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6

Yang-Mills thermodynamics: The confining phase

Ralf Hofmann

Institut fur Theoretische Physik

Universitat Frankfurt

Johann Wolfgang Goethe - Universitat

Robert-Mayer-Str. 10

60054 Frankfurt, Germany

Abstract

We summarize recent nonperturbative results obtained for the thermody-namics of an SU(2) and an SU(3) Yang-Mills theory being in its confining(center) phase. This phase is associated with a dynamical breaking of thelocal magnetic center symmetry. Emphasis is put on an explanation of theinvolved concepts.

Introduction. This is the last one in a series of three papers giving an abbreviatedpresentation of nonperturbative concepts and results for the thermodynamics of anSU(2) or an SU(3) Yang-Mills theory as obtained in [1, 2, 3]. Here we discuss theconfining or center phase.

The three unexpected results for the confining phase are the spin-1/2 nature ofthe massless (neutral, Majorana) and massive (charged) excitations, the Hagedornnature of the transition from the confining to the preconfining phase, and the exactvanishing of the pressure and energy density of the ground state in the confiningphase of an SU(2) or SU(3) Yang-Mills theory.

The first result clashes with the perception about bosonic glueballs being theobservable excitations of pure SU(3) Yang-Mills theory at zero temperature. Thisstatement seems to be supported by lattice simulations [4] and by analysis basedon the QCD-sum-rule method [5]. We have discussed in [1] why lattice simulationsof pure SU(2) and SU(3) Yang-Mills theory run into a severe finite-size problem atlow temperatures and thus are unreliable. QCD spectral sum rules [6], on the otherhand, assume the existence of a lowest resonance with finite coupling to the currentsof a given production channel (these currents are formulated as local functionals ofthe fundamental fields in the QCD Lagrangian). The resonance’s properties aredetermined subsequently by assuming the analyticity of the associated correlationfunction in the external momentum and by appealing to an operator product ex-pansion in the deep euclidean region. Analyticity, however, must break down acrosstwo phase boundaries (deconfining-preconfining, preconfining-center) provided thatthe effects arising due to the deviation from the thermodynamical limit can, on aqualitative level, be neglected in the production process. As a consequence, theQCD sum rule method is probably unreliable for the investigation of the spectrumof a pure SU(2) and SU(3) Yang-Mills theory. (We hasten to add that the situationis different for real-world hadronic resonances because the dynamical mixing of pureSU(3) and pure dual SU(3) gauge theories may restore a quasi-analytical behaviorof the relevant correlation functions [1]. After all the overwhelming phenomenolog-ical successes of QCD sum rules are a lot more more than coincidence. The term‘dynamical mixing’ includes the occurrence of the fractional Quantum Hall effect[7, 8] which renders quarks to be emerging phenomena, for a discussion see [1].)

The second result – the Hagedorn nature of the transition to the truly confiningphase – was suspected to occur for the real-world strong interactions a long time ago[9, 10]. Subsequently performed lattice simulations seemed to exclude a Hagedorntransition (diverging partition function above the critical point) even in the caseof a pure SU(2) or SU(3) Yang-Mills theory [12]. Again, this a consequence ofthe lattice’s failure to properly capture the infrared physics in thermodynamicalsimulations at low temperature, for an extended discussion see [1].

The third result, namely the excact vanishing of the ground-state pressure andenergy density of the Yang-Mills theory at zero temperature, commonly is used asa normalization assumption in lattice computations [11] and not obtained as a dy-namical result. In [1] we have shown the absoluteness, that is, the gravitational mea-

1

surablility of the finite and exactly computable energy-momentum tensor associatedwith the ground-state in the deconfining and preconfining phases: An immediateconsequence of the fact that these ground states are determined by radiatively pro-tected BPS equations. (Since these equations are first-order as opposed to second-order Euler-Lagrange equations the usual shift ambiguity in the corresponding po-tentials is absent.) Recall, that the finiteness of the ground-state energy densityand pressure in the deconfining and preconfining phases arises from averaged-overinteractions between and radiative corrections within solitonic field configurations.Being (euclidean) BPS saturated, classical configurations in the deconfining phasethe latter are free of pressure and energy density in isolation. The same applies tothe massless, interacting magnetic monopoles which, by their condensation, formthe ground state in the preconfining phase. In the confining phase configurationsthat are free of pressure and energy density do also exist (single center-vortex loops).In contrast to the other phases propagating gauge field fluctuations are, however,absent in the confining phase. Only contact interactions occur between the centersolitons, which, however, do not elevate the vanishing energy density of the isolatedsoliton to a finite value for the ensemble. The proof for this relies on computingthe curvature of the potential for the spatial coarse-grainined center-vortex conden-sate at its zeros and by comparing this curvature with the square of the maximalresolution that is allowed for in the effective theory [1], see below.

The outline of this paper is as follows: First, we discuss the occurence of iso-lated, instable, that is, contracting and collapsing center-vortex loops in the precon-fining phase. From the evolution of the magnetic coupling constant in this phasewe conclude that center-vortex loops become stable, particle-like excitations at thedeconfining-confining phase boundary. Second, we point out the spin-1/2 nature ofthese particles, and we derive a dimensionless parameter with discrete values describ-ing the condensate of pairs of single center-vortex loops after spatial coarse-graining.A discussion of the creation of center-fluxes (local phase jumps of the vortex conden-sate) by the decay of the monopole condensate in the preconfining phase is given.Third, we construct potentials for the vortex condensates which, in their physical ef-fects, are uniquely determined by the remaining local symmetry and by the positivesemi-definiteness of the energy density: Particle creation by local phase jumps ofthe order parameter may only go on so long as the energy density feeding into theircreation is nonvanishing. Fourth, we discuss in detail the remarkable result that forSU(2) and SU(3) Yang-Mills dynamics the confining phase’s ground-state energydensity is exactly nil. In particular, we stress the fact that radiative corrections tothe tree-level result are entirely absent. Fifth, we give an estimate for the density ofstatic fermion states and thus establish the Hagedorn nature of the transition fromthe confining to the preconfining phase. Finally, we summarize our results in viewof its implications for particle physics and for cosmology.

Instable center-vortex loops in the preconfining phase. Here we discuss the SU(2)case only, results for SU(3) follow by simple doubling. The ground-state of the

2

preconfining phase is a condensate of magnetic monopoles peppered with instabledefects: closed magnetic flux lines whose core regions dissolve the condensate locallyand thus restore the dual gauge symmetry U(1)D (for SU(3): U(1)2

D). It was shownin [1] that the magnetic flux carried by a given vortex-loop solely depends on thecharge of the monopoles and antimonopoles contributing to the explicit magneticcurrent inside the vortex core. Thus the various species of vortex-loops, indeed, aremapped one-to-one onto the nontrivial center elements of SU(2) or SU(3): They de-serve the name center-vortex loops. In the magnetic phase, center-vortex loops are,however, instable as we show now. To derive the classical field configuration associ-ated with an infinitely long vortex line one considers an Abelian Higgs model withno potential and a magnetic coupling g. (We need to discuss the energy-momentumtensor of the solitonic configuration relative to the ground state obtained by spatiallyaveraging over instable vortex loops. Thus we need to substract the temperaturedependent ground-state contribution which is reached far away from the consideredvortex core as a result of the applicable spatial coarse-graining, see [1] for details.)The following ansatz is made for the static dual gauge field aD

µ [13]:

aD4 = 0 , aD

i = ǫijkrjek A(r) (1)

where r is a radial unit vector in the x1x2 plane, r is the distance from the vortexcore, and e denotes a unit vector along the vortex’ symmetry axis which we chooseto coincide with the x3 coordinate axis. No analytical solution with a finite energyper vortex length is known to the system of the two coupled equations of motionhonouring the ansatz (1) and the Higgs-field decomposition ϕ = |ϕ|(r) exp[iθ]. Anapproximate solution, which assumes the constancy of |ϕ|, is given as

A(r) =1

gr− |ϕ|K1(g|ϕ|r) −→

1

gr− |ϕ|

√

π

2g|ϕ|r exp[−g|ϕ|r] , (r → ∞) . (2)

In Eq. (2) K1 is a modified Bessel function. Outside the core region the isotropicpressure Pv(r) in the x1x2 plane is, up to an exponentially small correction, givenas

Pv(r) = −1

2

Λ3Mβ

2π

1

g2r2. (3)

Notice that we have substituted the asymptotic value |ϕ| =

√

Λ3

Mβ

2π, (β ≡ 1

T) , as

it follows from the spatially coarse-grained action in the preconfining (or magnetic)phase [1, 14]. Notice also the minus sign on the right-hand side of Eq. (3): Theconfiguration in Eq. (2) is static due to its cylindrical symmetry but highly instablew.r.t. bending of the vortex axis. In particular, the pressure inside a center-vortexloop is more negative than outside causing the soliton to contract, and, eventually,to dissolve. Bending of the vortex axis occurs because there are no isolated magneticcharges in the preconfining phase which could serve as sources for the magnetic flux.An equilibrium between vortex-loop creation by the spatially and temporally corre-

lated dissociation of large-holonomy calorons and vortex-loop collapse is responsible

3

for the negative pressure of the ground state in the preconfining phase. The typical

core-size R of a center-vortex loop evaluates as R ∼ 1m

aDµ

= 1g

√

Λ3

M

βand its energy

as Ev ∼ πg

√

Λ3

Mβ

2π. (This takes into account an estimate for ϕ’s gradient contribution

to the total energy of the soliton.)Notice that core-size R, energy Ev, and pressure Pv(r) of a center-vortex vanish

in the limit g → ∞. This situation is reached at the critical temperature Tc,M wherethe magnetic coupling diverges in a logarithmic fashion: g ∼ − log(T − Tc,M) [1].At Tc,M the creation of single center-vortex loops at rest with respect to the heatbath (i) does not cost any energy and (ii) entails the existence of stable and masslessparticles. The latter do, in turn, condense pairwise into a new ground state.

Pairwise condensation of single center-vortex loops: Ground-state decay and change

of statistics. We consider a static, circular contour C(x) of infinite radius – an S1

– which is centered at the point x. In addition, at finite coupling g we consider asystem S of two single center-vortex loops, 1 and 2, which both are pierced by C(x)and which contribute opposite units of center flux Fv1

= 2πg

=∮

C(x)dzµ aD

1,µ = −Fv2

through the minimal surface spanned by C(x). Depending on whether 1 collapsesbefore or after 2 or whether 1 moves away from C before or after 2 the total centerflux F through C ′s minimal surface reads

F =

{±2πg

(either 1 or 2 is pierced by C(x))

0 (1 and 2 or neither 1 nor 2 are pierced by C(x)) .(4)

The limit g → ∞, which dynamically takes place at Tc,M , causes the center flux ofthe isolated system S to vanish and renders single center-vortex loops massless andstable particles. The center flux of the isolated system S does no longer vanish ifwe couple S to the heat bath. Although 1 and 2 individually are spin-1/2 fermionsthe system S obeys bosonic statistics. (Both, 1 and 2, come in two polarizations:the projection of the dipole moment, generated by the monopole current inside thecore of the center-vortex loop, onto a given direction in space either is parallel orantiparallel to this direction.) Thus, assuming the spatial momentum of 1 and 2 tovanish, the quantum statistical average flux reads

limg→∞

Fth = 4πF

∫

d3p δ(3)(p) nB(β|2 Ev(p)|)

= 0,± 8π

β|ϕ| = 0,±4 λ3/2c,M . (5)

According to Eq. (5) there are finite, discrete, and dimensionless parameter valuesfor the description of the macroscopic phase

ΓΦ

|Φ|(x) ≡ limg→∞

exp[i

⟨∮

C(x)

dzµ aDµ

⟩

] (6)

4

associated with the Bose condensate of the system S. In Eq. (6) Γ is an undeter-mined and dimensionless complex constant. Notice that taking the limit of vanishingspatial momentum for each single center-vortex loop is the implementation of spatialcoarse-graining. This coarse-graining is performed down to a resolution |Φ| which isdetermined by the (existing) stable solution to the equation of motion in the effectivetheory, see below.

For convenience we normalize the parameter values given by limg→∞ Fth asτ ≡ ±1, 0.

Coarse-grained action and center jumps. To investigate the decay of the monopolecondensate at Tc,M (pre- and reheating) and the subsequently emerging equilibriumsituation, we need to find conditions to constrain the potential VC for the macro-scopic field Φ in such a way that the dynamics arising from it is unique. Recallthat at Tc,M the dual gauge modes of the preconfining phase decouple. Thus theentire process of fermionic pre- and reheating in the confining phase is describedby spatially and temporally discontinuous changes of the modulus (energy loss) andphase (flux creation) of the field Φ. Since the condensation of the system S ren-ders the expectation of the ’t Hooft loop finite (proportional to Φ) the magneticcenter symmetry Z2 (SU(2)) and Z3 (SU(3)) is dynamically broken as a discretegauge symmetry. Thus, after return to equilibrium, the ground state of the confin-ing phase must exhibit Z2 (SU(2)) and Z3 (SU(3)) degeneracy. This implies thatfor SU(2) the two parameter values τ = ±1 need to be identified while each of thethree values τ = ±1, 0 describe a distinct ground state for SU(3). Let us now dis-cuss how either one of these degenerate ground states is reached. Spin-1/2 particlecreation proceeds by single center vortex loops being sucked-in from infinity. (Theoverall pressure is still negative during the decay of the monopole condensate thusfacilitating the in-flow of spin-1/2 particles from spatial infinity.) At a given pointx an observer detects the in-flow of a massless fermion in terms of the field Φ(x)rapidly changing its phase by a forward center jump (center-vortex loop gets piercedby C(x)) which is followed by the associated backward center jump (center-vortexloop lies inside C(x)). Each phase change corresponds to a tunneling transitioninbetween regions of positive curvature in VC . If a phase jump has taken place suchthat the subsequent potential energy for the field Φ is still positive then Φ’s phaseneeds to perform additional jumps in order to shake off Φ’s energy completely. Thiscan only happen if no local minimum exists at a finite value of VC . If the created sin-gle center-vortex loop moves sufficiently fast it can subsequently convert some of itskinetic energy into mass by twisting: massive, self-intersecting center-vortex loopsarise. These particles are also spin-1/2 fermions: A Z2 or Z3 monopole, constitutingthe intersection point, reverses the center flux [15], see Fig. 1.

If the SU(2) (or SU(3)) pure gauge theory does not mix with any other precon-fining or deconfining gauge theory, whose propagating gauge modes would coupleto the Z2 (or Z3) charges, a soliton generated by n-fold twisting is stable in isola-tion and possesses a mass n ΛC . Here ΛC is the mass of the charge-one state (one

5

Z 2 monopole

Figure 1: The creation of an isolated Z2 monopole by self-intersection of a center-vortex loop.

self-intersection). After a sufficiently large and even number of center jumps hasoccurred the field Φ(x) settles in one of its minima of zero energy density. Forward- and backward tunneling inbetween these minima corresponds to the spontaneouson-shell generation of a massless, single center-vortex loop of zero momentum. In aWKB-like approximation one expects the associated euclidean trajectory to have alarge action which, in turn, predicts large suppression. We conclude that tunnelingbetween the minima of zero energy density is forbidden.

Let us summarize the results of our above discussion: (i) the potential VC

must be invariant under magnetic center jumps Φ → exp[πi]Φ (SU(2)) and Φ →exp[±2π

3]Φ (SU(3)) only. (An invariance under a larger continuous or discontinu-

ous symmetry is excluded.) (ii) Fermions are created by a forward - and backwardtunneling corresponding to local center jumps in Φ’s phase. (iii) The minima of VC

need to be at zero-energy density and are all related to each other by center trans-formations, no additional minima exist. (iv) Moreover, we insist on the occurrenceof one mass scale ΛC only to parameterize the potential VC . (As it was the case forthe ground-state physics in the de - and preconfining phases.) (v) In addition, it isclear that the potential VC needs to be real.

SU(2) case:

A generic potential VC satisfying (i),(ii), (iii), (iv), and (v) is given by

VC = vC vC ≡(

Λ3C

Φ− ΛC Φ

) (

Λ3C

Φ− ΛC Φ

)

. (7)

The zero-energy minima of VC are at Φ = ±ΛC . It is clear that adding or subtractingpowers (Φ−1)2l+1 or Φ2k+1 in vC , where k, l = 1, 2, 3, · · · , generates additional zero-energy minima, some of which are not related by center transformations (violationof requirement (iii)). Adding ∆VC , defined as an even power of a Laurent expansionin ΦΦ, to VC (requirements (iii) and (v)), does in general destroy property (iii). A

6

-1

0

1-1

0

1

0

2

4

-1

0

1

-1

0

1-1

0

1

02468

-1

0

1

SU(2) SU(3)

Figure 2: The potential VC = vC(Φ)vC(Φ) for the center-vortex condensate Φ.Notice the regions of negative tangential curvature inbetween the minima.

possible exception is

∆VC = λ(

Λ2C − Λ

−2(n−1)C

(

ΦΦ)n

)2k

(8)

where λ > 0; k = 1, 2, 3, · · · ; n ∈ Z. Such a term, however, is irrelevant for the de-scription of the tunneling processes (requirement (ii)) since the associated euclideantrajectories are essentially along U(1) Goldstone directions for ∆VC due to the polein Eq. (7). Thus adding ∆VC does not cost much additional euclidean action andtherefore does not affect the tunneling amplitude in a significant way. As for thecurvature of the potential at its minima, adding ∆VC does not lower the value asobtained for VC alone. One may think of multiplying VC with a positive, dimen-sionless polynomial in Λ−2

C ΦΦ with coefficients of order unity. This, however, doesnot alter the physics of the pre - and reheating process. It increases the curvatureof the potential at its zeros and therefore does not alter the result that quantumfluctuations of Φ are absent after relaxation.

SU(3) case:

A generic potential VC satisfying (i),(ii), (iii), (iv), and (v) is given by

VC = vC vC ≡(

Λ3C

Φ− Φ2

) (

Λ3C

Φ− Φ2

)

. (9)

The zero-energy minima of VC are at Φ = ΛC exp[

±2πi3

]

and Φ = ΛC . Again,adding or subtracting powers (Φ−1)3l+1 or (Φ)3k−1 in vC , where l = 1, 2, 3, · · · andk = 2, 3, 4, · · · , violates requirement (iii). The same discussion for adding ∆VC toVC and for multiplicatively modifying VC applies as in the SU(2) case. In Fig. 2 plotsof the potentials in Eq. (7) and Eq. (9) are shown. The ridges of negative tangentialcurvature are classically forbidden: The field Φ tunnels through these ridges, and

7

a phase change, which is determined by an element of the center Z2 (SU(2)) or Z3

(SU(3)), occurs locally in space. This is the afore-mentioned generation of one unitof center flux.

No vacuum energy after relaxation. The action describing the process of relaxationof Φ to one of VC ’s minima is

S =

∫

d4x

(

1

2∂µΦ∂µΦ − 1

2VC

)

. (10)

Once Φ has settled into VC ’s minima Φmin there are no quantum fluctuations δΦ to

be integrated out anymore. Let us show this: Writing Φ = |Φ| exp[

i θΛc

]

, we have

∂2θVC(Φ)

|Φ|2∣

∣

∣

∣

Φmin

=∂2|Φ|VC(Φ)

|Φ|2

∣

∣

∣

∣

∣

Φmin

=

{

8 (SU(2))18 (SU(3))

. (11)

Thus a potential fluctuation δΦ would be harder than the maximal resolution |Φmin|corresponding to the effective action Eq. (10) that arises after spatial coarse-graining.Thus quantum fluctuations are already contained in the classical configuration Φmin:The cosmological constant in the confining phase of an SU(2) or SU(3) Yang-Millstheory vanishes exactly. Again, adding the term ∆VC of Eq. (8) to the potentials inEqs. (7) and (9) or performing the above multiplicative modification does not lowerthe value for the curvature as obtained in Eq. (11) and therefore does not changethis result.

Estimate for density of states, Hagedorn nature of the transition. That the transitionfrom the confining to the preconfining phase is of the Hagedorn nature is shown byan estimate for the density of massive spin-1/2 states. The multiplicity of massivefermion states, associated with center-vortex loops possessing n self-intersections, isgiven by twice the number Ln of bubble diagrams with n vertices in a scalar λφ4

theory. In [16] the minimal number of such diagrams Ln,min was estimated to be

Ln,min = n!3−n . (12)

The mass spectrum is equidistant. That is, the mass mn of a state with n self-intersections of the center-vortex loop is mn ∼ n ΛC. If we only ask for an estimateof the density of static fermion states ρn,0 = ρ(E = nΛC) of energy E then, byappealing to Eq. (12) and Stirling’s formula, we obtain [1]

ρn,0 >

√8π

3ΛC

exp[n log n](

log n + 1)

or

ρ(E) >

√8π

3ΛCexp[

E

ΛClog

E

ΛC](

logE

ΛC+ 1

)

. (13)

Eq. (13) tells us that the density of static fermion states is more than exponentiallyincreasing with energy E. The partition function ZΦ for the system of static fermions

8

thus is estimated as

ZΦ >

∫ ∞

E∗

dE ρ(E) nF (βE)

>

√8π

3ΛC

∫ ∞

E∗

dE exp

[

E

ΛC

]

exp[−βE] , (14)

where E∗ ≫ ΛC is the energy where we start to trust our approximations. ThusZΦ diverges at some temperature TH < ΛC . Due to the logarithmic factor inthe exponent arising in estimate Eq. (13) for ρ(E) we would naively conclude thatTH = 0. This, however, is an artefact of our assumption that all states with n

self-intersections are infinitely narrow. Due to the existence of contact interactionsbetween vortex lines and intersection points this assumption is the less reliable thehigher the total energy of a given fluctuation. (A fluctuation of large energy hasa higher density of intersection points and vortex lines and thus a larger likelihoodfor the occurrence of contact interactions which mediate the decay or the recom-bination of a given state with n self-intersections.) At the temperature TH theentropy wins over the Boltzmann suppression in energy, and the partition functiondiverges. To reach the point TH one would, in a spatially homogeneous way, needto invest an infinite amount of energy into the system which is impossible. Byan (externally induced) violation of spatial homogeneity and thus by a sacrifice ofthermal equilibrium the system may, however, condense densly packed (massless)vortex intersection points into a new ground state. The latter’s excitations exhibita power-like density of states and thus are described by a finite partition function.This is the celebrated Hagedorn transition from below.

Summary in view of particle physics and cosmology. The confining phase of anSU(2) and SU(3) pure Yang-Mills theory is characterized by a condensate of singlecenter-vortex loops and a dynamically broken, local magnetic Z2 (SU(2)) and Z3

(SU(3)) symmetry: No massless or finite-mass gauge bosons exist. Single center-vortex loops emerge as massless spin-1/2 particles due to the decay of a monopolecondensate. A fraction of zero-momentum, single center-vortex loops subsequentlycondenses by the formation of Cooper-like pairs. Protected from radiative correc-tions, the energy density and the pressure of this condensate is precisely zero in athermally equilibrated situation. The spectrum of particle excitations is a tower ofspin-1/2 states with equidistant mass levels. A massive state emerges by twistinga single center-vortex loop hence generating self-intersection point(s). This takesplace when single center-vortex loops collide. The process of mass generation thusis facilitated by converting (some of) the kinetic energy of a single center-vortexloop into the (unresolvable) dynamics of a flux-eddy marking the self-intersectionpoint, see Fig. (1). Due to their over-exponentially increasing multiplicity heavystates become instable by the contact interactions facilitated by dense packing. Ina spatially extended system (such as the overlap region for two colliding, heavy,and ultrarelativistic nuclei) there is a finite value in temperature, comparable to

9

the Yang-Mills scale ΛC , where a given, spatially nonhomogeneous perturbation in-duces the condensation of vortex intersections. This is the celebrated (nonthermal)Hagedorn transition.

The existence of a Hagedorn-like density of states explains why in an isolatedsystem, governed by a single SU(2) Yang-Mills theory, the center-flux eddy in aspin-1/2 state with a single self-intersection appears to be structureless for externalprobes of all momenta with one exception: If the externally supplied resolution iscomparable to the Yang-Mills scale ΛC , that is, close to the first radial excitationlevel of a BPS monopole [17] then the possibility of converting the invested energyinto the entropy associated with the excitation of a large number of instable andheavy resonances does not yet exist. As a consequence, the center of the flux eddy –a BPS monopole – is excited itself and therefore reveals part of its structure. For anexternally supplied resolution, which is considerable below ΛC , there is nothing tobe excited in a BPS monopole [17] and thus the object appears to be structurelessas well.

There is experminental evidence [18, 19, 20] that this situation applies to chargedleptons being the spin-1/2 states with a single self-intersection of SU(2) Yang-Millstheories with scales comparable with the associated lepton masses [1]. The corre-sponding neutrinos are Majorana particles (single center-vortex loops) which is alsosupported by experiment [21]. The weak symmetry SU(2)W of the Standard Model(SM) is identified with SU(2)e where the subscript e refers to the electron. The im-portant difference compared with the SM is that the pure SU(2)e gauge theory by

itself provides for a nonperturbative breakdown of its continuous gauge symmetryin two stages (deconfining and preconfining phase) and, in addition, generates theelectron neutrino and the electron as the only stable and apparently structurelesssolitons in its confining phase: No additional, fundamentally charged, and fluctuat-ing Higgs field is needed to break the weak gauge symmetry. The confining phase isassociated with a discrete gauge symmetry – the magnetic center symmetry – beingdynamically broken.

As far as the cosmological-constant problem is concerned the state of affairs isnot as clear-cut as it may seem. Even though each pure SU(2) or SU(3) gauge theorydoes not generate a contribution to the vacuum energy in its confining phase oneneeds to include gravity, the dynamical mixing of various gauge-symmetry factors,and the anomalies of emerging global symmetries in the analysis to obtain the com-plete picture on the Universe’s present ground state. We hope to be able to pursuethis program in the near future. Notice that today’s ground-state contribution dueto an SU(2) Yang-Mills theory of scale comparable to the present temperature of thecosmic microwave background is small as compared to the measured value [22]. ThisSU(2) Yang-Mills theory masquerades as the U(1)Y factor of the Standard Modelwithin the present cosmological epoch.

10

Acknowledgements

The author would like to thank Mark Wise for useful conversations and for thewarm hospitality extended to him during his visit to Caltech in May 2005. Financialsupport by Kavli Institute at Santa Barbara and by the physics department of UCLAis thankfully acknowledged.

References

[1] R. Hofmann, Int. J. Mod. Phys. A 20, 4123 (2005).

[2] U. Herbst, R. Hofmann, and J. Rohrer, Acta Phys. Pol. B 36, 881 (2005).

[3] U. Herbst and R. Hofmann, hep-th/0411214.

[4] C. J. Morningstar and M. J. Peardon, Phys. Rev. D 60, 034509 (1999).

[5] S. Narison, Z. Phys. C 26, 209 (1984).

[6] M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, Nucl.Phys. B 147, 448 (1979).M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, Nucl. Phys. B 147, 519 (1979).

[7] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

[8] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).

[9] R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965).R. Hagedorn and J. Ranft, Nuovo Cim. Suppl. 6, 169 (1968).

[10] G. Veneziano, Nuovo Cim. A 57, 190 (1968).

[11] G. Boyd et al., Phys. Rev. Lett. 75, 4169 (1995).G. Boyd et al., Nucl. Phys. B469, 419 (1996).J. Engels, F. Karsch, T. Scheideler, Nucl. Phys. B 564, 302 (1999).

[12] B. Lucini, M. Teper, and U. Wenger, JHEP 0401, 061 (2004).B. Lucini, M. Teper, and U. Wenger, JHEP 0502, 033 (2005).

[13] H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973).

[14] R. Hofmann, hep-th/0507122.

[15] H. Reinhardt, Nucl. Phys. B 628, 133 (2002).

[16] C. M. Bender and T. T. Wu, Phys. Rev. 184, 1231 (1969).

[17] P. Forgacs and M. Volkov, Phys. Rev. Lett. 92, 151802.

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[18] H. Alvensleben et al., Phys. Rev. Lett. 21, 1501 (1968).

[19] A. Ashkin, L. A. Page, and W. M. Woodard, Phys. Rev. 94, 357 (1953).

[20] M. B. Scott, A. O. Hanson, and E. M. Lyman, Phys. Rev. 84, 638 (1951).

[21] H. V. Klapdor-Kleingrothaus, I. V. Krivosheina, A. Dietz, and O. Chkvorets,Phys. Lett. B 586, 198 (2004).

[22] R. Hofmann, hep-ph/0508176.

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