructed, giving evidence of ?A2
µ? ?= 0, and as a consequence a nonvan-
Effective gluon masses have found application in phenomonological studies like [17, 18, 19].
Also lattice simulations of the gluon propagator revealed the need for massive parameters,
when the obtained form factors are fitted by means of functional forms [20, 21, 22, 23]. Other
approaches to dynamical gluon masses are e.g. [24, 25]. The estimates of these mass pa-
rameters are grosso modo all in the same ballpark, ranging from a few hundred MeV up to
1.2 GeV .
It is perhaps important to spend a few words at clearing up a common misconception. The
concept of a dynamically generated effective gluon mass does not necessarily entail that we
are considering massive gauge bosons that are belonging to the physical spectrum, i.e. that
are observable particles. At low energies, perturbative QCD expressed in terms of gluons
and quarks completely fails, and the effective degrees of freedom become the hadrons. The
phenomena we are interested in, in casu the study of the condensates and ensuing dynam-
ical mass generation, occur in a energy window located in between perturbative QCD and
the confined region. Perturbation theory still has its validity there, but it gets corrected by
nonperturbative effects like condensates. Due to the lack of an explicit knowledge of the
correct physical degrees of freedom (the hadrons), we continue to use the gluons as effective
degrees of freedom, although we are already out of the energy regime where these might be
considered as asymptotic observables. If we cross from high to low energies, the originally
massless and physical gluons will not stante pede become confined at the confinement scale,
but rather they will behave as a kind of massive quasi particles before getting confined, and
this happens at scales that are phenomenologically relevant. This also means that unitarity
3
Page 4
GaugeOperator
1
2Aa
2Aa
1
2Aβ
1
2Aa
linear covariant
Curci-Ferrari
maximal Abelian
nonlinear class
µAa
µ+ αcaca
µAβ
µAa
µ
1
µAa
µ+ αcβcβ
µ
Table 1: Gauges and their renormalizable dimension two operator
in terms of the gluons is not required or even desired. One expects that quasi particles do
have a finite lifetime and cannot be observed as asymptotically free particles.
We have already explained the preferred role of the Landau gauge, since in that case a gauge
invariant meaning can be assigned to ?A2
theory, the condensates influencing physical quantities should be at least gauge invariant.
Therefore, it would be nice to have a dimension 2 condensate that could also be treated in
other gauges. As the operator A2
minremains nonlocal, it falls beyond the applicability of the
OPE. It is also unclear how e.g. renormalizability or an effective potential approach could
be established for nonlocal operators. In most covariant gauges, we and collaborators have
discussed that other dimension two, renormalizable and local operators exist. We showed
that these operators condense and give rise to a dynamical gluon mass, see Table 1 and
[26, 27, 28, 29, 30, 31, 32, 33]. Quite recently, it has also been shown that a class of nonlin-
ear covariant gauges enjoys the fact that A2
µis multiplicatively renormalizable [34]. In the
maximal Abelian gauge, it was found that only the off-diagonal gluons Aβ
namical mass [31], a fact qualitatively consistent with the lattice results from [22, 23]. Let us
also mention that we have been able to make some connection between the various gauges
and their dimension two operators by constructing renormalizable interpolating gauges and
operators [31, 35]. These can be used to obtain a formal result on the gauge parameter in-
dependence of the nonperturbative vacuum energy due to the condensation, which is lower
than the perturbative (zero) vacuum energy [30].
µ?. Obviously, since we are working in a gauge
µ acquire a dy-
A certain disadvantage of the research so far is the explicit gauge dependence of the used
operator. We started looking for a gauge invariant dimension two operator, which a fortiori
needstobenonlocal. Wewouldlike todevelopaconsistent(calculational) framework, hence
we are almost forced to look for an operator that can be localized by introducing a suitable
set of extra fields. From this perspective, A2
minseems to be rather inadequate as it is a infinite
series of nonlocal terms. A perhaps more appealing operator is [1]
?
Thisoperatorfoundalready useinthestudyofadynamical massgenerationin3-dimensional
gauge theories [36]. When we add the operator O to the Yang-Mills action via
SY M−m2
4
O =
1
V T
d4xFa
µν
??D2?−1?abFb
µν.
(1.11)
?
d4xFa
µν
??D2?−1?abFb
µν,
(1.12)
we can localize it to
SY M+
?
d4x
?im
4
?B − B?a
µνFa
µν+1
4
?
Ba
µνDab
σDbc
σBc
µν− Ga
µνDab
σDbc
σGc
µν
??
,(1.13)
4
Page 5
at the cost of introducing a set of extra fields [1].
The action (1.13) as it stands is however not renormalizable, but we and collaborators have
shown that the generalized version (1.1) is renormalizable to all orders in the class of lin-
ear covariant gauges, implemented through Sgf, in [1, 2]. We have also calculated several
renormalization group functions to two loop order, confirming the renormalizability at the
practical level. Various consistency checks were at our disposal in order to establish the reli-
ability of these results, e.g. the gauge parameter independence of the anomalous dimension
of gauge invariant quantities like g2, λabcdor m, or the equality of others, in accordance with
the output of the Ward identities in [1]. We refer the reader to [1, 2] for all details concern-
ing the localization procedure or renormalizability analysis, as well as the need for the extra
couplings.
Furthermore, we have proven in [2] the perturbative equivalence of the model (1.1) with or-
dinary Yang-Mills theory in the case that m ≡ 0. We notice that this is a nontrivial statement
due to the presence of the quartic interaction ∼ λabcdin the extra fields. It has an interesting
corollary: because we employ a massless renormalization scheme, in casu MS, we can set the
mass m equal to zero to determine the renormalization group functions of e.g. the coupling
constant g2, the gauge parameter ξ or original Yang-Mills fields. Since both theories are per-
turbatively equivalent for m ≡ 0, the already mentioned renormalization group functions
must be identical. This has indeed been confirmed by the explicit results of [1, 2]. In particu-
lar, our model is thus asymptotically free at high energies, with or without a mass. At lower
energies, nonperturbative effects can set in, completely analogous to the Yang-Mills case.
Summarizing, we have thus found a classically gauge invariant action, which at the quan-
tum level can be renormalized to all orders in at least the class of linear covariant gauges,
and as a bonus it is perturbatively equivalent with ordinary Yang-Mills gauge theories for
vanishing mass. We can now ask ourselves two questions:
1. If we treat the mass m as a given classical input, can we consider our model as a candi-
date for a gauge theory with massive excitations? Therefore, we should prove that the
theory is unitary, containing massive particles in a suitably defined asymptotic phys-
ical subspace. The particles correspond to the elementary excitations of the original
fields. As it is well known, proving the unitarity of gauge theories is not a trivial job.
A well known proof in the case of Yang-Mills theories based on the BRST symmetry
[37, 38], is given in [39].
2. If we do not want to treat our model as one with a given classical mass m, can we
dynamically generate it in a selfconsistent way in this case? Said otherwise, can we de-
velop a method to find a reasonable gap equation for this mass? At high energies, the
model is massless and the same as Yang-Mills theory, but it might develop a dynamical
mass scale at lower energies, without spoiling the gauge invariance. We cannot add
mass terms to the Yang-Mills action without spoiling the gauge invariance or renor-
malizability, but we can add mass terms to our model. Just as for Yang-Mills theories,
we expect ourmodel to be confining at lower energies. As we have already mentioned,
the original fields can develop a behaviour different from the one expected from per-
turbation theory in the energy regime in between confinement and the perturbatively
accessible highenergyregion. Forexample, a gaugeinvariant mass parametercouldbe
dynamically generated, thereby modifying the propagators in a nonperturbative fash-
5
Page 6
ion, and this without the need that these describe asymptotically observable physical
particles.
In this paper, we shall provide an answer to the first questionby studying the massive gauge
model (1.1). More precisely, we shall quantize the model canonically to have a clear particle
interpretation of the quantum fields, and we shall find out whether it is possible to define
a physical subspace of states endowed with a positive norm. Naively, one might expect
the model to be unitary, because the action (1.1) enjoys a BRST symmetry, generated by the
nilpotent operator s,
sAa
µ
=
−Dab
g
2fabccacb,
gfabccbBc
gfabccbBc
µcb,
sca
=
sBa
sBa
µν
=
µν,
µν
=
µν,
sGa
sGa
µν
=
gfabccbGc
gfabccbGc
ba,
µν,
µν
sca
sba
s2
=
µν,
=
=0,
=0.
(1.14)
It should not come as a surprise that we shall heavily rely on this BRST symmetry to dis-
cuss the unitarity of the model. The paper is organized as follows. In section 2, we review
how a sensible physical subspace can be defined by using the free BRST charge [40, 41]. As
a warming up exercise, we apply the results of section 2 to the well known canonical quan-
tization of Yang-Mills gauge theories in section 3, before turning to the explicit quantization
of the gauge model (1.1) in section 4. In section 5, we discuss the presence of some extra
symmetries which allow to reduce the physical subspace further. Section 6 is devoted to the
(free) classical equations of motion and the Fourier decomposition of the solutions. We shall
encounter the problem of “multipoles”, since the free equations of motion couple different
fields to each other. This gives rise to higher derivative decoupled equations of motion. We
also pay attention to the BRST charge and Hamiltonian. In section 7, we discuss how to
derive the commutation relations between the creation and annihilation operators, without
using the brackets betweenthe fields and theirconjugate momenta, which we want to avoid,
since not all the Fourier components of the fields and momenta are independent. Once this
is done, we come to the conclusion that the massive gauge model (1.1) is not unitary, as we
end up with negative norm modes in the physical subspace. We are unaware of any step to
further reduce this subspace in a consistent way3to remove these unwanted modes. We end
with some conclusions in section 8.
3That means compatible with the interactions of the model.
6
Page 7
2A constructiveapproach tothequestion of unitarityin gauge the-
ories
In this section, we shall review how we can construct the action S for an interacting gauge
theory, if we have a free theory S0at our disposal, together with a nilpotent symmetry gen-
erator s0, so that s0S0= 0. The content of this section is mainly based on [40, 41], although
here and there we adapted the proofs. We shall only be concerned with the non-reducible
case in this paper.
Let us thus start from the free action S0. This action contains a set of fields, appearing
quadratically. It is given that S0enjoys a BRST symmetry s0, with corresponding nilpo-
tent charge Q0. As a standard example, we can consider the free part of a gauge theory
in a particular gauge, with its corresponding gauge fixing part. For example, in the linear
covariant gauge we have
?
The free BRST symmetry is generated by
S0
=
d4x
?
−1
4
?∂µAa
ν− ∂νAa
µ
?2+ ba∂µAµa+ ca∂2ca+ξ
2baba
?
.
(2.1)
s0Aa
s0ca
s0ca
s0ba
µ
=
−∂µca,
0,
ba,
=
=
=0,
s2
0
=0.
(2.2)
We mention that also the free “ghost part” has to be included in S0. We may define the
ghost charge Qgh. In [40, 41], the ghost charge is not used. We may use it anyhow in the
definition of the physical subspace. However, this requirement is a bit redundant. A BRST
cohomological analysis (see later) will eventually learn that a physical state counts neither
ghosts nor anti-ghosts in the case of Yang-Mills gauge theories.
Unitarity means that we start from a physical state space Hphys, which is a subspace of the
total Hilbert state space H. Hphysshould of course be endowed with a positive norm in order
to have a sensible probabilistic interpretation of the quantum theory. If we let the states
of Hphysinteract, we must end up again in the (same) subspace Hphys. Nonphysical states,
which can have negative norms, may contribute to the S-matrix in internal processes, but
they cannot appear in the observable sector (the “out”-space), unless perhaps in zero norm
combinations.
Consequently, two questions need to be answered:
1. How do we define the physical subspace Hphys?
2. Do the states in the physical subspace Hphyspossess a positive norm?
Let us first explain how we define our physical subspace, starting from the free action. A
state |ψp? is called physical if
|ψp? ∈ Hphys⇔ Q0|ψp? = 0 ,
|ψp? ?= |...? + Q0|...?.
(2.3)
7
Page 8
Physical states are thus defined from the “free” BRST charge Q0. Since Q0is supposed to
be nilpotent, states of the form Q0|...? are trivially annihilated by Q0. We notice that these
have zero norm4. We can identify them with the trivial state, more mathematically speaking
this amounts to consider the Q0cohomology. In the usual terminology, we define Q0-closed
and Q0-exact states by
|ψp? is Q0-closed
|ψp? is Q0-exact
Since Q2
can reexpress the condition (2.3) as
⇔
⇔
Q0|ψp? = 0 ⇔ |ψp? ∈ KerQ0,
|ψp? = Q0|φ? ⇔ |ψp? ∈ ImQ0.
(2.4)
(2.5)
0= 0, every exact state is trivially closed, meaning that ImQ0⊂ KerQ0. Hence, we
|ψp? ∈ Hphys
⇔|ψp? ∈ cohomQ0≡KerQ0
ImQ0
.
(2.6)
For the moment, we leave open the (key) question whether these states |ψp? have a positive
norm.
The next problem is whether we can construct an action S compatible with unitarity? Start-
ing from the free action S0, we can complement it order by order with terms in the coupling
constant(s), so that
S = S0+ S1+ S2+ ... .
(2.7)
ThequestionbecomeshowtodeterminetheinteractiontermsS1, S2, ..., suchthat S describes
a unitary model? More precisely, having defined a physical subspace by means of (2.6), we
would like to construct the action S such that the subspace defined by (2.6) is maintained
under time evolution. In the operator language, we must therefore require that the time
evolution operator S, given by
?
with T the usual time-ordering operation, commutes with the operator Q0. Then clearly the
S-matrix will be unitary, as states evolved w.r.t. S will asymptotically again belong to the
(same) physical subspace Hphys.
In order to solve the previous requirement, we prefer to work in the path integral language
rather than in the operator language. Let us thus rephrase the previous requirement in the
path integral language. From the LSZ reduction formulae, see [42] and [43] for the original
paper, we know that the S-matrix elements5are determined by the (connected) amputated
n-point Green functions, put on-shell. In a rough notation, we can write
S = T
e−iR+∞
−∞Hint(t)dt?
,
(2.8)
??k′
1,...,?k′
m|S|?k1,...,?kn? ∼ ?Θ|T [φmφn]|Θ?,
(2.9)
where ∼ symbolizes all the necessary prefactors, putting it on-shell, amputating and Fourier
transforming to momentum space. φmand φnare certain functionals of the fields, leading to
a (m + n)-point function.
Starting from a generic physical state |?k1,...,?kn? with the property
Q0|?k1,...,?kn? = 0,
(2.10)
4The BRST charge can be chosen to be Hermitian.
5We can restrict ourselves to the connected S-matrix elements.
8
Page 9
we are wondering which condition will assure that
??k′
1,...,?k′
m|Q0S|?k1,...,?kn?
?= ??k′
1,...,?k′
m|SQ0|?k1,...,?kn? = 0,
(2.11)
wherethelastequalityfollowsfrom(2.10). Asit iswellknown,wecanexpresstheT -ordered
product with the path integral, so that we find6
??k′
1,...,?k′
m|Q0S|?k1,...,?kn? ∼ ?Θ|T [(Q0φm)φn]|Θ? =
?
dX (B0φm)φneiS, (2.12)
where X represents all the fields. Now, we can write
?
since B0φn= 0 by virtue of (2.10).
Consider the path integral
dX (B0φm)φneiS
=
?
dXB0(φmφn)eiS,
(2.13)
?
dXΦeiS,
(2.14)
for an arbitrary functional Φ of the fields. We perform the transformation of the path integral
variables
X = X′+ ǫB0X′,
As B0induces a linear transformation, there is no associated Jacobian, and we find
?
Dropping the prime again, we find
?
Taking a look at (2.17), we can be certain that (2.13) holds when we impose
ǫ infinitesimal.
(2.15)
dXΦeiS=
?
dX′Φ′eiS′+ ǫ
?
dX′B0
?
Φ′eiS′?
.
(2.16)
dXB0
?ΦeiS?= 0.
(2.17)
B0eiS= 0.
(2.18)
In order to proceed, we notice that it is in principle sufficient that (2.18) is fulfilled on-shell
as the S-matrix is of course considered on-shell. At the level of the action however, we must
require that it holds off-shell. Let us introduce a (very) condensed notation for the action of
B0
B0= ∆φ0
∂
∂φ.
(2.19)
Implementing (2.18) at lowest order and making it valid off-shell means that
B0S1
=0
⇒
????
off−shell
∆φ0
∂
∂φS1= −∆φ1∂S0
∂φ.
(2.20)
6We shall use the notation Q0for the charge, eventually written in terms of creation/annihilation operators,
while B0represents the functional analog of Q0.
9
Page 10
We already see here that B0will get adapted, more precisely we can introduce the modified
operator B by
B0≡ ∆φ0
∂
∂φ→ B = B0+ ∆φ1
∂
∂φ,
(2.21)
so that
B(S0+ S1) = 0.
(2.22)
We remind here that all Q’s (B’s) are Grassmann operators.
Since B0is nilpotent, we can act with it on (2.20) to find that
0=
B2
0S1= B0
?
−∆φ1
∂
∂φS0
?
=
−∆φ0
∂
∂φ(∆φ1)∂
∂φS0+ ∆φ1∆φ0
∂2
∂φ2S0.
(2.23)
Acting with
∂
∂φon
0 = B0S0= ∆φ0
∂
∂φS0,
(2.24)
yields
∂
∂φ(∆φ0)∂
∂φS0+ ∆φ0
∂2
∂φ2S0= 0.
(2.25)
Combination of (2.23) and (2.25) learns
∆φ0
∂
∂φ(∆φ1)∂
∂φS0+ ∆φ1
∂
∂φ(∆φ0)∂
∂φS0= 0,
(2.26)
from which we infer that
∆φ0
∂
∂φ(∆φ1)∂
∂φ+ ∆φ1
∂
∂φ(∆φ0)∂
∂φ= 0.
(2.27)
The identity (2.27) expresses nothing more than the nilpotency of Q = Q0+ Q1, given by
(2.21) since, at lowest order
B2
= (∆φ0
∂
∂φ+ ∆φ1
∂
∂φ(∆φ0)∂
∂
∂φ(∆φ0)∂
∂
∂φ)(∆φ0
∂
∂φ+ ∆φ1
∂2
∂φ2+ ∆φ0∆φ1
∂
∂φ(∆φ1)∂
∂
∂φ)
=∆φ1
∂φ+ ∆φ1∆φ0
∂2
∂φ2+ ∆φ0
∂
∂φ(∆φ1)∂
∂φ+ HOT
=∆φ1
∂φ+ ∆φ0
∂φ= (2.27) = 0,
(2.28)
where we used for example the nilpotency of B0. We dropped the last term as it is of higher
order.
We notice that the potential solution of (2.20) is apparently restrained by the condition that
it is invariant under a nilpotent operator Q (B), which reduces to Q0(B0) in the free limit.
This construction can be continued at higher order. One proves that the action at order n,
S = S0+ S1+ ... + Sn,
(2.29)
10
Page 11
is the solution of
∆φ0
∂
∂φSn+ ∆φ1
∂
∂φSn−1+ ... + ∆φn
∂
∂φS0= 0,
(2.30)
where consistency demands that the BRST operator,
B = B0+ ... + Bn= ∆φ0
∂
∂φ+ ... + ∆φn
∂
∂φ,
(2.31)
is nilpotent at the considered order n, thus
B2= 0(or Q2= 0).
(2.32)
Byconstruction,thefinal action(2.29)shall beinvariant undertheBRSTsymmetrygenerated
by (2.31).
To make things a bit more comprehensible, let us work out the procedure at second order.
We hence demand that (2.7) is consistent with (2.18), and this extended to the off-shell level,
meaning that
iB0S2−1
2B0(S1S1) = −i?
∆φ2∂S0
∂φ.
(2.33)
The complex unity i in the r.h.s. as well as the ?-notation are merely introduced for later
i∆φ0∂S2
∂φS1= −i?
Next, using Wick’s theorem, we can write7
convenience. Using (2.20), we may rewrite (2.33) as
∂φ+ ∆φ1∂S0
∆φ2∂S0
∂φ.
(2.34)
∆φ1∂S0
∂φS1= i∆φ1∂S1
∂φ+ i?
∆φ1∂S0
∂φ,
(2.35)
since roughly said,∂S0
to a iD−1× D with D a free propagator. All other terms are taken together in?
∆φ0∂S2
∂φ∼ D−1×φ, and a “contraction” of this with a φ from S1will give rise
∆φ1.
Upon taking (2.34) and (2.35) into account, we come to the conclusion that
∂φ+ ∆φ1∂S1
∂φ
= −∆φ2∂S0
∂φ
(2.36)
in order to have the condition (2.33) fulfilled. We defined
∆φ2=?
∆φ2+?
∆φ1.
(2.37)
Analogously at is was proven in (9) to (14), the nilpotency of Q0+Q1leads to the nilpotency
of Q0+ Q1+ Q2as a consistency requirement.
Of course,there is noguaranteethat theforegoing“bottomtop”constructionofthecomplete
action will end at a finite order. Given that it ends at a finite order, it could still be a very
cumbersome job to actually get the nilpotent BRST charge and corresponding action. The
situation becomes much more appealing when we already have at our disposal a complete
7This operation is understood within the path integral.
11
Page 12
action, with a nilpotent charge generating a symmetry. If the interaction is switched off by
setting all coupling constants equal to zero, we obtain the free action, with a free nilpotent
charge. When the above “bottom top” machinery is unleashed, the complete original action
and its BRST symmetry generator shall quite evidently be a solution to the iterative proce-
dure. From this viewpoint, we have a “top bottom” approach to unitarity for actions with
nilpotent BRST charge, when they are “reduced” to their free counterpart.
3Unitarity of Yang-Mills gauge theories using the BRST charge
We should still provide an answer to question 2, namely do the states that are annihilated by
the free BRST charge have a positive norm? It is well known that this is the case for Yang-
Mills gauge theories. For completeness, let us nevertheless repeat the argument. This will
allow for a comparison with Yang-Mills theories when we start analyzing our generalized
model.
We shall base ourselveson [44] forthis particular job8. We optto work in the Feynmangauge
for simplicity (ξ = 1 in (2.1)). Let us first determine the conjugate momenta of all fields.
πa
i
=
−Fa
ba,
∂0ca,
−∂0ca,
0i,
π0,a
πa
πa
=
c
=
c
=
(3.1)
so that quantization requires
?
Aa
?
Aa
i(? x,t),πb
j(? y,t)
?
=
iδabgijδ(3)(? x − ? y),
iδabδ(3)(? x − ? y),
iδabδ(3)(? x − ? y),
iδabδ(3)(? x − ? y),
?
0(? x,t),π0,b(? y,t)
?
=
ca(? x,t),πb
?
c(? y,t)
?
=
ca(? x,t),πb
c(? y,t)
?
=
other (anti-)commutators trivial.
(3.2)
We mention that the classical equations of motion are
∂2Aa
∂2ca
µ
=0,
∂2ca= 0,
−∂µAa
=
ba
=
µ,
(3.3)
and that we use the hermiticity assignment
c†
=
c,c†= −c.
(3.4)
8We shall however use other conventions than those of [44].
12
Page 13
We propose the following Fourier decompositions9
A0(x)=
?
?
d3k
(2π)3
1
2ωk
?
?
?
?
?
a0(k)e−ikx+ a†
0(k)eikx?
i(?k)e−ikx+ a†
,
Ai(x)=
d3k
(2π)3
1
2ωk
3
m=1
?
am(k)εm
m(k)εm
i(?k)eikx?
3(k))eikx?
,
b(x) ≡ π0(x)=
i
?
?
?
d3k
(2π)3
d3k
(2π)3
d3k
(2π)3
1
2
1
(a0(k) − a3(k))e−ikx− (a†
η(k)e−ikx+ η†(k)eikx?
η(k)e−ikx− η†(k)eikx?
(?k)formanorthonormalset,ε(m)
0(k) − a†
,
c(x)=
2ωk
1
2ωk
,
c(x)=
.
(3.5)
Thepolarization vectorsε(m)
ki
|?k|. We shall assume that the particles move along the z-axis, so that εj
ii
(?k)ε(n)
i
(?k) = δmnwithε(3)
i= δij.
i(?k) =
Implementing (3.2), we must require the following (anti-)commutation rules
[a0(k),a0(q)]
[am(k),an(q)]
?
η(k),η†(q)
=
−2(2π)3ωkδ(3)(?k − ? q),
2(2π)3ωkδ(3)(?k − ? q)δmn,
−2(2π)3ωkδ(3)(?k − ? q),
−2(2π)3ωkδ(3)(?k − ? q) .
=
η(k),η†(q)
?
=
?
?
=
(3.6)
For later use, let us already introduce the operator [44]
?
which counts the unphysical modes.
N =
d3k
(2π)3
1
2ωk
?
−a†
0(k)a0(k) + a†
3(k)a3(k) − η†(k)η(k) − η†(k)η(k)
?
,
(3.7)
Weare nowreadytoexpresstheBRSTcharge intermsofthecreation/annihilation operators.
The BRST Noether current is given by
Jµ
0
=
Fµν,a∂νca− ba∂µca,
(3.8)
which leads to the charge
Q0
=
?
d3x?ca∂0ba− ba∂0ca?,
(3.9)
where use was made of the classical equation of motion
∂µFµν,a= ∂νba.
(3.10)
After substitution of (3.5) in (3.9), the BRST charge is expressed as
?
9We suppressed the global color indices.
Q0
=
d3k
(2π)3
??
a†
0(k) − a†
3(k)
?
η(k) + η†(k)
?
a0(k) − a3(k)
??
.
(3.11)
13
Page 14
If we define
R =1
2
?
d3k
(2π)3
1
4ω2
k
??
a†
0(k) + a†
3(k)
?
η(k) + η†(k)(a0(k) + a3(k))
?
,
(3.12)
then a little algebra yields
N =
?
Q0,R
?
.
(3.13)
The fact that the “nonphysical” counting operator N is BRST exact is a very powerful result
[44]. Assume that |ψp? is constrained by
Q0|ψp? = 0,
and that it contains n ?= 0 unphysical modes, i.e.
N|ψp? = n|ψp?,
then consequently
(3.14)
(3.15)
|ψp?
=
N
n|ψp? =1
n(Q0R + RQ0)|ψp? = Q0
?1
nR|ψp?
?
,
(3.16)
meaning that a state |ψp? annihilated by the BRST charge and containing nonphysical modes
is a fortiori BRST exact, and hence it is zero in the physical cohomology. Said otherwise,
physical states do not contain unphysical modes. The physical subspace Hphysof Yang-Mills
gauge theories does only contain the 2 transverse polarizations of the gauge field, whereas
the scalar and longitudinal polarizations cancel with the ghost degrees of freedom.
In [40], a different proof was presented of the fact that a state annihilated by a BRST charge
Q0of the form (3.11) containing nonphysical modes, must have zero norm. However, the
cohomological approach used in e.g. [44] is somewhat more elegant.
4 Application to the massive gauge model: preliminary remarks
Setting the couplings g and λabcdequal to zero in (1.1), we are considering the quadratic
action
?
+
4
m2λ3
32
S0
=
d4x
?
−1
4
?∂µAa
ν− ∂νAa
µν∂2Gµνa?−3
µν− Ba
µ
?2+im
4(B − B)a
µν(∂µAνa− ∂νAµa)
?Ba
2baba
1
?Ba
µν∂2Bµν,a− Ga
?Ba
8m2λ1
µνBµνa− Ga
?
µνGµνa?
+
µν
?2+ ba∂µAµa+ ca∂2ca+ξ
(4.1)
This action enjoys the free BRST symmetry generated by
s0Aa
s0ca
s0Ba
s0ca
s0ba
µ
=
−∂µca,
0,
s0Ba
ba,
=
µν
=
µν= s0Ga
µν= s0Ga
µν= 0,
=
=0,
(4.2)
14
Page 15
where clearly
s2
0= 0.
(4.3)
We can hence apply the results of section 2 to the action (1.1). The only thing left to prove
is that there exist a physical subspace with positive norm. This subspace is certainly anni-
hilated by the (free) BRST charge, but nothing prevents us from using other available sym-
metries to further reduce the physical subspace. In the next section, we shall introduce 2
extra symmetries with nilpotent generator of the complete action (1.1). We first determine
the BRST charge in functional form. We shall see that it remains unchanged compared to the
Yang-Mills case (3.9). The Noether current corresponding to the BRST transformation (4.2)
and action (4.1) is given by
Jµ
0
=
Fµν,a∂νca− ba∂µca−im
2
?B − B?µν,a∂νca,
(4.4)
which leads to the BRST charge
Q0
=
?
?
d3x
?
F0i∂ica− ba∂0ca−im
d3x?ca∂0ba− ba∂0ca?,
2
?B − B?0i,a∂ica
?
=
(4.5)
where we invoked the equation of motion
∂µFµν= ∂νba+im
2∂µ
?B − B?µν,a.
(4.6)
For what concerns the Faddeev-Popov ghosts c and c, it is immediately seen from the action
(4.1) that their quantization remains unchanged compared to the Yang-Mills case, given in
(3.1), (3.5) and (3.6). Therefore, since (4.5) must be time independent as a conserved charge,
we already infer that Q0will only act nontrivially on massless excitations. This shall be
confirmed later once we have found the excitations belonging to the ba-field (see section 6).
5A further reduction of the physical subspace
In the following sections, we shall make use of a cohomological result [45], summarized
here.
Doublet theorem Consider a transformation δ with the property that
δui= vi,
δvi= 0,
δu′
δv′
i= v′
i= 0,
i,
(5.1)
with ui,v′
doublets.
icommuting and u′
i,vianticommuting quantities. We call (ui,vi) and (u′
i,v′
i) δ-
Then it is a trivial exercise to show that δ is a nilpotent transformation. Moreover, (ui,vi)
and (u′
i) appear trivially in the δ-cohomology. This can be proven [45] by introducing the
“counting” operator
?
i,v′
P =
d4x
?
ui
δ
δui
+ vi
δ
δvi
+ u′
i
δ
δu′
i
+ v′
i
δ
δv′
i
?
,
(5.2)
15
View other sources
Hide other sources
- Available from Nele Vandersickel · May 27, 2014
- Available from ArXiv