Temporal Wilson loop in the Hamiltonian approach in Coulomb gauge

0.1 MB·English

by  Markus Quandt

Temporal Wilson loop in the Hamiltonian approach
in Coulomb gauge
3
1
0
2

n
a
J MarkusQuandt
∗
7 UniversitätTübingen
1
E-mail:[email protected]

]
h H. Reinhardt
t UniversitätTübingen
-
p E-mail:[email protected]
e
h
G. Burgio
[
UniversitätTübingen
1
E-mail:[email protected]
v
8
5
We investigatethe temporalWilson loopusingthe Hamiltonianapproachto Yang-Millstheory.
1
4 In simple cases such as the Abelian theory or the non-Abelian theory in (1+1) dimensions, the
.
1 known results can be derived using unitary transformations to take care of time evolution. Al-
0
ternatively,theexactsolutioncanalsobefoundinCoulombgaugeusingtheexactgroundstate
3
1 wavefunctionalwhichisknownexplicitlyinthesesimplecases. TheCoulombgaugetechnique
:
v can also be appliedto the morerealistic case of Yang-Millstheoryin (3+1)dimensions, where
i
X one has to rely on the approximatevacuum wave functionalobtained e.g. in recent variational
r approaches. WeusethisformulationtocomputethetemporalWilsonloopin(3+1)dimensional
a
Yang-Millstheory,andfindthattheWilsonandCoulombstringtensionagreewithinthisapprox-
imationscheme.Possibleimprovementsofthesefindingsarebrieflydiscussed.
XthQuarkConfinementandtheHadronSpectrum
8-12October2012
TUMCampusGarching,Munich,Germany
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
1. Introduction
Recent efforts to study Yang-Mills (YM) theory are largely based on Coulomb gauge, be-
cause theGribov-Zwanziger picture ofconfinement [1,2]becomes particularly transparent inthis
framework: theonlyconstraintonthephysicalstatesinaHamiltonianapproachbasedonCoulomb
gaugeisGauß’law,whichinturncanberesolvedexactlysothatanynormalizablewavefunctional
canbeusedasaphysicalstate.
Such a formulation naturally lends itself to variational methods, which are usually based on
(modified)GaussianAnsätzeforthevacuumwavefunctional,seee.g.[3,4]andreferencestherein.
The solutions agree with the Gribov-Zwanziger scenario and, when combined with the horizon
condition, leadtoacoherentpictureofstronglycoupledYMtheory,whichismainlydominatedby
adiverginggluonenergyintheinfrared,alinearlyrisingCoulombpotentialbetweenstaticquarks,
and a strongly enhanced Faddeev-Popov ghost propagator in the infrared. The latter condition
implies that the YM vacuum behaves as a perfect colour dia-electric medium [5], i.e. a dual su-
perconductor. Allthesefindingsagreequalitatively withtheresultsfromrecentlatticesimulations
[6],althoughtherearestillsomediscrepanciesinthequantitativedetails,inparticularfortheghost
sector[7].
However, this simple picture is subject to the caveat that the Coulomb potential is only an
upper bound for the true potential between heavy quarks [8]. For a complete description, one
has to address the physical potential from large Wilson loops directly. In the continuum, this is
complicated by path ordering and large self-energy contributions which obscure the extraction of
thetruepotential fromlargeWilsonloops[9].
In this talk, I report on a recent approach [10] to compute the Wilson potential in the Hamil-
tonian approach in Coulomb gauge. A careful analysis of Gauß’ law and the physical degrees
of freedom allows for an elegant way to isolate and remove the self-energy contributions to the
Wilson loop, while unitary transformations are applied to take care of time evolution. Asspecific
examples, westudy quantum electrodynamics (or rather Maxwell’s theory) in D=(3+1) space-
time dimensions, and YM theory in both D=(1+1) and D=(3+1). In the latter case, only an
approximate solutioncanbefoundandIbrieflydiscusspossible waystoimproveourfindings.
2. The Wilsonloopinthe Hamiltonianapproach
We study SU(N) YM theory in euclidean space using the Hamiltonian approach in Weyl
gauge, A = 0. The physical energy states of this formulation are stationary wave functionals
0
Y n[A(x)] e−iEnt whichobeytheSchrödingerequation basedontheHamiltonian
·
g2 1
H = dxP 2(x)+ dxB2. (2.1)
2 2g2
Z Z
Here, Aa(x)isthegaugeconnection, P a(x)= id /d Aa(x)itsconjugate momentum, gthe(bare)
−
coupling constant andBa=(cid:209) Aa 1fabcAb Ac thenon-Abelian magneticfield.
× −2 ×
To proceed, we must fix the residual time-independent gauge symmetry left after imposing
Weylgauge,forwhichwechoosetheCoulombcondition, (cid:209) A=0. Thisaddsgauge-fixingterms
·
totheHamiltonian(whichwediscussbelow)andimposesGauß’lawasaconstraintonthephysical
2
TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
states: Inthesectorwithnstaticchargeslocatedatpositionsx ...,x ,wehavenadditionalcolour
1 n
indices onthewavefunctionals andGauß’lawreads(withgenerators intherightrepresentation)
n
Gˆa(z)Y (A;x ,...,x )= (cid:229) (cid:229) Ta d (z x )(cid:213) d Y (A;x ,...x ). (2.2)
i1,...in 1 n − iℓkℓ − ℓ imkm · k1,...,kn 1 n
(ℓ=0 k m=ℓ )
{ ℓ} 6
ThecompletephysicalHilbertspacethusdecomposesintoorthogonalchargesectorscharacterised
bythedatafromtheexternal charges.
Nextweconsiderarectangular temporalWilson loopofextension (T R). SinceA =0,we
0
×
haveW =tr U[A](R,0;T)† U[A](0,0;0) ,whereU[A](y,x;t)istheparalleltransporter x yat
· →
fixed time t, in the background of the gauge potential A. Using euclidean time evolution and the
(cid:8) (cid:9)
abbreviationU(x) U[A](x,0;0), wefind
≡
W =tr 0 U(R)†e (H E0)TU(R) 0 =tr R e T(H E0) R . (2.3)
− − − −
h i h | | i h | | i
Thesalientpointisnowthatthevacuumisgauge-invariant, whiletheWilsonstate R =U(R) 0
| i | i
is in the qq¯-sector according to eq. (2.2). Inserting a complete set of energy eigenstates from all
sectors, onlytheqq¯-sectorwillthuscontribute andwehave
W =tr(cid:229) 0 U†(R) n e (H E0)T n U(R) 0 =(cid:229) e T(En(qq¯) E0) R n 2, (2.4)
qq¯ − − qq¯ − − qq¯
h i h | | i · · h | | i kh | i k
n n
where W 2 tr(W †W ). In the limit of large Euclidean time extensions, we project out the Wil-
k k ≡
son potential as the difference between the ground state energies with and without static quarks,
E0qq¯(R)−E0=−limT→¥ T−1lnhWi.
ThenextstepistoremovetheWilsonlinefromthestate R bymeansofaunitarytransforma-
| i
tionbasedontheparalleltransporterU(R). Undersuchatransformation,theconjugatemomentum
(andthustheHamiltonianeq.(2.1))acquiresanadditional term,P a(x) P a(x)=P a(x)+e a(x),
i → i i i
where
e
R
e a(x) dyd (x y)U(y)†iTaU(y). (2.5)
≡ 0 −
Z
Asaconsequence, theWilsonloopisnowcomputableasazero-charge vacuumexpectation value,
W =tr 0 e T(H E0) 0 (2.6)
− −
h i h | | i
e
whereH differsfromeq.(2.1)bythemomentumshift1 e (x).
To illustrate the problems in extracting the true potential, let us consider the contribution to
theWilesonloopeq.(2.6)whichisquadratic intheinduced electricfielde (x),
g2 g2
V (R)= tr dx[e a(x)]2 = C d (2)(0) R , (2.7)
ind 2
2 2 ·| |
Z ⊥
whereC is the quadratic Casimir operator for the colour group, and the (quadratically) divergent
2
prefactor d (0) is due to the infinitely thin Wilson lines. Although V is formally confining, it
ind
⊥
1ThisshiftcomplicatesthetreatmentofHconsiderably,since[P a(x),e b(x)]=0.
6
e
3
TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
must be spurious because it exists even for G=U(1). On the other hand, it is also not correct to
simplydropV ,because thetruepotential ishiddenunderneath thedivergence: ForG=U(1),
ind
g2 g2
V (R)= tr dx[e (x)]2+ tr dx e (x) 2, (2.8)
ind
2 ⊥ 2 k
Z Z
(cid:2) (cid:3)
wherethefirstfactorrenormalizestheWilsonloop,andthesecondfactorgivesthetrue(Coulomb)
potential, cf.section3.1.
Theunitarytransformation leadingtoeq.(2.6)onlyassumesWeylgaugeandthusworkswith
orwithoutCoulombgaugefixing. Incaseswheretheexactgauge-invariantgroundstateisknown,it
isnotnecessarytofixtheresidualgaugefreedomsincetheWilsonloopisgaugeinvariant. Inmore
realistic cases, however, the gauge invariant ground state willnot be known and wehave to resort
to approximations based on Coulomb gauge. The required gauge-fixing is virtually impossible to
effect directly in eq. (2.6), except for the Abelian model. A much better strategy is to go back
to eq. (2.3), perform the standard gauge fixing and resolve Gauß’ law exactly, which induces in
H H thewell-knownCoulombterm,2
fix
→
g2
HC= 2 d3(x,y)J−1[r a(x)+r dayn(x)]J Fab(x,y)[r b(y)+r dbyn(y)], (2.9)
Z
whereJ istheFaddeev-PopovdeterminantandFabthetheso-calledCoulombkernel. TheWilson
loopnowreads
W =tr R e T(Hfix E0) R (2.10)
− −
h i h ⊥| | ⊥i
wherethenewWilson-likestate R =U[A ](R)0 isnolongerfromtheqq¯sector,butratherhas
| ⊥i ⊥ | i
overlapwiththezero-charge vacuum. Anydivergences associatedwiththepoint-likecrosssection
of of the Wilson lines in R must therefore be unrelated to external charges: as we will see,
the extra divergences renor|m⊥aliize the composite operator W W Z (L ) W, but they do not
R W
→ ≡ ·
affect the physical potential. Whether or not the state R in eq. (2.10) is subsequently removed
| ⊥i
from theWilson loop by aunitary transformation U (R)asdescribed earlier becomes amatterof
⊥
computational convenience.
3. Applications
3.1 QuantumElectrodynamics
In the Abelian case G =U(1), the exact ground state is known and we can first attempt a
gauge-invariant treatmentbasedoneq.(2.6). Theinduced electricfield
R
e (x)= dyd (3)(x y). (3.1)
−
Z
0
isnowfield-independent, sothattheinduced potential (2.7)could bepulled outoftheexpectation
value W . Inviewofthediscussion above,itismoreconvenient tofirstsplite =e +e andpull
⊥ k
h i
2Asaconsequenceoftheparalleltransport,thenon-Abelianchargesr a(x)=d (x R)iTa d (x)U(R)iTaU(R)†
− −
arefield-dependent.
4
TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
outthelongitudinal pieceonly,hWi=e−TVk(R)·h0|e−T(HQ′ED−E0)|0iwhere
e
1 1
H = d3x g2 P (x)+e (x) 2+ (cid:209) A (x) 2 . (3.2)
Q′ED 2 ⊥ ⊥ g2 × ⊥
Z (cid:26) (cid:27)
(cid:2) (cid:3) (cid:2) (cid:3)
Itisnowconvenienettoreversetheunitarytransformationthatledtoeq.(2.7),butthistimewiththe
transversal gauge connection A only. As a result, the shift in the electric field is removed from
⊥
the Hamiltonian (3.2) and reshuffled into the state R =U[A ](R) 0 U (R) 0 . Although
⊥ ⊥ ⊥
| i | i≡ | i
this new state resembles the initial Wilson state, it is gauge-invariant as it depends on A only.
⊥
Inserting a complete set of eigenstates of the standard QED Hamiltonian eq. (3.2) (without e )
⊥
yields
hWi=e−TVk(R)·(cid:229) |h0|U⊥(R)|ni|2e−T(En−E0)T−→→¥ |h0|U⊥(R)|0i|2·e−TVk(R), (3.3)
n
provided that R has non-vanishing overlap with the true vacuum. For Maxwell’s theory the
⊥
| i
ground state isGaussian and the relevant matrix element can be computed exactly. Using aO(3)-
invariantUVcutoffL forthemomentumintegration,wefindthatZ formallyvanishesinthelimit
W
L ¥ because R haspooroverlapwiththetruegroundstate[9]. Ontheotherhand,Z isalso
⊥ W
→ | i
independent of the temporal extension T of the loop and hence cannot contribute to the physical
potential.
The correct interpretation is given by the operator product expansion (OPE): Since the loop
operatorW contains productsoffieldoperatorsatarbitrarily closepoints, weexpectshortdistance
(L ¥ ) divergences associated with W, on top of the counter terms necessary to render Green
→
functions finite. In fact, it has been known fora long time [11, 12]that one overall multiplication
W W Z 1 W is sufficient to render W =e TV finite. From eq. (3.1), the true physical
→ R ≡ W− · h Ri − ⊥
potential becomes
g2 g2 g2
V(R)=V (R)= dx[e (x)]2= . (3.4)
k 2 k 4p R−4p 0
Z | |
ThisisjusttheusualCoulombpotential including theself-energy ofthecharges.
TheentirederivationcouldalsoberepeatedinaCoulombgaugefixedformulation. Ifwestart
from eq. (2.6), the gauge fixing gives H H =H +V , cf. eq. (3.2), and the computation
→ fix Q′ED k
becomes identical to the gauge-invariant treatment above. Alternatively, we could also start from
eq.(2.3)andresolveGauß’lawforthesetate eR explicitly, whichinduces theCoulombterm H
| i →
H =H [A ]+H in theHamiltonian. Thistime, the Coulomb potential comes directly from
fix QED C
⊥
H ,whilethefirstterminH onlyrenormalizes theWilsonloop.
C fix
3.2 Yang-MillsTheoryinD=1+1
This model is (almost) topological and requires anon-contractible spactime manifold to give
nontrivial results. Since time must be unrestricted in the Hamiltonian formalism, we compactify
space toaninterval [0,L]withperiodic boundary conditions andtake spacetime tobethe cylinder
M= S1. Forthismodel,theinducedpotential eq.(2.7)givesthecorrectstringtension
R×
s = g2C SU=(2) 3g2. (3.5)
1+1 2
2 8
5
TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
In view of the above discussion, it is, however, unclear why this should be the correct answer, in
particular since V =s R is not periodic or invariant under R L R as expected from the
ind 1+1
→ −
compactification.
To address this issue, we go back to eq. (2.3) and fix the minimal Coulomb gauge ¶ Aa =0
1 1
and Aa =diag. Taking the colour group G=SU(2) for simplicity, we are thus left with only one
1
scalardegreeoffreedom
1
J A3L [0,p ]. (3.6)
≡ 2 1 ∈
The restriction to the compact interval (the fundamental modular region in this case) eliminates
all residual gauge symmetries. In order to fix eq. (2.3), we have to determine the gauge fixed
Hamiltonian intheqq¯-sectortowhichtheWilsonstate R belongs. Thishastwopieces
| i
g2
H =H0 +H =H + d(x,y)r a(x)Fab(x,y)r b(y), (3.7)
fix fix C fix 2
Z
wherethenon-AbelianchargeinteractioncomesfromtheresolutionofGauß’law. Thezero-charge
Hamiltonian H0 and its spectrum have been studied thoroughly in ref. [13]. As for the Coulomb
fix
term, we have to take into account that the external charges r a are field-dependent due to the
paralleltransport. Expandingallcolourvectorsinaapolarbasis(e =e ,e =(e e )/√2),we
0 3 1 2
± ±
find
HC= g2 (cid:229) 1 ts ts† 2Fs (0;J ) e−2is (R/L)J Fs (R;J )+cc , (3.8)
8 −
s = 1
− n h io
where ts are the polar Pauli matrices. The Coulomb kernel Fs (R;J ) in the polar basis has also
been determined in ref. [13] by a Fourier expansion which is strictly periodic in R. In the first
period R <L,theseriescanactually beresummedtogivethesimpleresult
| |
g2 R2
H = 3R V (R) ; (3.9)
C C
8 | |− L 1≡ 1
(cid:20) (cid:21)
for R >L it must be continued periodically. Quite surprisingly, this turns out to be independent
| |
of the gauge field J and can thus be pulled out of the gauge-fixed expectation value eq. (2.3).
Inserting againthecompletesetofeigenstates J n =Y (J )ofH0 ,wearriveat
h | i n fix
W =tr R e−T(Hfix+HC−E0) R T→¥ tr 0U(R)0 2 e−TVC(R). (3.10)
h i h | | i −→ |h | | i| ·
The prefactor Z = tr 0U(R)0 2 can be computed explicitly and turns out to T-independent
W
|h | | i|
(and finite). Again, Z isthe OPErenormalization constant forthecomposite operatorW and we
W
obtain the renormalized Wilson loop WR =e−TVC(R). SinceVC(R) is periodically continued for
h i
R >L,wehavetogotothelimitL Rtoavoidfinitesizeeffects,andtheeffectivestringtension
| | ≫
becomes
dV (R) 3
s = C = g2. (3.11)
1+1
dR 8
(cid:12)R=0
(cid:12)
Thisagreeswitheq.(3.5)andisalsotheknownexp(cid:12)ressionfromtheliterature. Itremainstoexplain
(cid:12)
why the physical potentialV (R) is periodic in R, but not invariant under R L R as it should
C
→ −
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TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
be on a compactified [0,L] S1. A careful analysis [10] along the lines in sec. 2 reveals that the
≃
physical potential between static quarks is not given by any single Wilson loop once the space
direction iscompactified. Instead, alltheequivalent loopswithR R+mL(m )compete and
→ ∈Z
the one giving the minimal potential dominates in the limit T ¥ . Taking this prescription into
→
account, thelevelcrossing between R and R L atR=L/2yieldsapotentialV(R)thatisboth
| i | − i
invariant andsymmetricabout R=L/2.
3.3 Yang-MilsTheoryinD=3+1
ForthephysicallymostinterstingcaseofYMtheoryinD=3+1,westartagainfromeq.(2.3)
andintroducetheusualCoulombgaugefixingintheHamiltonian,combinedwiththeresolutionof
Gauß’lawintheqq¯sectortowhichtheWilsonstate R belongs,
| i
W =tr R e−T(HYfixM+HC[rdyn+r ]−E0) R (3.12)
h i h ⊥| | ⊥i
Inviewoftheapproximationsbelow,itisexpedienttoperformtheunitarytransformationfromsec-
tion 2andreshuffle the parallel transporter U[A ](R)from the Wilson state into theHamiltonian.
⊥
Asaresult, wehave
hWi=trh0|e−T(HYfixM P →P +e+HC[rdyn+rind+r ]−E0)|0i≡trh0|e−T(Hfix−E0)|0i, (3.13)
(cid:12)
where the momentum P in the g(cid:12)auge-fixed vacuum Hamiltonian is shifteed by the induced elec-
tric field eq. (2.5), and the Coulomb term eq. (2.9) will now contain an induced charge r =
ind
[ A ,e ], in addition to the dynamic charge r = [ A ,P ] of the gluon and the external
⊥ ⊥ dyn ⊥ ⊥
− −
charge r of the static quarks. To proceed, we decompose the complete Hamiltonian H from
fix
eq.(3.13)intothreepieces,H =H0 +V +D H,whereH0 istheusualCoulombHamiltonianin
fix fix C fix
the absence of external charges, V is the non-Abelian Coulomb interaction between theeexternal
C
charges, andD H containingetheremaininge(indirect) chargeinteractions.
In order to evaluate eq. (3.13e) with this decomposition, we have resort to a sequence of ap-
proximations, whicharenotallundergoodcontrol:
1. Jensen’s inequality: W trexp T 0Hfix E 0 =trexp T 0 V +D H 0
0 C
h i≥ − h | − | i − h | | i
h i h i
2. Abelization: [U (R),Ta] 0
⊥ e e
≈
3. Factorization: 0QaFabQb 0 0QaQb 0 0Fab 0 0QaQa 0 F¯
′ ′ ′
h | | i≈h | | ih | | i≡h | | i
Approximation (1.) allowstodrop H0 ,while(2.) removestermsinvolving theconjugate momen-
fix
tum P and e , and (3.) simplifies the remaining pieces in D H and, in particular, the Coulomb
⊥
interaction, V . After a lengthy calculation [10], the Wilson loop is thus reduced to a form
C
which requires, as only input from the ground state, the gluon propagator 0Aa(x)Ab(y)0 =
h | i j | i
d abt D(x ey)andthevev.oftheCoulombkernel, 0Fab(x y)0 =d abF¯(x y):
ij
− h | − | i −
g2
W exp T N d3(x,y)e (x) N 1d (x y)+F(x y)D(x y) e (y)
h i≈ − 2 C ⊥a C− − − − ⊥a ×
( Z )
h i
exp TC g2 F(R) F(0) . (3.14)
2
× − −
( )
h i
7
TemporalWilsonloopintheHamiltonianapproachinCoulombgauge MarkusQuandt
The first term in the exponent is again the self-energy of the Wilson lines which gives rise to
a renormalization Z of the loop operator. The large distance behaviour of the second term
W
is dominated by the small momentum form of the correlators, for which we use the relations
D(k) [k2+M4/k2]−12 and F¯(k) k−4 suggested by both variational calculations [4] and recent
∼ ∼
lattice simulations [7]. It can then be shown that the second piece in the exponent of eq. (3.14)
is subdominant ( R 1) at large distances. The dominant contribution to the Wilson potential is
−
∼
given, in our approximation, by the explicit Coulomb interaction, and the Wilson string tension
equalstheCoulombstringtension, s s .
W C
≈
4. Conclusions
In this talk, I have presented a recent calculation of the Wilson potential in the Hamiltonian
approach to Yang-Mills theory. A careful analysis of Gauß’ law and the physical degrees of free-
dom combined withunitary transformations totake careoftimeevolution revealthat thespurious
divergencesintheWilsonpotentialcanbeabsorbedinOPErenormalizationsofthecompositeloop
operator. Usingthisformalism,thecorrectresultcouldbe derivedincaseswheretheexactground
state isknown. Forthephysically mostinteresting case ofYang-Mills theory in D=(3+1), only
anapproximate solution could beobtained inwhich theWilson and Coulombstring tension equal
(while s /s 0.3...0.5 in lattice simulations). The main reason for this discrepancy is most
W C
≃
likely theapproximate factorization usedinthecalculation, whichneglectsthescreening ofexter-
nal charges by gluons. One-loop calculations of this contribution indeed indicate a reduction of
s ,andfurtherinvestigations ofthisissueareinpreparation.
W
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