Hamiltonian approach to Yang-Mills theory in Coulomb gauge - revisited

0.6 MB·English

by  H. Reinhardt

Hamiltonian approach to Yang-Mills theory in Coulomb
gauge - revisited¶
H. Reinhardt∗, D. R. Campagnari∗, M. Leder∗, G. Burgio∗, J. M. Pawlowski†,
M. Quandt∗ and A. Weber∗∗
1
1
0 ∗InstitutfürTheoretischePhysik,UniversitätTübingen,AufderMorgenstelle14,72076Tübingen,Germany
2 †InstitutfürTheoretischePhysik,UniversitätHeidelberg,Philosophenweg16,69120Heidelberg,Germany
∗∗InstitutodeFísicayMatemáticas,UniversidadMichoacanadeSanNicholásdeHidalgo,EdificioC-3,Ciudad
n
Universitaria,58040Morelia,Michoacán,Mexico
a
J
6 Abstract. I briefly review resultsobtained within the variational Hamiltonian approach toYang-Mills theory in Coulomb
2 gauge and confront them with recent lattice data. The variational approach is extended to non-Gaussian wave functionals
including three- and four-gluon kernels inthe exponential of the vacuum wave functional and used to calculate the three-
] gluonvertex.AnewfunctionalrenormalizationgroupflowequationforHamiltonianYang–MillstheoryinCoulombgauge
h issolvedforthegluon andghost propagator under theassumption ofghost dominance. Theresultsarecompared tothose
t obtainedinthevariationalapproach.
-
p
e
h INTRODUCTION of Gauss’ law, resulting in the gauge fixed Yang–Mills
[
Hamiltonian[6]

1 In recent years there have been substantial efforts de-
1
v voted to non-perturbative studies of continuum Yang– H = dDx J−1[AAA]PPP J[AAA]PPP +BBB2 +H , (1)
8 YM 2 c
Mills theory. Among these are variational solutions of Z
09 theYang–MillsSchrödingerequationinCoulombgauge H = g2 dD((cid:0)x,x′)J−1[AAA]r a(x)J[AAA](cid:1)Fab(x,y)r b(x),
c
[1–3]. In this approachthe energydensityis minimized 2
5 Z
1. usingGaussian-typeansätzeforthevacuumwavefunc- where P a(x)=d /id Aa(x) is the canonical momentum
tional. In this talk I will first briefly review results ob-
0 (electricfield)operatorand
tained within the Tübingen approach [2] using a modi-
1
1 fiedGaussianwavefunctional.ThenIwillpresentanew J[AAA]=Det(−DDD(cid:209)(cid:209)(cid:209) ) (2)
: approachtoHamiltonianquantumfieldtheory,whichal-
v
lowstousenon-Gaussianwavefunctionalsinvariational istheFaddeev–Popovdeterminant.Furthermore
i
X calculations [4]. The approach is illustrated by using a
r a(x)=−fabcAAAbPPP c (3)
r wavefunctionalwithcubicandquarticgluonictermsin
a
the exponentialand appliedto calculate the three-gluon
isthecolorchargeofthegluonsand
vertex.Finally,Iwilldiscussanewfunctionalrenormal-
izationgroup(FRG)approachtotheHamiltonianformu- Fab(x,y)=hx,a|(−DDD(cid:209)(cid:209)(cid:209) )−1(−(cid:209)(cid:209)(cid:209) 2)(−DDD(cid:209)(cid:209)(cid:209) )−1|y,bi (4)
lationof Yang–Millstheory[5] andpresentresultssup-
portingthefindingsof the variationalapproach[2, 3]. istheso-calledCoulombkernel.Inthepresenceofmat-
ter fields with color charge density r a(x), the gluon
m
charge r a(x) in the Coulomb term is replaced by the
total charge r a(x)+r a(x) and the vacuum expectation
m
VARIATIONALAPPROACH valueofFab(x,y)acquiresthemeaningofthestaticnon-
AbelianCoulombpotential,seeeq.(16)below.
Instead ofworkingwith gaugeinvariantwave function- The gauge fixed Hamiltonian eq. (1) is highly non-
als, it is usually more convenient to fix the gauge. A local due to the Coulomb kernel Fab(x,y), eq. (4), and
particularlyconvenientchoice of gaugeis the Coulomb due to the Faddeev–PopovdeterminantJ[AAA], eq. (2). In
gauge (cid:209)(cid:209)(cid:209) ·AAA = 0, which allows an explicit resolution addition,the latteralso occursinthe functionalintegra-
tion measure of the scalar product of Coulomb gauge
wavefunctionals
¶ Invited talk given by H. Reinhardt at “T(r)opical QCD 2010”, hy |O|y i= DAJ[AAA]y ∗[AAA]Oy [AAA]. (5)
September26–October1,2010,Cairns,Australia. 1 2 1 2
Z
Whiletheeliminationofunphysicaldegreesoffreedom I (kkk)= Nc d3q 1−(kkkˆ·qqqˆ)2 d(qqq−kkk) .
via gauge fixing is often beneficial (and sometimes un- d 2 (2p )2 w (qqq)(qqq−kkk)2
Z
avoidable) in practical calculations, the price to pay is h i (12)
theincreasedcomplexityofthegaugefixedHamiltonian
eq. (1). The non-trivialFaddeev–Popovdeterminantre- Theghostformfactord(kkk)measuresthedeviationfrom
flectstheintrinsicallynon-linearstructureofthespaceof theQEDcase,whered(kkk)=1.Furthermore,d−1(kkk)rep-
gaugeorbitsanddominatestheIRbehaviorofthetheory. resentsthedielectricfunctionoftheYang–Millsvacuum,
OnceCoulombgaugeisimplemented,anyfunctionalof andtheso-calledhorizoncondition
the(transverse)gaugefieldisaphysicalstate.
d−1(0)=0 (13)
TreatingtheHamiltonian(1)intheordinaryRayleigh-
Schrödingerperturbationtheoryonefindstoleadingor-
ensuresthattheYang–Millsvacuumisadualsupercon-
dertheusualone-loopb functionwithb =−11N [7].
0 3 c ductor [9]. Minimization of the vacuum energy hH i
Our main interest lies, however, in the IR sector of the YM
withrespecttothekernelw (kkk)leadstothegapequation,
theory,whichrequiresanon-perturbativetreatment.
whichafterrenormalizationreads[2],[10]
In ref.[2] the vacuumenergydensitywas minimized
usingthefollowingansatzforthevacuumfunctional
w 2(kkk)=kkk2+c 2(kkk)+c +D I(2)(kkk)+
2
Y [AAA]= N exp −1 d3k w (kkk)Aa(kkk)Aa(−kkk) . +2c (kkk)[D I(1)(kkk)+c1], (14)
J[AAA] 2 (2p )3 i i
(cid:20) Z (cid:21) where
(6)
p
For this wave functional the static gluon propagator is
D I(n)(kkk)=I(n)(kkk)−I(n)(0), w (kkk)≡w (kkk)−c (kkk),
givenby1
N d3q
hAa(−kkk)Ab(qqq)i=d abt (kkk)(2p )3d (kkk−qqq) 1 , (7) I(n)(kkk)= 4c (2p )3 1+(kkkˆ·qqqˆ)2 V(qqq−kkk)
i j ij 2w (kkk) Z (cid:16)w n(qqq)−w n((cid:17)kkk)
× , (15)
sothatw (kkk)representsthe(quasi-)gluonenergy. w (qqq)
To the order considered in ref. [2], i.e., up to two
and
loopsintheenergy(whichcorrespondstooneloopinthe d2(kkk)
associated Dyson–Schwinger equations), it was shown V(kkk)≡g2hFi= (16)
kkk2
[8]thattheFaddeev–Popovdeterminant,eq.(2),canbe
representedas isthenon-AbelianCoulombpotential,wheretheapprox-
imation
J[AAA]=exp − d3xd3yAa(x)c ab(x,y)Ab(y) , (8)
i ij j (−DDD(cid:209)(cid:209)(cid:209) )−1·(−(cid:209)(cid:209)(cid:209) 2)·(−DDD(cid:209)(cid:209)(cid:209) )−1
(cid:20) Z (cid:21)
where (cid:10) ≈ (−DDD(cid:209)(cid:209)(cid:209) )−1 ·(−(cid:209)(cid:209)(cid:209) 2(cid:11))· (−DDD(cid:209)(cid:209)(cid:209) )−1 (17)
1 d 2lnJ[AAA] has been used(cid:10). In the gap(cid:11)equation(cid:10)(14), c1 and(cid:11)c2 are
c iajb(x,y)=−2*d Aai(x)d Abj(y)+=d abtij(x)c (x,y) (tifionni,tew)hreerneoormneailmizaptoiosenscthoenshtoanritzso.nFcoorntdhieticornit(i1ca3l),sbooluth-
(9) w (kkk)andc (kkk)areinfrareddivergent,whichimpliesthat
istheghostloopandrepresentsameasureofthecurva-
thetransversegluonpropagatorvanishesatk→0,while
ture[2]oftheCoulombgaugefixedconfigurationspace.
ExpressingtheghostGreen’sfunctionas w (0)≡lim(w (kkk)−c (kkk))=c . (18)
1
k→0
d
h(−D(cid:209)(cid:209)(cid:209) )−1i= g(−D ), (10) A perimeter law of the ’t Hooft loop requires c1 = 0
andthisvalueisalsofavoredbythevariationalprinciple
[10]. Furthermore, for c =0 the IR limit of the wave
whered(kkk)istheghostformfactor,itsDyson–Schwin- 1
functional(6)is
gerequationreadstoone-looporder
d−1(kkk)=g−1−I (kkk), (11) y [AAA]=N (cid:213) y (kkk), y (kkk=0)=1, (19)
d
kkk
which is the exact vacuum wave functional in 1+1
1 Hereandinthefollowing,tij(kkk)≡dij−kikj/kkk2 denotesthetrans- dimensions [11]. The renormalization parameter c2, on
verseprojectorinmomentumspace. theotherhand,hasnoinfluenceontheIRorUVbehavior
0.45
lattice
0.4
Gaussian functional
0.35 Non-Gaussian functional
0.3
p) 0.25
D( 0.2
0.15
0.1
0.05
0
0 2 4 6 8 10
p
FIGURE 1. Gluon propagator obtained with a Gaussian
FIGURE 2. The numerical solution for the static non-
(dashed line) and a non-Gaussian functional (straight line),
AbelianCoulombpotential[3].
comparedtothelatticedatafromRef.[12].
an inverse gluon propagator which in the UV behaves
ofthesolutionsofthegapequation(14).Onlythemid-
like the photon energy but diverges in the IR, signal-
momentum regime of w (kkk) is weakly dependent on c
2 ingconfinementinagreementwiththeIRanalysis(20),
[2]. Since we are mainly interested in the IR properties
(21),(22).Figures1and2showtheresultinggluonen-
wewillputc =0.
2 ergy w (kkk) and non-Abelian Coulomb potential(16) for
The coupled integral equations (11) and (14) can be
the solution b =1. Their IR behavior is in agreement
solved analytically in the IR (for the critical solution
with the results of the IR analysis. The obtained prop-
satisfying the horizon condition (13)) using power law
agator also compares favorably with the available lat-
ansätze[2,13]
ticedata[12].Thereare,however,deviationsinthemid-
w (kkk)∼k−a , d(kkk)∼k−b . (20) momentum regime (and minor ones in the UV) which
can be attributed to the missing gluon loop, which es-
Under the assumption of a trivial scaling of the ghost- capes the Gaussian wave functionals. These deviations
gluonvertexonefindsthatinDspacedimensionstheIR arepresumablyirrelevantfortheconfinementproperties,
exponentssatisfythesumrule whicharedominatedbytheghostloop(whichisfullyin-
cludedundertheGaussianansatz),butarebelievedtobe
a =2b +2−D. (21) importantforacorrectdescriptionofspontaneousbreak-
ingofchiralsymmetry[16].
Thissumruleensuresthatc (kkk)hasthesameIRbehavior
asw (kkk),cf.eq.(18).Furthermore,inD=3onefindstwo
solutions[13] NON-GAUSSIAN WAVEFUNCTIONALS
b ≃0.8 and b =1. (22)
Inref.[4]thevariationalapproachtoYang–Millstheory
in Coulomb gaugewas extendedto non-Gaussianwave
The latter givesrise to a Coulomb potential(16) which
functionalsoftheform
is strictly linear at large distances. In D = 2 only one
solutionb = 25 isfound,implyinga = 45 andaCoulomb |y [AAA]|2=exp(−S[AAA]), (24)
potential (16) rising as V(r) ∼ r4/5 [14]. Furthermore
1 1
in D=2 onlythe critical solutionsatisfying(13)exists S[A]= w AAA2+ g AAA3+ g AAA4, (25)
3 4
[14]. On the lattice, on the other hand,one finds a =1 Z 3!Z 4!Z
inD=3andalinearlyrisingpotential[15].IntheUV- wherew ,g ,g arevariationalkernels.
3 4
regimethecoupledequations(11)and(14)canbesolved Representing the Faddeev–Popov determinant by a
perturbativelyandonefindsatlargemomentak≫1[2] functionalintegraloverghostfieldsfromtheidentity
w (kkk)∼k, d(kkk)∼1/ log(k). (23) d
0= DA J[AAA]e−S[AAA]K[AAA] , (26)
d A
Thefullnumericalsolutionsoftheabpoveequationswere Z (cid:16) (cid:17)
given, for D=3 space dimensions, in refs. [2] and [3], withK[AAA]anarbitraryfunctionalofthegaugefield,one
where the critical solutionsb =0.8 and b =1, respec- canderive,inthestandardfashion,DSEsforthevarious
tively,werefound,andforD=2inref.[14].Onefinds gluon and ghost Green functions. These DSEs are the
= −2 1
0.5
0
FIGURE3. TruncatedDSEforthethree-gluonvertex,under
theassumptionofghostdominance. 2,0) -0.5
p -1
(A
f3 -1.5
usual DSEs of Landau gauge Yang–Mills theory, how- -2
ever,in D=3 dimensionsandwith the bare verticesof -2.5 this work
the usual Yang–Millsaction replaced by the variational lattice
-3
kernelsw ,g ,g .ItshouldbestressedthattheseHamilto- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
3 4
nianDSEsarenotequationsofmotionintheusualsense, p
butratherrelationsbetweentheGreenfunctionsandthe
FIGURE 4. Form factor f of the three-gluon vertex for
so far undeterminedvariationalkernels. By using these 3A
orthogonalmomentaandcomparisontolatticedataforthed=
DSEs, theenergydensitycan bewritten asa functional
3Landau-gaugevertex[20].Themomentumscaleisarbitrary
ofthevariationalkernels, and has been adjusted to make the sign change occur at the
samepoint.ThelatticedataareshownbycourtesyofA.Maas.
hH i=E[w ,g ,g ]. (27)
YM 3 4
Byusingaskeletonexpansion,thevacuumenergycanbe
gluonvertex
writtenatthedesiredorderofloops.Confiningourselves G ·G (0)
totwoloops,thevariationofthevacuumenergyEq.(27) f := 3 3 , (29)
withrespecttothekernelg fixesthelatterto 3A G (0)·G (0)
3 3 3
g abc(ppp,qqq,kkk)=2igfabc where G (30) is the perturbativevertex, given by Eq. (28)
ijk with W (ppp) replaced by |ppp|. Furthermore,we consider a
d (p−q) +d (q−k) +d (k−p)
ij k jk i ki j particular kinematicconfiguration,where two momenta
× . (28)
W (ppp)+W (qqq)+W (kkk) havethesamemagnitude
Equation(28)is reminiscentof the lowest-orderpertur- ppp2=ppp2=p2, ppp ·ppp =cp2. (30)
1 2 1 2
bativeresult[17],withtheperturbativegluonenergy|ppp|
replacedbythenon-perturbativeoneW (ppp). To evaluatetheformfactor f (p2,c), we use theghost
3A
Withthisresult,variationofhH iwithrespecttow and gluon propagators obtained with a Gaussian wave
YM
yieldsthe gapequation(14),which now,however,con- functional[3] as input. The IR analysis of the equation
tains on the r.h.s. in addition the gluon loop Ig(kkk) [4], for f3A(p2,c) [Eq. (29)] performed in Ref. [13] shows
which escapes the Gaussian wave functional (6). The thatthisformfactorshouldhaveapowerlawin theIR,
presenceofthegluonloopIg(kkk)inthegapequationmod- withanexponentthreetimestheoneoftheghostdress-
ifiestheUV behaviorandallowsustoextract,fromthe ingfunction;thisisconfirmedbyournumericalsolution
non-renormalizationof the ghost-gluon vertex, the cor- [4].Theresultforthescalarformfactor f fororthogo-
3A
rect first coefficient of the b function. In order to esti- nalmomenta, f(p2,0),isshowninFig.4,togetherwith
matethesizeofthegluon-loopcontributiontothegluon lattice results for d =3 Landau gauge Yang–Mills the-
propagator,weuse thegluonandghostpropagatorsob- ory.Ourresultandthelatticedatacomparefavorablyin
tained with a Gaussian wave functional[3] to calculate the low-momentum regime. In particular, in both stud-
the gluon loop. The result is shown in Fig. 1, together ies,thesignchangeoftheformfactoroccursroughlyat
withlatticedatafromRef.[12].Theagreementbetween thesamemomentumwherethegluonpropagatorhasits
the continuumand the lattice results is improvedin the maximum.(ThescaleinFig.4isarbitrary.)
mid-momentum regime by the inclusion of the gluon
loop,i.e.,thethree-gluonvertex,asobservedalsoinLan-
daugauge[18]. HAMILTONIANFLOW
ThetruncatedDSEforthethree-gluonvertexG under
3
the assumption of ghost dominance is represented dia- TheadvantageofthevariationalapproachtotheHamil-
grammaticallyinFig.3.Possibletensordecompositions tonianformulationisitscloseconnectiontophysics.The
ofthethree-gluonvertexaregiveninRef.[19].Forsake price to pay is the apparent loss of manifest renormal-
of illustration, we confine ourselves to the form factor ization groupinvariance.Renormalizationgroupinvari-
corresponding to the tensor structure of the bare three- ance is naturally built-in in the functional renormaliza-
tion group approach to the Hamiltonian formulation of
k∂ −1= − −
Yang–Millstheoryproposedinref.[5]. k
IntheFRG approachthequantumtheoryofafieldj
isinfraredregulatedbyaddingtheregulatorterm
FIGURE5. Truncatedflowequationofthegluonpropagator.
1 1 Thebareverticesatk=L aresymbolizedbysmalldotsandthe
D Sk[j ]= 2j ·Rk·j ≡ 2 j Rkj (31) regulatorinsertionk¶kRbyacrossedsquare.
Z
to the classical action, which in the Hamiltonian ap-
proachis givenby thelogarithmofthe wave functional
k∂ −1= +
k
(24). The regulator function R (p) is an effective mo-
k
mentumdependentmasswiththeproperties
FIGURE6. Truncatedflowequationoftheghostpropagator
lim R (p)>0, lim R (p)=0, (32)
k k
p/k→0 k/p→0
which ensures that R (p) suppresses propagation of
k
and the perturbative initial conditions at the large mo-
modeswith p.kwhilethosewith p&kareunaffected
mentumscalek=L ,
and the full theory at hand is recovered as the cut-off
scale k is pushed to zero. Wetterich’s flow equation for
theeffectiveactionG [f ]ofafieldf isgivenby dL (p)=dL =const., w L (p)=p+a. (38)
k
1 1 With these initial conditions, the flow equations for the
k¶ G [f ]= Tr k¶ R , (33)
k k 2 G (2)[f ]+R k k ghost and gluon propagators are solved under the con-
k k straintofinfraredscalingfortheghostformfactor.The
d nG [f ]
G (n) [f ]= k (34) resultingfullflowoftheghostdressingfunctionisshown
k,1...n df 1...df n inFig.7.
aretheone-particleirreduciblen-pointfunctions(proper AstheIRcut-offmomentumkisdecreased,theghost
vertices).Thegenericstructureoftheflowequation(33) formfactordk(ppp)(constantatk=L )buildsupinfrared
isindependentofthedetailsoftheunderlyingtheory,but strengthandthefinalsolutionatk=kminisshowninFig.
isamereconsequenceoftheformoftheregulatorterm 9togetherwiththeoneforthegluonenergyw kmin(ppp)in
(31), being quadratic in the field. By taking functional Fig.8.ItisseenthattheIRexponents,i.e.,theslopesof
derivatives of Eq. (33) one obtains the flow equations the curvesdkmin(ppp), w kmin(ppp) do notchange as the min-
for the (inverse) propagators and proper vertices. The imal cut-off kmin is lowered. Let us stress that we have
generalflowequation(33)stillholdsfortheHamiltonian assumed infrared scaling of the ghost form factor but
formulation of Yang–Mills theory provided that f is not the horizon condition d−1 (p = 0)= 0. The latter
k=0
interpretedasthesuperfieldf =(AAA,c,c¯). was obtained from the integration of the flow equation
TheFRG flow equationsembodyan infinitetowerof but notput in by hand(the same is also true for the in-
coupledequationsfortheflowofthepropagatorsandthe frared analysis of the Dyson–Schwinger equations fol-
propervertices.Theseequationshavetobetruncatedto lowing from the variational Hamiltonian approach, i.e.,
get a closed system. We shall use the following trun- assumingscalingtheDSEsyieldthehorizoncondition).
cation: we only keep the gluon and ghost propagators, The infrared exponentsextracted from the numerical
whichwewriteintheform solutions of the flow equations are a =0.28 and b =
p2
G (2) =2w (ppp), G (2) = . (35)
k,AA k k,c¯c d (ppp)
k
Inaddition,wekeeptheghost-gluonvertexG (3) ,which 1e+04
k,Ac¯c
we assume to be bare, i.e., we do not solve its FRG
1e+03
flowequation.Moreover,weshallassumeinfraredghost
dominance and discard gluon loops. Then the resulting 1e+02
flowequationsoftheghostandgluonpropagatorreduce
1e+01
totheonesshowninFigs.5and6.
Theseflowequationsaresolvednumericallyusingthe 1e+00
regulators 1e-05
1e-02
RA,k(ppp)=2prk(ppp), Rc,k(ppp)=p2rk(ppp), (36) p 1e+011e+04 1e+04 1e+02 1e+00 1e-02 1e-04
k
k2 p2
r (ppp)=exp − , (37)
k p2 k2 FIGURE7. Flowdk(p)oftheghostformfactor.
(cid:20) (cid:21)
approaches and highlights the inclusion of a consistent
-3
k ~10 UV renormalizationprocedure in the present approach.
min
k ~10-4 The above results encourage further studies, which in-
100 min -5 cludes the flow of the potential between static color
k ~10
min sourcesaswellasdynamicquarks.
10
ACKNOWLEDGMENTS
We wish to thank P. Watson for useful discussions.
-7 -5 -3 -1 1 3
10 10 10 10 10 10 Financial support by the DFG under contracts No.
p [GeV]
Re856/6-2,3andbytheCusanuswerkisgreatlyacknowl-
FIGURE8. Inversegluonpropagatorw atthreeminimalcut- edged.
offvalueskmin.
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2 Insteadofthecompleteright-handsideofEq.(14)onereallyobtains 21. C.S.Fischer, andJ. M. Pawlowski,Phys. Rev. D80,
onlytheIR-dominantcontributionc 2.However,itturnsout[5]thatthe 025023(2009),0903.2193.
numericalsolutionoftheequationsisthesameasinRef.[2]overthe
wholemomentumrange.