Semilocal string formation in two dimensions

9 Pages·1.3 MB·

by  Ana Achucarro

Semilocal string formation in two dimensions
Ana Achu´carro
Department of Theoretical Physics, UPV-EHU, Bilbao, Spain
Institute for Theoretical Physics, University of Groningen, The Netherlands
Department of Mathematics, Tufts University, Medford, MA 02155
Julian Borrill
Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755
♠Center for Particle Astrophysics, University of California, Berkeley, CA 94720
♠National Energy Research Scientific Computing Center, Lawrence Berkeley National Laboratory,
University of California, Berkeley, CA 94720
Andrew R. Liddle
8 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
9
(February 1, 2008)
9
1 We present a toy model for the investigation of the formation of semilocal strings, where a planar
n symmetry is employed to reduce the system to two dimensions. We approximate the symmetry
a breaking using an extension of the Vachaspati–Vilenkin algorithm, where we throw down random
J phases for the scalar fields and then find the gauge field configuration which minimizes gradient

3 energy in this fixed scalar background. We show this procedure reproduces the standard estimate
1 for the formation rate of cosmic strings. For semilocal strings the configurations generated by this
method are ambiguous, and we numerically evolve the configurations forward in time to identify

2 which regions form strings. We find a significant rate of formation, depending on the ratio of
v couplingsβ. Forlowβ theformation rateisaboutonequarterthatofcosmicstrings;thisfalls asβ
8
isincreasedandaboveβ =1,wherethestringsolutionisdynamicallyunstable,nosemilocalstrings
6
form. Weshowtheresultsarerobustbyexaminingdifferentinitialconditionswhere,asexpectedin
3
a thermal environment,the initial scalar field need not be on the vacuum. We discuss implications
2
for thethree-dimensional case.
0
7
PACS numbers: 11.27.+d, 11.15.Ex
9
/ Preprint: EHU-FT/9702, DART-HEP-97/02, SUSSEX-AST97/2-1, hep-ph/9702368
h
p
- I. INTRODUCTION of whether the defects are global or local (though for
p
global strings or monopoles the defect cores themselves
e
h Astheuniverseexpandsandcools,oneexpectsittogo provenot particularlyimportantfor the dynamics ofthe
v: through a variety of phase transitions. At each, there is field ordering).
i the possibility of the formationof topologicaldefects [1], Wewishtoconsidertheformationofsemilocalstrings.
X the precise types being dictated by the physics of the This is a considerably harder problem, because the ex-
r breaking (see Ref. [2] for reviews). The standard ex- istence of string cores depends crucially on the way in
a
amples are domain walls, cosmic strings, monopoles and which the scalar and gauge fields interact. The semilo-
texture, where for strings and monopoles one may have cal string scenario is characterized by the gauge fields
either a global or a local symmetry breaking. More re- having insufficient degrees of freedom to be able to com-
cently, a new type of defect has been introduced which pletely cancel the scalar field gradients even away from
possesses both local and globalsymmetries, the simplest the core of the string, and the final configuration de-
example being the semilocal string [3,4]. pends very much on which of the gradients the gauge
Ofgreatinterestisthenumberdensityofdefectswhich field chooses to cancel. Recently an analysis was made
form. The standard technique for making such esti- by Nagasawa and Yokoyama [6] for a related type of de-
mates is a discretization of space into correlation vol- fect, the electroweakstring[7]formedatthe electroweak
umes, which may or may not be accompanied by a dis- phase transition, using probabilistic arguments applied
cretization of the vacuum manifold as well. This ap- to thermal configurations. They concluded a very low
proachwas pioneered by Vachaspati and Vilenkin [5] for production rate, though their analysis may significantly
the case of cosmic strings, and is easily generalized to underestimate the extent to which configurations ‘near’
monopoles, and, with a bit of further work, to global string configurations might relax to form proper strings
textures. These cases provide no particular difficulty since they do not consider the effect of the gauge fields
because the presence or absence of defects is dictated in driving the Higgs values towards zero. Note also that
solely by the configuration of the scalar field, regardless such strings are unstable [8]. We adopt the alternative
1
strategy of carrying out numerical simulations of initial alone;when the symmetry breaks anindependent choice
configurations in order to determine those regions which of vacuum state is to be made in each correlation vol-
prefer to accumulate magnetic flux to form strings. In ume, in accordance with the Kibble mechanism [1]. At
this paper we shall only consider semilocal string forma- this point, the theory with two complex scalars is ex-
tion in two dimensions, in order to test the method we actly that of a global texture theory. We then fix the
propose. We shall turn to the three-dimensional case in scalarfields, and choose the gaugefield to be in the con-
afuture publication. We specialize totwodimensions by figuration which minimizes the total energy in this fixed
imposing a planar symmetry. inhomogeneous scalar field background.
We estimate the number density by an extension of We work in flat space-time throughout. The Lagrang-
theVachaspati–Vilenkinalgorithm[5]. Firstwegenerate ian for the simplest semilocal string model is
an initial configuration by choosing a fixed scalar field
background, upon which we find the gauge field config- L=(∂µ−iAµ)φ†1(∂µ+iAµ)φ1
uration which minimizes the energy associated with the +(∂ −iA )φ†(∂µ+iAµ)φ
µ µ 2 2
(covariant) gradients and magnetic field energy. Strings
1
are identified by the location of flux tubes of Nielsen– − F Fµν −V(φ ,φ ), (1)
µν 1 2
4
Olesen type [9]. For cosmic strings we show that this
accurately reproduces the standardresult. For semilocal where φ and φ are two complex scalar fields, A is the
1 2 µ
strings, however, the initial configurations thus gener- gaugefield,the gaugecouplinghasbeensettounityand
atedareambiguous,featuringacomplicatedstructureof F is the gauge field strength. The potential for the
µν
magnetic flux. We resolvethis ambiguity by numerically scalar fields is chosen to be of symmetry breaking type
evolvingthe configurationforwardintime, allowingcon-
figurations in the ‘basin of attraction’ of the semilocal V(φ ,φ )= β |φ |2+|φ |2−1 2 , (2)
1 2 1 2
string solution to relax unambiguously into that config- 2
(cid:0) (cid:1)
uration. where the vacuum expectation value of the breaking,
In orderto confirmthatour results arenot anartifact like the gauge coupling, has been scaled to unity as
ofchoosingparticularinitialconditions,wealsocarryout in Ref. [3]. The remaining parameter β, which cannot
asetofsimulationsinwhichthescalarfieldisnotinitially be scaled away, measures the relative strength of the
onthevacuummanifold. Inatrulythermalenvironment scalar and gauge coupling constants (more specifically,
representingasecond-orderphasetransition,oneexpects β ≡ m2/m2 is the ratio between the squared scalar and
s v
the field to be out of the vacuum as the phase transition vectormasses). Itsvaluedeterminesthestabilityofanin-
completes, while the vacuum initial conditions are the finite semilocal string with a Nielsen–Olesen profile: the
expectedoutcomeofafirst-orderphasetransition. Inany semilocal string is stable for β < 1, neutrally stable for
case, we find that the semilocal string density obtained β =1 and unstable for β >1 [4,10].
in these simulations is extremely similar to that of the In this paper we shall only consider the two-dimen-
vacuum initial conditions. sional situation obtained by imposition of a planar sym-
metryinthe ‘3’direction,D ≡0. Workinginthe gauge
3
A =0,thisreducestothemorefamiliar∂ ≡0. Thereis
3 3
II. THE MODEL
a residualgauge freedomwhich allowsthe further choice
A =0. It is then clear that the energy-minimizing con-
0
Semilocalstringsariseinmodelswith scalarfields and figurationswillhave∂ =0,sincetimederivativesappear
0
gaugefields,inwhichthereareinsufficientgaugedegrees quadratically. Splitting up the scalarfields into four real
of freedom to completely cancel off the gradient energy scalars via φ = ψ +iψ , φ = ψ +iψ , the energy
1 1 2 2 3 4
associated with an inhomogeneous scalar field distribu- density of a static configuration can be written
tion [3]. The simplest such theory is one which con-
1
tainstwocomplexscalarfieldscoupledtoasingleabelian ρ=A2+A2+ (∂ A −∂ A )2
gaugefield. Werethereonlyonecomplexscalarfield,one 1 2 2 1 2 2 1
would have local cosmic strings, where the scalar gradi- −2[A1ψ1∂1ψ2−A1ψ2∂1ψ1+A2ψ1∂2ψ2−A2ψ2∂2ψ1
ents canbe completely cancelledby the gaugefieldaway +A ψ ∂ ψ −A ψ ∂ ψ +A ψ ∂ ψ −A ψ ∂ ψ ]
1 3 1 4 1 4 1 3 2 3 2 4 2 4 2 3
fromthestringcore. Weshallusethisstandardsituation +(∂ ψ )2+(∂ ψ )2+(∂ ψ )2+(∂ ψ )2
as a test of our method. 1 1 1 2 1 3 1 4
The main question to be addressed in the formation +(∂2ψ1)2+(∂2ψ2)2+(∂2ψ3)2+(∂2ψ4)2
of semilocal strings is which of the scalar gradients the +V(ψ ,ψ ,ψ ,ψ ). (3)
1 2 3 4
gauge field chooses to cancel. As in any physical sys-
tem, one expects the driving force will be the desire of Having thrown down the scalar field values and fixed
the systemto minimize its energy. We therefore propose them everywhere, the only terms required to minimize
the following model for determining conditions after the the energy are those depending on the remaining two
phasetransition. First,weshallconsiderthescalarfields components of the gauge field. This contribution to the
energy density can be written as
2
1
ρ =A2+A2+2A f +2A f + [∂ A −∂ A ]2, (4) Ref. [12] to set up initial conditions for dynamical tex-
A 1 2 1 1 2 2 2 1 2 2 1 ture evolution. The simplest interpolation is somewhat
discontinuousatthe boundariesofthe initialgrid,which
where
didnotmatterinRef.[12]astheywerequicklysmoothed
by dynamical evolution. Here we use a cubic interpola-
f =−ψ ∂ ψ +ψ ∂ ψ −ψ ∂ ψ +ψ ∂ ψ , (5)
i 1 i 2 2 i 1 3 i 4 4 i 3
tion in order to provide a smoother configuration that
givesthe sourcetermsforthegaugefields,whichdepend allow examination of the initial configuration itself. The
only on the (fixed) scalar field configuration. The re- gaugefieldsaresubsequentlychosentominimize(covari-
maining gauge terms are coupled only through the final ant)gradients. Weimposeperiodicboundaryconditions;
magnetic flux term. Were it absent, there would be an these enforce zero net flux through the grid, but this
analytic minimization of the energy given by turns out to be unimportant provided the grid is suffi-
ciently large.
A (x)=−f (x) ; A (x)=−f (x). (6) We minimize the energyvia a Monte Carloprocedure,
1 1 2 2
startingwith the analytic minimumEq.(6), by choosing
This solutionprovidesa gooditerative starting point for a grid-point and updating the gauge field values to new,
findingthetrueminimumnumerically,whichwillinvolve energy-minimizing,values. Aftereachpointhasbeenvis-
tradingoffsomeofthefluxenergyintheanalyticsolution itedmany times,the totalenergyis found tostabilize —
forenergyintheotherterms. Consequently,theanalytic we experimented with variation of the random sequence
solution provides an upper limit to the amount of flux ofpointsvisitedandalsotheinitialconditionstoconfirm
energy possible in the minimum energy configuration. that the system did not get trapped in a false minimum.
Thisprocedureissufficienttoobtainresultsforcosmic
strings, reported in the next Section, simply by identi-
III. INITIAL CONDITIONS AND EVOLUTION fying visually (or automatically via a numerical analy-
sis routine) the location of flux tubes in the configura-
A. Initial conditions tion. For semilocal strings it provesimpossible to decide
fromtheiteratedconfigurationwhichregionswillbecome
semilocalstrings,asaverycomplexfluxstructureresults
ThereisnohopeofananalyticminimizationofEq.(4)
due to the more complicated scalar field background. In
for a general scalar field configuration, so the problem
order to resolve this ambiguity, it proves necessary to
must be dealt with via discretization. The problem be-
evolve the initial configuration forward in time to see
ing two-dimensional, we break up the space into a grid
what happens. It is also clear that this is necessary to
of dimension N ×N. The first stage is to lay down ran-
distinguish the situations of stable and unstable semilo-
dom phases for the scalar field to form the background
calstrings,determinedbythechoiceofβ. Theprocedure
for the energy minimization. The scalar field problem
thus far has been independent of β, but for β >1 we ex-
is just that of a global texture theory, which was exam-
pect the flux to dissipate, while for β < 1 some stable
ined in Ref. [11]. However, in that paper the vacuum
strings might form.
manifoldwasdiscretized,whichisunnecessary(andeven
disadvantageous) here; instead random locations on the
vacuum manifold are chosen. Because the manifold has
B. Evolution
three-sphere topology, the scalar field can be in vacuum
everywhere,incontrasttotheusualcaseswheretopology
may force the scalar field to leave the vacuum manifold. Both analytical [13–15] and numerical [10] methods
However,once the gaugefields are introduced, they may have already been used to study issues concerned with
make it energetically favourable for the scalar field to stability of the string configurations for general β, and
move away from the vacuum manifold, for example by the cosmologicalimplications,buthavenotbeenusedto
allowing the magnetic flux to concentrate in areas of re- examinetheevolutionofinitialconfigurationssuchaswe
storedsymmetry,andinsuchlocationsasemilocalstring have generated here.
will have formed. Withourgaugechoicesandwithplanarsymmetry,the
Inorderthatthefieldscanbetreatedasfairlycontinu- equations of motion from the Lagrangianof Eq. (1) are
ousonthegrid,oneshouldnotthrowdownacompletely
random value at each grid-point. Instead, we choose the ψ¨i−∇2ψi+β ψ12+ψ22+ψ32+ψ42−1 ψi (7)
correlation length to be some number of grid-points and +A2ψ +(−(cid:0)1)j(2A.∇+∇.A)ψ =(cid:1)0,
i j
throwdownphases onthat coarsergrid. The correlation
length is chosen to be 2p grid lengths, where p is an in- (where j is the complement of i — 1 ↔ 2, 3 ↔ 4, and
teger. The field is then interpolated recursively onto the dots are of course time derivatives) for the scalar fields
full grid by bisection. Thus our two-dimensional square and
grid with 2m grid-points on a side contains 22(m−p) ini-
↔ ↔
tial correlation volumes. Such a procedure was used in A¨ −∇2A +∂ ∇.A+2 ψ ∂ ψ +ψ ∂ ψ (8)
i i i 1 i 2 3 i 4
(cid:18) (cid:19)
3
+2A ψ2+ψ2+ψ2+ψ2 =0, probability of string formation can be computed analyt-
i 1 2 3 4
(cid:0) (cid:1) ically. For example, take one corner to be at zero phase,
for the gauge fields, together with the constraint (from take the diagonally opposite corner to be uniformly dis-
the gauge choice) tributed in phase and compute the probability that the
remaining two corners have phases in the right place to
↔ ↔
2 ψ ∂ ψ +ψ ∂ ψ +∂ A˙ =0, (9) give a winding. The probability is given by
1 0 2 3 0 4 i i
(cid:18) (cid:19)
2 2π 1
whichisusedtotestthestabilityofthecode. Thearrows nVcsV = (2π)3 Z (2π−θ)θdθ = 3, (11)
indicate asymmetric derivatives. 0
This system is discretized using a standard staggered inagreementwithwhatwefoundabove. Thegoodagree-
leapfrog method; however, to reduce its relaxation time ment arises because the simulations are themselves car-
we also add an ad hoc dissipation term to each equa- riedoutonasquarelattice,andconfirmsthatouridenti-
tion — ηψ˙i and ηA˙i respectively. In an expanding Uni- ficationofcosmicstringsviathefluxtubestructureworks
versetheexpansionratewouldplaysucharole,thoughη extremely well. The greatest uncertainty in the esti-
wouldtypically not be constant. We tested a wide range matecomesfromthechoiceofdiscretization;forexample
of strengths of dissipation, and checked that it did not had we instead used a triangular lattice the Vachaspati–
affect the end result, only the time the system took to Vilenkin algorithm would give a formation rate of 1/4
adoptadistinctivestructure. Thesimulationswedisplay per triangle rather than 1/3 per square [16], though one
laterusedη =0.5. Fortunatelyweexpectthe long-range would also have to account for the difference in area.
interactionsbetweenstaticparallelsemilocalstringstobe Note that although the flux tubes in the upper panel
negligible[10]sowecantakethenumberoffluxtubesre- of Fig. 1 are very clearly defined, they are not Nielsen–
mainingattheendoftheevolutiontobeagoodestimate Olesenvortices[9],becausethe scalarfieldis everywhere
of the initial number of strings (and in any case a lower invacuum. Theirwidthisgovernedbytheinversevector
limit). Thisexpectationisalsoborneoutbyexamination mass. Once the scalar fields are allowed to relax, the
of the flux tubes’ evolution over time. flux tubes become Nielsen–Olesen vortices. The result
of dynamical evolution, for one hundred time units, is
showninthelowerpanelofFig.1,withβ chosenas0.05.
IV. RESULTS With that value, the inverse scalar mass is larger than
the inverse vector mass (recall that β ≡ m2/m2) and so
s v
A. Cosmic strings the total flux, while remaining 2π, spreads over a larger
area. During the evolution between the upper and lower
panels, two pairs of flux tubes annihilate leaving 18 out
To check our general strategy, we first test the per-
of the original 22. Such annihilations are expected as
formance of our codes for the case of cosmic strings.
the fields evolve and develop longer-range correlations,
Thissimplyinvolvesignoringoneofthe(complex)scalar
though this is inhibited by our artificial viscosity.
fields, setting ψ = ψ = 0. This makes the defects
3 4
topological, and the flux tubes formed now map out the
locations of winding in the scalar field configurations.
B. Semilocal strings
In this case the flux tube structure is very well de-
finedaftertheiterationhasbeenusedtoproducetheini-
tial conditions, andwe candirectly estimate the number For semilocal strings, the initial configurations gener-
densityofstringsformedwithoutneedinganydynamical ated by iteration prove much more ambiguous, because
evolution. Using a 1282 grid and smoothing over every thereisnotopologicalconstraint. Oneseesacomplicated
16grid-pointswehave64initialcorrelationvolumes. An flux structure with extrema ofdifferent heights, andit is
iteratedconfiguration,with22fluxtubes,isshowninthe farfromclearwhichofthese,ifany,mightevolvetoform
upperpanelofFig.1. Examining10differentinitialfield semilocalstrings;anexampleisshowninthetoppanelof
configurations we obtain a cosmic string number density Fig.2. Inany case,something hasto be doneto account
for the influence of the parameter β, which determines
n =0.32±0.02 (10) the stability of the string configurations. We do this by
cs
evolvingthe systemforwardin time, including a dissipa-
percorrelationvolume. Theerrorquotedisthestandard tion term as described above to aid the relaxation.∗ In
error on the mean, but note that the uncertainty from
discretization is somewhat larger.
This result agrees perfectly with the standard
Vachaspati–Vilenkin algorithm on a 2-dimensional ∗A 100 frame mpeg movie (0.5Mb) of the entire simulation
square lattice with a continuous field. Assuming the
can beviewed at http://star-www.cpes.susx.ac.uk/
geodesicrule(i.e.thatthefieldalwayscoverstheshortest
people/arl recent.html and we shall maintain it there for
pathonthevacuummanifoldalongagridedge),thenthe
as long as we can.
4
1
0.5
x 0
u
fl
−0.5
−1
120
100
80
60
120
40 100
80
20 60
40
20
1
0.5
x 0
u
fl
−0.5
−1
120
100
80
60
120
40 100
80
20 60
40
20
FIG. 1. The flux tube structure of a cosmic string network. The flux tubes tend to line up on the sub-grid formed by
the initial correlation length of 16 gridpoints. The spikes in the upper panel are taller and narrower than the corresponding
Nielsen–Olesen vortices which result after evolution forward in time with β = 0.05, as shown in the lower panel at t = 100.
Two fluxpairs annihilate duringthis evolution.
5
0.4
0.3
0.2
0.1
x 0
u
fl
−0.1
−0.2
−0.3
−0.4
120
100
80
60
120
40 100
80
20 60
40
20
0.4
0.3
0.2
0.1
x 0
u
fl
−0.1
−0.2
−0.3
−0.4
120
100
80
60
120
40 100
80
20 60
40
20
FIG.2. The flux tube structure in a semilocal string simulation. The upperpanel (t=0) shows the initial condition after
iteration, which is much more complicated than in the cosmic string case and not readily interpreted. The lower panel shows
the configuration resolved into fiveflux tubes by a short period of dynamical evolution (t=100). Note that the vertical scale
differs from that of Fig. 1; the finalflux tubeshavethesame profile as in that case.
6
fact, carrying out this dynamical evolution renders the
energy-minimization process redundant, since the early
stages of evolution carry out this role anyway.
As anticipated, in the unstable regime β > 1 the flux
quicklydissipatesleavingnostrings. Inthestableregime
β <1stringlikefeaturespersist. ThelowerpanelofFig.2
shows the result of evolving the configuration of the up-
per panel forward in time for one hundred time units
(whichis abouta factoroftwolongerthanthe timescale
foundinRef.[10]fortherelaxationintosemilocalstrings
ofnearbyconfigurationswithnon-zerowindingnumber).
Comparing the two panels indicates that some, but not
all,ofthemoreprominentfluxfeaturesintheinitialcon-
ditions do indeed relax into semilocal strings, that is,
Nielsen–Olesen vortices identical to those shown in the
FIG.3. The number of semilocal strings formed per ini-
bottom panel of Fig. 1 for the cosmic string case (note
tial correlation volume. Each point is an average over ten
the different vertical scales of Figs. 1 and 2).
simulations. Squares indicate that the vacuum initial condi-
Inordertoautomatethecountingoffluxtubesforthe
tionsofSectionIVwereused,whileopencirclesindicatethat
largenumberofsimulationswehaveobtained,weadopta
thenon-vacuuminitialconditionsdescribedinSectionVwere
criterionthatanyisolatedextrenuminthemagneticflux
used.
which is at least half that of a perfect Neilsen–Olesen
vortex is identified as a string. We tested this criterion
on selected simulations which we studied in detail our- obtained is consistent with Fig. 3.
selves,andconcludeditgivesanaccuratecounting. This Notice that the finalconfigurationillustratedin Fig. 2
enabled us to carry out a large number of runs to bring does not have the same number of flux tubes of upward
down the statistical errors on our results. and downward pointing flux, despite the zero total flux
Foreachofsevendifferentvaluesofβ,wetake10initial boundarycondition. Theextrafluxresidesinsmall‘nod-
configurations on a 1282 grid smoothed over every 16 ules’ of flux that can be seen in the figure; although the
grid-points. For β < 1 we find a formation rate which maximumfluxdensityistiny,theyareneverthelessquite
dependsonβ,andwhichisalwayssignificant,thoughnot diffuseandcontainasignificantamountoftotalflux. The
aslargeasthenumberdensityfoundinthecosmicstring structure ofthese nodules changesif the artificialviscos-
case. At the smallest β we were able to simulate, 0.05, ity is varied, indicating that they are in fact an artifact
the formationrateis aboutonequarterofthatofcosmic made long-lived by the viscous term. However, we note
strings. (If β is lowered further, the scalar string cores that if expansion was taken into account it would have
become too wide to fit into a correlation volume.) Note some properties similar to our viscosity,and may permit
that the lower number density means that annihilations such‘skyrmionic’configurationsto be long-lived. Such a
duringtheearlystagesofevolutionarelesslikelythanin suggestion has been made by Benson and Bucher [15].
the cosmic string case, so would lead to an even smaller
correction here.
Support for our results comes from an analysis [10] in V. RELAXING THE INITIAL CONDITIONS
whichsimple configurationswerepermitted to relax into
semilocal strings. It was found that configurations some In order to confirm the robustness of our conclusions,
wayfromamaximalcircle(aboutonequarteroftheway we also carriedout the same number of simulations with
fromthegreatcircleforlowvaluesofβ,decreasingtozero different initial conditions. For a second-order phase
as β approaches one) would relax into semilocal strings transition, one expects the scalar field to be out of the
if β <1. Consider the values of the Higgs in four points vacuumasthetransitiontakesplace. Thishasbeenstud-
onthelatticedefiningagridsquare. Allvalueslieonthe ied by Ye and Brandenberger [17], and we closely follow
three sphere. If we project all four points onto the most theirstrategy. Theychosethescalarfieldmagnituderan-
favourable maximal circle, then the probability of wind- domly from a uniform distribution; we adopt a similar
ing is identical to the Vachaspati–Vilenkin calculation, approach though using a gaussian distribution instead.
i.e. 1/3. Since the points are away from the maximal We examined different choices for the width of the gaus-
circle the suppression to the probability that they will sian, and any reasonable choice makes no difference to
stillformasemilocalstringcanbe estimatedby whether the results. We shalldisplaythoseforwhenthe gaussian
the point which is furthest from the maximal circle is has a dispersion equal to the vacuum expectation value,
closer thana certaincutoff distance Dc, andwill be pro- which is the natural value at the phase transition.
portional to the volume of a fringe of width 2Dc around We find startlingly good agreement with the vacuum
the equator (over the total volume of the three sphere). initial conditions, as shown in Fig. 3. The same general
Ref. [10] gives an estimate of D , and the probability
c
7
shape for the β-dependence is found, and the number of possibilitythatthestringsmightendinadiffusecloudof
strings identified per simulation is extremely close, with flux resembling a globalmonopole. Further, the analytic
the error bars overlapping. For cosmic strings we found infinitesemilocalstringsolutioncorrespondstoawinding
that the agreement between the two sets of simulations ofthegaugefieldsroundafixedgreatcircleoftheS3vac-
is not quite asgood, thoughstill impressive;some differ- uummanifold,whereasinarealisticsituationevenifthe
ences are expected as the initial correlation length will winding condition continues to be obeyed as one moves
have changed, and we conclude that these changes are along a putative string, the winding may correspond to
less significant for semilocal strings than cosmic strings. different great circles at different points along the string
and whether or not such a situation can be dynamically
stableremainsunexplored. Consequently,asyetwehave
VI. CONCLUSIONS no guidance as to how easily a configuration may find
itself within a ‘basin of attraction’ of a semilocal string
In this paper we have proposed a method of investi- configuration. A complete picture of the formation rate
gating semilocal string formation by following the mag- must awaitfull three-dimensionalsimulations, which are
netic flux, and tested it by restricting the problem to currently underway.
two dimensions by imposing a planar symmetry. We
have shown that our method can accurately reproduce
the standard result for cosmic strings. We find a very ACKNOWLEDGMENTS
significant rate of semilocal string formation; about one
quartertherateofcosmicstringformationforthelowest AAispartiallysupportedbyNSFgrantPHY-9309364,
β we tested, falling towards zero as the unstable regime CICYT grant AEN96-1668 and UPV grant 063.310-
is reached at β = 1. This result is robust to significant EB225/95. JB is supported by NSF grantPHY-9453431
changes in the initial conditions. and ARL by the Royal Society. This work was also
Although superficially our result for the semilocal supported in part by the European Commission under
string density appears to be in disagreement with the the Human Capital and Mobility programme, contract
verylowprobabilityofformationfoundbyNagasawaand number CHRX-CT94-0423, and by a NATO Collabora-
Yokoyamaforelectroweakstrings[6],adirectcomparison tiveResearchGrant‘CosmologicalPhaseTransitions’no.
cannot be made at this point. Their computation con- SA-5-2-05 (CRG 930904) 1082/93/JARC-501. ARL is
cerns the initial configurationonly. It does not take into grateful to the Kapteyn Institute and the University of
account the effect of the A2φ2 term in the Lagrangian Groningen for their hospitality, and AA and ARL thank
(which is usually responsible for the Meissner effect in the Isaac Newton Institute where this work was begun.
superconductors), which will tend to drive the value of We thank Mark Hindmarsh, Konrad Kuijken and Tan-
the Higgs field towards zero in regions where there is a may Vachaspati for many useful discussions.
highconcentrationof magneticflux, incompetition with
the potential term which favours the vacuum value of
the Higgs. Our simulations seem to indicate that the
backreaction of the gauge fields is substantial, at least
in the case of semilocal strings, and we expect this to
be true also for electroweak strings as long as the SU(2)
couplingissufficientlysmalltoallowstablestrings. This ♠ Present addresses
may lead to string formationduring the subsequent evo- [1] T. W. B. Kibble, J. Phys. A9, 1387 (1976).
lution. Further,theelectroweakstringstheyconsiderare [2] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and
infactunstable[8],andwerewetocarryoutsimulations Other Topological Defects, Cambridge University Press,
of that case we would conclude a zero formation rate.† Cambridge (1994); M. Hindmarsh and T. W. B. Kibble,
However, our simulations are only two-dimensional, Rep. Prog. Phys.58, 477 (1995).
and the three-dimensional implications are not straight- [3] T. Vachaspati and A. Achu´carro, Phys. Rev D 44, 3067
(1991).
forward. For cosmic strings, the two-dimensional forma-
[4] M. Hindmarsh,Phys. Rev.Lett. 68, 1263 (1992).
tion rate can be used more or less directly to estimate
[5] T. Vachaspati and A. Vilenkin, Phys. Rev D 30, 2036
the three-dimensional formation rate, because one can
(1984).
rely on the topology to ensure that the strings cannot
[6] M. Nagasawa and J. Yokoyama, Phys. Rev. Lett. 77,
end. Semilocal strings need not be infinite; there is the
2166 (1996).
[7] Y.Nambu,Nucl.Phys.B130,505(1977);T.Vachaspati,
Phys. Rev. Lett. 68, 1977 (1992), (E) 69, 216 (1992),
Nucl. Phys. B397, 648 (1993).
†Wealsonotethatinthecaseofafirst-orderphasetransition [8] M. James, L. Perivolaropoulos and T. Vachaspati, Nucl.
theuseofthegeodesicrulehasbeenquestionedbySaffinand Phys. B395, 534 (1993); W. Perkins, Phys. Rev. D 47,
Copeland [18]. R5224 (1993).
8
[9] H.B.NielsenandP.Olesen,Nucl.Phys.B61,45(1973).
[10] A. Achu´carro, K. Kuijken, L. Perivolaropoulos and T.
Vachaspati, Nucl.Phys. B388, 435 (1992).
[11] J. Borrill, E. J. Copeland and A. R. Liddle, Phys. Lett.
B258, 310 (1991).
[12] J. Borrill, E. J. Copeland and A. R. Liddle, Phys. Rev.
D 47, 4292 (1993).
[13] J.Preskill,Phys.Rev.D46,4218(1992);G.W.Gibbons,
M.Ortiz,F.Ruiz-RuizandT.Samols,Nucl.Phys.B385,
127 (1992).
[14] M. Hindmarsh, Nucl.Phys. B392, 461 (1993).
[15] K.BensonandM.Bucher,Nucl.Phys.B406,355(1993).
[16] R.LeeseandTProkopec,Phys.Rev.D44,3749(1991);
T. Prokopec, Phys.Lett. B 262, 215 (1991).
[17] J. Ye and R. H. Brandenberger, Nucl. Phys. B346, 149
(1990).
[18] P. M. Saffin and E. J. Copeland, Phys. Rev.D 56, 1215
(1997).
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