The unphysical nature of the SL(2,R) symmetry and its associated condensates in Yang-Mills theories

0.15 MB·English

by  Mboyo Esole

The unphysical nature of the SL(2,R) symmetry and its associated condensates in
Yang-Mills theories
Mboyo Esole and Filipe Freire
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
BRSTcohomologymethodsareusedtoexplaintheoriginoftheSL(2,R)symmetryinYang-Mills
theories. Clear evidence is provided for the unphysical nature of this symmetry. This is obtained
from theanalysis ofalocal functionalofmass dimension twoand constitutesano-gostatement for
giving a physical meaning to condensates associated with thesymmetry breaking of SL(2,R).
PACSnumbers: 11.15.-q
Inrecentyearstherehasbeenagrowinginterestincon- the antifield BV formalism while BRS is used for the
4 densates of functionals of mass dimensiontwo in SU(N) original Becchi-Rouet-Stora transformation[9].
0 Yang-Mills (YM) theories in four dimensions involving
We present a clear picture of why no physical con-
0 ghostfields[1,2,3]. Thecondensatesaresignaledbynon-
tent should be attributed to condensates linked to the
2
vanishing expectation values of the functionals. These breaking of SL(2,R), e.g., hfABCCBCCi, hfABCCBCCi
n studies have been done in the maximal Abelian (MA) and hfABCCBCCi[3]. This is obtained by studying the
a gauge and the generalizedLorentz or Curci-Ferrari(CF)
J local functional A0 given by (1). We note that A0 is de-
gaugeandweremotivatedbytheprospectthatthesecon-
fined without specifying any particular gauge, but it is
6 densates may control the infrared divergences in pertur-
knownthatitisonly intheCFgaugethatA0 isBRSand
bation theory and consequently shed light on the origin
2 anti-BRSclosedmodulo the equations ofmotion(EOM)
of the mass gap for the gluon excitations.
v of the gauge-fixed action. However, this does not imply
Theinterestincondensatesinvolvingghostsgoesback
2 the existence of an observable associated to A0. Indeed,
5 to the suggestion that ghost-antighost pairs condense in forlocalfunctionalstheon-shellBRSinvariancemustbe
1 the MA gauge following the symmetry breaking of a
supplemented by appropriate conditions that guarantee
5 global SL(2,R) symmetry of the gauge-fixed action[1].
the isomorphismbetweenthe cohomologyofthe on-shell
0 AlsointheCFgaugethebreakingofanSL(2,R)symme-
3 BRS symmetry and the BRST operator[10, 11]. It fol-
try has been associated with ghost condensates[3]. The
0 lows from[11, 12] that the non-existence of a local ob-
mass generated by this condensation scenario was later
/ servableassociatedtoanon-shell(anti-)BRSclosedlocal
h shown to be tachyonic[4]. The origin and physical rele-
functionalisduetoglobalsymmetriesofthecorrespond-
p-t vuasendceanofdtthheelSinLk(s2,bRet)wseyemnmtheterycornedmeanisnaetdesu,ntchleeagra.uges icniagtgeadutgoe-tfihxeendoanc-tdioiang.oFnoarlAge0n,etrhaetsoerssyomfmSLet(r2i,eRsa).reTahsessoe-
e
It has then been suggested that in SU(N) YM the
h generators can not have any physical meaning in YM
mass dimension two local functional
: as they always involve trivial elements of the cohomol-
v
Xi A0 =R dnx(21BµABAµ+αCACA), A=1 ··· N2−1, (1) otrgyy[b8r]e.aTkihnegreofofrSe,Lc(o2n,dRe)ncsaantesnaostsboceialitnekdetdottohethsyemmmases-
r might play an important role in the mass generation for generation in YM contrary to the suggestions in[3, 13].
a gluons because it is “BRS-closed”[2]. In (1) BA is the
µ Before starting our study of A0 we summarize the
gauge potential, CA and CA are, respectively, the ghost
salient features of the BV formalism in the context of
and antighost fields and α is a gauge-fixing parameter.
YM theories[5, 6, 7, 8]. The antifield formalism starts
By local it is meant that the functional depends on the
by enlarging the original space to contain not only the
fieldsandafinitenumberoftheirderivativesallofwhich gauge fields BA and the ghosts CA, but also sources
µ
are evaluated at the same point in space-time.
for their BRST variations[14] denoted respectively by
InthisletterweuseBRSTcohomologymethodsinthe B∗µ and C∗. They are the antifields in the BV formal-
A A
Batalin-Vilkovisky (BV) antifield formalism[5, 6, 7, 8]
ism and each of them has a Grassmann parity opposite
to explain the links between the functional A0, the CF to the corresponding field. An odd symplectic struc-
gauge and the SL(2,R) symmetry. It is important to
ture (., .) called the antibracket is defined on the ex-
realize that BRST cohomology though it is a perturba- tended phase space, so that the fields Φi ={BA,CA} are
µ
tive method can provide important information on non canonically conjugate to the antifields Φ∗ = {B∗µ,C∗}
perturbative phenomena. This is well illustrated in the i A A
in the sense that (Φi,Φ∗) ≡ δi. The antibracket is ex-
classificationofanomalies[8]. Likeanomalies,theforma- j j
tended to any functionals A =Rdnxa and B =Rdnxb,
tion of condensates is associated with symmetry break-
ing. ThedifferencehereisthatBRSTcohomologyisused as (A,B) = Rdnx(δδRΦai δδΦL∗b − δδΦRa∗ δδΦLbi), where δLδz,R, de-
i i
to study the physical relevance of the broken symmetry note respectively left and right Euler-Lagrange deriva-
associatedwith the condensate. For convenience,we use tives. In the context of the gauge independent formu-
the terminology BRST only for the s-transformations in lation no antighosts or auxiliary fields are needed. The
2
classical action S0 is extended to a local functional S, fix the gauge freedom. To construct Ψ it is necessary to
which includes terms involving ghosts and antifields and extend the phase space to include non-minimalvariables
is a proper solution of the master equation (S,S) = 0. {CA,bA} and their respective antifields. The C’s have
This equation contains all the information about the ac- thenecessarynegativeghandarethe familiarantighosts
tion, the infinitesimal gauge transformations and their of gauge-fixed actions and should not be confused with
algebra. For YM the minimal proper solution is C∗, the antifield of the ghost. These variables are triv-
ial in the BRST cohomology, i.e., sCA = bA, sbA = 0,
S =S0+R dnx(BA∗µDµABCB + 21CA∗fABCCCCB), (2) sb∗ = −C∗ and sC∗ = 0. To implement these transfor-
A A A
mations a new term −C∗bA is added to S in (2).
swthruercetuSre0 cisontshteanYtsManadctDioAn,Bf=ABδACBa∂re−thfeAgBaCuBgeC.group Let sΨ be the BRST tAransformationgeneratedbySΨ,
The BRST operator isµcanonicallµy generatedµ by S sΨA=(SΨ,A). SinceSΨ isobtainedfromS byacanon-
through the antibracket in the sense that sA = (S,A). ical transformation the cohomology of s and sΨ are iso-
morphic. However, this isomorphism is only guaranteed
To analyze the BRST cohomology different gradings
in the space of local functionals as long as all antifields
are introduced[6, 8]: the antifield number (antif), the
are kept. For YM,
pureghost number (puregh) and the usual (total) ghost
nanumtifb(eBr∗µ(g)h)=. 1T, haenytifar(eC∗g)iv=en 2b,y,puarnetgihf((BΦAi)) == 00,, SΨ =S0Ψ+R dnx(BA′∗µ γΨBAµ+CA′∗ γΨCA−C′A∗bA) (5)
A A µ
gpuhr=egphu(rCeAg)h=−1a,nptiufreagnhd(Φcl∗ie)ar=ly0g.hT(Sh)e=gh0o.stTnhuemasbseirgnis- whereS0Ψ =S0+Rdnx(δδBΨµA γΨBAµ+δδCΨA γΨCA−δδCΨAbA)
mentofanantif toeachvariableisanimportantfeature isafunctionofBµA,CA,CA andbA. Themorefamiliaron-
oftheformalism[10,15]thatwillplayacentralrolelater. shellBRStransformationina fixedgaugespecifiedby Ψ
For a given S, the BRST differential s can be expanded is defined in this formalism by sBRS(Φi)= sΨ(Φi)|Φ∗=0.
i
according to the antifield number[6, 15, 16, 17]. In the It corresponds to a global symmetry of the gauge-fixed
case of YM (2) we have actionS0Ψ andthereforedoesnotrequirethefullantifield
formalism and should not be confused with sΨ.
s=δ+γ, (3) Finally, we discuss the determination of the set of in-
tegrated observables, i.e., on-shell gauge invariant func-
whereδdecreasesantif byoneandγleavesitunchanged. tionals[6, 18, 19]. Let A = A(Φi,Φ∗) be a local func-
δisknownastheKoszul-Tatedifferentialandisrelatedto i
tional with gh(A) = 0. If A is BRST closed, sA = 0,
othfefuEnOctMionosfSor0[f6u,n1c6ti,o1n7a]l,swahloilnegγtmheeagsauurgeestohrebivtasraiantdioint by expanding it according to antif, A=A0+Pk≥1Ak,
reduces to the usual off-shell BRS transformation when with antif(Ak) = k, we have γA0 +δA1 = 0. Hence,
it acts in the space of fields Φi. For s to be nilpotent, γA0 vanishes on-shell (γA0 ≈ 0 w.r.t. S0)[17]. As for
we must have δ2 = γ2 = δγ +γδ = 0. The differential the converse, if γA0 ≈ 0 w.r.t. S0 then there exists a
BRST-closed functional A, sA= 0, with an antifield in-
δ acts only non-trivially on the antifields. Due to the
choiceofantif itgivestheEOMwhenitactsonB∗µ,i.e., dependent part that equals A0. In this case, A is called
δBA∗µ =δS0/δBµA,whilewhenitactsonCA∗ itgivesAδCA∗ = aproBpReSrtTy-[c1l9o]seadsietxmteenasniosnthoaftAa0g.iveTnhBisRiSsTacvoecryycluesAefuisl
DµABBB∗µ whichensuretheacyclicityofδ,i.e.,Hn>0(δ)≡ completely determined by its antifield independent part
0. This property is central to the computation of the
A0. Therefore, the set of local gauge invariant function-
BRST cohomology[8, 15, 17]. δ implements the EOM
als canbe determined by the cohomologyof the off-shell
in its cohomology in the following sense. A functional F
BRSoperatorγ modulotheEOMforthegaugeinvariant
vanishes when the EOM of S0 hold if and only if it can action S0 (sA=0 ⇐⇒ γA0 ≈0).
bewrittenasaδ-exacttermF =δG,forsomefunctional
One would wish that similarly, the cohomology of the
G. Then F is said to vanish on-shell (w.r.t. S0) and we on-shell BRS differential can be used directly to iden-
denote it by F ≈0. Changing the antif of the variables
tify the on-shell gauge invariant operators. However, as
mayalter the acyclic propertyofδ andthe EOM[10,12]
first noticed by Henneaux[10], this is not the case in
that are implemented in the cohomology and therefore
the space of local functionals as it requires extra condi-
the observables of the theory.
tions[11]. Indeed, in a given gauge Ψ, there is no guar-
In the BV formalism, a change of gauge corresponds
antee that a generic local on-shell BRS-closedfunctional
taoddaacanneoxnaiccatltterramns(foSr,mΨa)ti=onR[5d,n7x]suΨsintog tthhee fsroeleudtoiomn toof A0 (sBRSA0 ≈0 w.r.t. S0Ψ) with gh(A)=0 possesses a
BRST-closed extension A. Hence, we have to check ex-
the master equation. Then, S →SΨ with
plicitlyforeachA0whetheritispossibletofindtheterms
SΨ =S−(S,Ψ)=S[Φi =Φ′i,Φ∗ =Φ′∗+ δΨ ], (4) Ak≥1, that make its extension BRST closed, sΨA=0.
i i δΦi Whenever we have a BRST extension of an on-shell
where Ψ is the gauge-fixing fermion which must have BRSclosedlocalfunctionalA0(Φ)inagivengauge,then
gh(Ψ)=−1 as the antibracketincreases gh by one. The in any other gauge Ψ there is an on-shell BRS-closed
Ψ
antifield-independent partof S is the “gauge-fixed”ac- functional which corresponds to A0 and can be written
tion S0Ψ. The Ψ’s of interest are those that completely as AΨ(Φ) = A(Φ,Φ∗ = δδΨΦ). This illustrates the advan-
3
tages of the antifield formalism as it enables us to study new choice of antifield number, say antifΨ, such that
the observables of a gauge theory in a manifestly gauge antifΨBA∗µ=antifΨCA∗ =antifΨC∗A=antifΨb∗A=1 [10].
invariantway. Furthermore,Φ∗ = δΨ providesa“dictio- Were-emphasizethatitistheBRSToperatorstogether
δΦ
nary” to interpret the results in any given gauge Ψ. with the originalantif gradingthat is relevant for ques-
Next,wefocusonanexplicitstudyofA0 givenby (1). tionsaboutgaugeinvarianceandrenormalizationoflocal
UsingStokes’theorem,localfunctionalscanbeidentified operators[8].
withtheir integrandsmodulo totaldivergences[8]. Here- In order to have a better understanding about what
after, we denote the integrand of A0 by a0. Using the prevents A0 to be gauge invariant, we now try to con-
BRSTsymmetry,weshowthatA0 isnotgaugeinvariant struct its BRST-closed extension[10] by assuming that
andcannotevenbe extendedtoanon-shellgaugeinvari- the extension exists, sΨA = 0. In line with (6), we have
ant local functional (w.r.t. S0). Now, A0 =Rdnxa0 for the integrand sΨa+dm = 0. This equation can be
is BRST closed if and only if it can be extended to decomposedinto asystemby insertingthe expansionsin
A=Rdnxa in such a way that it satisfies the condition antifΨ, sΨ =δΨ+γΨ, a=Pk≥0ak and m=Pk≥0mk,
astheresultingtermsateachorderinantifΨmustvanish
sa+dm=0, (6) separately providing a method of solving it iteratively.
The first two lowest order non-trivial equations are
where a = Pk≥0ak with antif(ak) = k and dm is the
exterior derivative of some form m. However, a0 cannot γΨa0+δΨa1+dm0=0, (10)
be extended tosatisfy (6). This wouldhaveimpliedthat
γΨa1+δΨa2+dm1=0. (11)
γa0 ≈0, but it follows from the integrand of (1) that
γa0 =∂µ(CABµA)+∆a0, (7) dFδrΨom=(101)waendseebythuastinγgΨtah1e insilδpΨo-tcelnocsyedofmδoΨduanloddδ,Ψid.e+.,
δΨ(γΨa1) = d(δΨm1). Moreover, by re-expressing (11)
where∆a0 =−CA(∂µBµA−αfABCCBCC+αbA). Clearly, as γΨa1 = δΨ(−a2)+d(−m0), γΨa1 is indeed δΨ-exact
this last term does not vanish modulo the EOM of the
modulo d. Therefore, γΨa1 must be a trivial element of
gluons and it is not a total derivative because its Euler-
the homology of δΨ in the space of local functionals.
Lagrange derivatives do not vanish (see Theorem 4.1
With this necessary condition for a1 in mind we now
of[8]). Hence,A0doesnothaveaBRST-closedextension try to extend A0 given by (1). From (7) and (10) it
because of the “obstruction term” ∆a0 of antifield num- follows that δΨa1 =CA(∂µBµA−αfABCCBCC +αbA).
ber zero. Next, we look at the possibility of deforming
AsδΨ actsonlyontheantifields,δΨa1 mustbealinear
the theory in order that ∆a0 vanishes modulo the modi- combination of the gauge-fixed EOM. In this case the
fiEeOdMEOfoMr .thTehaisuxcailniabryefiaeclhdiebvAedarifeintatkheenntoewbeaction,the onlypossibilityistoinvoketheEOMofbA,δΨb′A∗ = δδSbA0Ψ,
given by (8), where b′∗ are the antifields in the base Ψ
a
∂µBA−αfABCCBCC +αbA =0. (8) as in (4). This is achieved with the choice of antifΨ
µ mentioned above. Therefore we have a1 =−CAb′A∗ and
These equations constrain the fields BA and completely
fix their gauge freedom. Therefore, thµe deformation we −γΨa1 = 21b′A∗fABCCBCC +C′A∗CA. (12)
areseekingisagaugefixingprocedure. Infact,theEOM
The r.h.s. of (12) does not involve any derivatives of
(8) can only be obtained if we choose
thefields{Aµ,C,C}soitcannotbeδΨ-exactasrequired
Ψ=CA(∂µBA− 1αfABCCBCC + 1αbA). (9) for the extension to exist. Thus, we conclude that (12)
µ 2 2
constitutes an obstruction for A0 to be extended into
This is the gauge-fixing fermion that corresponds to the a local BRST-closed functional of sΨ. In fact, γΨa1 in
CF gauge. Hence, it is only in this gauge that A0 is on- (12) is a non-trivial element of the homology of δΨ with
shell BRS closed with respect to the gauge-fixed action. antifΨ=1 and therefore δΨ mod d is no longer acyclic.
The restriction to work in a specific gauge is a natural There is a global symmetry of the gauge-fixed action
consequence of the gauge dependence of A0. This has associated to the obstruction (12). From the r.h.s. of
been a serious cause of confusion in the literature on the (12),weeasilyidentifythegeneratorsofthesymmetryas
gauge invariant status of A0. they couple linearly to the antifields. Indeed, any linear
Some care has to be taken on how to interpret this functionofthe antifieldsisnaturallyviewedasatangent
deformation[12]. This is not simply a canonical trans- vectortofieldspace[17,20]. Thegeneratorofthisglobal
formation in the sense of (4) where the theory remains symmetry is
YM. As we need to use explicitly the EOM of the aux-
iliary fields, the new deformed theory is to be seen as a 1fABCCBCC δL +CA δL . (13)
2 δbA δCA
theory where ghosts and antighosts have now their own
dynamics, which is governed by the action SΨ that now IntheBVformalismamorefamiliarwaytoarriveatthis
0
plays the role of a “classical action” (5). The EOM of symmetryisto expressitascanonicallygeneratedinthe
{C,C,b} can be implemented in the cohomology with a antibracket,i.e.,δˆ (Φi)=(τ,Φi)withτ givenby(12). δˆ
τ τ
4
isoneofthetwonon-diagonalgeneratorsoftheSL(2,R) SL(2,R). These obstructions are not symmetries of YM
symmetry of the gauge-fixed action (5). The functional as they do not involve gauge fields of S0 and their exis-
A0 isalsoon-shellinvariantfortheanti-BRStransforma- tenceisonlyassociatedtothespecificchoiceofthegauge-
tions¯Ψ intheCFgauge. IfweattempttoextendA0 into fixing where A0 is on-shell BRS closed, namely the CF
the cohomologyof s¯Ψ[21] we also find an obstruction,in gauge. Infact,theSL(2,R)generatorsδˆτ andδˆτ¯ (13,14)
this case it corresponds to the infinitesimal symmetry that constitute the obstruction always involve variables
of the non-minimal sector of the phase space. Hence,
1fABCCBCC δL +CA δL , (14) these symmetries are trivial for YM[8, 17] and therefore
2 δbA δCA
can not be of any physical relevance. As SL(2,R) is not
which is the other non-diagonal generator δˆτ¯ of the a symmetry of YM there is no physical justification to
SL(2,R) symmetry, with τ¯ = 1b′∗fABCCBCC +C′∗CA. impose to the quantum theory Ward identities associ-
2 A A
The third generator, δˆ , is diagonal and is given by ated with it[3]. The expectation that there is a physical
the commutator [δˆτ,δˆτ¯]FoPf the two non-diagonal genera- meaning attached to the symmetry breaking of SL(2,R)
tors. δˆ is the generator of the ghost number which is by ghost condensates loses all its support in view of this
FP analysis. We alsonotethatasA0 isnoton-shell(BRST)
trivially a global symmetry of the gauge-fixed action as
Ψ gauge invariant it can not be used as a mass term to be
gh(S )=0.
0 added to the action.
Evenifthe auxiliaryfieldsarereplacedby theirEOM,
The SL(2,R) symmetry was originally discovered in
the obstructions are still present. They will only involve
theghostsandtheantighosts(δˆτ →CAδδCLA,δˆτ¯ →CAδδCLA tahnealgyasiusgteo-fitxheedoancetipornesfeonrtethdehMereA, cgaanugbee[1d]o.nAe bsyimciolanr-
andδˆFP →δˆFP),buttheiralgebrawillstillbeSL(2,R). sidering the operator A˜0 =R(12BµaBaµ+αCaCa), i.e., A0
The analysis of that case is found in the work of Brandt restrictedtothecontributionfromtheoff-diagonalfields.
[12] where the CF mass term (1) has been studied from We havealsocheckedthatifwesetα=0in(1), Landau
the perspective of the deformations within the extended gauge, we encounter the same obstruction.
BRST formalism which implements not only the gauge
Finally, we point out that the on-shellBRS invariance
but also the global symmetries of a given action [22]. In of A0 does not correspond to a residual U(1)N−1 sym-
this context,the introduction ofthe CF mass termleads
metry due to a partial gauge fixing as claimed in [23]
tothelossofnilpotencyoftheon-shell(anti-)BRSopera-
as there is no gauge freedom left in the CF gauge. The
tor due to the modificationof the EOM.More explicitly,
broaderimplicationof our resultis that mass generation
s2BRS =δˆτ 6=0 and s¯2BRS =δˆτ¯ 6=0. This is also linked to inYM cannotbe linkedto the condensationofthe local
the lost of unitarity of the resulting theory [12].
functional A0 of dimension two.
To summarize, A0 is on-shell BRS and anti-BRS in-
variant only in the CF gauge, but if we try to extend
this property to any other gauge we encounter obstruc- ACKNOWLEDGMENTS
tions that are global symmetries of the gauge fixed ac-
tion. More precisely, the obstructions (13) and (14) to We thank A. Achu´carro and P. van Baal for discussions
the local extension A of A0 to be, respectively, BRST and encouragement. ME also thanks G. Barnich for dis-
andanti-BRSTclosed,aretwoofthe three generatorsof cussions. The research of FF is supported by FOM.
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