Physical unitarity of a massive Yang-Mills theory without the Higgs field from a viewpoint of confinement

0.09 MB·English

by  Kei-Ichi Kondo

Physical unitarity of a massive Yang-Mills theory
without the Higgs field from a viewpoint of
confinement
3
1
0
2
Kei-IchiKondo∗
n
a DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
J E-mail:[email protected]

1
1 KentaSuzuki

DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
]
h
t
-
p HitoshiFukamachi
e DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
h
[

1 ShogoNishino
v
DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
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8
4
2 ToruShinohara
. DepartmentofPhysics,GraduateSchoolofScience,ChibaUniversity,Chiba263-8522,Japan
1
0
3
1
Inthistalk,weexaminethephysicalunitarityinamassiveYang-MillstheorywithouttheHiggs
:
v field in which the color gauge symmetry is not spontaneouslybroken and kept intact. For this
i
X purpose,we usea newframeworkproposedoneoftheauthorsbasedona nonperturbativecon-
r structionofanon-Abelianfielddescribingamassivespin-onevectorbosonfield,whichenables
a
us to performthe perturbativeand nonperturbativestudies on the physicalunitarity. Moreover,
wepresentanewperturbativetreatmentforthephysicalunitarityaftergivingthegeneralproper-
tiesofthemassiveYang-Millstheory. Thenwereproducetheviolationofphysicalunitarityina
transparentway. Thisworkis apreliminaryworktothe subsequentworksin whichwe present
anonperturbativeframeworktoproposeapossiblescenario ofrestoringthephysicalunitarityin
theCurci-Ferrarimodel.Wediscusstheimplicationsforthelow-energyQCDinrelationtocolor
confinement,glueballmassandBRST-invariantdimension-twocondensate.
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
PhysicalunitarityofamassiveYang-MillstheorywithouttheHiggsfield Kei-IchiKondo
1. Introduction
In this talk we reconsider [1, 2] a massive Yang-Mills theory [3] without the Higgs field [4].
Amotivationofthisresearchstemsfromsomenonperturbative phenomenacausedbystronginter-
actions.
(i) Confinement and Green functions— The deep infrared behaviors of the gluon and ghost
Greenfunctionsarebelievedtobeintimatelyconnected tocolorconfinementinQCD[5,6].
IntheLandaugauge,thedecoupling solution [7,8,9]forthegluonandghostpropagators is
currently supported ratherthan thescaling solution [10]byrecent numerical simulations on
large latticesinthreeandfourspacetime dimensions [11]. Quiterecently, ithasbeenshown
[12]thatthedecouplingsolutionforthegluonandghostpropagatorscanbewellreproduced
from a low-energy effective model of a massive Yang-Mills theory, which is a special case
of theCurci-Ferrari (CF)model [13]. Thisfeature isnot restricted tothe Landau gauge and
is common to manifestly Lorentz covariant gauges, e.g., the maximal Abelian gauge [14],
as pointed out and demonstrated in [15]. We can ask how color confinement in QCD is
understood fromtheCFmodel.
(ii) Glueball mass spectrum— A glueball should be constructed from the fundamental degrees
of freedom of QCD, i.e., quark, gluon and ghost. For instance, the potential model of [16]
identifiesglueballswithboundstatesofmassivegluons. Theyaredescribedsimplybyintro-
ducinganaivemasstermforgluons, 1M2Am ·Am ,whichhoweverbreakstheBecchi-Rouet-
2
Stora-Tyutin (BRST)symmetry. Weaskhowwecanintroduce aBRST-invariant massterm
for gluons to establish a firm field theoretical foundation for treating glueballs, which will
enable ustoanswerhowprecisely themassandspin oftheresulting glueballs arerelated to
thoseoftheconstituent gluons.
(iii) Vacuum condensates— Besides gauge-invariant vacuum condensates represented by hyy¯ i
with mass dimension-three and hFmn2 i with mass dimension four, which are very impor-
tant to characterize the nonperturbative vacuum of QCD, there might exist an extra dimen-
sion two condensate. In fact, such a lower dimensional vacuum condensate is needed from
the phenomenological point of view. However, such a condensate cannot be constructed
fromgauge-invariant localcompositeoperators intheframeworkofthelocalfieldtheory. A
BRST-invariantvacuumcondensateofmassdimensiontwohasbeenconstructedin[17,18].
However, it is just on-shell BRST invariant. Can we construct an off-shell BRST invariant
versionofvacuumcondensate ofmassdimensiontwo?
Another motivation of studying the CF model comes from the field theoretical interest, since
the massive Yang-Mills theory without the Higgs field has an unsatisfactory aspect as a quantum
field theory. Renormalizability [19, 20] is an important criterion for a quantum field theory to be
a calculable and predictable theory. In addition, physical unitarity [19, 20, 21, 22, 23, 24, 25, 26,
27,28,29,30]isanotherimportant criterion foraquantum theoryofgaugefieldstobeameaning
theory, whichpreventsunphysical particles frombeingobserved.
Inviewofthis,weremindthereadersofthewell-knownfacts:
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PhysicalunitarityofamassiveYang-MillstheorywithouttheHiggsfield Kei-IchiKondo
(i) The massless Yang-Mills theory satisfies both renormalizability and physical unitarity [19,
21].
(ii) Themassive Yang-Mills theory inwhich local gaugeinvariance isspontaneously broken by
theHiggsfieldandthegaugefieldacquires themassthrough theHiggsmechanism satisfies
bothrenormalizability andphysicalunitarity [20].
In fact, the unified theory of Glashow-Weinberg-Salam for the electromagnetic and weak inter-
actions based on the spontaneous symmetry breaking: SU(2) ×U(1) →U(1) predicted the
L Y EM
massive gauge bosons W+,W−, and Z0 which have been discovered in the mid-1980s, and the
remaining Higgsparticleisabouttobediscovered.
However,inallthemodelsproposedsofarasthemassiveYang-MillstheorywithouttheHiggs
fields (in which the local gauge symmetry is not spontaneously broken), it seems that renormal-
izability and physical unitarity are not compatible with each other. See [29, 30] for reviews and
[31]for laterdevelopments. Indeed, the CF model has been shown to be renormalizable [25, 27],
whereas the CF model does not seem to satisfy physical unitarity according to [25, 26, 27]. Al-
thoughtheCFmodelisnotinvariantundertheusualBRSTtransformation,itcanbemadeinvariant
bymodifying theBRSTtransformation. But,themodifiedBRSTtransformation isnotnilpotent.
Itisknownthatnilpotency isthekeypropertytoshowphysicalunitarityintheusualmassless
Yang-Millstheory, sincetheunphysical statesformtheBRSTquartets andthecancellations occur
among the quartets (Kugo-Ojima quartet mechanism) [5, 21]. It is not so clear if nilpotency is
necessary torecoverphysicalunitarityinthemassivecase. ThephysicalunitarityoftheCFmodel
willbediscussed intheperturbative andanonperturbative frameworkinforthcoming papers[1].
2. The Curci-Ferrari model andthe modified BRST transformation
Inorderto look foracandidate ofthe massive Yang-Mills theory without theHiggs field, we
start from the usual massless Yang-Mills theory in the most general Lorentz gauge formulated in
a manifestly Lorentz covariant way. The total Lagrangian density iswritten in terms of the Yang-
Mills fieldAm ,the FPghost fieldC,the antighost fieldC¯and theNLfieldN . Asacandidate of
themassiveYang-MillstheorywithouttheHiggsfield,weaddthe“massterm”L :
m
Ltot =L +L +L , (2.1a)
mYM YM GF+FP m
1
mn
LYM=− Fmn ·F , (2.1b)
4
a b b
LGF+FP= N ·N + N ·N +N ·¶ m Am − gN ·(iC¯×C)
2 2 2
b
+iC¯·¶ m Dm [A]C + g2(iC¯×C)·(iC¯×C)
4
b a
=N ·¶ m Am +iC¯·¶ m Dm [A]C + (N¯·N¯+N ·N )+ N ·N , (2.1c)
4 2
1
Lm= M2Am ·Am +b M2iC¯·C, (2.1d)
2
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PhysicalunitarityofamassiveYang-MillstheorywithouttheHiggsfield Kei-IchiKondo
where a and b are parameters corresponding to the gauge-fixing parameters in the M →0 limit,
Dm [A]C(x):=¶ m C(x)+gA(x)×C(x),and
N¯ :=−N +giC¯×C. (2.2)
The a = 0 case is the CF model with the coupling constant g, the mass parameter M and
the parameter b . In the Abelian limit with vanishing structure constants fABC =0, the FP ghosts
decouple andtheCFmodelreducestotheNakanishimodel[32].
In what follows, werestrict our considerations to the a =0 case. In the a =0 case, L +
YM
L is constructed so as to be invariant under both the usual BRST transformation and anti-
GF+FP
BRSTtransformation:
ddd Am (x)=Dm [A]C(x) ddd¯Am (x)=Dm [A]C¯(x)
ddd C(x)=−gC(x)×C(x) ddd¯C¯(x)=−gC¯(x)×C¯(x)
 2 ,  2 . (2.3)
ddd C¯(x)=iN (x) ddd¯C(x)=iN¯(x)
ddd N (x)=0 ddd¯N¯(x)=0
 
 
 
Indeed, itischecked that
ddd L =0, ddd L =0, ddd¯L =0, ddd¯L =0. (2.4)
YM GF+FP YM GF+FP
Thisisnotthecaseforthemassterm L ,i.e.,
m
ddd L 6=0. (2.5)
m
Even in the presence of the mass term L , however, the total Lagrangian Ltot can be made
m mYM
invariant bymodifying theBRSTtransformation [13]: d ′ =ld dd ′ withaGrassmannian number
BRST
l and
ddd ′Am (x)=Dm [A]C(x)
ddd ′C(x)=−g2C(x)×C(x) . (2.6)
ddd ′C¯(x)=iN (x)
ddd ′N (x)=M2C(x)



ThemodifiedBRSTtransformation deformstheBRSTtransformation oftheNLfieldandreduces
to the usual BRST transformation in the limit M → 0. It should be remarked that ddd ′Ltot =0
mYM
followsfrom
0=ddd ′(L +L ), (2.7)
GF+FP m
while
ddd ′L 6=0, ddd ′L 6=0. (2.8)
m GF+FP
Similarly, the total action is invariant under a modified anti-BRST transformation ddd¯′ defined
by
ddd¯′Am (x)=Dm [A]C¯(x)
ddd¯′C¯(x)=−g2C¯(x)×C¯(x) , (2.9)
ddd¯′C(x)=iN¯(x)
ddd¯′N¯(x)=−M2C¯(x)



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PhysicalunitarityofamassiveYang-MillstheorywithouttheHiggsfield Kei-IchiKondo
whichreducestotheusualanti-BRSTtransformation inthelimitM→0. Itissometimesusefulto
giveanotherform:
ddd ′N¯(x)=gN¯(x)×C(x)−M2C(x), ddd¯′N (x)=gN (x)×C¯(x)+M2C¯(x). (2.10)
Moreover,thepath-integralintegrationmeasure DADCDC¯DN isinvariantunderthemod-
ified BRST transformation. Indeed, it has been shown in [1] that the Jacobian associated to the
change of integration variables F (x) → F ′(x) = F (x)+ld dd ′F (x) for the integration measure is
equaltoone.
However,themodifiedBRSTtransformation violatesthenilpotency whenM6=0:
ddd ′ddd ′Am (x)=0,
ddd ′ddd ′C(x)=0,
 (2.11)
ddd ′ddd ′C¯(x)=iddd ′N (x)=iM2C(x)6=0,
ddd ′ddd ′N (x)=M2ddd ′C(x)=−M2g2C(x)×C(x)6=0.


The nilpotency is violated also for the modified anti-BRST transformation when M 6=0: In the
limit M →0, the modified BRST and anti-BRST transformations reduce to the usual BRST and
anti-BRSTtransformations andbecomenilpotent.
3. Defining a massiveYang-Millsfield
Werequirethefollowingpropertiestoconstructanon-Abelianmassivespin-onevectorboson
fieldKm (x)inanonperturbative way:
(i) Km hasthemodifiedBRSTinvariance (offmassshell):
ddd ′Km =0. (3.1)
(ii) Km isdivergenceless (onmassshell):
¶ m Km =0. (3.2)
(iii) Km obeystheadjointtransformation underthecolorrotation:
Km (x)→UKm (x)U−1, U =exp[ie AQA], (3.3)
whichhastheinfinitesimalversion:
d Km (x)=e ×Km (x). (3.4)
The field Km is identified with the non-Abelian version of the physical massive vector field with
spin one, as ensured by the above properties. Here (i) guarantees that Km belong to the physical
fieldcreating aphysical state withpositive norm. (ii)guarantees that Km havethecorrect degrees
of freedom as a massive spin-one particle, i.e., three in the four-dimensional spacetime, i.e., two
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PhysicalunitarityofamassiveYang-MillstheorywithouttheHiggsfield Kei-IchiKondo
transverse and one longitudinal modes, excluding one scalar mode. (iii) guarantees that Km obey
thesametransformation ruleasthatoftheoriginalgaugefieldAm
We observe that the total Lagrangian of the CF model is invariant under the (infinitesimal)
globalgaugetransformation orcolorrotation definedby
d F (x):=[e CiQC,F (x)]=e ×F (x),for F =Am ,N ,C,C¯, (3.5)
dj (x):=[e CiQC,j (x)]=−iej (x), (3.6)
where j is a matter field. The conserved Noether charge QA := d3xJ0,A obtained from the
R color
colorcurrent J0 iscalledthecolorchargeandisequaltothegeneratorofthecolorrotation.
color
Ithasbeenshown[1]thatsuchafieldKm isobtained byanonlinear butlocaltransformation
fromtheoriginalfields Am ,C,C¯andN oftheCFmodel:
Km :=Am −M−2¶ m N −gM−2Am ×N
+gM−2¶ m C ×iC¯+g2M−2(Am ×C)×iC¯. (3.7)
IntheAbelianlimitorthelowestorderof g,Km reduces totheProcafieldformassivevector:
1
Km →Am − ¶ m N :=Um . (3.8)
M2
It should be remarked thatUm is invariant under the Abelian version of the modified BRST,but it
isnotinvariant underthenon-Abelian modifiedBRSTtransformation.
ThenewfieldKm isconverted toasimpleform:
Km (x)=Am (x)+ 1 iddd ′ddd¯′Am (x). (3.9)
M2
Ithasbeenexplicitly shownin[1]thatthefieldKm definedby(3.7)or(3.9)satisfiesalltheabove
properties. ThefieldKm playstheroleofthenon-Abelianmassivevectorfieldandisidentifiedwith
a non-Abelian version of the spin-one massive vector field. Equation (3.7) gives a transformation
from Am ,N ,C andC¯toKm .
Asanapplication of the above result, wecan construct amass term which is invariant simul-
taneously underthemodifiedBRSTtransformation, Lorentztransformation andcolorrotation:
1
M2Km (x)·K m (x). (3.10)
2
This can be useful as a regularization scheme for avoiding infrared divergences in non-Abelian
gauge theories. Moreover, we can obtain a dimension-two condensate which is modified BRST
invariant, Lorentzinvariant, andcolor-singlet:
m
hKm (x)·K (x)i. (3.11)
This dimension-two condensate is off-shell (modified) BRST invariant and should be compared
withthedimension-two condensate proposed in[17,18]whichisonlyon-shell BRSTinvariant:
1
Am (x)·Am (x)+b C(x)·C¯(x) . (3.12)
D2 E
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PhysicalunitarityofamassiveYang-MillstheorywithouttheHiggsfield Kei-IchiKondo
4. Perturbative violationofphysicalunitarity
In[2],wehavecheckedinanewperturbativetreatmentwhetherornottheCFmodelsatisfies
the physical unitarity. Then we have confirmed the violation of the physical unitarity in the per-
turbative treatment and wehave clarified the reason in the massive Yang-Mills theory without the
Higgs field. The perturbative analysis for the physical unitarity imposes a restriction on the valid
energy together with the parameter of the CF model: E2 <4b M2 in order to confine unphysical
modes(ghost,antighost, scalarmode). However,b =0isnotallowedinthisscenario.
It should be remarked that even the modified BRST (and anti-BRST) invariant quantity de-
pendsonaparameterb intheM6=0case. ThisshouldbecomparedwiththeM=0case,inwhich
b isagauge-fixingparameterandtheBRST-invariantquantitydoesnotdependonb ,whichmeans
thatthephysicsdoesnotdependonb intheM=0case. ThisisnotthecaseforM6=0[33].
5. Possiblenonperturbative restorationofphysicalunitarity
Theconclusionobtainedinthisworkstillleavesapossibilitythatthenonperturbativeapproach
can modify the conclusion. In a subsequent paper, indeed, we will propose a scenario in which
the physical unitarity can be recovered in the CF model thanks to the FP conjugation invariance.
Indeed, we will show that the norm cancellation is automatically guaranteed from the Slavnov-
Taylor identities if the ghost-antighost bound state exists. In this way, the physical unitarity can
be recovered in a nonperturbative way. To show the existence of the bound state of ghost and
antighost, theNambu-Bethe-Salpeter equation istobesolved. Thisisahardworktobetackledin
subsequent papers.
Acknowledgements:ThisworkissupportedbyGrant-in-AidforScientificResearch(C)24540252
fromtheJapanSocietyforthePromotionofScience(JSPS).
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