Magnetic charge of finite lifetime in SU(2) gluodynamics

12 Pages·0.25 MB·English

by  Ralf Hofmann

EPJ manuscript No.
(will be inserted by the editor)
Magnetic charge of finite lifetime in SU(2) gluodynamics
Ralf Hofmann
1
0
Theoretical Physics Institute, Universityof Minnesota, Minneapolis, MN 55455, USA
0
2
present address: Max-Planck-Institutfu¨r Physik,F¨ohringer Ring6, 80805 Mu¨nchen, Germany

n
a
J
Received: date/ Revised version: date
9
2

Abstract. Aself-dual,localizedsolutiontotheclassicalSU(2)Yang-MillsequationinEuclideanspacetime,
1
v
7 which formally possesses infinite action, is investigated in view of its U(1) charge content after Abelian
1
3 projection.Thisissuggestedbynotingthatthesolutionsatisfies’tHooft’sdifferentialprojectioncondition
1
0 away from the singularities. As a result the existence of dynamical, magnetic charge of finite lifetime is
1
0 established. A covariant cutoff for the action is introduced by demanding the solution to be close to an
/
h
p instanton topologically. This is in analogy to the calculation of the mass of a point charge in classical
-
p electrodynamics or the subtraction of diverging self-energies of magnetic monopoles as discussed in the
e
h
literature. The Wilson loop is evaluated in the background of a dilute gas. Assuming identical integrals
:
v
i over size distributions, the corresponding static quark/anti-quark potential at infinite spatial separation
X
r can be seizably higher than thepotential in a dilute instanton gas.
a
PACS. ? ? – ? ?
1 Introduction tialsuccessesinexplainingthelowenergyphenomenology
of strong interactions, such as spontaneous chiral sym-
Quantum Chromodynamics (QCD) [1] is widely accepted
metry breaking and the axial U(1) problem, have been
as the theory of strong interactions due to its experi-
achievedthisway(seeref.[4]andreferencestherein).How-
mentally verifyable feature of asymptotic freedom in the
ever,the basicobservationthatnofreecolorchargesexist
UV momentum regime (see for example ref.[2] and ref-
in nature is understood only poorly so far.
erences therein). A perturbative treatment of the theory
in the IR domain is excluded by a large coupling con- Thedualsuperconductorpictureofthenon-perturbative
stant.Non-perturbativetechniquesarenecessaryandpar- QCD vacuum dates back to the mid seventies [3]. It pro-
2 Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics
vides an appealing mechanism for color confinement due loopisinfinitesimal[11,12].Theinvestigationoffieldcon-
tothe formationofcolorelectricfluxtubes linking colored figurations, which are not exact solutions to the Yang-
objects immersed in a condensate of magnetic charges. Mills equations (instanton/anti-instanton pair), are usu-
The resulting potential, linear in the (large) separation ally carried out on the lattice. Since the lattice provides
betweenthecharges,isdeterminedbyaneffective,univer- a UV cutoff and thereby has a limited resolution and
sal scale the string tension. The occurence of magnetic since only approximative solutions have been looked at
−
chargesinSU(n)Yang-Millstheorywasmadetransparent the claimed detection of finite monopole loops is not well
inthe processofthe so-calledmaximalAbeliangaugefix- established.
ing (MAGF) [6].Due to this fixing gaugetransformations
The purpose of this work is to point out the presence
belonging to the maximal Abelian subgroup U(1)n 1 of
−
of long-lived magnetic charge due to exact, self-dual so-
SU(n) are unconstrained and the resulting Abelian gauge
lutions to the SU(2) Yang-Mills equations and to inves-
theory has electric and magnetic charges. A rather suc-
tigate their effect on the static quark potential at infi-
cessful model the dual Ginzburg-Landau theory has
− −
nite separation. The fact that a solution possessing infi-
been employed to effectively mimic the low energy sector
nite action has a vanishing contribution to the partition
ofSU(n)Yang-MillstheoryafterMAGF[7].Toclassically
function is delt with by an appropriatesubtractionin the
describethedeconfinementphasetransitioninducedbyfi-
spiritofref.[5].Thedivergenceoftheactionistracedback
nitetemperatureanon-renormalizableversionofthedual
to singularies of the action density on a hypersurface. In
Ginzburg-Landau theory has been put forward recently
ref.[6] the appearance of magnetic monopoles in SU(n)
[8].
gauge theory was made manifest after imposing a non-
Yetnoanalyticalderivationofmagneticmonopolecon- propagatinggaugewhichleavesthemaximalAbeliansub-
densation has been given although rather impressive re- group U(1)n 1 invariant. This gauge condition demands
−
sults were obtained from the lattice [9]. However,the lat- that a matrix quantity, say the field strength component
tice detection of monopole trajectories has its inherent F12, transforming homogeneously under gauge transfor-
uncertainties (see below). Exact classical solutions to the mations, be diagonal. In particular, for the case of SU(2)
Yang-MillsequationssubjectedtoAbelianprojectionhave it is easy to shown that at points, where this condition is
been investigatedmore recently in view of their magnetic noconstraintonthecorrespondinggaugetransformations,
chargecontent[10,11,12].Concerningchargesoffinitelife- theinhomogeneouspartofthegaugetransformationofA
µ
timetheresultsarenotconclusiveinthecaseofinstantons is singular and contains an Abelian, magnetic monopole
since either the gauge fixing functional diverges for non- [13]. Due to the Coulomb-like behavior of the magnetic
infinitesimal monopole trajectories [10] or the monopole field strength in the vicinity of such a point the action
Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics 3
of the field configuration formally diverges, and hence it found a long time ago in ref.[15]. These solutions have
mustberegularizedphenomenologically.Thesolutiondis- finite classical action S = 1/(4g2) d4xFa Fa = 8π2/g2
µν µν
R
cussedobeys the differentialAbelian projectioncondition and spontaneously break eight of the symmetries of the
away from the singularities, and hence the restriction to classical theory. They are (anti)self-dual, that is
its diagonal component is suggested.
Fa = ε Fa , (1)
In the present work magnetic charge is identified by µν ± µνκλ κλ
means of the Coulomb-like behavior of the field strength
and are known as instantons. As first suggested by Gri-
ofafictitiouschargemovingtoward(awayfrom)aspatial
bov instantons can be interpreted as tunneling trajecto-
point which turns into a singularity at a certain pair of
ries linking topologically distinct vacua [16]. In the last
time slices. The definition of the physical action is per-
20yearsor so there has been anenormousindustry relat-
formed in analogy to defining the mass of a static point
ingthepresenceofthesesolutionsinthenon-perturbative
chargein classicalelectrodynamics.A covariantcutoff for
QCD vacuum to low energy phenomenology, namely the
theactionfunctionalisintroducedwhich,inprinciple,can
U(1) problem and spontaneous chiral symmetry break-
A
be determined phenomenologically.
ing(seeref. [4]andreferencestherein).However,theissue
Summarizing, the line of reasoning in this work is as
of color-confinement, which should be an important con-
follows:Classical,self-dual,infinite actionsolution,which
sequence of QCD, has not been satisfactorily settled by
is regular almost everywhere, possesses a converging ac-
the instanton calculus [17,19].
tion integral in the infinite, and is in Abelian gauge (Sec-
ForgaugegroupSU(2),insingulargauge,andforcen-
tion 2) presence of magnetic charge infinite ac-
→ ⇒ ter z =0 the instanton solution is given as
tion (Section 3) phenomenological action regulariza-
→
x
tion (Section 4). In addition, the static quark/anti-quark Aaµ =2η¯aµνρ2x2(x2ν+ρ2) , (2)
potential in a dilute gas is calculated for infinite spatial
with η¯ i tr(τ τ τ+), τ = (τ, i), and τ the
separation in Section 5. Thereby, the integral over size aµν ≡ −2 a µ− ν µ± ∓ a
Paulimatrices.Thesolution(2)canbeobtainedasfollows
distribution is assumed to be identical to that of the in-
[20]. Making the ansatz
stanton case. The result is compared to the potential due
to a dilute instanton gas. Aa =η¯ x 1+φ(x2) , (3)
µ aµν ν x2
the action takes the form
2 Classical solution
8π23 1 1 x2
Aclassofwinding,BPSstaturatedsolutionstothe SU(2) S = dτ (φ)2+ (φ2 1)2 , τ ln ,
g2 2 2 ′ 8 − ≡ ρ2
Z (cid:26) (cid:27) (cid:18) (cid:19)
classicalYang-MillsequationsinEuclideanspacetimewas (4)
4 Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics
where ρ is some length scale. A BPS saturated vacua in- Thesurfaceofdivergenceof(8)isthe3-spherex2 =ρ2.
terpolation of the theory (4) is For this reason it is refered to as spheron throughout the
τ remainder of the paper. Writing the spheron as
φ= tanh . (5)
− 2
(cid:16) (cid:17) ρ2
Aa = η¯a ∂ log 1 , (10)
Substituting this into eq.(3), one obtains the solution of µ − µν ν − x2
(cid:18) (cid:19)
eq.(2). Indeed, evaluating the action yields itisobviousthatawayfromthesingularitiesthefollowing
8π23 τ 8π2 two sets of conditions are fulfilled
4
S = g2 8 dτcosh− (2)= g2 . (6)
Z
AnotherBPSsaturatedsolutiontotheequationofmotion ∂µ±iA3µ A±µ =0 , ∂µAaµ =0 , (11)
(cid:0) (cid:1)
corresponding to (4) is obtained by letting τ →τ +τ0 in where A±µ ≡ A1µ ±iA2µ. The first set expresses the fact
(5). Thereby, τ0 may be complex as long as the new solu- that the spheron is in ’t Hoofts maximal Abelian gauge
tionisreal.Thisisnecessarytohavearealaction.Admis-
(MAG)[6]whichleavesaU(1)subgroupofSU(2)unfixed.
sible complex shifts are τ0 = inπ with n ∈ Z. Choosing Globally,this is mirroredby a minimum ofthe functional
τ0 =iπ yields
1
τ G [A]= d4x A1(x)A1(x)+A2(x)A2(x)
φ coth . (7) M 4 µ µ µ µ
→− 2 Z
(cid:16) (cid:17) = 2 d4x (cid:8) ρ4 (cid:9)(12)
Note that (5) and (7) behave identically in the infinite. x2(x2 ρ2)2
Z −
For the Yang-Mills potential eq.(7) implies undergaugetransformations[11]ifthedivergenceonx2 =
A A = 2η¯ ρ2 xν . (8) ρ2 is cut out (see Section 4). Obeying of the second set
µ → µ − aµν x2(x2 ρ2)
− ofconditions in(11)indicatesthat the spheronis Lorentz
Thecorrespondinganti-solution(negativetopologicalcharge
gauged. Restricting oneself to the diagonal component,
density) is obtained by substituting η¯ η . For-
aµν aµν
→ which is suggested by MAG and Abelian dominance (see
mally, the action of solution (8) reads
for example [21]), the Lorentz gauge guarantees the min-
8π23 τ
S = g2 8 dτsinh−4(2) . (9) imization of the functional
Z
1 ρ4
It diverges due to the quartic pole of the integrand at G [A]= d4xA3(x)A3(x)= d4x
L 4 µ µ x2(x2 ρ2)2
Z Z −
τ =0andnotduetoadivergingintegralintheinfinite.An (13)
infinite action of a classicalsolution generally is sufficient under U(1) gauge transformations [14].
causetodisregardthissolutionongroundsofitsvanishing Backtothe problemofadivergingaction:Itisargued
contribution to the partition function of the theory. As it inthefollowingthatourpoorunderstandingofthenature
will be explained below there is physical information in ofchargedparticlesinclassicalelectrodynamicsmustnec-
the solution (8), and hence it will not be dismissed. essary carry over to a classical, Abelian projected SU(2)
Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics 5
(and more general SU(n)) gauge theory in which the ap- x4 ρ there is a 2-sphere of divergence with radius
| | ≤
pearance of localized U(1) charges is expected [3,6]. The ρ˜(x4) = ρ2 x24. At x4 = ρ a pointlike singularity in
− −
p
term”poorunderstanding”referstothefactthattheclas- 3-space emerges which starts expanding with Minkowski-
sical self-energy of a static point charge is divergent. In space velocity v(x4 = ρ) = 1/√2. The 2-sphere reaches
−
classical electrodynamics the usual procedure is to con- its maximal radius ρ˜= ρ at x4 = 0 with velocity v(x4 =
strain the corresponding energy functional to regions of 0) = 0, and shrinks to a point at x4 = +ρ. At times
spacewith a regularbehaviorofthe integrandto produce x4 > ρ there is no 2-sphere, and the field is regular in
| |
the observedmass ofthe associatedparticle.In the quan- 3-space.
tum theoryofelectrodynamics(QED) the mere introduc-
tion of a mass parameter into the Lagrangian accounts
in an effective fashion for the aforementioned process in
Theunderlyingtheoryismatter-freeandnon-Abelian.
the limitofzerocoupling1.Atnonzerocouplingthe mass
The luxury of placing a static δ-source on the right-hand
startstorun(albeitslowlyinQEDduetoanIRconformal
side of Maxwell’s equation to obtain a Coulomb law for
point)with the momentumprobingit.Goingfromthe IR
the corresponding field strength is absent. If attributed
regime up to large momenta (yet below the Landau pole)
to the spatial divergences of the spheron field the life of
the gross contribution to the mass is of a classical origin
a U(1) charge is highly dynamic and of limited duration.
astheweaklogarithmicdependenceonmomentumshows.
Hence, there are strong retardation and threshold effects
FromtheviewpointofAbelianprojectionitisthusnotim-
which necessarily blurr the Coulomb picture. Moreover,
perative to disregarda classicalsolution of infinite action
in the case of accelerated magnetic charge the conserva-
due to divergences of the action density on hypersurfaces
tion of magnetic flux can be realized by a smeared flux
if these divergences can be attributed to U(1) charge in
distribution in contrast to the static Dirac string. Thus,
the Abelian projected theory (see ref.[5] where the sub-
at x4 = ρ there may be divergences of B2 higher than
±
tractionof aninfinite self energywasmade explicitin the
quartic in x. It is then not justified to read off the mag-
| |
case of 2+1 dimensional QCD).
netic charge from the ”residue” belonging to a fourth-
order Coulomb pole x = 0 of B2 at x4 = ρ. However,
| | ±
3 Magnetic charge
if there is charge a Euclidean space observer placed at
x =0 measures squares of field strengths that are quar-
In this Section the consequences of Abelian projection | |
applied to the spheron are investigated. For fixed times ticallydivergentasx4 →−ρ.Closeto the singularitythis
1 In principle, the mass of the particle can then still be de- behavior can be attributed to a fictitious charge moving
termined gravitationally. uniformely at Euclidean ”speed-of-light” along one of the
6 Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics
spatial coordinate axis toward the observer2. At x4 = ρ the electric charge e is read off as
−
the hypothetical charge materializes, that is, there is a
4π
e = . (17)
| | g
genuinesingularityatx=0.After Abelianprojectionthe
squares of electric and magnetic chargescan thus be read For B2 one finds
off as the ”residues” of the fourth-order Coulomb pole at
E2 ε ∂ A3 ∂ A3 2 =4 (∂ A3)2 ∂ A3∂ A3 ,
x4 = ρ (x = 0) in E2 and B2, respectively. At x4 = ρ ≡ ijk j k− k j m n − m n n m
− (cid:0) (cid:2) (cid:3)(cid:1) (cid:0) (cid:1)
(18)
(x=0) a similar argument applies. The situation is sum-
where the expression in terms of derivatives of h(x2) has
marized in Fig. 2.
beenomitted.The”residue”oftheCoulombpolemturns
Usingeq.(10)andthepropertiesoftheη-symbols,one
out to vanish. Eqs.(17) signals that e fulfills the Dirac
easily obtains
condition with respect to the SU(2) coupling g. Thus it
E2 ≡ ∂4A3i −∂iA34 2 = ∂4A3i 2−2∂4Ai∂iA34+ ∂iA34 2 should be interpreted as a magnetic charge3. This can be
= (cid:0)∂1∂4h(x2) 2+(cid:1) ∂2(cid:0)∂4h(x(cid:1)2) 2+ ∂42h(x2) 2(cid:0)+ (cid:1) mademanifestbyexploitingthefollowingchargerotation
(cid:0)∂1∂3h(x2)(cid:1)2+(cid:0)∂2∂3h(x2)(cid:1)2+(cid:0)∂32h(x2)(cid:1)2 symmetry of Maxwell’s equations [6]
−
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) ρ2
−2∂42h(x2)∂3∂4h(x2) , h(x2)≡log(1− x2)(1.5) m →mcosα+esinα ,
e msinα+ecosα , (19)
For x4 ρ and x = 0 eq.(15) reduces asymptotically →−
→ ±
to
which allows to rotate e into m for α = π. In Fig. 3 the
2
1
E2x4→±ρ,x=0 = x4 ρ4 . (16) angular distribution of E2 at three different instants is
| ∓ |
Convertingfromthenon-perturbativetotheperturbative depicted.
definition of A by virtue of A 1A and comparing The concentration of field strength in the ”northern”
µ µ → g µ
with the standard Coulomb expression, the magnitude of hemisphere at x4 = ρ evolves to a mirror symmetric
−
2 TheLorentzboostconnectingtherest-frameofthischarge distribution atx4 =0.At x4 =ρ field strength is concen-
to that of the observer is a O(4) rotation T of angle π2. For a trated in the ”southern” hemisphere.
charge moving along the1-axis one has
At this point it is worth comparing the above results
0 10 0 to the work of ref.[11]. There, an infinitesimal magnetic
 
−100 0 monopole loop was obtained by looking at singular gauge
T =  . (14)
 
 0 01 0 transformationsappliedtoaninstantoninsingulargauge.
 
 
 
 0 00 1 Thesetransformationswereparametrizedbyaradiusscale
 
 
It is easy to check that the boost T leaves the field strength 3 The corresponding quark doublet would have electric
components E1 and B1 invariant. charge ±21g [6].
Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics 7
R deviding spacetime into ”outer” and ”inner” regions, electrodynamicsthecutofffortheenergyfunctionalofthe
withthegaugetransformedinstantoninsingularandreg- electron’s Coulomb field is determined from its measured
ular gauge, respectively. The MAG functional turned out mass. Similarily, estimating the mean topological suscep-
to be stabilized at R = 0, that is, back at the instanton. tibilityofthevacuum ε Fa Fa intermsofthemea-
µνκλ κλ µν
(cid:10) (cid:11)
Now, the instanton has finite action. Does this contradict suredmassesmη′,mK,mπ,mηbymeansoftheVeneziano-
the things said before? It does not because the identified Wittenrelation[24]orestimatingthegluoncondensateby
magnetic charge lives only infinitesimally short in such a means of QCD sum rules [25], one could, in principle, ap-
way as to keep the action finite. In ref.[11] further in- proximate the action cutoff for the spheron phenomeno-
vestigations toward the detection of finite-size monopole logically. For now the covariant cutoff τ is simply as-
m
loopswerecarriedoutonthelatticeforaninstanton/anti- sumed to produce the classical action S = 8π2/g2 of the
instanton(I-A)pair.Sincethelatticenaturallyprovidesa instanton. The cut off action integral for the spheron is
UV cutoff field strength divergences remain hidden. Fur- evaluated as
thermore, the I-A system is not an exact solution to the 8π2 1 1
3
Ss = g2 2 − 8[cosh(3τm/2)−3cosh(τm/2)]sinh− (τm/2) ,
equation of motion and the results of an abelian projec- (cid:18) (cid:19)
τ log(x2 /ρ2) . (20)
tion may heavily depend on the chosen ansatz for a vari- m ≡ m
ational principle approach4. Thus, the claimed detection Setting S = 8π2/g2, one obtains τ = log3. In units of
s m
of monopole loops in the I-A system using the lattice [11] ρ the minimal distance ξ to the sphere consequently is
remains questionable. ξ exp(τ ) 1 0.73. For this particular value the
m
≡ − ∼
p
MAG functional of eq.(12) yields
4 Regularized action 1
G [A]=4π2ρ2 =2π2ρ2 (21)
M
exp(τ ) 1
m
−
InthisSectionacovariantdefinitonofthephysicalspheron asopposedtoG [A]=4π2ρ2fortheinstantoninsingular
M
action is suggested. For the spheron to be of relevance its
gauge. In Section 5 the cutoff will be varied to test the
contributiontothepartitionfunction(ortunnelingampli-
sensitivity of the static quark potential.
tude in Minkowski space) must be seizable. Moreover,its
topological charge is up to the factor 8π2/g2 equal to the
5 Wilson loop in a dilute spheron gas
action (for the anti-spheron equal minus the action) and
is strictly positive (negative) for finite cutoff. In classical InthisSectionthestaticquark/anti-quarkpotentialE(R)
4 Inthestreamlineapproachthereisnostabilizationatfinite isestimatedfromthe rectangularWilsonloop5 inadilute
distance between the centers [22] whereas in the simple sum 5 Time-likelinesofintegrationseparatedbyspatialdistance
ansatz there is [23]. R and runningfrom −∞ to +∞.
8 Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics
spherongas backgroundandcomparedto that of a dilute putation of the x4 integral of eq.(23) yields
instanton gas for infinite spatial separation R . All
→ ∞ f (u)=
s
conventionsandgeneralresults,whichdoonlyexploitthe
localization property of the instanton in singular gauge, −tr exp −2iρ2σ·xZ√∞ρ2εm−x2dx4 (x24+x2)(x124+x2−ρ2)!−1!
√ε u2
m
can be taken from ref.[17]. There, E(R) was obtained as = 2 cos π 2arctan − +
− − u
(cid:18) (cid:26) (cid:18) (cid:19)(cid:27)
u ε u2
dρ 1 m
E(R)=−Z d3xρ5Dn(ρ)ntr U+(x)U−(x+R)−1 , θ(u−1)×cos(√u2−1"π+2arctan r u2−−1 !#)+
(cid:0) (cid:1)
(22) u √ε u2 √1 u2
m
θ(1 u) cos log − − − 1 . (25)
− × (√1 u2 √εm u2+√1 u2!)− !
where − − −
Using ε exp(τ ) = 3 from Section 4 and performing
m m
U± exp i dx4A4(x) , (23) ≡
≡P ±
(cid:18) Z (cid:19) the remaining space integral of eq.(24), one has
demandspath-ordering,D (ρ)denotesthespheronsize
P n dρ 1 √εm
lim E (R)= D (ρ) duu2f (u)
distribution in a dilute gas for gauge group SU(n), where R→∞ s −Z ρ2 n n Z0 s
dρ 1
quantum fluctuations are included in a semiclassical way. ∼ ρ2 nDn(ρ)×8π×1.52 . (26)
Z
ForR onehasU (x+R) +1.Inthislimiteq.(22)
−
→∞ → The result for the instanton in singular gauge is
reduces to6
u
f (u) tr U+ 1 =2 cos π 1 1 ,
i
Rlim Es(R)=− d3xdρρ5 Dn(ρ)n1 tr U+(x)−1 ≡− (cid:0) − (cid:1) (cid:18) (cid:26) (cid:20) − √1+u2(cid:21)(cid:27)−(27(cid:19))
→∞ Z
dρ 1 (cid:0) (cid:1)
≡− d3xρ5 Dn(ρ)nfs(u) , (24) which implies
Z
dρ 1
where u ≡ x2/ρ2. It is assumed here that the integral Rlim Ei(R)= −4π ρ2 nDn(ρ)
→∞ Z
over spheropn and instanton size distributions are identi- ∞duu2f (u) dρ 1D (ρ) 8π 1.10 . (28)
s ∼ ρ2 n n × ×
0
Z Z
cal7. Implementation of the covariantcutoff τ and com-
m
Hence, for a cutoff chosen to give spheron and instan-
6 Note that the authors of ref.[17] work in regular gauge
tonidenticaltopologicalchargetheρ-independentpartsof
whereasherethespheronbehaveslikeaninstantoninsingular
the integrandsare comparable.Setting ξ =1 andthereby
gauge in the infinite, thus limR→∞U−(x+R) = +1 and not
avoidingthe problemof slightly different scalesin the cut
−1 as in [17].
7 Thespheroncutoffisclosetoρfortopologicalchargeofor- off solution, the number 1.52 in eq.(26) changes to 2.41.
derone.Aconstantfactornottoofarfromunitydistinguishes ing amplitude [18,17]). There is no easy estimate of these ef-
instanton and spheron distributions since there are slightly fects, and onewould haveto perform theone-loop calculation
different scales involved and also due to the low lying non- directly along the lines of ref.[18]. Furthermore, the cutoff for
zero mode eigenvalues of gaussian operators in the respective the size integrations may phenomenologically turn out to be
backgrounds being different (one-loop contribution to tunnel- different for theinstanton as opposed to thespheron case.
Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics 9
Forevenhighercutoff,ξ =√5 1 1.24,oneobtains3.52. spheron’s topological charge takes values around unity it
− ∼
Dependingonthephenomenologicallyfavorablecutoffthis is necessary to calculate the numerical factor distinguish-
might have implications for color confinement would it ing instanton and spheron size distributions for decisive
qualitativlycarryovertothespheronliquid.Inref.[19]the statements. Moreover, it is not quite clear how much the
static quark/anti-quark potential in the instanton liquid interaction between anti-instantons with spherons (and
wasevaluatedfortwoaverageinstantonsizesρ¯.Thevalues viceversa)andtheinteractionbetweenspheronsandanti-
ρ¯=600MeV 1andρ¯=400MeV 1yieldedlim E spheronswouldalterwhatwassaidabove.Itishopedthat
− − R i
→∞ ∼
140MeV and lim E 460MeV, respectively. This theseuncertaintiescanbepositivelyeliminatedinthenear
R i
→∞ ∼
asymptotic behavior is approached rapidly (at R 2fm future.Atthisstagetheaboveresultsstillshouldbetaken
∼
one has E 110,MeV ρ¯ = 400MeV 1). Concentrating asanotherseriousindicationthatthephysicsofcolorcon-
i −
∼
on the case ξ = 1, 3.2 times these values (instanton plus finementmaybelinkedtotheexistenceoflong-livedmag-
spheron contributions) are lim (E +E ) 450MeV netic charges in the vacuum of the corresponding SU(n)
R s i
→∞ ∼
andlim (E +E ) 1.5GeVwhichincludestherange gauge theory.
R s i
→∞ ∼
of spectral continuum thresholds √s0 1.2...1.4 GeV
∼
successfully used in light-quark channel QCD sum rules 6 Summary
[25]. This means that the out-of-the-vacuum creation of
Inthisworktheidentificationoflong-livedmagneticcharges
a light pair of current quark and anti-quark eventually
in the the vacuum of SU(2) Yang-Mills theory has been
turning into the massive constituents of hadrons (con-
persued. The existence of a self-dual, infinite action solu-
ceivably not without exciting more current quark/anti-
tionto the classicalYang-Millsequation,whichobeysthe
quark pairs) would happen to a good approximation in
differential Abelian projection conditions away from its
the asymptotically free regime and thereby be supported
singularities, proved essential for this identification. Fur-
byalargephasespace.Inthispicturethenotionofabsolut
thermore, the Abelian component of this solution is in
quarkconfinementduetoaninfinitestaticquarkpotential
Lorentz gauge which suggests an interesting connection
for infinite separation seems too restrictive. It would be
to the recently discussed appearance of a mass dimension
sufficient to demonstrate that the integrated probability
twocondensateinQCD[14].Forthesolutiontocontribute
of the aforementioned pair creation process is practically
to the partition function in an essential way its physical
unity for separations comparable to or smaller than typi-
action must be small. In analogy to the definition of the
cal hadron sizes. At this point the somewhat speculative
massofastaticpointchargeinclassicalelectrodynamicsa
natureofthe aboveconsiderationsisstressed:Apartfrom
covariantcutofffortheactionhasbeenintroduced.Inthis
the question of how physical the assumption is that the
work it was chosen such as to give topological charge of
10 Ralf Hofmann: Magnetic charge of finitelifetime in SU(2) gluodynamics
orderone to the solution.The Wilson loopin a dilute gas 1. H.Fritzsch,M.Gell-Mann, and H.Leutwyler,Phys.Lett.
was evaluated, and the static quark/anti-quark potential B47, 365 (1973).
forinfinite spatialseparationwasextracted.Itturnedout D. J. Gross and F. Wilczek, Phys.Rev. D8,3633 (1973).
that this number can be seizably higher than the one ob- H. D.Politzer, Phys.Rev.Lett. 30, 1346 (1973).
2. M. Beneke, Phys. Rept. 317, 1 (1999), hep-ph/9807443.
tained in a dilute instanton gas if identical integrals over
3. S. Mandelstam, Phys.Rept. C23, 245 (1976).
size distributions are assumed.
4. T. Sch¨afer and E. V. Shuryak, Rev. Mod. Phys. 70, 323
Thequestionwhetheronecandescribelowenergyphe-
(1998)
nomena in QCD in a semi-classical way starting from its
5. G. ’t Hooft, Nucl. Phys.B138, 1 (1978).
classical Lagrangian remains. It is well possible that the
6. G. ’t Hooft, Nucl. Phys.B190, 455 (1981).
influence of quantum fluctuations generates a quantum
7. T. Suzuki, Prog. Theor. Phys. 80, 929 (1988); 81, 752
effective action at low scales which is even qualitatively
(1989).
different from the classical theory. However, the hope is
S. Maedan and T. Suzuki, Prog. Theor. Phys. 81, 229
there that at least the relevant degrees of freedom in the
(1989).
lowenergydomaincanbeidentifiedintheclassicaltheory.
8. R. Hofmann, Phys. Rev. D62, 105021 (2000), hep-
ph/0006163
9. M.N.Chernodub,S.Fujimoto,S.Kato,M.Murata,M.I.
Acknowledgements
Polikarpov and T. Suzuki,
Phys. Rev.D62, 094506 (2000) , hep-lat/0006025.
The author thanks G. Gabadadze and M. Pospelov for
10. M.N.ChernodubandF.V.Gubarev,JETPLett.62,100
stimulatingdiscussionsatanearlystageofthiswork.Use-
(1995).
fulconversationswithT.terVeldhuisareappreciated.The
11. R. C. Brower, K. N. Orginos, and C.-I. Tan, Phys. Rev.
fertileatmosphereatTheoreticalPhysicsInstituteandthe
D55, 6313 (1997).
hospitality extended to the author, in particular by M.
12. H. Reinhardt and T. Tok, hep-th/0009205.
Shifman, are gratefully acknowledged.The author thanks
13. H. Suganuma, S. Sasaki, and H. Toki, Nucl. Phys.B435,
KarinThierforherunderstandingandherendurancedur-
207 (1995).
ing the past year’s hardships of spatial separation. This
14. F.V. Gubarev and V.I.Zakharov, hep-ph/0010096.
work was funded by Deutscher Akademischer Austausch-
F.V. Gubarev, L. Stodolsky, and V.I. Zakharov, hep-
dienst (DAAD).
ph/0010057.
15. A. A.Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S.
Tyupkin,
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