Cosmic Strings on the Lattice

arXiv:hep-lat/9212023v1 18 Dec 19921Cosmic Strings on the LatticeA. K. Bukenova, A. V. Pochinsky, M. I. Polikarpovb, L. Polleyc and U.-J.

WiesedaUniversity of Alma–Ata, KazakhstanbITEP Moscow, 117259, RussiacUniversit¨at Oldenburg, 2900 Oldenburg, GermanydHLRZ J¨ulich, 5170 J¨ulich, GermanyWe develop a formalism for the quantization of topologically stable excitations in the 4-dimensional abelianlattice gauge theory. The excitations are global and local (Abrikosov-Nielsen-Olesen) strings and monopoles.

Theoperators of creation and annihilation of string states are constructed; the string Green functions are representedas a path integral over random surfaces. Topological excitations play an important role in the early universe.

Inthe broken symmetry phase of the U(1) spin model, closed global cosmic strings arise, while in the Higgs phaseof the noncompact gauge-Higgs model, local cosmic strings are present. The compact gauge-Higgs model alsoinvolves monopoles.

Then the strings can break if their ends are capped by monopoles. The topology of theEuclidean string world sheets are studied by numerical simulations.1.

INTRODUCTIONDuring cooling, the early universe has under-gone different phase transitions. Depending onthe symmetry that is spontaneously broken, var-ious excitations, for example, domain walls, cos-mic strings or monopoles, arise as topologicallystable objects [1,2].

The U(1) symmetric scalarfields in the 4-dimensional field theories give riseto string-like topological excitations.In a bro-ken U(1) symmetry phase, the strings are sta-ble solutions of the classical equations of motion.One distinguishes two types of cosmic strings:global and local ones. Global strings arise whena global U(1) symmetry breaks spontaneously,whereas local strings are due to the breakdown ofthe U(1) gauge symmetry.

A well known exam-ple is the local Abrikosov-Nielsen-Olesen stringpresent in the Higgs phase of the noncompactabelian gauge-Higgs model with a local U(1) sym-metry [3,4]. An even simpler example is the 4-dimensional global U(1) scalar field theory witha field Φ = |Φ| exp(iϕ).

In two dimensions, ze-ros of the complex scalar field, Φ(x) = 0, arelocated at isolated points x. Going around x, thephase ϕ may change by 2πn, where n ∈ZZ isthe topological characteristic of the vorticity ofthe scalar field.

The relevant homotopy group isΠ1[U(1)] = ZZ. In three dimensions, the zeros ofthe scalar field are located on lines (the global cos-mic strings), and in four dimensions, on surfaces(the world sheets swept out by the strings duringtheir time evolution).

The strings are topologi-cally stable excitations, i.e., they are insensitiveto small deformations of the field Φ.The dynamics of cosmic strings and monopolesis usually investigated in semiclassical physics.In the early universe, however, quantum effectsplayed a crucial role.The classical descriptionis inadequate when complicated dynamical sit-uations arise in which strings or monopoles arecondensed. The description of phase transitionsmay also require the inclusion of quantum effects.Due to the topological nature of the problem, itis essential to formulate it in a nonperturbativeframework, as provided by the lattice regulariza-tion.

For global cosmic strings, we start with acomplex scalar field Φ = |Φ|2 exp(iϕ) with a quar-tic potential V (Φ) = λ(|Φ|2 −v2)2, and considerthe limit as λ →∞.Then only the compactvariable ϕ ∈[−π, π] remains and the theory is re-duced to the global U(1) lattice spin model (the

2four-dimensional XY model).A cosmic stringthen manifests itself by the vorticity of a pla-quette and the corresponding dual plaquette be-longs to the Euclidean string world sheet. Thelocal cosmic strings results from the gauging ofthe U(1) symmetry.

On the lattice, this leads tothe noncompact and the compact abelian gauge-Higgs models. The theory with compact gaugefields has monopoles as additional topological ex-citations.

Since the string theory is formulatedon the lattice, one has complete nonperturbativecontrol of the dynamics. In particular, one canstudy the string tension or the question of stringcondensation in the early high temperature phaseby standard lattice techniques like numerical sim-ulations or strong coupling expansions.In the present publication, we only give the ex-plicit form of the creation operators of the strings,the details of the derivation will be published inthe subsequent paper.

The quantization of theglobal cosmic strings is discussed in ref.[5].2. GLOBAL STRINGSLet us consider the 4-dimensional U(1) latticespin model in the Villain formulation [1].

Its par-tition function is given byZ =Xl∈ZZ(C1)Z +π−πDϕ exp(−κ2 ∥dϕ + 2πl∥2). (1)We use the notations of the calculus of differ-ential forms on the lattice [6].

Dϕ denotes theintegral over all site variables, ϕ; d is the exte-rior differential operator, dϕ is the link variableconstructed as usual in terms of the site anglesϕ. The scalar product is defined in the standardway, e.g., if ϕ and ψ are the site variables, then(ϕ, ψ) = Ps ϕ(s)ψ(s), where Ps is the sum overall sites s. The norm is defined as: ∥a∥2 = (a, a);therefore ∥dϕ + 2πl∥2 implies summation over alllinks.Pl∈ZZ(C1) denotes the sum over all con-figurations of the integers l attached to the linksC1.

It occurs that the partition function (1) canbe represented as follows [5]:ZK =X∗k∈ZZ(C2)δ∗k=0exp(−2π2κ(∗k, ∆−1∗k)). (2)Where ∗x denotes the object dual to x, the cod-ifferential δ = ∗d∗satisfies the rule for partialintegration (ϕ, δψ) = (dϕ, ψ).The summationin (2) is carried over the integer variables k,which belong to plaquettes C2.The conditionδk = 0 means that the summation is performedover the closed two-dimensional objects definedby k.The surface elements interact with eachother via long range forces described by the in-verse Laplacian.

Physically, these forces are dueto the massless Goldstone bosons in the brokenphase of the original spin model. The action ofthe random surface model is nonlocal and differsfrom the Nambu action which is proportional tothe surface area.

It can be expected that the ran-dom surface model is equivalent to a lattice the-ory of closed strings, and that the closed surfacesare actually the string world sheets. To justifythis statement one must construct creation andannihilation operators of string states.3.

STRING CREATION OPERATORSThe creation of a cosmic string (as a nonlo-cal object) requires the use of nonlocal opera-tors. Global (local) cosmic strings are surroundedby a cloud of Goldstone (gauge) bosons, just ascharged particles are surrounded by their photoncloud.Creation operators for charged particleswere first constructed by Dirac [7], whose idea wasto compensate the gauge variation of a chargedfield, Φ(x)′ = Φ(x) exp(iα(x)), by a contributionof the gauge field representing the photon cloud:Φc(x) = Φ(x) exp(iZd3yBi(x −y)Ai(y)),(3)where ∂iBi(x) = δ(x), and Ai(x)′ = Ai(x) +∂iα(x) is the photon field.

The gauge invariantoperator Φc(x) creates a scalar charged particle atthe point x, together with the photon cloud sur-rounding it. Our construction of string creationoperators [5] is based on the same idea, and it isvery similar to the construction of soliton creationoperators suggested by Fr¨ohlich and Marchetti[8].In fact, it is a generalization of their con-struction of monopole sectors in 4-dimensionalU(1) lattice gauge theory.

We perform the dualitytransformation of the original theory defined by

3the partition function (1), and obtain the hyper-gauge theory of integer valued fields on the duallattice. The Wilson loop W(C), constructed fromthe auxiliary gauge fields, is gauge invariant, butnot hypergauge invariant.

We make W(C) hyper-gauge invariant by surrounding it by the cloud ofthe hypergauge field. This hypergauge invariantoperator can be shown to create the closed stringon the curve C. In terms of the variables k (2), theexpectation value of the string creation operatorhas the form:< UC >=1ZK ×(4)Xk∈ZZ(C2)δ∗k=−δCexp{−2π2κ(∗(k + DC), ∆−1∗(k + DC))};DC, being the counterpart of the function Bi(x)in eq.

(3), depends on the curve C: δ(3)∗DC = δC,since the operator of the creation of the stringshould act at a definite time slice, we use thethree-dimensional operator of the codifferentia-tion δ(3); δC is the lattice delta function which isequal to unity on the links belonging to the curveC, and equal to zero on the other links. It is im-portant that because of the condition δ∗k = −δC,the summation in (4) is performed over closedsurfaces, and those bounded by the curve C. Thisis exactly what one would expect intuitively: astring world sheet opens up on curve along whichthe string is created.In the general case, thecurve C may consist of several closed loops: δC =Pi δCi; then DC = Pi DCi.

Moreover, if we setδCj = −1 on several closed loops, then the stringis annihilated on these loops. Placing the loops Ciand Cj on different time slices, we may constructoperators corresponding to string scattering anddecay processes.The expectation value of the string creation op-erator in terms of the original fields ϕ is:< UC >= 1ZXl∈ZZ(C1)Z π−πDϕ(5)exp(−β∥dϕ + 2πδ∆−1(DC −ρC) + 2πl∥2).Here the integer valued field ρC satisfies the equa-tion δ(3)(∗DC −∗ρC) = 0.

It can be shown that∗ρC is the analog of the (invisible) Dirac string.The Dirac string connected to the monopole is aone-dimensional object, while ∗ρC, being definedon the plaquettes, is a two-dimensional object.4. LOCAL COSMIC STRINGSLocal cosmic strings are topological excitationsin the abelian gauge-Higgs model whose partitionfunction in the Villain form is given by:Z =Xn∈ZZ(C2)ZDθXl∈ZZ(C1)Z +π−πDϕ(6)exp(−β∥dθ + 2πn∥2 −κ2 ∥dϕ −θ + 2πl∥2).If the gauge field is noncompact (the integrationover θ is from −∞to +∞), then the excitationsare closed local strings.

Below we discuss a moreinteresting case, when the gauge fields are com-pact (−π < θ ≤π) and the monopoles are presentin addition to strings. It occurs that the stringsmay be open, with monopoles at their ends.

Thegeneral Green function for this case consists ofthe creation and annihilation operators of stringsand monopoles:⟨YiUCiYjΦxjYk¯Φxk⟩=(7)1ZXn∈ZZ(C2)Z +π−πDθXl∈ZZ(C1)Z +π−πDϕexp(−β∥dθ + 2πn + 2πδ∆−1(B −ω)∥2)exp(−κ2 ∥dϕ + 2πl −θ + 2πδ∆−1(D −ρ)∥2).It follows from the derivation of this formula(which is skipped) that δ Pi δCi + Pj δxj −Pk δxk=0;therefore openstrings carrymonopoles and antimonopoles at their ends. Inthe case of creation of monopoles at points xj andantimonopoles at points xk, the field B satisfiesthe equation: δ(3)∗B = Pj δxj −Pk δxk; Diracstring ∗ω ∈ZZ is such that δ(3)(∗B −∗ω) = 0; Dand ρ are defined by the equations: δ(3)∗D =δC + ∗B; d(3)(D −ρ) = B −ω.It is easy

4to show that the Green function (7) is invari-ant under the deformation of the Dirac string:ω′ = ω + dξ, ρ′ = ρ + ξ.5. COSMIC STRING DYNAMICS ANDNUMERICAL SIMULATIONSIn the high temperature phase of the earlyuniverse, the U(1) symmetry was unbroken andstrings were condensed.This was possible be-cause the string tension vanished and string cre-ation required no energy.

As a consequence, oneexpects that the strings formed clusters perco-lating through the universe.After the phasetransition, the U(1) symmetry gets spontaneouslybroken and the string network freezes. Becausethe string tension no longer vanishes, the stringsbecome massive.Small strings shrink and de-cay, whereas large cosmic strings survive as sta-ble massive structures catalyzing the formation ofgalaxies.

Such scenario can be studied within theformalism discussed above.The string condensate can be calculated nu-merically by direct measurement of the expecta-tion value < UC >.A similar measurement ofthe monopole condensate in the compact electro-dynamics was performed in refs.[9,10]. Anotherquantity that can be studied numerically is thestring tension of the cosmic strings.

We are per-forming these calculations at present.The simplest objects related to the cosmicstrings are the defects in the four-dimensionalXY model. In the given configuration of the spinsϕ, each plaquette carries vorticity ∗j =12π([ϕ1 −ϕ2]2π + [ϕ2 −ϕ3]2π + [ϕ3 −ϕ4]2π + [ϕ4 −ϕ1]2π),where ϕk are the angles on the corners of the pla-quette, and [α]2π is α mod 2π.

If ∗j ̸= 0, thenwe may ascribe vorticity j to the plaquette dualto the original one. The surfaces formed by thesedislocations are closed.

The proof is very simple:∗j =12πd([dϕ]2π) =12πd(dϕ + 2πp) = dp, wherethe integer p is such that (dϕ + 2πp) ∈(−π, π].Now, using equalities d = ∗δ∗and δ2 = 0, we getδj = 0. These world sheets are characterized bythe Euler number: NE = np −nl + ns, where np,nl and ns are the number of the plaquettes, thelinks and the sites belonging to the world sheetrespectively.

The number of handles is defined byg = 1 −NE/2.We have studied numerically the topology ofthe world sheets in the four-dimensional XYmodel on the lattices 64 −84 for the different val-ues of κ. Usually, there are several disconnectedclosed world sheets in each configuration of thespins.

It occurs that below the phase transition(κ < κC), in the region of the expected conden-sation of the strings, there exist one world sheetof the size comparable to that of the lattice; thisworld sheet contains a lot of handles. There arealso satellites: small disconnected objects with asimple topology (no handles).After the phasetransition (κ > κC), almost all world sheets haveno handles.

It turns out that the number of han-dles gi taken into account with the weight propor-tional to the area Si of the corresponding worldsheet i (i.e. < g >= Pi giSi/ Pi Si) is the orderparameter of the system.

Numerical simulationsshow that < g ≯= 0 for κ < κC and < g >= 0for κ > κC.Two of the authors (MIP and LP) would liketo thank the HLRZ in J¨ulich for hospitality. Thework of MIP and AVP has been partially sup-ported by the grant of the American Physical So-ciety.REFERENCES1.A.

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