Off-diagonal mass generation for Yang-Mills theories in the maximal Abelian gauge

0.28 MB·English

by  D. Dudal

LTH-645
Off-diagonal mass generation for Yang-Mills theories in the maximal Abelian gauge
D. Dudala,∗ J.A. Graceyb,† V.E.R. Lemesc,‡ M.S. Sarandyd,§ R.F. Sobreiroc,¶ S.P. Sorellac,∗∗ and H. Verscheldea††
a Ghent University
Department of Mathematical Physics and Astronomy
Krijgslaan 281-S9
B-9000 Gent, Belgium
b Theoretical Physics Division
Department of Mathematical Sciences
University of Liverpool
P.O. Box 147, Liverpool, L69 3BX, United Kingdom
c UERJ - Universidade do Estado do Rio de Janeiro
5 Rua S˜ao Francisco Xavier 524, 20550-013 Maracan˜a
0 Rio de Janeiro, Brazil
0
2 d Chemical Physics Theory Group, Department of Chemistry,

n University of Toronto, 80 St. George Street, Toronto, Ontario, M5S 3H6, Canada
a
J Weinvestigateadynamicalmassgeneration mechanismfor theoff-diagonalgluonsandghosts in
8 SU(N) Yang-Mills theories, quantized in the maximal Abelian gauge. Such a mass can be seen as
2 evidence for the Abelian dominance in that gauge. It originates from the condensation of a mixed
gluon-ghost operator of mass dimension two, which lowers the vacuum energy. We construct an

1 effective potential for this operator by a combined use of the local composite operators technique
v withalgebraicrenormalizationandwediscussthegaugeparameterindependenceoftheresults. We
7 alsoshowthatitispossibletoconnectthevacuumenergy,duetothemassdimensiontwocondensate
2 discussed here, with the non-trivial vacuum energy originating from the condensate A2 , which
µ
2 has attracted muchattention in theLandau gauge.
1 (cid:10) (cid:11)
0 PACSnumbers: 11.10.Gh,12.38.Lg
5
0
/
h
t
- I. INTRODUCTION.
p
e
h An unresolved problem of SU(N) Yang-Mills theory is color confinement. A physical picture that might explain
: confinement is based on the mechanism of the dual superconductivity [1, 2], according to which the low energy
v
i regime of QCD should be described by an effective Abelian theory in the presence of magnetic monopoles. These
X
monopoles should condense, giving rise to the formation of flux tubes which confine the chromoelectric charges.
r
a
Let us provide a very short overview of the concept of Abelian gauges, which are useful in the search for
magnetic monopoles, a crucial ingredient in the dual superconductivity picture.
Abelian gauges.
We recall that SU(N) has a U(1)N−1 subgroup, consisting of the diagonal generators. In [2], ’t Hooft proposed the
idea of the Abelian gauges. Consider a quantity X(x), transforming in the adjoint representation of SU(N).
X(x) U(x)X(x)U+(x) with U(x) SU(N). (1)
→ ∈
∗TalkgivenbyD.Dudalat“XXVEncontroNacionaldeF´isicadePart´iculaseCampos”,Caxambu,MinasGerais,Brasil,24-28Aug2004;
ResearchAssistantoftheFundForScientificResearch-Flanders(Belgium);Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected]
¶Electronicaddress: [email protected]
∗∗Work supported by FAPERJ, Fundac¸a˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro, under the program Cientista do Nosso
Estado, E-26/151.947/2004.; Electronicaddress: [email protected]
††Electronicaddress: [email protected]
2
FIG. 1: A chromoelectric flux tubebetween a static quark-antiquarkpair.
The transformation U(x) which diagonalizes X(x) is the one that defines the gauge. If X(x) is already diagonal,
then clearly X(x) remains diagonal under the action of the U(1)N−1 subgroup. Hence, the gauge is only partially
fixed because there is a residual Abelian gauge freedom.
In certain space time points x , the eigenvalues of X(x) can coincide, so that U(x ) becomes singular. These
i i
possible singularities give rise to the concept of (Abelian) magnetic monopoles. They have a topological meaning
since π SU(N)/U(1)N−1 =0 and we refer to [3, 4] for all the necessary details.
2
6
(cid:0) (cid:1)
The dual superconductor as a mechanism behind confinement.
Let us give a simplified picture of the dual superconductor to explain the idea. If the QCD vacuum contains
monopoles and if these monopoles condense, there will be a dual Meissner effect which squeezes the chromoelectric
field into a thin flux tube. This results in a linearly rising potential, V(r) = σr, between static charges, as can
be guessed from Gauss’ law, EdS = cte or, since the main contribution is coming from the flux tube, one finds
E∆S cte, hence V = Edr cte r
≈ − R ≈ ×
R
An example of an Abelian gauge: the maximal Abelian gauge (MAG).
Let A be the Lie algebra valued connection for the gauge group SU(N), whose generators TA, satis-
µ
fying TA,TB = fABCTC, are chosen to be antihermitean and to obey the orthonormality condition
Tr TATB = T δAB, with A,B,C = 1,..., N2 1 . In the case of SU(N), one has T = 1. We de-
(cid:2) (cid:3)− F − F 2
compose the gauge field into its off-diagonal and diagonal parts, namely
(cid:0) (cid:1) (cid:0) (cid:1)
A =AATA =AaTa+AiTi, (2)
µ µ µ µ
where the indices i, j, ...label the N 1 generatorsof the Cartansubalgebra. The remaining N(N 1) off-diagonal
− −
generators will be labeled by the indices a, b, .... The field strength decomposes as
F =FATA =Fa Ta+Fi Ti , (3)
µν µν µν µν
with the off-diagonal and diagonal parts given respectively by
Fa = DabAb DabAb +gfabcAbAc , (4)
µν µ ν − ν µ µ ν
Fi = ∂ Ai ∂ Ai +gfabiAaAb ,
µν µ ν − ν µ µ ν
where the covariant derivative Dab is defined with respect to the diagonal components Ai
µ µ
Dab ∂ δab gfabiAi . (5)
µ ≡ µ − µ
For the Yang-Mills action one obtains
1
S = d4x Fa Fµνa+Fi Fµνi . (6)
YM −4 µν µν
Z
(cid:0) (cid:1)
The maximal Abelian gauge (MAG), introduced in [2, 3, 4], corresponds to minimizing the functional
[A]= d4x AaAµa (7)
R µ
Z
(cid:2) (cid:3)
3
One checks that [A] does exhibit a residual U(1)N−1 invariance.
R
The MAG can be recast into a differential form
DabAµb =0 (8)
µ
Although we have introduced the MAG here in a functional way, it is worth mentioning that the MAG does
correspond to the diagonalization of a certain adjoint operator, see e.g. [5].
The renormalizability in the continuum of the MAG was proven in [6, 7], at the cost of introducing a quartic
ghost interaction. The corresponding gauge fixing term turns out to be [6, 7]
α α α
S =s d4x ca DabAbµ+ ba gfabicacbci gfabccacbcc , (9)
MAG µ 2 − 2 − 4
Z (cid:16) (cid:16) (cid:17) (cid:17)
where α is the MAG gauge parameter and s denotes the nilpotent BRST operator, acting as
sAa = Dabcb+gfabcAbcc+gfabiAbci , sAi = ∂ ci+gfiabAacb ,
µ − µ µ µ µ − µ µ
g g
sca = gf(cid:0)abicbci+ fabccbcc, (cid:1) sci = f(cid:0)iabcacb, (cid:1)
2 2
sca = ba , sci =bi ,
sba = 0, sbi =0. (10)
Hereca,ci aretheoff-diagonalandthediagonalcomponentsoftheFaddeev-Popovghostfield,whileca,ba arethe off-
diagonalantighostandLagrangemultiplier. WealsoobservethattheBRSTtransformations(10)havebeenobtained
by their standard form upon projection on the off-diagonal and diagonal components of the fields. We remark that
the MAG (9) can be written in the form
1 α
S =ss d4x AaAµa caca , (11)
MAG 2 µ − 2
Z (cid:18) (cid:19)
with s being the nilpotent anti-BRST transformation, acting as
sAa = Dabcb+gfabcAbcc+gfabiAbci , sAi = ∂ ci+gfiabAacb ,
µ − µ µ µ µ − µ µ
g g
sca = gf(cid:0)abicbci+ fabccbcc, (cid:1) sci = f(cid:0)iabcacb, (cid:1)
2 2
sca = ba+gfabccbcc+gfabicbci+gfabicbci , sci = bi+gfibccbcc ,
− −
sba = gfabcbbcc gfabibbci+gfabicbbi sbi = gfibcbbcc . (12)
− − −
It can be checked that s and s anticommute.
Expression (9) is easily worked out and yields
α
S = d4x ba DabAµb+ ba +caDabDµbccc+gcafabi DbcAµc ci+gcaDab fbcdAµccd
MAG µ 2 µ µ µ
Zαgfabi(cid:16)bacb(cid:16)ci g2fabifcdic(cid:17)acdAbAµc αgfabcbacbcc (cid:0) αg2fab(cid:1)ifcdicacbcccd(cid:0) (cid:1)
− − µ − 2 − 4
α α
g2fabcfadicbcccdci g2fabcfadecbcccdce . (13)
− 4 − 8
(cid:17)
We note that α =0 does in fact correspond to the “real” MAG condition, given by eq.(8). However, one cannot set
α = 0 from the beginning since this would lead to a nonrenormalizable gauge. Some of the terms proportional to
α would reappear due to radiative corrections, even if α = 0. See, for example, [30]. For our purposes, this means
thatwehavetokeepαgeneralthroughoutandleavetotheendtheanalysisofthelimitα 0,torecovercondition(8).
→
In order to have a complete quantization of the theory, one has to fix the residual Abelian gauge freedom by
means of a suitable further gauge condition on the diagonal components Ai of the gauge field. A common choice for
µ
the Abelian gauge fixing, also adopted in the lattice papers [5, 8], is the Landau gauge, given by
S =s d4x ci∂ Aµi = d4x bi∂ Aµi+ci∂µ ∂ ci+gfiabAacb , (14)
diag µ µ µ µ
Z Z
(cid:0) (cid:0) (cid:1)(cid:1)
4
where ci,bi are the diagonal antighost and Lagrange multiplier.
Abelian dominance.
According to the concept of Abelian dominance, the low energy regime of QCD can be expressed solely in terms of
Abelian degrees of freedom [9]. Lattice confirmations of the Abelian dominance can be found in [10, 11]. To our
knowledge, there is no analytic proof of the Abelian dominance. Nevertheless, an argument that can be interpreted
as evidence of it, is the fact that the off-diagonalgluons would attain a dynamical mass. At energies below the scale
set by this mass, the off-diagonal gluons should decouple, and in this way one should end up with an Abelian theory
at low energies.
A lattice study of such an off-diagonal gluon mass reported a value of approximately 1.2GeV [5]. More re-
cently, the off-diagonalgluon propagator was investigated numerically in [8], reporting a similar result.
There have been several efforts to give an analytic description of the mechanism responsible for the dynami-
cal generation of the off-diagonal gluon mass. In [12, 13], a certain ghost condensate was used to construct an
effective, off-diagonal mass. However, in [14] it was shown that the obtained mass was a tachyonic one, a fact
confirmedlaterin[15]. Anothercondensation,namelythatofthemixedgluon-ghostoperator(1AaAµa+αcaca)[39],
2 µ
that could be responsible for the off-diagonalmass, was proposed in [16]. That this operator should condense can be
expected on the basis of a close analogy existing between the MAG and the renormalizable nonlinear Curci-Ferrari
gauge [17, 18]. In fact, it turns out that the mixed gluon-ghost operator can be introduced also in the Curci-Ferrari
gauge. Adetailedanalysisofits condensationandofthe ensuing dynamicalmassgenerationcanbe foundin[19, 20].
Here, we shall report on the results of [21]. It was investigated explicitly if the mass dimension two operator
(1AaAµa + αcaca) condenses, so that a dynamical off-diagonal mass is generated in the MAG. The pathway we
2 µ
intend to follow is based on previous research in this direction in other gauges. In [22], the local composite operator
(LCO) technique was used to construct a renormalizable effective potential for the operator AAAµA in the Landau
µ
gauge. As a consequence of AAAµA =0, a dynamical mass parameter is generated [22]. The condensate AAAµA
µ 6 µ
has attracted attention from theoretical [23, 24] as well as from the lattice side [25]. It was shown by means of the
algebraic renormalization te(cid:10)chnique (cid:11)[26] that the LCO formalism for the condensate AAAµA is renorma(cid:10)lizable to(cid:11)
µ
all orders of perturbation theory [27]. The same formalism was successfully employed to study the condensation of
(1AAAµA+αcAcA) in the Curci-Ferrarigauge [19, 20]. We would like to note that the(cid:10)Landau(cid:11)gauge correspondsto
2 µ
α=0. Lateron,the condensationofAAAµA wasconfirmedinthe linear covariantgauges[28, 29], whichalsopossess
µ
the Landau gauge as a special case. It was proven formally that the vacuum energy does not depend on the gauge
parameterin these gauges. As such, the linear,Curci-FerrariandLandau gaugesare allconnected to eachother. We
managedto connectalsothe MAG withthe Landaugauge,andas suchwith the linearandCurci-Ferrarigauges[21].
II. RENORMALIZABILITY OF SU(N) YANG-MILLS THEORIES IN THE MAG IN THE PRESENCE
OF THE LOCAL COMPOSITE OPERATOR (1AaAµa+αcaca) .
2 µ
To prove the renormalizability to all orders of perturbation theory, we shall rely on the algebraic renormalization
formalism [26]. In order to write down a suitable set of Ward identities, we first introduce external fields Ωµi, Ωµa,
Li, La coupled to the BRST nonlinear variations of the fields, namely
S = d4x Ωµa Dabcb+gfabcAbcc+gfabiAbci Ωµi ∂ ci+gfiabAacb
ext − µ µ µ − µ µ
Z
+ La gf(cid:0)abicbci(cid:0)+ gfabccbcc +Lig fiabcacb ,(cid:1) (cid:0) (cid:1) (15)
2 2
(cid:16) (cid:17) (cid:17)
with
sΩµa = sΩµi =0, (16)
sLa = sLi =0.
Moreover,in order to discuss the renormalizability of the gluon-ghost operator
1
= AaAµa+αcaca , (17)
OMAG 2 µ
5
we introduce it in the starting action by means of a BRST doublet of external sources (J,λ)
sλ=J , sJ =0, (18)
so that
1 λJ
S = s d4x λ AaAµa+αcaca +ζ (19)
LCO 2 µ 2
Z (cid:18) (cid:18) (cid:19) (cid:19)
1 J2
= d4x J AaAµa+αcaca +ζ αλbaca
2 µ 2 −
Z (cid:18) (cid:18) (cid:19)
g
+ λAµa Dabcb+gfabiAbci +αλca gfabicbci+ fabccbcc ,
µ µ 2
(cid:0) (cid:1) (cid:16) (cid:17)(cid:17)
where ζ is the LCO parameter accounting for the divergences present in the vacuum correlator (x) (y) ,
MAG MAG
which are proportional to J2. Therefore, the complete action hO O i
Σ=S +S +S +S +S , (20)
YM MAG diag ext LCO
is BRST invariant
sΣ=0. (21)
As noticed in [16, 31], the gluon-ghost mass operator defined in eq.(17) is BRST invariant on-shell. We have written
downin[21]allthe Wardidentities,whicharesufficienttoprovethatthe mostgenerallocalcounterterm,compatible
with the symmetries of the model, can always be reabsorbed by means of multiplicative renormalization. As an
interestingby-product,wehavebeenabletoestablisharelationbetweenthe anomalousdimensionofthe gluon-ghost
operator and other, more elementary, renormalization group functions. Explicitly, it holds to all orders of
MAG
O
perturbation theory that
β(g2)
γ (g2)= 2 γ (g2) , (22)
OMAG − 2g2 − ci
(cid:18) (cid:19)
where β(g2)=µ∂g2 and γ (g2) denotes the anomalous dimension of the diagonal ghost field.
∂µ ci
A. The effective potential via the LCO method.
We present here the main steps in the construction of the effective potential for a local composite operator. A
more detailed account of the LCO formalism can be found in [32, 33].
To obtain the effective potential for the condensate , we set the sources Ωi, Ωa, La, Li and λ to
hOMAGi µ µ
zero and consider the renormalized generating functional
exp( i (J)) = [Dϕ]expiS(J),
− W
Z
1 J2
S(J) = S +S +S +S + d4x Z J Z AaAµa+Z Z αcaca +(ζ+δζ) ,(23)
YM MAG diag count J 2 A µ α c 2
Z (cid:18) (cid:18) (cid:19) (cid:19)
where ϕ denotes the relevant fields and S is the usual counterteerm contributioen, i.e. the part without the
count
composite operator. The quantity δζ is the counterterm accounting for the divergences proportional to J2. Using
dimensional regularizationthroughout with the convention that d=4 ε, one has the following identification
−
ζ J2 =µ−ε(ζ +δζ)J2 . (24)
0 0
The functional (J) obeys the renormalizationgroup equation (RGE)
W
∂ ∂ ∂ δ ∂
µ +β(g2) +αγ (g2) γ (g2) d4xJ +η(g2,ζ) (J)=0, (25)
∂µ ∂g2 α ∂α − OMAG δJ ∂ζ W
(cid:18) Z (cid:19)
6
where
∂
γ (g2) = µ lnα,
α
∂µ
∂
η(g2,ζ) = µ ζ . (26)
∂µ
¿From eq.(24), one finds
η(g2,ζ)=2γ (g2)ζ +δ(g2,α), (27)
OMAG
with
∂ ∂
δ(g2,α)= ε+2γ (g2) β(g2) αγ (g2) δζ . (28)
OMAG − ∂g2 − α ∂α
(cid:18) (cid:19)
Up to now, the LCO parameter ζ is still an arbitrary coupling. As explained in [32, 33], simply setting ζ =0 would
give rise to an inhomogeneous RGE for (J)
W
∂ ∂ ∂ δ J2
µ +β(g2) +αγ (g2) γ (g2) d4xJ (J)=δ(g2,α) d4x , (29)
∂µ ∂g2 α ∂α − OMAG δJ W 2
(cid:18) Z (cid:19) Z
and a non-linear RGE for the associated effective action Γ for the composite operator . Furthermore, mul-
MAG
O
tiplicative renormalizability is lost and by varying the value of δζ, minima of the effective action can change into
maxima or can get lost. However, ζ can be made such a function of g2 and α so that, if g2 runs according to β(g2)
and α according to γ (g2), ζ(g2,α) will run according to its RGE (27). This is accomplished by setting ζ equal to
α
the solution of the differential equation
∂ ∂
β(g2) +αγ (g2,α) ζ(g2,α)=2γ (g2)ζ(g2,α)+δ(g2,α). (30)
∂g2 α ∂α OMAG
(cid:18) (cid:19)
Doing so, (J) obeys the homogeneous renormalization group equation
W
∂ ∂ ∂ δ
µ +β(g2) +αγ (g2) γ (g2) d4xJ (J)=0. (31)
∂µ ∂g2 α ∂α − OMAG δJ W
(cid:18) Z (cid:19)
To lighten the notation, we will drop the renormalization factors from now on. One will notice that there are terms
quadraticinthe sourceJ presentin (J), obscuringthe usualenergyinterpretation. This canbe curedby removing
the terms proportional to J2 in theWaction to get a generating functional that is linear in the source, a goal easily
achieved by inserting the following unity,
1 1 σ 2
1= [Dσ]exp i d4x ζJ , (32)
MAG
N Z " Z −2ζ (cid:18)g −O − (cid:19) !#
with N the appropriate normalization factor, in eq.(23) to arriveat the Lagrangian
1 1 σ2 1 1
(A ,σ)= Fa Fµνa Fi Fµνi+ + + gσ ( )2 , (33)
L µ −4 µν − 4 µν LMAG Ldiag− 2g2ζ g2ζ OMAG− 2ζ OMAG
while
exp( i (J)) = [Dϕ]expiS (J), (34)
σ
− W
Z
σ
S (J) = d4x (A ,σ)+J . (35)
σ µ
L g
Z (cid:18) (cid:19)
¿From eqs.(23) and (34), one has the following simple relation
δ (J) σ
W = = , (36)
MAG
δJ −hO i − g
(cid:12)J=0 (cid:28) (cid:29)
(cid:12)
meaning that the condensate MAG is direct(cid:12)ly related to the expectation value of the field σ, evaluated with the
actionS = d4x (A ,σ). AshOit is obivious from(cid:12) eq.(33), σ =0 is sufficient to have a tree leveldynamical mass for
σ µ
L h i6
the off-diagonal fields. At lowest order (i.e. tree level), one finds
R
gσ gσ
moff−diag. = , moff−diag. = α . (37)
gluon ζ ghost ζ
r 0 r 0
Meanwhile, the diagonal degrees of freedom remain massless.
7
III. GAUGE PARAMETER INDEPENDENCE OF THE VACUUM ENERGY.
Webeginthissectionwithafewremarksonthedeterminationofζ(g2,α). Fromexplicitcalculationsinperturbation
theory, it will become clear [40] that the RGE functions showing up in the differential equation (30) look like
β(g2) = εg2 2 β g2+β g2+ ,
0 1
− − ···
γOMAG(g2) = γ0(α)g2+(cid:0)γ1(α)g4+··· , (cid:1)
γ (g2) = a (α)g2+a (α)g4+ ,
α 0 1
···
δ(g2,α) = δ (α)+δ (α)g2+ . (38)
0 1
···
As such, eq.(30) can be solved by expanding ζ(g2,α) in a Laurent series in g2,
ζ (α)
ζ(g2,α)= 0 +ζ (α)+ζ (α)g2+ . (39)
g2 1 2 ···
More precisely, for the first coefficients ζ , ζ of the expression (39), one obtains
0 1
∂ζ
0
2β ζ +αa = 2γ ζ +δ ,
0 0 0 0 0 0
∂α
∂ζ ∂ζ
1 0
2β ζ +αa +αa = 2γ ζ +2γ ζ +δ . (40)
1 0 0 1 0 1 1 0 1
∂α ∂α
Noticethat,inordertoconstructthen-loopeffectivepotential,knowledgeofthe(n+1)-loopRGEfunctionsisneeded.
The effective potential calculated with the Lagrangian (33) will explicitly depend on the gauge parameter α.
The question arises concerning the vacuum energy E , (i.e. the effective potential evaluated at its minimum); will
vac
it be independent of the choice of α? Also, as it can be seen from the equations (40), each ζ (α) is determined
i
through a first order differential equation in α. Firstly, one has to solve for ζ (α). This will introduce one arbitrary
0
integration constant C . Using the obtained value for ζ (α), one can consequently solve the first order differential
0 0
equation for ζ (α). This will introduce a second integration constant C , etc. In principle, it is possible that these
1 1
arbitrary constants influence the vacuum energy, which would represent an unpleasant feature. Notice that the
differential equations in α for the ζ are due to the running of α in eq.(30), encoded in the renormalization group
i
function γ (g2). Assume that we wouldhavealreadyshownthatE does notdepend onthe choice ofα. If we then
α vac
set α = α∗, with α∗ a fixed point of the RGE for α at the considered order of perturbation theory, then equation
(30) determining ζ simplifies to
∂
β(g2) ζ(g2,α∗)=2γ (g2)ζ(g2,α∗)+δ(g2,α∗), (41)
∂g2 OMAG
since
γ (g2)α =0. (42)
α α=α∗
This will lead to simple algebraic equations for the ζ (α∗(cid:12)). Hence, no integration constants will enter the final result
i (cid:12)
for the vacuum energy for α = α∗, and since E does not depend on α, E will never depend on the integration
vac vac
constants,evenwhencalculatedforageneralα. Hence,wecanputthemequaltozerofromthebeginningforsimplicity.
Summarizing, two questions remain. Firstly, we should prove that the value of α will not influence the ob-
tained value for E . Secondly, we should show that there exists a fixed point α∗. We postpone the discussion
vac
concerning the second question to the next section, giving a positive answer to the first one. In order to do so, let
us reconsider the generating functional (34). We have the following identification, ignoring the overall normalization
factors
2
1 1 σ
exp( i (J))= [Dϕ]expiS (J)= [DϕDσ]expi S(J)+ d4x ζJ , (43)
σ MAG
− W Z N Z " Z −2ζ (cid:18)g −O − (cid:19) !#
where S(J) and S (J) are given respectively by eq.(23), and eq.(35). Obviously,
σ
d 1 1 σ 2 d
[Dσ]exp i d4x ζJ = 1=0, (44)
MAG
dαN Z " Z −2ζ (cid:18)g −O − (cid:19) !# dα
8
so that
d (J) 1
W = s d4xs caca +terms J , (45)
dα − 2 ∝
(cid:28) Z (cid:18) (cid:19)(cid:29)(cid:12)J=0
(cid:12)
(cid:12)
which follows directly from (cid:12)
dS(J) 1
=ss d4x caca +terms J . (46)
dα 2 ∝
Z (cid:18) (cid:19)
We see thatthe firsttermin the righthand side of(46) is anexactBRST variation. As such,its vacuumexpectation
value vanishes. This is the usual argument to prove the gauge parameter independence in the BRST framework
[26]. Note that no local operator ˆ, with sˆ = , exists. Furthermore, extending the action of the BRST
MAG
O O O
transformation on the σ-field by
α
sσ =gs = AµaDabcb+αbaca αgfabicacbci gfabccacbcc (47)
OMAG − µ − − 2
one can easily check that
s d4x (A ,σ)=0, (48)
µ
L
Z
so that we have a BRST invariant σ-action. Thus, when we consider the vacuum, corresponding to J = 0, only the
BRST exact term in eq.(45) survives. The effective action Γ is related to (J) through a Legendre transformation
W
Γ σ = (J) d4yJ(y)σ(y), while the effective potential V(σ) is defined as
g −W − g
(cid:16) (cid:17) R
σ
V(σ) d4x=Γ . (49)
− g
Z (cid:18) (cid:19)
If σ is the solution of dV(σ) =0, then it follows from δ Γ= J, that
min dσ δ(σ) −
g
σ =σ J =0, (50)
min
⇒
and hence,
d d
V(σ) d4x= (J) , (51)
dα dαW
(cid:12)σ=σminZ (cid:12)J=0
(cid:12) (cid:12)
(cid:12) (cid:12)
or, due to eq.(45), (cid:12) (cid:12)
d
V(σ) =0. (52)
dα
(cid:12)σ=σmin
(cid:12)
(cid:12)
We conclude that the vacuum energy Evac should be ind(cid:12)ependent from the gauge parameter α.
A completely analogous derivation was performed in the case of the linear gauge [29]. Nevertheless, in spite
of the previous argument, explicit results in that case showed that E did depend on α. In [29] it was argued that
vac
thisapparentdisagreementwasduetoamixingofdifferentordersofperturbationtheory. Weproposedamodification
of the LCO formalism suitable circumventing this problem and obtaining a well defined gauge independent vacuum
energyE ,withouttheneedofworkingatinfiniteorder[29]. Insteadoftheaction(23),letusconsiderthefollowing
vac
action
ζ
S(J) = S +S +S + d4x J (g2,α) + 2(g2,α)J2 , (53)
YM MAG diag MAG
F O 2F
Z (cid:20) (cid:21)
where, for the momeent,e (g2,α) is an arbitrary function of αeof the form e
F
(g2,α)=1+f (α)g2+f (α)g4+O(g6), (54)
0 1
F
9
and J is now the source. The generating functional becomes
e exp( i (J))= [Dφ]expiS(J). (55)
− W
Z
f e e e
Taking the functional derivative of (J) with respect to J, we obtain
W
f e δ (J) e
W = (g2,α) . (56)
MAG
δJ (cid:12) −F hO i
f e (cid:12)J=0
(cid:12)
Once more, we insert unity via e (cid:12)(cid:12)e
2
1 1 σ
1= [Dσ]exp i d4x ζJ (g2,α) , (57)
N Z " Z −2ζ (cid:18)gF(g2,α) −OMAG− F (cid:19) !#
e
e e
to arrive at the following Lagrangian
1 1 σ2 1 1
(A ,σ)= Fa Fµνa Fi Fµνi+ + + gσ ( )2.(58)
L µ −4 µν − 4 µν LMAG Ldiag− 2g2 2(g2,α)ζ g2 (g2,α)ζ OMAG− 2ζ OMAG
F F
e
¿Ferom thee generating functional e
exp( i (J)) = [Dφ]expiS (J), (59)
σ
− W
Z
fS (eJ) = d4x (A e,σe)+Jσ . (60)
σ µ
L g
Z (cid:18) (cid:19)
it follows that e e e ee
δ (J) σ
W = σ =g (g2,α) , (61)
MAG
δJ (cid:12) − g ⇒h i F hO i
f e (cid:12)(cid:12)J=0 (cid:28)e(cid:29)
(cid:12) e
The renormalizability of the actione(35(cid:12))eimplies that the action (60) will be renormalizable too. Notice indeed that
both actions are connected through the transformation
J
J = . (62)
(g2,α)
F
The tree level off-diagonal masses are now providedeby
gσ gσ
moff−diag. = , moff−diag. = α , (63)
gluon sζ0 ghost s ζ0
e e
while the vacuum configuration is determined by solving the gap equation
dV(σ)
=0, (64)
dσ
e e
withV(σ)the effectivepotential. Minimizing V(σ)will leadto a vacuumenergyE (α) whichwilldependonαand
vac
e
the hitherto undetermined functions f (α) [41]. We will determine those functions f (α) by requiring that E (α) is
i i vac
α-indeepeendent. More precisely, one has e e
dE
vac
=0 first order differential equations in α for f (α). (65)
i
dα ⇒
Of course, in order to be able to determine the f (α), we need an initial value for the vacuum energy E . This
i vac
correspondstoinitialconditionsforthef (α). Inthecaseofthelineargauges,tofixtheinitialconditionweemployed
i
the Landau gauge [29], a choice which would also be possible in case of the Curci-Ferrari gauges, since the Landau
gauge belongs to these classes of gauges. This choice of the Landau gauge can be motivated by observing that
10
the integrated operator d4xAAAµA has a gauge invariant meaning in the Landau gauge, due to the transversality
µ
condition ∂ AµA =0, namely
µ R
(VT)−1 min d4x AA U AµA U = d4x(AAAµA) in the Landau gauge, (66)
µ µ
UǫSU(N)
Z h(cid:0) (cid:1) (cid:0) (cid:1) i Z
with the operator on the left hand side of eq.(66) being gauge invariant. Moreover, the Landau gauge is also an
all-order fixed point of the RGE for the gauge parameter in case of the linear and Curci-Ferrari gauges. At first
glance, it could seem that it is not possible anymore to make use of the Landau gauge as initial condition in the case
ofthe MAG,sincethe Landaugaugedoesnotbelongtothe classofgaugeswearecurrentlyconsidering. Fortunately,
we shall be able to prove that we can use the Landau gauge as initial condition for the MAG too. This will be the
content of the next section.
Before turning our attention to this task, it is worth noticing that, if one would work up to infinite order,
the expressions (53) and (60) can be transformed exactly into those of (23), respectively (35) by means of eq.(62)
and its associated transformation
σ = (g2,α)σ , (67)
F
so that the effective potentials V(σ) and V(σ) are exactly the same at infinite order, and as such will give rise to the
e
same, gauge parameter independent, vacuum energy.
e e
IV. INTERPOLATING BETWEEN THE MAG AND THE LANDAU GAUGE.
In this sectionwe shallintroduce a generalizedrenormalizablegauge whichinterpolates between the MAG and the
Landau gauge. This will provide a connection between these two gauges, allowing us to use the Landau gauge as
initial condition. An example of such a generalizedgauge,interpolating between the Landauand the Coulomb gauge
was already presented in [34]. Moreover, we must realize that in the present case, we must also interpolate between
the composite operator 1AAAµA of the Landau gauge and the gluon-ghost operator of the MAG. Although
2 µ OMAG
this seems to be a highly complicated assignment, there is an elegant way to treat it.
Consider again the SU(N) Yang-Mills action with the MAG gauge fixing (11). For the residual Abelian
gauge freedom, we impose
S′ = d4x bi∂ Aµi+ci∂2ci+ci∂ gfiabAµacb +κgfiabAa ∂ ci cb+κg2fiabficdcacdAbAµc
diag µ µ µ µ µ
Z
κgfiab(cid:0)AiAµa(bb gfjbccccj)+(cid:0)κgfiabAµi(D(cid:1)accc)cb+κg2(cid:0)fabif(cid:1)acdAiAµccdcb , (68)
− µ − µ µ
where κ is an additional gauge parameter. The gauge fixing (68) can be rewritten as a BRST(cid:1)exact expression
1
S′ = d4x (1 κ)s ci∂ Aµi +κss AiAµi . (69)
diag − µ 2 µ
Z (cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)
Next, we will introduce the following generalized mass dimension two operator,
1 κ
= AaAµa+ AiAµi+αcaca , (70)
O 2 µ 2 µ
by means of
λJ
S′ = s d4x λ +ζ (71)
LCO O 2
Z (cid:18) (cid:19)
J2 g
= d4x J +ζ αλbaca+λAµaDabcb+αλca gfabicbci+ fabccbcc
O 2 − µ 2
Z (cid:18) (cid:16) (cid:17)
κλci∂ Aµi+κgfiabλAaAµicb ,
− µ µ
(cid:19)
with (J,λ) a BRST doublet of external sources,
sλ=J , sJ =0. (72)