Induced Vacuum Decay by Topological Solitons by B. Kumar, M.B. Paranjape, U. Yajnik Phys.Rev.D82:025022,2010 Bum-Hoon Lee , Wonwoo Lee , Richard MacKenzie, M.B. Paranjape , U.A. Yajnik, Dong-han Yeom Phys.Rev. D88 (2013) 10, 105008; Phys.Rev. D88 (2013) 085031 É. Dupuis, Y. Gobeil, R. MacKenzie, L. Marleau, M.B. Paranjape, Y.Ung Phys.Rev. D92 (2015) no.2, 025031 M. Haberichter, Richard MacKenzie, M. B.Paranjape, Y. Ung J.Math.Phys. 57 (2016) no.4, 042303 Jennifer Ashcroft, Minoru Eto, Mareike Haberichter, Muneto Nitta, M.B. Paranjape arXiv:1604.08413 [hep-th] • False vacuum decay is an interesting physical phenomena. • Decay of a super-heated or super-cooled liquid to the gas or solid phase are common examples from tangible physical systems. • The decay corresponds to quantum tunnelling transition.
V(φ*φ)
1 -0.1 |φ|
FIG. 2. The rescaled potential (3) with ✏ =0.1.
=0and =1). The potential energy density of the false vacuum = v vanishes, | | | | | | while that of the true vacuum is V (0) = ✏v6.Instanton-typesolutionscorrespondingto true vacuum bubbles (in a sea of the false vacuum) will then have finite (Euclidean) action. After rescaling by appropriate powers of v and so that all fields, constants and the space- time coordinates are dimensionless, the Lagrangian density is still given by (1), multiplied 1/2 by an overall factor of , where now the potential is
2 2 2 V ( ⇤ )=( ✏)( 1) . (3) | | | | The overall factor does not affect the equations of motion and for the quantum theory is absorbed into an appropriate redefinition of ~. The potential for a value of ✏ in the range of interest is exhibited in Fig. 2. As mentioned in the introduction, in a false-vacuum universe topologically nontrivial field configurations (vortices) exist. These configurations may or may not be classically stable, but even if classically stable they can tunnel quantum mechanically to configuration of the same energy with a large core of true vacuum which will then expand rapidly.
We will look for rotationally-symmetric solutions for and Aµ in polar coordinates (r, ✓, t). We use the following time-dependent ansatz for a vortex of winding number n: ij in✓ n " rj (r, ✓,t)=f(r, t)e ,Ai(r, ✓,t)= a(r, t), (4) e r2 where "ij is the two-dimensional Levi-Civita symbol. The energy functional for the vortex has the form
2 1 1 E[Aµ, ]= d x + F0iF0i + FijFij +(D0 )⇤(D0 )+(Di )⇤(Di )+V ( ⇤ ) . (5) 2 4 Z 6 In supersymmetric theories, a symmetry broken phase is required on phenomenological grounds, but is generically metastable on theoretical grounds.
String based cosmology yields a de Sitter expanding universe that is generically meta-stable and its phenomenological viability depends on the decay rate being sufficiently slow.
Generally speaking the false vacuum can decay via tunnelling to the true vacuum, and this tunnelling is mediated by instantons which correspond to Euclidean trajectories where a bubble of true vacuum forms inside the false vacuum, grows to a maximum size and then bounces back to the false vacuum. • The spontaneous decay of a translationally invariant false vacuum is generically very slow. • The existence of topological objects can provide new sources of metastability. • There can exist solutions which interpolate between distinct false vacua, necessarily passing through the true vacuum. • These solutions can induce the decay of the false vacuum in a much more rapid fashion. Cartoon of Vacuum Bubble
False Vacuum
True vacuum
False Vacuum • The amplitude for such a decay is suppressed by the usual exponential of the classical action: e−SE /h¯ • This suppression can be reduced or even removed by the presence of perturbations or disturbances which can seed the transition. • Such cases are called induced vacuum decay. • We consider the situation that the false vacuum corresponds to the symmetry broken vacuum, and the true vacuum has good symmetry. • In such a situation, any local region where the true vacuum has appeared should be unstable to growth without bound. • Topological solitons seem ideally suited to provoke this kind of decay, generally they have unbroken symmetry inside the core and broken symmetry outside. • It can happen that the topological soliton is metastable, that classically it will be stable, however, it could be unstable to quantum tunnelling decays. Topological defects • We considered three types of topological defects: monopoles, cosmic strings and domain walls • In each, the true vacuum will be trapped behind a wall from the false vacuum outside. • Generally, the energy will be lowered by the region occupied by the true vacuum expanding, however, there can be a counteracting energy increase from the wall between the true and false vacua. 2 of vacuum decay has been rather elusive although the and we determine the existence of the instanton for this ideas have been adequately explicated in [15–18]. More tunneling within the same thin wall approximation. In recently, the relevance of the mechanism has been demon- section V we determine the Euclidean action for this in- 2 strated in specific examples, in [19] for the mediating sec- stanton, the so called bounce B which determines2 the tor of a hidden sector scenario of supersymmetry break- tunneling rate for the appearance of the large radius un- of vacuum decay has been rather elusive although the and we determine the existence of the instanton for this ingof and vacuum in [20 decay] in has a GUT been modelrather elusive with O’Raifeartaigh although the stable monopole. In section VI we relate our findings ideas have been adequately explicated in [15–18]. More andtunneling we determine within the the existence same thin of the wall instanton approximation. for this In2 typeideas direct have supersymmetry been adequately breaking. explicated in In [15 this–18 paper]. More we tunnelingto a previous within the study same of thin classical wall approximation. monopole instability In in explorerecently,recently, a the model the relevance relevance that is of of amenable the the mechanism mechanism to an has has analytical been been demon- demon- treat- sectionsectionsupersymmetricV weV we determine determine GUT the models. Euclideanthe Euclidean In action section action forVII this forwe in- this discuss in- of vacuum decay has been rather elusive although the and we determine the existence of the instanton for this mentstratedstrated within in in specific specific the techniques examples, examples, developed in in [ [19]] for for thein the [21 mediating mediating]. In doing sec- sec- so stanton,stanton,our results the the so and calledso compare called bounce bounce ourB tunnelingwhichB which determines rate determines formula the the with ideas have been adequately explicated in [15–18]. More tunneling within the same thin wall approximation. In wetortor provide of of a ahidden hidden a transparent sector sector scenario scenario model of in supersymmetry supersymmetry which the generic break- break- ex- tunnelingtunnelingthat of rate the rate for homogeneous for the the appearance appearance bubble of the offormation large the large radius case radius un- without un- recently, the relevance of the mechanism has been demon- section V we determine the Euclidean action for this in- inging and and in in [20 [20]] in in a a GUT GUT model model with with O’Raifeartaigh O’Raifeartaigh stablestable monopole. monopole. In section In sectionVI weVI relatewe relate our findings our findings pectationsstrated raised in specific in [15– examples,18] can be in realised [19] for the and mediating a specific sec- monopoles.stanton, the We so show called that bounce in additionB which to determines our tunneling the typetype direct direct supersymmetry supersymmetry breaking. breaking. In In this this paper paper we we toto a previous a previous study study of classical of classical monopole monopole instability instability in in formulator can of be a hidden derived. sector scenario of supersymmetry break- ratetunneling being significantly rate for the appearance faster, it also of the indicates large radius a regime un- exploreexplore a a model model that that is is amenable amenable to to an an analytical analytical treat- treat- supersymmetricsupersymmetric GUT GUT models. models. In section In sectionVII weVII discusswe discuss We constructing and in an [20SU] in(2) a GUT gauge model model with with O’Raifeartaigh a triplet instable which monopole. the monopoles In section becomeVI unstable,we relate hence our findings showing mentment within within the the techniques techniques developed developed in in [ [2121].]. In In doing doing so so ourour results results and and compare compare our tunnelingour tunneling rate formula rate formula with with type direct supersymmetry breaking. In this paper we thatto a the previous putative study non-trivial of classical vacuum monopole indicated instability by in the scalarwe provide field with a transparent two possible model translationally in which the generic invariant ex- that of the homogeneous bubble formation case without we provideexplore a transparent a model that model is amenable in which to an the analytical generic ex- treat- thate supersymmetricective of the potential homogeneous GUT is in models. fact bubble unstable. In formation section VII casewe without discuss vacua,pectations one with raisedSU in(2) [15–18 broken] can be to realisedU(1) and a the specific other monopoles. We show that in addition to our tunneling pectationsment raised within in the [15 techniques–18] can be developed realised in and [21]. a specific In doing so monopoles.our results We and show compare that our in tunneling addition rate to our formula tunneling with withformula the original can be derived. gauge symmetry intact. The former rate being significantly faster, it also indicates a regime formulawe can provide be derived. a transparent model in which the generic ex- ratethat being of the significantly homogeneous faster, bubble it also formation indicates case a without regime phaseWe permits construct the an existenceSU(2) gauge of monopoles. model with By a triplet appro- in which the monopoles become unstable, hence showing We constructpectations raisedan SU in(2) [15 gauge–18] can model be realised with and a triplet a specific in whichmonopoles.II. UNSTABLEthe monopoles We show MONOPOLES that become in addition unstable, to IN hence our A FALSEtunneling showing priatescalar choice field with of potential two possible for the translationally triplet it can invariant be ar- that the putative non-trivial vacuum indicated by the scalarformula field with can two be derived. possible translationally invariant thatrate the being putative significantly non-trivialVAC faster, U vacuumU it M also indicates indicated a regime by the rangedvacua, that one the with phaseSU(2) of broken unbroken to U(1) symmetry and the is other lower e ective potential is in fact unstable. vacua, one with SU(2) broken to U(1) and the other e ectivein which potential the monopoles is in fact become unstable. unstable, hence showing in energywith theWe and original construct represents gauge an the symmetrySU(2) true gauge vacuum intact. model of The the with former theory. a triplet with thescalar original field withgauge two symmetry possible intact. translationally The former invariant that the putative non-trivial vacuum indicated by the Thephase monopoles permits interpolate the existence between of monopoles. the true vacuum By appro- and Consider an SU(2) gauge theory with a triplet scalar phase permitsvacua, one the with existenceSU(2) of broken monopoles. to U(1) By and appro- the other II.e ective UNSTABLE potential MONOPOLES is in fact unstable. IN A FALSE thepriate false choice vacuum. of potential For a wide for the range triplet of the it can parameters, be ar- field ⇧ with the Lagrangian density given by priatewith choice the of original potential gauge for symmetry the triplet intact. it can The be former ar- II. UNSTABLEVAC MONOPOLES U U M IN A FALSE theseranged monopoles that the are phase in fact of unbroken classically symmetry stable. is In lower previ- VAC U U M rangedin energyphase that and thepermits represents phase the of existence the unbroken true vacuum of symmetrymonopoles. of the theory. is By lower appro- 1 1 ous work [15, 16] the dissociation of such monopoles was =II. UNSTABLEFMagnetica F µ a + MONOPOLES( DMonopoles⇧a)(Dµ⇧a) IN AV ( FALSE⇧a⇧a) (1) inThe energy monopolespriate and choice represents interpolate of potential the between true for vacuum the the true triplet of vacuum the it can theory. and be ar- Consider an SUµ (2) gauge theoryµ with a triplet scalar considered, varying the parameters of the theory to criti- L 4 2VAC U U M Thethe monopoles falseranged vacuum. that interpolate the For phase a wide between of range unbroken the of true the symmetry parameters, vacuum is and lowerfield Consider⇧ with the an LagrangianSU(2) gauge density theory given with by a triplet scalar calthethese values falsein monopoles energyvacuum. where and the are For represents monopoles in a fact wide classically the range were true classically of vacuum stable. the parameters, In of unstable previ- the theory. fieldwhere• ⇧Considerwith the a Lagrangian Georgi-Glashow density model, given by with The monopoles interpolate between the true vacuum and Consider1 an SU1(2) gauge theory with a triplet scalar duetheseous to monopoleswork infinite [15, dilation.16] are the in dissociation This fact can classically occur of such for stable. monopoles example In previ- in was the = ‘tHooftF a F µPolyakov a + (D monopoles⇧a)(Dµ⇧a) V (⇧a⇧a) (1) the false vacuum. For a wide range of the parameters, L field 4⇧ with1µ a the Lagrangian2 1aµ densitya givenabc b by c earlyousconsidered, work Universe [15, varying16 where] the the the dissociation parameters high temperature of of such the theory monopoles phase to prefers criti- was Faµ =µ ⌃a µA ⌃ Aa µ +µe⇥a AµA ,a a (2) these monopoles are in fact classically stable. In previ- = Fµ F + ( Dµ⇧ )(D ⇧ ) V (⇧ ⇧ ) (1) oneconsidered,cal vacuum values varying where in which the the monopoles the parameters system were starts, of classically the theory but with unstable to criti- adia- whereL 4 1 21 ous work [15, 16] the dissociation of such monopoles was = F a F µ a + (D ⇧a)(Dµ⇧a) V (⇧a⇧a) (1) baticcaldue values reduction to infinite where in dilation. the temperature, monopoles This can a were occur di erent classically for example phase unstable becomes in the and L 4 µ 2 µ earlyconsidered, Universe where varying the the high parameters temperature of the phase theory prefers to criti- where a a a abc b c moredue to favorable. infinite dilation. The Universe This can is then occur liable for example to simply in theroll Fµ = ⌃µA ⌃ Aµ + e⇥ AµA , (2) one vacuumcal values in where which the the monopoles system starts, were but classically with adia- unstable where D ⇧a = ⌃ ⇧a + e⇥abcAb ⇧c. (3) overearly, by Universe classical where evolution, the high to temperature the true vacuum. phase prefers a µ a µ a abcµ b c baticdue reduction to infinite in temperature, dilation. This a di can erent occur phase for example becomes in theand Fµ = ⌃µA ⌃ Aµ + e⇥ AµA , (2) oneIt vacuum wasearly however, Universe in which overlooked where the system the high that starts, temperature these but monopoles with phase adia- prefers are a a a abc b c more favorable. The Universe is then liable to simply roll The potentialFµ we = use⌃µA is a polynomial⌃ Aµ + e⇥ ofAµ orderA , 6 in ⇧(2)and inbatic fact reductionone unstable vacuum in temperature, due in which to quantum the a system di erent tunneling starts, phase but becomes well with be- adia- and a a abc b c over, by classical evolution, to the true vacuum. may convenientlyDµ⇧ = be⌃µ⇧ written+ e⇥ asAµ⇧ . (3) foremore the favorable.batic parameters reduction The Universe in reach temperature, their is then critical a liable di erent to values. simply phase becomesroll We and It was however, overlooked that these monopoles are a a abc b c dubover, such bymore classical monopoles favorable. evolution,false The Universe monopoles to the istrue then. vacuum. Working liable to simply in therollThe potential we useDµ⇧ is a= polynomial2⌃µ⇧2 + e2⇥ of2 orderAµ⇧2 6.2 in ⇧ and (3) in fact unstable due to quantum tunneling well be- V (⇧)=⌅⇧a (⇧ a a )abc+ b ⇧c ⇥ (4) over, by classical evolution, to the true vacuum. may conveniently beD writtenµ⇧ = ⌃ asµ⇧ + e⇥ Aµ⇧ . (3) thinItfore wallwas the however, limit parameters for overlooked the reach monopoles their that critical these [15], monopoles we values. show We that are It was however, overlooked that these monopoles are The potential we use is a polynomial of order 6 in ⇧ and suchindub fact monopoles such unstable monopoles undergo duefalse to quantum quantum monopoles tunneling tunneling. Working to well in larger the be- whereThe potential⇥ is defined we2 use so2 that is a2 polynomial the2 potential2 2 of order vanishes 6 in ⇧ atand the in fact unstable due to quantum tunneling well be- may convenientlyV (⇧)=⌅⇧ be(⇧ writtena ) as+ ⇧ ⇥ (4) monopoles,forethin the wall parameters whichlimit for are the reach then monopoles their classically critical [15], unstable we values. show by that We ex- meta-stablemay conveniently vacua. be The written vacuum as energy density di er- fore the parameters reach their critical values. We pandingdubsuch such monopoles indefinitely, monopoles undergo consequentlyfalse quantum monopoles converting tunneling. Working to all larger in space the whereence⇥ isis then defined equal so that to ⇥2. the Such2 potential a2 potential2 vanishes2 2 was at numerically the dub such monopoles false monopoles. Working in the V (⇧)=⌅⇧ (2⇧ 2 a 2) 2+ 2⇧ 2 ⇥ (4) tothinmonopoles, the wall true limit vacuum, which for are the the then monopoles phase classically of unbroken [15], unstable weSU show(2) by ex-sym- that meta-stable vacua.V (⇧ The)=⌅⇧ vacuum(⇧ energya ) + density⇧ ⇥ di er- (4) thin wall limit for the monopoles [15], we show that analyzed by [22] as a toy model for the dissociation of metry.panding Further, indefinitely, the formula consequently we derive converting also recovers all space the ence is then equal to ⇥. Such a potential was numerically such monopolessuch monopoles undergo undergo quantum quantum tunneling tunneling to larger to larger wheremonopoles.⇥ is defined Here we so that obtain the explicit potential analytical vanishes formulae at the to the true vacuum, the phase of unbroken SU(2) sym- analyzedwhere by⇥ [22is] defined as a toy so model that the for potential the dissociation vanishes of at the regimemonopoles, ofmonopoles, parameter which which are space, then are within then classically the classically thin unstable wall unstable monopole by ex-by ex- meta-stablefor the quantum vacua. tunneling The vacuum decay energy of the monopoles. density di er- The metry. Further, the formula we derive also recovers the monopoles.meta-stable Here vacua. we obtain The explicit vacuum analytical energy formulaedensity di er- limit,panding wherepanding indefinitely, no indefinitely, tunneling consequently is consequently required converting for converting the decay all space all but space encepotential is then has equal a minimum to ⇥. Such at a⇧ potentialT ⇧ = 0 was which numerically for =0 regime of parameter space, within the thin wall monopole for theence quantum is then tunneling equal to ⇥ decay. Such of a potentialthe monopoles. was numerically The theto the monopoleto true the vacuum, true is simply vacuum, the classically phase the phase of unbrokenunstable of unbroken asSU previously(2)SU(2) sym- sym- analyzedis degenerate by [22 with] as the a toy manifold model for of vacua the dissociation at ⇧T ⇧ = ofa2. limit, where no tunneling is required for the decay but potentialanalyzed has a by minimum [22] as a at toy⇧T model⇧ = 0 for which the for dissociation =0 of treatedmetry.metry. [ Further,15, 16 Further,]. the formula the formula we derive we derive also also recovers recovers the the the monopole is simply classically unstable as previously ismonopoles. degenerateWhenmonopoles. we with set Here Here the= we we manifold 0, obtain obtain we get explicitof explicit vacua a manifold analytical analytical at ⇧T ⇧ of= degenerate formulaea formulae2. regime of parameter space, within the thin wall monopole ⌅ T 2 Thetreated restregime [15 of, of16 the]. parameter paper is space, organised within as the follows. thin wall In monopole sec- Whenformetastablefor the we the quantum set quantum vacua= 0, tunneling attunneling we⇧ get⇧ a= decay decay manifold⇤ (where of of the the of the monopoles. degeneratemonopoles. exact value The The of TT tionlimit,TheII wherelimit,we rest specify where noof the tunneling no paperthe tunneling model is is organised required under is required as consideration for follows. the for thedecay In decay sec- but and butmetastablepotentialthepotential VEV, vacua has⇤, has is⌅ a at calculable a minimum⇧ minimumT ⇧ = ⇤2 and at(where at⇧⇧ satisfies⇧⇧ the== exact 0 0⇤ which which valuea for for for of small =0=0 ), ⇥ TT 22 thethetion monopoletheII we monopole specify is ansatz simply is the simply along classically model withclassically under unstable the consideration unstable equations as previously as of previously and mo- theisand VEV, degenerateis the degenerate⇤, minimum is calculable with with at the the and⇧ manifold= manifold satisfies 0 becomes of⇤ of vacua vacuaa thefor true small at ⇧⇧ vacuum. ⇧⇧),==aa.. A tion.treatedthe In monopoletreated [15 section, 16 [].15 ansatz,III16].we along delineate with the the equations conditions of under mo- andWhenplot theWhen of minimum we the we set potential set at =⇧== 0, 0, 0 for we becomes we small get get a a the manifold manifoldas⇥ true a function vacuum. of degenerate degenerate Aof one of ⌅ ⌅ T 2 inThetion. which rest InThe there section of rest the should of paperIII thewe exist paper is delineate organised a is metastable organised the as conditions follows. as monopole follows. under In sec- In so- sec-plotmetastablethe ofmetastable components the potential vacua vacua of for at at⇧⇧ smallT⇧is⇧⇧ shown== ⇤as⇤2 (where a(where in function figure the the1 of exact.Asupersym- one value valueof of of lutiontionin whichII withtionwe thereII aspecify largewe should specify radius the exist model the and a model metastable under a thin under consideration wall. monopole consideration We find so- and the andthethemetry componentsthe VEV, VEV, breaking⇤,⇤ is, of is calculable⇧ calculable modelis shown [23 and and in] containing figure satisfies satisfies1.Asupersym-⇤⇤ monopolesa for small small and ),), a the monopole ansatz along with the equations of mo- and the minimum at ⇧ = 0 becomes the⇥⇥ true vacuum. A thinthelution monopole wall with monopole a ansatz large solutions radius along and with and a thin also the wall. equations justify We their find of the exis- mo- metryandscalar breakingthe potential minimum model similar at [23⇧]= containing to 0 thebecomes one monopoles given the true in Eqn. and vacuum. a (4) was A tion.thin In walltion. section monopole In sectionIII solutionsweIII delineatewe and delineate also the justify conditions the conditions their exis- under underscalarplotplot potential of of the the potential similar potential to for the for small smallone given asas in a a Eqn. function function (4) was of of one one of of tence. Inin section which thereIV we should use the exist thin a metastable wall approximation monopole so- studiedthe components in [20]. of ⇧ is shown in figure 1.Asupersym- intence. which In there section shouldIV we exist use the a metastable thin wall approximation monopole so- studiedthe components in [20]. of ⇧ is shown inT figure 1.Asupersym-2 which permitslution with a treatment a large radius of the and solution a thin wall. in terms We find of the Themetry manifold breaking model of vacua [23]T containing at ⇧ ⇧ 2 = monopoles⇤ is topologi- and a lutionwhich with permits a large a treatment radius and of a the thin solution wall. in We terms find of the metryThe manifold breaking2 of model vacua [23 at ]⇧ containing⇧ = ⇤ monopolesis topologi- and a a singlea singlethin collective collective wall monopole coordinate, coordinate, solutions the and radius radius alsoRR justifyofof the the their thin thin exis-callycallyscalar an anS2 potentialSandand as spatial similar as spatial infinity to the infinity one is topologically given is topologically in Eqn. also (4) was also thin wall monopole solutions and also justify their exis- scalar2 potential similar to the one given in Eqn. (4) was wall.wall. Wetence. We argue argue In section that that the theIV we monopole monopole use the is thin is unstable unstable wall approximation to to tun- tun- S2,S thestudied, the appropriate appropriate in [20]. homotopy homotopy group group of the of manifold the manifold of of nelingtence. towhich In sectiona new permits configurationIV we a treatment use the of thin of a the muchwall solution approximation larger in radius terms of studiedthe vacuaThe in manifold [20 of]. the symmetry of vacua at breaking⇧T ⇧ =SU⇤2(2)is topologi-U(1) is neling to a new configuration of a much larger radius the vacua of the symmetry breakingTSU(2) 2 U(1) is which permitsa single collective a treatment coordinate, of the solution the radius inR termsof the of thin Thecally manifold an S2 and of as vacua spatial at infinity⇧ ⇧ is= topologically⇤⇤ is⇤ topologi- also 2 a singlewall. collective We argue coordinate, that the the monopole radius isR unstableof the thin to tun- callyS2, an theS appropriateand as spatial homotopy infinity group is of topologically the manifold also of wall. Weneling argue to a that new the configuration monopole of is a unstable much larger to tun- radius S2,the the vacua appropriate of the symmetry homotopy breaking group ofSU the(2) manifoldU(1) is of neling to a new configuration of a much larger radius the vacua of the symmetry breaking SU(2) ⇤ U(1) is ⇤ • The potential in this theory is of the form: 3
1.2
Wall 1
0.8 True Vacuum False Vacuum ) -->
! 0.6
0.4
0.2 ! ""> Potential V( −$ $ h(r)/! K(r) 0 R
"# R - "/2 R + "/2
-0.2 r -->
FIG. 1: The potential V (⌅) for = 0 as a function of one of FIG. 2: The monopole profile under the thin wall approxima- 6 the components of the field ⌅, shifted by an additive constant tion. so that ⌅ = ⇤ has vanishing V and the true vacuum has V = ⇥. III. THIN WALLED MONOPOLES
2(SU(2)/U(1)) which is Z. This suggests the existence When the di⇥erence between the false and true vacuum of topologically non-trivial solutions of the monopole energy densities ⇤ is small, the monopole can be treated type which are classically stable. The presence of the as a thin shell, the so called thin wall approximation. global minimum at ⌥ = 0 allows for the possibility that Within this approximation, the monopole can be divided the monopole solution although topologically non-trivial, into three regions as shown in figure 2. There is a region could be dynamically unstable. of essentially true vacuum extending from r =0uptoa A time independent spherically symmetric ansatz for radius R.Atr = R, there is a thin shell of thickness ⇥ the monopole can be chosen in the usual way as in which the field value changes exponentially from the true vacuum to the false vacuum. Outside this shell the ⌥a =ˆra h(r) monopole is essentially in the false vacuum, and so we 1 K(r) have Aa = ⇤ rˆ µ µab b er ⇥ A0 =0 (5) h 0 ,K 1 r 1.2 Wall 1 0.8 True Vacuum False Vacuum ) --> ! 0.6 0.4 0.2 ! ""> Potential V( −$ $ h(r)/! K(r) 0 R "# R - "/2 R + "/2 -0.2 r --> FIG. 1: The potential V (⌅) for = 0 as a function of one of FIG. 2: The monopole profile under the thin wall approxima- 6 the components of the field ⌅, shifted by an additive constant tion. so that ⌅ = ⇤ has vanishing V and the true vacuum has V = ⇥. III. THIN WALLED MONOPOLES 2(SU(2)/U(1)) which is Z. This suggests the existence When the di⇥erence between the false and true vacuum of topologically non-trivial solutions of the monopole energy densities ⇤ is small, the monopole can be treated type which are classically stable. The presence of the as a thin shell, the so called thin wall approximation. global minimum at ⌥ = 0 allows for the possibility that Within this approximation, the monopole can be divided the monopoleThin solution Wall although False topologicallyMonopoles non-trivial, into three regions as shown in figure 2. There is a region could be dynamically unstable. • We search for soliton solutions to this of essentially true vacuum extending from r =0uptoa A time independent spherically symmetric ansatz for radius R.Atr = R, there is a thin shell of thickness ⇥ the monopoletheory can which be chosen would exist in the in usualthe false way as vacuum: in which the field value changes exponentially from the true vacuum to the false vacuum. Outside this shell the ⌥a =ˆra h(r) monopole is essentially in the false vacuum, and so we 1 K(r) have Aa = ⇤ rˆ µ µab b er ⇥ A0 =0 (5) h 0 ,K 1 r 1.2 Wall 1 0.8 True Vacuum False 3Vacuum ) --> ! 0.6 1.2 Wall 1 0.4 0.8 True Vacuum False Vacuum 0.2 ) --> ! ""> Potential V( ! −$ $ 0.6 h(r)/! K(r) 0 R "# 0.4 R - "/2 R + "/2 -0.2 r --> 0.2 ! ""> Potential V( −$ $ h(r)/! K(r) FIG. 1: The potential V (⌅) for = 0 as a function of one of 0 FIG. 2: The monopoleR profile under the thin wall approxima- "# 6 R - "/2 R + "/2 the components of the field ⌅, shifted by an additive constant tion. -0.2 so that ⌅ = ⇤ has vanishing V and the true vacuum has r --> V = ⇥. FIG. 1: The potential V (⌅) for = 0 as a function of one of FIG. 2: The monopole profile under the thin wall approxima- 6 the components of the field ⌅, shifted by an additive constant tion. so that ⌅ = ⇤ has vanishing V and the true vacuum has III. THIN WALLED MONOPOLES V = ⇥. 2(SU(2)/U(1)) which is Z. This suggests the existence When the di⇥erence between the false and true vacuum III. THIN WALLED MONOPOLES of topologically non-trivial solutions of the monopole energy densities ⇤ is small, the monopole can be treated type which are classically stable. The presence of the (SU(2)/U(1)) which is Z. This suggests the existence as a thin shell, the so called thin wall approximation. global2 minimum at ⌥ = 0 allows for the possibility thatWhen the di⇥erence between the false and true vacuum of topologically non-trivial solutions of the monopole energyWithin densities this⇤ is small, approximation, the monopole the can monopole be treated can be divided thetype monopole which are solution classically although stable. topologically The presence non-trivial, of the into three regions as shown in figure 2. There is a region could be dynamically unstable. as a thin shell, the so called thin wall approximation. global minimum at ⌥ = 0 allows for the possibility that Withinof this essentially approximation, true the vacuum monopole extending can be divided from r =0uptoa theA time monopole independent solution although spherically topologically symmetric non-trivial, ansatz forinto three regions as shown in figure 2. There is a region could be dynamically unstable. radius R.Atr = R, there is a thin shell of thickness ⇥ the monopole can be chosen in the usual way as of essentiallyin which true the vacuum field extending value changes from r =0uptoa exponentially from the A time independent spherically symmetric ansatz for radius R.Atr = R, there is a thin shell of thickness ⇥ the monopole can be chosen in the usual way as true vacuum to the false vacuum. Outside this shell the ⌥a =ˆra h(r) in which the field value changes exponentially from the true vacuummonopole to the is false essentially vacuum. Outside in the this false shell vacuum, the and so we • The functions⌥ =ˆr h ( r ) and1 K ( r ) should have aa a 3 monopolehave is essentially in the false vacuum, and so we Aµ = ⇤µab rˆb the followinga behaviour:1 K(err) have Aµ = ⇤µab rˆb 1.2 A =0 er (5) ⇥ 0 h 0 ,K 1 ⇥ r 1.2 Wall 1 0.8 True Vacuum False Vacuum ) --> ! 0.6 0.4 0.2 ! ""> Potential V( −$ $ h(r)/! K(r) 0 R "# R - "/2 R + "/2 -0.2 r --> FIG. 1: The potential V (⌅) for = 0 as a function of one of FIG. 2: The monopole profile under the thin wall approxima- 6 the components of the field ⌅, shifted by an additive constant tion. so that ⌅ = ⇤ has vanishing V and the true vacuum has V = ⇥. III. THIN WALLED MONOPOLES 2(SU(2)/U(1)) which is Z. This suggests the existence When the di⇥erence between the false and true vacuum of topologically non-trivial solutions of the monopole energy densities ⇤ is small, the monopole can be treated type which are classically stable. The presence of the as a thin shell, the so called thin wall approximation. global minimum at ⌥ = 0 allows for the possibility that WithinThin this Walled approximation, False the monopole Monopoles can be divided the monopole solution although topologically non-trivial, into three regions as shown in figure 2. There is a region could be dynamically unstable. of essentially true vacuum extending from r =0uptoa A time independent spherically symmetric ansatz for •radiusWe firstR.At justr = assumeR, there the is aexistence thin shell ofof thickness thin ⇥ the monopole can be chosen in the usual way as inwalled which themonopoles. field value changes This means exponentially the from the true vacuum to the false vacuum. Outside this shell the ⌥a =ˆra h(r) monopolefunctions is essentiallyh(r) and K(r) in the have false the vacuum, following and so we 1 K(r) have Aa = ⇤ rˆ profiles: µ µab b er ⇥ A0 =0 (5) h 0 ,K 1 r 2. Undershoot Thus we get the the inequality sandwich 6 2 ⌅ 2 2 To prove the undershoot we use the expression Eqn. This function is plotted in figure 4.Thereisaminimum and at R = R1 and this corresponds to the classically stable 1 1 dK 2 monopole solution. This solution has a bubble of true A˙ a A˙ a = R˙ 2. (52) 4 µ µ 2e2r2 dr vacuum in its core and the radius R1 of this bubble is ⇤ ⌅ obtained by solving dE/dR = 0. However, this monopole configuration can tunnel quantum mechanically through The Lagrangian can then be expressed as the finite barrier into a configuration with R = R2 where 2 2 ⇤ 2 dh 2 1 dK 2 E(R1)=E(R2). Once this occurs, the monopole can L =2⇥ r R˙ + R˙ dr E(R). dr e2 dr continue to lose energy through an expansion of the core 0 ⇤ ⌅ ⇤ ⌅ ⇧ ⇥ (53) since the barrier which was present at R1 is no longer able to prevent this. From (8), for large r, the equation of motion of h can be Graph of the energy written as ⇧V (h) h⇥⇥ =0. (54) ⇧h Multiplying both sides by h⇥ and integrating by parts with respect to r, one obtains h⇥ = 2V (h). (55) Instanton Tunneling Furthermore, since dh/dr⌃is non-vanishing only in the ENERGY --> thin-wall, the value of r in the first integral in (53) can be replaced by R and we have R1 R2 2 ⇤ dh ⇤ dh dr r2 R˙ 2 = R2R˙ 2 dr 2V (h) dr dr RADIUS OF BUBBLE --> ⇧0 ⇤ ⌅ ⇧0 ⇤ ⌅ 2 2 ⌃ = R R˙ S1 (56) FIG. 4: The function E(R) plotted versus bubble radius. The where classically stable monopole solution has R = R1. This solu- tion can tunnel quantum mechanically to a configuration with R = R2 and then expand classically. S1 = dh 2V (h). (57) 0 ⇧ ⌃ Defining We now proceed to determine the action of the instan- 2 ton describing the tunneling from R = R1 to R = R2.In 1 ⇤ dK S2 = 2 dr , (58) the thin wall approximation, the functions h and K can e 0 dr be written as ⇧ ⇤ ⌅ the Lagrangian (53) becomes h = h(r R) ˙ 2 2 K = K(r R) (48) L =2⇥R (S1R + S2) E(R) (59) and the exact forms of the functions h and K will not be and the action can be written as required in the ensuing analysis. The only requirement ⇤ ˙ 2 2 is that both h and K change exponentially when their S = dt 2⇥R (S1R + S2) E(R) . (60) argument (r R) is small. An example of a function with ⇧ ⇤ ⇥ this type of behaviour is the hyperbolic tangent function. In Euclidean space, the expression for the action becomes The time derivative of ⌅ can be written as ⇤ ˙ 2 2 dh SE = d⇤ 2⇥R (S1R + S2)+E(R) (61) ⌅˙a =ˆr R.˙ (49) a dR ⇧ ⇤ ⇥ where ⇤ = it is the Euclidean time and R˙ is the derivative From (48), since (dh/dR)2 =(dh/dr)2,wehave with respect to ⇤. The instanton solution R(⇤)whichwe 1 1 dh 2 1 dh 2 are seeking obeys the boundary conditions R = R1 for ⌅˙a⌅˙a = R˙ 2 = R˙ 2. (50) ⇤ = , R = R for ⇤ = 0, and dR/d⇤ = 0 for ⇤ = 0. 2 2 dR 2 dr ±⇥ 2 ⇤ ⌅ ⇤ ⌅ It can be obtained by solving the equations of motion Similarly, derived from (61). However, the exact form for R(⇤) will not be of interest here since the decay rate of the ˙ a 1 dK ˙ monopole is determined ultimately from SE [13]. The Aµ = µabrˆb R (51) er dR calculation of SE will be the subject of the next section. ⇤ ⌅ 7 This function is plotted in figure 4.Thereisaminimum and at R = R1 and this corresponds to the classically stable 1 1 dK 2 monopole solution. This solution has a bubble of true A˙ a A˙ a = R˙ 2. (52) 4 µ µ 2e2r2 dr vacuum in its core and the radius R1 of this bubble is ⇤ ⌅ obtained by solving dE/dR = 0. However, this monopole configuration can tunnel quantum mechanically through The Lagrangian can then be expressed as the finite barrier into a configuration with R = R2 where 2 2 ⇤ 2 dh 2 1 dK7 2 E(R1)=E(R2). Once this occurs, the monopole can L =2⇥ r R˙ + R˙ dr E(R). dr e2 dr continue to lose energy through an expansion of the core 0 ⇤ ⌅ ⇤ ⌅ This function is plotted in figure 4.Thereisaminimum and ⇧ ⇥ (53) since the barrier which was present at R1 is no longer at R = R1 and this corresponds to the classically stable From (8), for large r, the equation of motion of h can be able to prevent this. 1 1 dK 2 7 monopole solution. This solution has a bubble of true A˙ a A˙ awritten= as R˙ 2. (52) vacuum in its core and the radius R of this bubble is 4 µ µ 2e2r2 dr 1 ⇤ ⌅ ⇧V (h) obtainedThis by solving functiondE/dR is plotted= 0. However, in figure this4.Thereisaminimum monopole and h⇥⇥ =0. (54) The Lagrangian can then be expressed as ⇧h configurationat R = canR1 tunneland this quantum corresponds mechanically to the through classically stable 2 1 a a 1 dK 2 the finitemonopole barrier into solution. a configuration This solution with R = hasR2 awhere bubble of true Multiplying2 ˙ ˙ both sides2 by h⇥ and˙ integrating by parts ⇤ 2 dh 2 Aµ1AµdK= 2 22 R . (52) E(R1)=vacuumE(R2). in Once its core this and occurs, the the radius monopoleR of this can bubbleL =2 is⇥ r withR˙ respect4+ to r,2 onee Rr˙ obtainsdrdr E(R). 1 dr e2 dr ⇤ ⌅ continueobtained to lose energy by solving throughdE/dR an expansion= 0. However, of the core this monopole ⇧0 ⇤ ⌅ ⇤ ⌅ The Lagrangian can then be expressed⇥ (53) as since theconfiguration barrier which can was tunnel present quantum at R1 is mechanically no longer through h⇥ = 2V (h). (55) able to prevent this. Instanton Tunneling From (8), for large r, the equation of motion of h can be the finite barrier into a configuration with R = R2writtenwhere as 2 ⌃ 2 Furthermore,⇤ 2 dh since2 dh/dr1 isdK non-vanishing2 only in the E(R )=E(R ).ENERGY --> Once this occurs, the monopole can ˙ ˙ 1 2 L =2⇥ thin-wall,r the valueR + of r2in the firstR integraldr inE( (R53).) can 0 ⇧V (h)dr e dr continue to lose energy through an expansion of the core ⇧h⇥⇥be replaced⇤ =0 by⌅ .R and we have⇤ ⌅(54) R1 R2 ⇥ (53) since the barrier which was present at R1 is no longer ⇧h From (8), for large r, the2 equation of motion of h can be able to prevent this. ⇤ dh ⇤ dh Multiplyingwritten both sides as by drh⇥ rand2 integratingR˙ 2 = R by2R˙ parts2 dr 2V (h) with respect to r, one obtains dr dr RADIUS OF BUBBLE --> ⇧0 ⇤ ⌅ ⇧0 ⇤ ⌅ 2 2 ⌃ ⇧V (h=) R R˙ S1 (56) h⇥ = 2V (hh)⇥⇥. =0. (55) (54) FIG. 4: The function E(R) plotted versus bubble radius. The ⇧h Instanton Tunneling where classically stable monopole solution has RFurthermore,= R1. This solu- since dh/dr⌃is non-vanishing only in the Multiplying both sides by h⇥ and integrating by parts ENERGY --> tion can tunnel quantum mechanically to athin-wall, configuration the with value of r in the first integral in (53) can with respect to r, one obtainsS = dh 2V (h). (57) R = R2 and then expand classically. be replaced by R and we have 1 R1 R2 0 ⇧ ⌃ 2 h⇥ = 2V (h). (55) ⇤ dh Defining ⇤ dh WeInstanton now proceed Tunneling to determine the action ofdr the r2 instan- R˙ 2 = R2R˙ 2 dr 2V (h) 0 Furthermore,dr since dh/dr0 ⌃isdr non-vanishing2 only in the RADIUSton OF describing BUBBLE --> the tunneling from R = R1⇧ to R = R⇤2.In⌅ ⇧ ⇤ 1 ⌅ ⇤ dK ENERGY --> 2 2 S2 = ⌃ dr , (58) thin-wall, the= R valueR˙ S of1 r in the2 first integral(56) in (53) can the thin wall approximation, the functions h and K can e 0 dr be written as be replaced by R and we have ⇧ ⇤ ⌅ FIG. 4: The functionRE1(R) plotted versus bubble radius.R2 The classically stable monopole solution has R = R . This solu- where the Lagrangian (53) becomes 1 2 tion can tunnel quantum mechanically to a configurationh = h(r withR) ⇤ dh ⇤ dh dr r2 R˙ 2 = R˙22R˙ 2 2 dr 2V (h) R = R2 and then expand classically. K = K(r R) (48) S1 = dh 2VL(h=2). ⇥R (S1R + (57)S2) E(R) (59) 0 dr 0 dr RADIUS OF BUBBLE --> ⇧ ⇧⇤0 ⌅ ⇧ ⇤ ⌅ and the action⌃ can be2 written˙ 2 as ⌃ and the exact forms of the functions h andDefiningK will not be = R R S1 (56) required in the ensuing analysis. The only requirement We nowFIG. proceed 4: The to function determineE(R the) plotted action versus of the bubble instan- radius. The ⇤ 2 2 where S = dt 22⇥R˙ (S1R + S2) E(R) . (60) ton describingclassically the stable tunnelingis thatmonopole from bothR solutionh=andR1 Kto hasRchange=RR=2.In exponentiallyR . This solu- when their 1 ⇤ dK 1 S = dr , (58) the thintion wall can approximation, tunnelargument quantum the mechanically(r functionsR) is small.h toand a An configurationK examplecan of a with function with2 e2 ⇧ ⇤dr ⇥ ⇧0 ⇤ ⌅ be writtenR = asR2 and thenthis expand type of classically. behaviour is the hyperbolic tangent function. In EuclideanS1 = space, thedh expression2V (h). for the action(57) becomes The time derivative of ⌅ can be writtenthe as Lagrangian (53) becomes ⇧0 h = h(r R) ⇤ ⌃˙ 2 2 dh Defining ˙ 2 SE2 = d⇤ 2⇥R (S1R + S2)+E(R) (61) K = K(r R) ⌅˙a =ˆr (48)R.˙ L(49)=2⇥R (S1R + S2) E(R) (59) We now proceed to determine the actiona of the instan- ⇧ ⇤ dR 2 ⇥ Effectiveton describing dynamics the tunneling of from theR = radiusR1 to R = andR2.In the action can be written as 1 ⇤ dK ˙ and the exact forms of the functions h and K will2 not be 2 where ⇤S= =it is the Euclideandr time, and R is the derivative(58) the thin wall approximation,From (48), since the(dh/dR functions) =(dh/drh and) ,wehaveK can 2 2 required• In in the the thin ensuing wall analysis. approximation, The only the requirement only with respect toe ⇤.0 The instantondr solution R(⇤)whichwe ⇤ ˙ 2 2 ⇧ ⇤ ⌅ is thatbe both writtenh and asK change exponentially when2 their 2 S = dt 2⇥areR (S seeking1R + S obeys2) E the(R boundary) . (60) conditions R = R1 for dynamical degree of1 freedom˙a ˙a 1 is dhthe radius.˙ 2 1 dh ˙ 2 argument (r R) is small. An example⌅ ⌅ = of a functionR with= R . ⇧the ⇤(50) Lagrangian⇤ = (53, R)= becomesR2 for ⇤ = 0, and dR/d⇤ = 0 for ⇤ = 0. 2 2 dR 2 dr ±⇥ ⇥ this type• We of behaviour can write: is the hyperbolich = h(r tangent⇤R) ⌅ function. ⇤ ⌅ It can be obtained by solving the equations of motion 7In Euclidean space, the expression˙ for2 the action2 becomes The time derivative ofSimilarly,⌅ canK be written= K(r as R) (48) derivedL =2 from⇥R (61(S).1R However,+ S2) theE(R exact) form (59) for R(⇤) ⇤ will˙ 2 not be2 of interest here since the decay rate of the This function is plotted in figure 4.Thereisaminimum and dh 1 dK SE = andd the⇤ 2 action⇥R (S1 canR + beS2 written)+E(R as) (61) and the exact⌅˙ formsa =ˆr of theR.˙ functions˙ a h and(49)K will not˙ be monopole is determined ultimately from SE [13]. The • This gives: a Aµ = µabrˆb R ⇤(51) at R = R1 and this corresponds to the classically stable dR 2 ⇧ calculation of S will be⇥ the subject of the next section. required in the ensuing analysis. The onlyer requirementdR ⇤ E 1 a a 1 dK 2 ⇤ ⌅ 2 2 monopole solution. This solution has a bubble of true A˙ A˙ = R˙ . (52)where ⇤ = it is theS Euclidean= dt time2⇥ andR˙ (SR˙1isR the+ derivativeS2) E(R) . (60) From (48is), that since both (dh/dRhµ and)µ2 =(Kdh/dr2change2 )2,wehave exponentially when their vacuum in its core and the radius R1 of this bubble is 4 2e r dr argument (r R) is small.⇤ An example⌅ of a functionwith with respect to ⇤. The⇧ instanton ⇤ solution R(⇤)whichwe obtained by solving dE/dR = 0. However, this monopole • and consequently: ⇥ Thethis Lagrangian type of behaviour can then2 be is the expressed hyperbolic2 as tangent function.are seeking obeys the boundary conditions R = R1 for configuration can tunnel quantum mechanically through 1 ˙a ˙a 1 dh ˙ 2 1 dh ˙ 2 In Euclidean space, the expression for the action becomes The⌅ time⌅ = derivativeR of =⌅ can be writtenR . as(50) ⇤ = , R = R2 for ⇤ = 0, and dR/d⇤ = 0 for ⇤ = 0. the finite barrier into a configuration with R = R2 where 2 2 dR 2 2 dr 2 ±⇥ ⇤ ⇤ dh⌅ 1⇤ dK⌅ It can be obtained by solving the equations of motion E(R )=E(R ). Once this occurs, the monopole can L =2⇥ r2 R˙ 2 + R˙ 2 dr E(R). ⇤ ˙ 2 2 1 2 2 dh derived from (61S).E However,= d⇤ the2⇥ exactR (S1 formR + forS2)+R(⇤E) (R) (61) Similarly, 0 dr ˙a e dr continue to lose energy through an expansion of the core ⇧ ⇤ ⌅ ⌅ =ˆra⇤ R.⌅˙ (49) dR ⇥ (53)will not be of interest here⇧ ⇤ since the decay rate of the ⇥ since the barrier which was present at R1 is no longer From (8), for˙ a large r, the equation1 dK of˙ motion of h can bemonopole is determined ultimately from SE [13]. The˙ able to prevent this. Aµ = µabrˆb 2 R 2 (51) where ⇤ = it is the Euclidean time and R is the derivative From (48), since (dh/dRer ) dR=(dh/dr) ,wehave calculation of SE will be the subject of the next section. written as ⇤ ⌅ with respect to ⇤. The instanton solution R(⇤)whichwe ⇧V (h)2 2 are seeking obeys the boundary conditions R = R1 for 1 ˙a ˙a h⇥⇥1 dh =0˙ 2. 1 dh ˙ (54)2 ⌅ ⌅ = ⇧h R = R . (50) ⇤ = , R = R2 for ⇤ = 0, and dR/d⇤ = 0 for ⇤ = 0. 2 2 dR 2 dr ±⇥ ⇤ ⌅ ⇤ ⌅ It can be obtained by solving the equations of motion Multiplying both sides by h⇥ and integrating by parts withSimilarly, respect to r, one obtains derived from (61). However, the exact form for R(⇤) will not be of interest here since the decay rate of the ˙ ah⇥ = 2V (h). 1 dK ˙ (55) monopole is determined ultimately from SE [13]. The Instanton Tunneling Aµ = µabrˆb R (51) er dR calculation of SE will be the subject of the next section. Furthermore, since dh/dr⌃is non-vanishing⇤ ⌅ only in the ENERGY --> thin-wall, the value of r in the first integral in (53) can be replaced by R and we have R1 R2 2 ⇤ dh ⇤ dh dr r2 R˙ 2 = R2R˙ 2 dr 2V (h) dr dr RADIUS OF BUBBLE --> ⇧0 ⇤ ⌅ ⇧0 ⇤ ⌅ 2 2 ⌃ = R R˙ S1 (56) FIG. 4: The function E(R) plotted versus bubble radius. The where classically stable monopole solution has R = R1. This solu- tion can tunnel quantum mechanically to a configuration with R = R2 and then expand classically. S1 = dh 2V (h). (57) 0 ⇧ ⌃ Defining We now proceed to determine the action of the instan- 2 ton describing the tunneling from R = R1 to R = R2.In 1 ⇤ dK S2 = 2 dr , (58) the thin wall approximation, the functions h and K can e 0 dr be written as ⇧ ⇤ ⌅ the Lagrangian (53) becomes h = h(r R) ˙ 2 2 K = K(r R) (48) L =2⇥R (S1R + S2) E(R) (59) and the exact forms of the functions h and K will not be and the action can be written as required in the ensuing analysis. The only requirement ⇤ ˙ 2 2 is that both h and K change exponentially when their S = dt 2⇥R (S1R + S2) E(R) . (60) argument (r R) is small. An example of a function with ⇧ ⇤ ⇥ this type of behaviour is the hyperbolic tangent function. In Euclidean space, the expression for the action becomes The time derivative of ⌅ can be written as ⇤ ˙ 2 2 dh SE = d⇤ 2⇥R (S1R + S2)+E(R) (61) ⌅˙a =ˆr R.˙ (49) a dR ⇧ ⇤ ⇥ where ⇤ = it is the Euclidean time and R˙ is the derivative From (48), since (dh/dR)2 =(dh/dr)2,wehave with respect to ⇤. The instanton solution R(⇤)whichwe 1 1 dh 2 1 dh 2 are seeking obeys the boundary conditions R = R1 for ⌅˙a⌅˙a = R˙ 2 = R˙ 2. (50) ⇤ = , R = R for ⇤ = 0, and dR/d⇤ = 0 for ⇤ = 0. 2 2 dR 2 dr ±⇥ 2 ⇤ ⌅ ⇤ ⌅ It can be obtained by solving the equations of motion Similarly, derived from (61). However, the exact form for R(⇤) will not be of interest here since the decay rate of the ˙ a 1 dK ˙ monopole is determined ultimately from SE [13]. The Aµ = µabrˆb R (51) er dR calculation of SE will be the subject of the next section. ⇤ ⌅ 3 1.2 Wall 1 0.8 True Vacuum False Vacuum ) --> ! 0.6 0.4 0.2 ! ""> Potential V( −$ $ h(r)/! K(r) 0 R "# R - "/2 R + "/2 -0.2 r --> FIG. 1: The potential V (⌅) for = 0 as a function of one of FIG. 2: The monopole profile under the thin wall approxima- 6 the components of the field ⌅, shifted by an additive constant tion. so that ⌅ = ⇤ has vanishing V and the true vacuum has V = ⇥. III. THIN WALLED MONOPOLES 2(SU(2)/U(1)) which is Z. This suggests the existence When the di⇥erence between the false and true vacuum of topologically non-trivial solutions of the monopole energy densities ⇤ is small, the monopole can be treated type which are classically stable. The presence of the as a thin shell, the so called thin wall approximation. global minimum at ⌥ = 0 allows for the possibility that Within this approximation, the monopole can be divided the monopole solution although topologically non-trivial, into three regions as shown in figure 2. There is a region could be dynamically unstable. of essentially true vacuum extending from r =0uptoa A time independent spherically symmetric ansatz for radius R.Atr = R, there is a thin shell of thickness ⇥ the monopole can be chosen in the usual way as in which the field value changes exponentially from the true vacuum to the false vacuum. Outside this shell the ⌥a =ˆra h(r) monopole is essentially in the false vacuum, and so we 1 K(r) have Aa = ⇤ rˆ µ µab b er ⇥ A0 =0 (5) h 0 ,K 1 r This function is plotted in figure 4.Thereisaminimum and at R = R1 and this corresponds to the classically stable 1 1 dK 2 monopole solution. This solution has a bubble of true A˙ a A˙ a = R˙ 2. (52) vacuum in its core and the radius R of this bubble is 4 µ µ 2e2r2 dr 1 ⇤ ⌅ 7 obtained by solving dE/dR = 0. However, this monopole The Lagrangian can then be expressed as configuration can tunnelThis function quantum is plotted mechanically in figure through4.Thereisaminimum and 2 2 the finite barrier intoat aR configuration= R1 and this with correspondsR = R to2 where the classically stable ⇤ 2 dh 1 2 1 dK1 dK 2 2 E(R1)=E(R2). Oncemonopole this solution. occurs, the This monopole solution has can a bubbleL =2 of true⇥ r R˙ ˙ a+˙ a R˙ dr˙ 2 E(R). AµAµ =2 R .7 (52) 0 dr 4 e 2e2drr2 dr continue to lose energyvacuum through in its an core expansion and the radius of theR core1 of this bubble is⇧ ⇤ ⌅ ⇤ ⇤⌅ ⌅ obtained by solving dE/dR = 0. However, this monopole ⇥ (53) since the barrierThis which function was is present plotted in at figureR1 4is.Thereisaminimum no longer and configuration can tunnel quantum mechanicallyFrom through (8), forThe large Lagrangian r, the equation can then be of expressed motion of as h can be able to prevent this.at R = R1 and this corresponds to the classically stable 2 7 the finite barrier into a configuration with R = R2 where 1 1 dK2 2 monopole solution. This solution has a bubble of truewritten as A˙ a A˙⇤a = dh R˙ 2.1 dK (52)7 E(R )=E(R ). Once this occurs, the monopole can L =2⇥µ µ r2 2 2 R˙ 2 + R˙ 2 dr E(R). vacuumThis in1 its function core and2 is plotted the radius in figureR1 4of.Thereisaminimum this bubble is and 4 2e r dr 2 0 dr⇤ ⌅ e dr obtainedcontinueat byRThis solving= toR function loseanddE/dR energy this is plotted corresponds= through 0. However, in figure to an4 the.Thereisaminimum expansion this classically monopole of stable the coreand ⇧ ⇧⇤V (h⌅) ⇤ ⌅ 1 The Lagrangian canh then⇥⇥ be expressed=02 as. ⇥ (54)(53) configurationsince the can barrier tunnel which quantum was mechanically present at R through1 is no longer 1 a a 1 dK 2 monopoleat R = solution.R1 and this This corresponds solution to has the a classically bubble of stable true A˙ A˙ = ⇧h 2 R˙ . (52) From (81),µ forµ large12 r,2 thedK equation of motion of h can be the finiteablevacuum to barriermonopole prevent in into its solution. this. a core configuration and This the solution radius withRR has1 =of aR this bubble2 where bubble of true is 4A˙ a A˙ a2 = 2e r dr R2˙ 2. (52) written⇤ 2 4dh asµ µ 22e2r12 ⇤drdK ⌅ 2 E(R1)=obtainedEvacuum(R2). by in Once solving its core thisdE/dR and occurs, the= 0. radius the However, monopoleR1 of this this monopole can bubble isL =2⇥ r R˙ + R˙ dr E(R). MultiplyingThe Lagrangian both sidesdr can then by beeh2 expressed⇥⇤ anddr⌅ integrating as by parts continueconfiguration toobtained lose energy by can solving through tunneldE/dR quantum an expansion= 0. However, mechanically of the this core monopole through ⇧0 ⇤ ⌅ ⇤ ⌅ configuration can tunnel quantum mechanically throughwith respectThe Lagrangian to r, one can obtainsthen be expressed⇧V as(h)⇥ (53) the finite barrier into a configuration with R = R2 where 2 h⇥⇥ 2 =0. (54) since the barrier which was present at R1 is no longer ⇤ dh 1 dK the finite barrier into a configuration with R = R2 whereFrom (8), for large2 r, the2 equation˙ 2 of motion⇧2h ˙ 2 of h can be able to preventE(R1)= this.E(R2). Once this occurs, the monopole can L =2⇥ ⇤ r dh R +1 dK R dr E(R). E(R )=E(R ). Once this occurs, the monopole can L =2⇥ r2 dr R˙ 2 + e2 dr R˙ 2 dr E(R). continue1 to lose energy2 through an expansion of the corewritten as ⇧0 ⇤ h⇥⌅= 22V (⇤h). ⌅ (55) continue to lose energy through an expansion of the core Multiplying0 dr both sidese dr by h⇥ and⇥ integrating(53) by parts Instantonsince Tunneling the barrier which was present at R1 is no longer ⇧ ⇤ ⌅ ⇤ ⌅ ⇥ since the barrier which was present at R is no longer From (8), for large r, the⇧V equation(h) of motion of h can(53) be able to prevent this. 1 with respecth to r, one=0 obtains. (54) able to prevent this. Furthermore,writtenFrom (8 as), since for largedh/dr⇥⇥ r, the equation⌃is non-vanishing of motion of h can only be in the written as ⇧h ENERGY --> thin-wall,Multiplying the valueboth sides of r byin⇧hV the(andhh)⇥ first= integrating2 integralV (h). by parts in (53) can(55) Instanton Tunneling h⇥⇥ ⇧V⇥(h) =0. (54) be replacedwith respect by toRr,and oneh obtains we⇥⇥ have⇧h =0. (54) R1 R2 Furthermore, since⇧h dh/dr⌃is non-vanishing only in the ENERGY --> Multiplying both sides by h⇥ and integrating by parts Multiplyingthin-wall, both2 the sides value by h⇥ ofandr integratingin the first by integral parts in (53) can with respect to r,h one⇥ = obtains2V (h). (55) Instanton Tunneling ⇤ with2 respectbedh replaced to r, one2 by obtainsR and2 we2 have⇤ dh R1 R2 dr r R˙ = R R˙ dr 2V (h) Furthermore, since dh/dr⌃is non-vanishing only in the 0 dr h⇥ = 2V (h). 0 dr (55) RADIUSENERGY --> OF BUBBLE --> h⇥ = 22V (h). (55) InstantonInstanton Tunneling Tunneling ⇧ thin-wall,⇤ the⇤ value⌅ of rdhin the first⇧ integral⇤ in⇤ (53⌅) candh 2 2˙ 2˙ 2 2 ˙ 2 ⌃ be replacedFurthermore,Furthermore, by Rdr since sinceand r wedh/drdh/dr= have⌃R⌃isisR non-vanishingR non-vanishing=S1R R only only indr in the the (56)2V (h) R1 R2 dr dr ENERGY --> ENERGY --> 0 0 RADIUS OF BUBBLE --> thin-wall,thin-wall,⇧ the value value⇤ of ofrrin⌅in the the first first integral integral in⇧ in (53 (53) can)⇤ can ⌅ FIG. 4: The function E(R) plotted versus bubble radius. The 2 2 2 ⌃ ⇤bebe replaced replaceddh by by RR andand we we have have ⇤= R Rdh˙ S1 (56) R1 R1 R2R2 where • Then2 the first˙ 2 term becomes:2 ˙ 2 classically stable monopole solution has R = R1. This solu- dr r R = R R dr 2V (h) FIG. 4: The function E(R) plotted versus bubble radius. The0 dr 22 0 dr RADIUS OF BUBBLE --> ⇤ ⇤ ⌅dh ⇤ ⇤dh ⌅ tion can tunnel quantum mechanically to a configuration with ⇧ ⇤where2 dh ˙˙22 22˙ 2˙⇧2 ⇤ dh classically stable monopole solution has R = R1. This solu- drdr r RR===RR2RR˙R2RS drdr ⌃2V2(Vh(56))(h) 0 drdr 10 drdr R = R2 and then expandtion canclassically. tunnel quantumRADIUSRADIUS mechanically OF OFBUBBLE BUBBLE --> to--> a configuration with ⇧0 ⇤ S1⌅ = dh ⇧ 20V (h⇤). ⌅ (57) ⇧ ⇤ ⌅ 2 2 ⇧ ⇤ ⌅⌃ FIG. 4: The function E(R) plotted versus bubble radius. The =0 R 2R˙ ˙S2 1 ⌃ (56) R = R2 and then expand classically. where ⇧ = SR1 R=S1 dh 2V (h). (56) (57) classically stableFIG. 4: monopole The function solutionE(R) plotted has R versus= R1. bubble This radius. solu- The ⌃ 0 tion canFIG. tunnel 4: quantum The function mechanicallyE(R) plotted to a versus configuration bubble radius. with The where ⇧ classically stable monopole solution has R = R1. ThisDefining solu- where ⌃ classically stable monopole solution has R = R1. This solu- • with R = R2 andtion then can expand tunnel quantum classically. mechanically to a configuration with DefiningS1 = dh 2V (h). (57) We now proceed to determinetion can tunnel the quantum action mechanically of the instan- to a configuration with 0 We nowR = R proceed2 and then to expand determine classically. the action of the instan- S1 =⇧ dh 2V (h). 2 (57) R = R2 and then expand classically. S1 = dh⌃ 2V (h). (57) ton describing the tunneling from R = R1 to R = R2.In 1 ⇧0 ⇤ dK 2 ton describing the tunneling from R = R1 to R = R2Defining.In 0 ⌃ 1 ⇤ dK • and definingS2 = ⇧S =dr⌃ dr , , (58)(58) the thin wall approximation,Wethe now thin proceed wall the approximation, to functions determine theh and the action functionsK ofcan the instan-h and K canDefining e2 2 2 dr We now proceed to determine the action of the instan- Defining 0 e 0 2 dr be written as ton describingWe now the proceed tunneling to determine from R = theR1 actionto R = ofR the2.In instan- 1 ⇧⇤ dK⇤⇧ 2 ⌅⇤ ⌅ be writtenton describing as the tunneling from R = R1 to R = R2.In S = 1 ⇤dr dK , (58) the thin wall approximation, the functions h and K can 2 S =2 dr ,2 (58) tonthe describing thin wall the approximation, tunneling from theR functions= R1 to hRand= RK2.Incan the Lagrangian2 e 210 (53⇤) becomesdrdK the Lagrangian• yields: (53S) becomes= e 0 dr⇤ dr ⌅ , (58) be writtenthe as thin wall approximation, the functions h and K can 2 ⇧2⇧ ⇤ ⌅ h = beh( writtenr R as) h = h(r R) e 0 dr be written as the Lagrangianthe Lagrangian (53 ()53 becomes) becomes⇧ ˙ 2⇤ ⌅2 ˙ 2L =2⇥2R (S1R + S2) E(R) (59) K = K(r Rh )= Kh(r=h =RK)h((rr RR)) (48) (48)the LagrangianL =2⇥ (53R) becomes(S1R + S2) E(R ) (59) ˙ 2 ˙ 2 2 2 K = Kh(Kr= =hR(Kr) (rR)R) (48) (48) L =2L =2⇥R⇥R(S(1SR1R++SS22)) EE((R)) (59) (59) and the exact forms of the functions h and K will not be and theL =2 action⇥R˙ 2( canS R2 be+ S written ) E(R as) (59) K = K(r R) and(48) the action can be written1 as2 and the exact formsandrequired of the the exactand functions in forms the the exact of ensuing the formsh functionsand of analysis. theK functionswillh and The notKh andwill onlybeK notwill requirement be not beand theand actionthe action can can be be written written as as and the action can be⇤ written as˙ 2 2 required in the ensuingand analysis.required the exact in forms The the ensuing of only the analysis.functions requirement Theh and onlyK requirementwill not be S⇤ = dt 2⇥R (S1R + S2) E(R) . (60) requiredis that in the both ensuingh and analysis.K change The exponentially only requirement when their ⇤⇤ ˙ 2 2 requiredis that in both the ensuingh and K analysis.change exponentially The only requirement when their S =S = dt dt2⇥2R˙⇥˙2R2(S(SR1R2 +2+SS2)) EE(RR)) .. (60)(60) is that both h andis thatKargumentchange both h (andr exponentiallyRK) ischange small. exponentially An when example their when of a function their withS = dt⇤ 2⇥⇧R ⇤˙(2S1 1 R2 +2 S 2) E(R) . ⇥ (60) is thatargument both (hr andR) isK small.change An exponentially example of a function when their with S = ⇧ ⇤ dt 2⇥R (S1R + S2 ) E (R⇥) . (60) argumentthis type (r ofR) behaviour is small. An is example the hyperbolic of a function tangent with function. In⇧ ⇤ Euclidean space, the expression ⇥ for the action becomes argument (r R) is small.argumentthis An type example (r ofR behaviour) is small. of a is An function the example hyperbolic with of tangent a function function. with In Euclidean⇧ ⇤ ⇧ ⇤ space, the expression for the action becomes⇥ thisThe type time of behaviour derivative is the of hyperbolic⌅ can be writtentangent function. as In Euclidean space, the expression for the action⇥ becomes this type of behaviour isthis theThe type hyperbolic time of behaviour derivative tangent is of the⌅ can hyperbolic be function. written tangent as function.In EuclideanIn Euclidean space, space, the the expression expression for for the the action action becomes becomes The time derivative of ⌅ can be written as ⇤ ⇤2 2 2 2 The time derivative of ⌅ can be written as SE = d⇤ 2⇥R˙ (S1R +˙S2)+E(R) (61) a dh ⇤SE = d⇤ 2⇥R (S1R + S2)+E(R) (61) The time derivative of ⌅ can be writtena as⌅˙ =ˆrdh R.˙ (49) ⇤ ˙ 2 2 ⌅˙ dh=ˆr a R.˙ (49) SE = ⇧ ⇤d⇤ 2⇥R (˙S21R +2 S2)+E(R) (61) ˙a ˙ adhdR SE⇤= d⇤ ⇧2 ⇤⇥R (S1R + S2)+E(R⇥) (61) ⌅ =ˆr⌅a˙a =ˆR.r dRR.˙ (49) (49) ˙ 2 2 ˙ ⇥ dR 2a 2 SEwhere= ⇤⇧= ⇤it⇧d ⇤is⇤ the 2 Euclidean⇥R (S1 timeR and+ SR2is)+ the⇥E derivative(R) (61) a Fromdh (48), since (dh/dR) =(dRdh/dr) ,wehave where ⇤ = it is the Euclidean time and⇥ R˙ is the derivative ⌅˙ =ˆra R.˙ 2 (49)2 with respect to ⇤. The instanton solution˙ ˙ R(⇤)whichwe From (48), since (dh/dR2 )2 =(dh/dr2 2) ,wehave wherewhere⇤ =⇧⇤ ⇤it=isit theis the Euclidean Euclidean time time and andRRisis the the derivative derivative From (48From), since (48dR), (dh/dR since ()dh/dR=(dh/dr) =(2 dh/dr) ,wehave) ,wehave2 are seekingwith respect obeys the to boundary⇤. The conditions instantonR solution= R1 for⇥ R(⇤)whichwe 1 ˙a ˙a 1 dh ˙ 2 1 dh ˙ 2 withwith respect respect to ⇤ to. The⇤. The instanton instanton solution solutionRR(⇤(⇤)whichwe)whichwe ⌅ ⌅ = 2 R = R2 . (50) ⇤ = , R = R2 for ⇤ = 0, and dR/d⇤ = 0 for ⇤ = 0. 2 22 dR2 22 dr 2 whereare⇤ seekingare= seekingit±are⇥is obeys the seeking obeys Euclidean the the obeys boundary boundary the time conditions boundary conditions and R˙R conditionsRis== theRR for derivativefor R = R1 for 1 2 1 ˙1a ˙1a dh112 dhdh⇤ ⌅1˙ 2 dh1 1 ⇤dhdh⌅ ˙ 2 It can be obtained by solving the equations of motion11 From (48), since (dh/dR)⌅˙a=(⌅˙a⌅=dh/dr⌅⌅˙a⌅˙a==) ,wehaveR˙ 2 = RR˙ 2 == R˙ 2. R˙ 2.R(50). (50)⇤(50)=⇤ = ,⇤R=,=RR= R,forRfor⇤==⇤R= 0,2 0, andfor and⇤dR/d=dR/d 0,⇤ ⇤= and= 0 0 fordR/d for⇤⇤=⇤= 0.= 0. 0 for ⇤ = 0. 2 Similarly,2 2 2 dR22 dRdR 2 dr2 2 drdr with respectderived±⇥ to from⇤±.⇥2 (61 The).2 However, instanton the exact solution form forR(R⇤()whichwe⇤) ⇤⇤ ⌅ ⇤⇤ ⌅ ⌅ It canIt±⇥ can beIt obtained be can obtained be by obtained by solving solving by the the solving equations equations the of of equations motion motion of motion 2 ⇤ ⌅ 2 ⇤ ⌅ will not be of interest here since the decay rate of the are seekingderived obeys from (61 the). boundary However, the conditions exact form forRR(=⇤) R1 for 1 a a Similarly,1 Similarly,dhSimilarly,2 1 ˙ adh 2 1 dK ˙ derivedmonopole fromderived is (61 determined). from However, (61 ultimately). the However, exact from formS theE [13 for exact]. TheR(⇤ form) for R(⇤) ˙ ˙ ˙ Aµ = µabrˆb˙ R (51) ⌅ ⌅ = R = R .er dR(50) ⇤ = willwillcalculation not, R not bewill= of beR not of interest of2S interestEfor bewill of⇤ here be= interest here the since 0, subject since and the here the ofdR/d decay the decay since next⇤ rate rate the= section. of 0 of decay the for the ⇤ rate= 0. of the 2 2 dR 2 dr ⇤1 ⌅dK ±⇥monopole is determined ultimately from S [13]. The a ˙ a 1 dK1 dK˙ It canmonopole be obtainedmonopole is determined by is determined solving ultimately the from ultimately equationsSEE[13 from]. The ofS motion[13]. The ⇤ ⌅ A˙ = ˙Aa µ⇤rˆ= µab ⌅rˆb R˙ R ˙ (51) (51) E µ Aµabµ =b µabrˆb er dR R calculation(51)calculation of S of SwillE will be be the the subject subject of of the the next next section. section. er ⇤dRer⌅ dR derived fromcalculation (61E). However, of SE will be the the exact subject form of the for nextR( section.⇤) Similarly, ⇤ ⌅⇤ ⌅ will not be of interest here since the decay rate of the ˙ a 1 dK ˙ monopole is determined ultimately from SE [13]. The Aµ = µabrˆb R (51) er dR calculation of SE will be the subject of the next section. ⇤ ⌅ 7 This function is plotted in figure 4.Thereisaminimum and at R = R1 and this corresponds to the classically stable 1 1 dK 2 7 monopole solution. This solution has a bubble of true A˙ a A˙ a = R˙ 2. (52) 4 µ µ 2e2r2 dr vacuum in its core and the radius R1 of this bubble is ⇤ ⌅ obtainedThis by function solving isdE/dR plotted= in 0. figure However,4.Thereisaminimum this monopole and The Lagrangian can then be expressed as configurationat R = R can1 and tunnel this corresponds quantum mechanically to the classically through stable 2 monopole solution. This solution has a bubble of true 1 ˙ a ˙ a 1 dK ˙ 2 the finite barrier into a configuration with R = R2 where AµAµ =2 2 2 R 2. (52) vacuum in its core and the radius R of this bubble is ⇤ 4 dh 2e r 1 dr dK E(R )=E(R ). Once this occurs, the1 monopole can L =2⇥ r2 R˙ 2 + ⇤ ⌅ R˙ 2 dr E(R). 1 obtained by2 solving dE/dR = 0. However, this monopole 2 The Lagrangian0 candr then be expressede dr as continueconfiguration to lose energy can tunnel through quantum an expansion mechanically of the through core ⇧ ⇤ ⌅ ⇤ ⌅ ⇥ since the barrier which was present at R is no longer (53) the finite barrier into a configuration with1 R = R2 where 2 2 From (8),⇤ for large2 dh r, the2 equation1 dK of motion2 of h can be able toE( preventR1)=E this.(R2). Once this occurs, the monopole can L =2⇥ r R˙ + R˙ dr E(R). written as dr e2 dr continue to lose energy through an expansion of the core 0 ⇤ ⌅ ⇤ ⌅ ⇧ ⇥ (53) since the barrier which was present at R1 is no longer ⇧V (h) able to prevent this. From (8), for large r,h⇥⇥ the equation=0 of motion. of h can be(54) written as ⇧h Multiplying both sides by⇧V (hh⇥) and integrating by parts h⇥⇥ =0. (54) with respect to r, one obtains⇧h Multiplying both sidesh⇥ by= h⇥2Vand(h) integrating. by parts(55) Instanton Tunneling with respect to r, one obtains Furthermore, since dh/dr⌃is non-vanishing only in the ENERGY --> h⇥ = 2V (h). (55) Instanton Tunneling thin-wall, the value of r in the first integral in (53) can be replaced by R and we⌃ have R1 R2 Furthermore, since dh/dr is non-vanishing only in the ENERGY --> thin-wall, the value2 of r in the first integral in (53) can be⇤ replaced2 bydhR and2 we have2 2 ⇤ dh R1 R2 dr r R˙ = R R˙ dr 2V (h) dr dr RADIUS OF BUBBLE --> 0 2 0 ⇧ ⇤ ⇤ dh ⌅ ⇧⇤ dh⇤ ⌅ dr r2 R˙ 2 == RR2R2˙R2˙ 2S dr 2⌃V (h) (56) dr 1 dr RADIUS OF BUBBLE --> ⇧0 ⇤ ⌅ ⇧0 ⇤ ⌅ FIG. 4: The function E(R) plotted versus bubble radius. The 2 2 ⌃ where = R R˙ S1 (56) classically stable monopole solution has R = R1. This solu- tion canFIG. tunnel 4: The quantum function mechanicallyE(R) plotted to versus a configuration bubble radius. with The classically stable monopole solution has R = R . This solu- where R = R2 and then expand classically. 1 S1 = dh 2V (h). (57) tion can tunnel quantum mechanically to a configuration with ⇧0 R = R2 and then expand classically. S1 = dh ⌃2V (h). (57) Defining ⇧0 We now proceed to determine the action of the instan- ⌃ Defining 2 ton describingWe now the proceed tunneling to determine from R the= R action1 to R of= theR instan-2.In 1 ⇤ dK S2 = dr 2 , (58) the thinton wall describing approximation, the tunneling the from functionsR = RhtoandR =KRcan.In 1e2 ⇤ dKdr 1 2 S = 0 dr , (58) be written as 2 2 ⇧ ⇤ ⌅ the thin wall approximation, the functions h and K can e 0 dr be written as the Lagrangian (53) becomes⇧ ⇤ ⌅ h = h(r R) the Lagrangian (53) becomes Effective Euclidean2 2 Action h = h(r R) L =2⇥R˙ (S1R + S2) E(R) (59) K = K(r R ) (48) ˙ 2 2 K = K (r R) (48) L =2⇥R (S1R + S2) E(R) (59) and• theThen action the Minkow can be writtenski action as is: and the exact forms of the functions h and K will not be and the action can be written as requiredand in the the exact ensuing forms analysis. of the functions The onlyh and requirementK will not be required in the ensuing analysis. The only requirement ⇤ ˙ 2 2 is that both h and K change exponentially when their S = ⇤ dt 2⇥R˙ 2 (S1R2 + S2) E(R) . (60) is that both h and K change exponentially when their S = dt 2⇥R (S1R + S2) E(R) . (60) argument (r R) is small. An example of a function with ⇧ ⇤ argument (r R) is small. An example of a function with ⇧ ⇤ ⇥ this type of behaviour is the hyperbolic tangent function. ⇥ this type of behaviour is the hyperbolic tangent function. InIn• Euclidean Euclideanyielding space, space,the Euclidean the the expression expression action: for for the the action action becomes becomes The timeThe derivative time derivative of ⌅ ofcan⌅ can be written be written as as ⇤ ⇤ ˙22 2 2 SSEE == dd⇤⇤ 22⇥⇥RR˙ (S(S1R1R++S2S)+2)+E(ER()R) (61)(61) a a dh dh ⌅˙ =ˆ⌅˙ ra=ˆr R.˙ R.˙ (49)(49) dRa dR ⇧⇧ ⇤ ⇤ ⇥ ⇥ • wherewhereThe⇤⇤ = =instantonitit isis the Euclidean Euclideanequations time timeof motion and andR˙ Ris˙ theis the derivative derivative FromFrom (48), ( since48), since (dh/dR (dh/dR)2 =()2 dh/dr=(dh/dr)2,wehave)2,wehave withwithcorrespond respect respect to to ⇤⇤ .to. The The motion instanton instanton in -E(R) solution solution withR (anR⇤)whichwe( ⇤R)whichwe 2 2 2 2 areare seekingdependent seeking obeys obeys mass the the boundary boundary conditions conditionsR R= =R1 Rfor for 1 1 ˙a ˙1a dh1 dh ˙ 21 1dhdh ˙ 2 1 ⌅˙a⌅˙a⌅=⌅ = R˙ 2 =R = R˙ 2R. . (50)(50) ⇤⇤== ,, RR == R2 forfor⇤⇤== 0, 0, and anddR/ddR/d⇤ =⇤ = 0 for 0 for⇤ =⇤ 0.= 0. 2 2 2 dR2 dR 2 2dr dr ±±⇥⇥ 2 ⇤ ⇤⌅ ⌅ ⇤ ⇤⌅ ⌅ ItIt can can be be obtained obtained by by solving solving the the equations equations of of motion motion Similarly, derivedderived from from (61).). However, However, the the exact exact form form for forR(⇤R)(⇤) Similarly, will not be of interest here since the decay rate of the will not be of interest here since the decay rate of the 1 dK monopole is determined ultimately from S [13]. The A˙ a = rˆ R˙ (51) E ˙ a µ µab b 1 dK ˙ monopolecalculation is of determinedS will be the ultimately subject of from the nextSE section.[13]. The Aµ = µabrˆb er dRR (51) E er⇤ dR⌅ calculation of SE will be the subject of the next section. ⇤ ⌅ 8 V. BOUNCE ACTION substituting for E0 in terms of the solution for R1, we get 8 a cubic equation for R3, which can be exactly factored,8 giving R = 2R . Finally, to solve for R , we solve In this section,V. we will BOUNCE derive an ACTION expression for bounce substituting3 for E1 0 in terms of the solution2 for R1, we get action SE for theV. monopole BOUNCE tunneling ACTION and compare it substitutingJ a= cubic 0 neglecting equation for E0 thein for terms constantR3 of, which the and solution canlinear be for term exactlyR1, in weR get factored,since with the bounce action for the tunneling of the false vac- aR cubic2givingis large, equationR obtaining= for2RR3.,R which2 Finally,4⌥ can/ to be=3 solve exactly /⌅ for. factored,R , we solve In this section, we will derive an expression for bounce 3 1 ⇥ 2 uumIn to this the section, true vacuumwe will derive as discussed an expression in [13 for]withno bounce givingJFactoring=R 03 neglecting= J,wehave2R1. the Finally, constant to solve and for linearR2, term we solve in R since action SE for the monopole tunneling and compare it monopolesaction SE present.for the monopole From (61 tunneling), the equation and compare of motion it J =R 0 neglectingis large, obtaining the constantR and4⌥ linear/ =3 term / in⌅. R since with the bounce action for the tunneling of the false vac- 2 R2 2 ⇥ forwithR can the bounce be written action for the tunneling of the false vac- RS2 isFactoring= large,A⇤ obtaining J,wehavedRR⇤2R 4⌥ (R/ =3R )/2(⌅R. R )(R R ) uum to the true vacuum as discussed in [13]withno FactoringE J,wehave ⇥ 1 2 3 uum to the true vacuum as discussed in [13]withno R1 monopoles present. From (61), the equation of motion ⌥ R Instanton equations of1 motion⌦E R 2 monopoles2 present. From¨ (61),˙ 2 the equation of motion R2 2 for R can(R beS1 written+ S2)R + S1RR =0. (62) ⇤ ⇤ 2 SE = ⇤A dR⇤ R (R 2 R1) (R R2)(R R3) for R•canThe be equation written of motion for R4 ⌥is:⌦ R SE ==AA⇤ dRdR⇤RR(R(R R1R)1 ) (R(R RR2)( 2)(RR R3R) 3) R1 R1R⌥1 2 2 1 ⌦E ⌥⌥ R2 ¨ ˙ ˙ 1 ⌦E 5/2 Multiplying((RR both2SS1 ++ sidesSS2))R¨R by++SRS,1RR theR˙R2 equation=0 of=0 motion. . (62) as-(62) R72/2 2 R1 R1 R3 1 2 1 4⌥ ⌦R ==A⇤A ⇤R ⇤dR1⇤R(R R1I) ,(R R2)(R (69)R3) sumes the form 4⌥ ⌦R = A⇤ 2 dR R(R R1 ) (R R2)( R R3) R1051 R2 R 2 R2 ⌥R1 ⌥ ⇤ ⌅ ⇤ ⌅ Multiplying• but lack both of sidesexplicit by time˙R˙ , the dependence equation gives of motion a as- 5/2 5/2 5/2 Multiplyingd both1 sides by R, the equationE(R) of motion as- 7/2 72/2 2 R71/2 R21 R1 RR1R31 R3 R1 R3 2 ˙ 2 == A32⇤⌥ SR1 4⌥⌅/3R1 1 I , I , (69). sumessumes the theconstant form( Sof2 the+ R motion:S1)R =0. (63) = A⇤ R2 2 1 2 I , (69) dt 2 4⌥ 105 105 R2 105R2 R 2 RR2R22 R2 R2 R2 ⇧ ⌃ ⇤ ⇤ ⌅ ⌅⇤⇤ ⇤ 5/⌅2⌅ 5/⇤2⌅ ⌅ d 1 E(R) 7/2 27/2 2 R1 R1 R1 RR3 1 R3 d 1 22 2˙ 2 E(R) Here I=is a dimensionless32⌥S 4⌥⌅/3 functionR of1R1/R2 andIR3/R,2 . The term in the square((SS22++RR bracketsSS11)R)˙R is a constant=0=0. of. motion(63)(63) = 32⌥S1 41⌥⌅/3R2 2 1 I , . dt 22 4⌥4⌥ which is finite everywhere in105 the105 domainR2 R [R2 ,RR2 ]R andR2 2 isR2 and can be taken⇧⇧ to be zero with loss of⌃ generality.⌃ Set- ⇤ ⌅ 1⇤ 2 ⌅ obtained from the integral defined⇤ in Eqn,⌅ (69)remov-⇤ ⌅ ting this• which constant we tocan zero take gives equal to zero: HereHereI isI ais dimensionless a dimensionless function function of R1 of/RR2 1and/R2Rand3/R2R3/R2 TheThe term term in the the square square brackets brackets is is a a constant constant of of motion motion (5/2) 7/2 whichingwhich the is finite factor is finite everywhere of everywhere (1 (R in1/R the2 in)) domain the domainand [RR1,R2 [R2]and1 and,R2 some] is and is andand can can be be taken to to be be zero zero with with loss loss2 of2 of generality. generality. Set- Set- E(R)=2⌥(S2 + S1R )R˙ . (64) obtainednumericalobtained from factors. from the the integral It is integral expressible defined defined in in Eqn, terms in Eqn, (69 of)remov- elliptic (69)remov- in- tingting this this constant constant to to zero zero gives gives 7/2 ingtegrals the factor and its of explicit (1 (R expression/R ))(5/2) is(5and/ not2) R illuminating.and7/2 some As ing the factor of (1 1 (R21/R2)) and2 R and some Substituting this in (61), we have 2 ˙ 2 numericalS has dimensions factors. It of is expressibleµ 3 and ⌅ has in dimensions terms of elliptic2 of µ4 in-,the EE((RR)=2)=2⌥⌥((SS2 ++SS1RR)2R)R˙.2. (64)(64) 1numerical factors. It is expressible in terms of elliptic in- 2 1 tegralsexpression and its is explicit dimensionless, expression as is expected. not illuminating. Substituting As tegrals and its explicit3 expression is not illuminating.4 As Substituting this in⇤ (61), we have 2 ˙ 2 Sthe1 has value dimensions of R2 in ofSµE, and3 ⌅ has dimensions of µ ,the4 SubstitutingSE this= in (61d ),4 we⌥(S have2 + S1R )R . (65) S1 has dimensions of µ and ⌅ has dimensions of µ ,the expression is dimensionless,7 as expected. Substituting ⌥ ⇤ expression is dimensionless, as5/ expected.2 Substituting ⇤ 2 2 the value of144R2⌥ in SE, 2 R1 R1 R3 SE = ⇤ d 4⌥(S2 + S1R )R˙ . (65) 2 ˙ 2 theSE value= of R22 inS1 S3E, 1 I , . (70) Solving for R˙ SfromE = (64) andd 4 using⌥(S2 + thisS1 inR the)R above. equa-(65) 35 7⌅ R2 R2 R2 ⌥ ⇤ 5/2 144 ⌥ 2 7 R1 R1 R3 tion yields ⌥ ⇤ ⇤ ⌅ 5/⇤2 ⌅ SE = 144 ⌥2 S1 3 1 2 R1I , R1 . R3(70) Solving for R˙ from (64) and using this in the above equa- For smallS =35⌅, the term2⌅S containing R1 2 ⇥ ˜ inR the2I R potential2 , . (10(70)) ˙ E 1 3 Solvingtion yields for R from (64) and using this in the above equa- can be neglected.35 Using⇤⌅ equation ⌅R2 (57⇤) andR the2⌅ R fact2 that ⇤ dR 2 ˙ For small ⌅, the term containing⇥ ˜ in the potential (10) tion yieldsSE = d ( )4⌥(S2 + S1R )R ⇧ =˜aµ when⇥ ˜ = 0, ⇤ ⌅ ⇤ ⌅ d canFor be smallneglected.⌅, the Using term equation containing (57)⇥ ˜ andin the fact potential that (10) ⌥ ⇤⇤ dR 2 S = d (R )4⌥(S + S R )R˙ E ⇤ 2dR 2 1 ⇧ =˜canaµ bewhen neglected.⇥ ˜ = 0, Using˜ aµ˜ equation (57) and the fact that ⇤ d 22 ˙ 2⌃ 2 2 2 SE == ⌥32 ⇤⌥ d ( dR)4⌥((SS22 ++SS11RR ))ER(R). (66) ⇧ =˜aµ whenS1 =⇥ ˜ = 0, dh h(h µ a˜ ) (71) Rd2 aµ˜ R1 µ˜ 0 ⇤⌥ ⇤ ⌥ 2 2⌃ ⌥ 2 2 2 = 32⌥ R2dR (S2 + S1R )E(R). (66) S1 = dh h(h µ a˜ ) (71) 2⌃˜ aµ˜ ⇥ ⇤ R1 2 µ ˜ 0 2 2 2 Using the expression= 32⌥⌥ for E(dRR) given(S2 + inS (147R)) andE(R neglect-). (66) S1 = ⌃⌥ 4 3 dh h(h µ a˜ ) (71) 2 = a˜µ µ . ⇥ (72) ing S2 in comparison to⌥RS11R , the euclidean action of the ˜ 8 0 Using the expression for E(R) given in (47) and neglect- ⌃ 4 3 ⌥ bounce solution can be written2 = a˜ µ . ⇥ (72) ing S2 in comparison to S1R , the euclidean action of the 8 ˜ Using the expression for E(R) given in (47) and neglect- The value of can be⌃ obtained4 3 from equation (46)by bounce solutionR can be written2 = a˜ µ . (72) ing S2 in comparison2 to S1R , the euclidean action of the 2 5 4 2 Thenoting value that of the can terms be obtainedmultiplying8 fromr equationare large ( compared46)by bounceSE = solutionA R2dR can be( writtenR 4⌥ R CR + E0R ) 2 R 5 4 2 notingto the that terms the independent terms multiplying of r andr are the large term compared multiplying SE = A⌥ 1 dR ( R 4⌥ R CR + E0R ) The2 value of can be obtained from equation (46)by RR2 (67) to1/r the.Since terms independent⇤ is small, we of canr and write ther term2= R multiplyingand equation ⌥ 1 noting2 that the terms multiplying r are large compared whereS A= =A ⇤32⌥dRS . In( derivingR5 4⌥ theR4 aboveCR expression,+ E R(67)2) 1(/r46).Since becomes⇤ is small, we can write r = R and equation E 1 0 to the terms independent of r and the term multiplying thewhere constantA = ⇤RE132⌥=S1E.(R In) deriving was subtracted the above from expression, the ex- (46) becomes ⌥ 0 1 2 R+ (67) 1/r .Since⇤ is small,2 we can write r = R and equation pressionthe constant for E(ER0)in(= E47(R)1 so) was that subtracted the bounce from has the a finite ex- 1 2 = R+ 2 dr (h⇥) + V (h) . (73) where A = ⇤32⌥S1. In deriving the above expression, (46) becomes 1 action.pression Pulling for E( outR)in( a factor47) so of thatR from the bounce the square has a root finite in 2 2 = R 2dr (h⇥) + V (h) . (73) the constant E = E(R ) was subtracted from the ex- ⌥ theaction. integrand, Pulling we0 out have a factor1 of R from the square root in R R+ 2⇤ ⌅ ⌥ 2 2 pression for E(R)in(47) so that the bounce has a finite ⇤ 1 2 ⌅ the integrand, we have Substituting for =h⇥ from equationdr (h (⇥)55),+ V (becomesh) . (73) action. Pulling out a factorR2 of R from the square root in Substituting for h⇥ from equation 2 (55), becomes R 2 R2 ⌥ R+ SE = A dR⇤R⇤ J (68) ⇤2 ⌅ the integrand, we have R+ 2 SE = A dR⇤R⇤ J (68) 2 R1 Substituting for h⇥=from equationdr (h2⇥) (55), becomes(74) ⌥ R1 = dr (h⇥) (74) ⌥ R2 R 2 4 3 R⌥ 2 R+ where J = R 4 S4E⌥ =RA3 C +dRE0⇤RR.⇤ TheJ function J has(68) ⌥ 2 where J = R 4⌥ R C + E0R. The function J has 2 a double root at R = R , aR positive1 root at R = R , and = = dh (h⇥dr) (h⇥) (75)(74) 1 ⌥ 2 = dh (h⇥) (75) a double root at R = R1, a positive root at R = R2, and 0 R a negative root at R = R3. Since we are working with ⌅ 0⌥ ⌥ 2 a negative root4 at R = R3 3. Since we are working with ⌅ ⌥ where J = R 4⌥ R C + E0R. The function J has smallsmall and and =4=4⌥⌅⌥⌅ //3,3, we we can can neglect neglect the term term containing containing = dh 2V (h) (76) a double root at R = R , a positive root at R =2 R 1,/ and3 = = dh dh2V((hh⇥)) (76) (75) while solving dE/dR =1 0 and obtain R (4e2 ) 21/3 . 0 while solving dE/dR = 0 and obtain R11 (4e ) . 0⌥ 0 a negative root at R = R3. Since we are⇥⇥ working with ⌅ ⌥ ⌥ ToTo find findRR33wewe also also neglect neglect the the term term containing containing ,, and and ==S1S. 1. (77)(77) small and =4⌥⌅/3, we can neglect the term containing = dh 2V (h) (76) 2 1/3 while solving dE/dR = 0 and obtain R (4e ) . 0 1 ⇥ ⌥ To find R3 we also neglect the term containing , and = S1. (77) 8 8 V. BOUNCE ACTION substituting for E0 in terms of the solution for R1, we get a cubic equation for R3, which can be exactly factored, V. BOUNCE ACTION substituting for E0 in terms of the solution for R1, we get In this section, we will derive an expression for bounce giving R3 = 2R1. Finally, to solve for R2, we solve a cubic equation for R3, which can be exactly factored, action SE for the monopole tunneling and compare it J = 0 neglecting the constant and linear term in R since In this section, we will derive an expression for bounce givingR isR3 large,= obtaining2R1. Finally,R to4⌥ solve/ =3 for R/⌅2., we solve with the bounce action for the tunneling of the false vac- J = 02 neglecting the constant2 ⇥ and linear term in R since actionuumS toE for the the true monopole vacuum as tunneling discussed and in compare [13]withno it Factoring J,wehave R2 is large, obtaining R2 4⌥ / =3 /⌅. withmonopoles the bounce present. action for From the ( tunneling61), the equation of the false of vac-motion ⇥ Factoring J,wehaveR2 uum to the true vacuum as discussed in [13]withno 2 for R can be written SE = A⇤ dR⇤R (R R1) (R R2)(R R3) monopoles present. From (61), the equation of motion R 2R1 for R can be written 1 ⌦E ⌥ ⇤ 2 2 ¨ ˙ 2 SE = A⇤ dRR2 R (R R1) (R R2)(R R3) (R S1 + S2)R + S1RR =0. (62) ⇤R1 ⇤ 1 4⌦⌥E⌦R = A ⌥ dR R(R R1) (R R2)(R R3) 2 ¨ ˙ 2 R2R (R S1 + S2)R + S1RR =0. (62) ⌥ 1 ˙ 4⌥ ⌦R = A⇤ dR⇤R(R R1) 5/2 (R R2)(R R3) Multiplying both sides by R, the equation of motion as- 7/2 2 R1 R1 R3 R sumes the form = A⇤⌥ 1R2 1 I , (69) ˙ 105 R52/2 R2 R2 Multiplying both sides by R, the equation of motion as- 7/2 2 R1 R1 R3 ⇤ ⇤ ⌅ ⇤ 5/2⌅ sumes the form = A R2 1 7/2 2I , R1 R(69)1 R3 d 1 2 2 E(R) 105 R2 R2 R2 (S2 + R S1)R˙ =0. (63) = 32⌥S1 4⌥⌅/3R2 1 I , . ⇤ ⌅ 105⇤ R52/2⌅ R2 R2 dt 2 4⌥ 7/2 2 R1 R1 R3 d 1⇧ 2 2 E(R) ⌃ ⇤ ⌅ ⇤ ⌅ (S2 + R S1)R˙ =0. (63) = 32 ⌥S1 4 ⌥⌅/3R2 1 I , . Here I is a dimensionless105 function R of2 R /R Rand2 RR2 /R The termdt in2 the square brackets 4 is⌥ a constant of motion 1 2 3 2 ⇧ ⌃ which is finite everywhere in⇤ the domain⌅ [R⇤,R ] and⌅ is and can be taken to be zero with loss of generality. Set- Here I is a dimensionless function of R /R and1 R2/R The term in the square brackets is a constant of motion obtained from the integral defined in1 Eqn,2 (69)remov-3 2 ting this constant to zero gives which is finite everywhere in the domain [R1,R2] and is and can be taken to be zero with loss of generality. Set- ing the factor of (1 (R /R ))(5/2) and R7/2 and some obtained from the integral defined1 2 in Eqn, (692)remov- ting this constant to zero gives 2 ˙ 2 numerical factors. It is expressible in terms7/2 of elliptic in- E(R)=2⌥(S2 + S1R )R . (64) ing the factor of (1 (R /R ))(5/2) and R and some tegrals and its explicit 1 expression2 is not2 illuminating. As E(R)=2⌥(S + S R2)R˙ 2. (64) numerical factors. It is expressible in terms of elliptic in- Substituting this in (61),2 we have1 3 4 Euclidean Action tegralsS1 has and dimensions its explicit of expressionµ and ⌅ ishas not dimensions illuminating. of µ As,the expression is dimensionless,3 as expected. Substituting4 Substituting this in (61), we have S1 has dimensions of µ and ⌅ has dimensions of µ ,the • This gives the Euclidean⇤ action: 2 2 the value of R2 in SE, SE = d 4⌥(S2 + S1R )R˙ . (65) expression is dimensionless, as expected. Substituting ⇤ 2 2 the value of R in S , 7 S = ⌥ ⇤d 4⌥(S + S R )R˙ . (65) 1442 ⌥ E 2 R 5/2 R R E 2 1 S = 2 S 1 1 I 1 , 3 . (70) ˙ E 71 3 Solving for R from⌥ ⇤ (64) and using this in the above equa- 144 ⌥35 2 ⌅ R R52/2 R RR2 R2 S = 2 S 1 ⇤ 1 ⌅I 1⇤, 3 . ⌅(70) tion• yieldswhich˙ becomes E 1 3 Solving for R from (64) and using this in the above equa- For small35 ⌅, the term⌅ containing R2 ⇥ ˜ inR2 theR potential2 (10) tion yields ⇤ ⌅ ⇤ ⌅ ⇤ dR Forcan small be⌅ neglected., the term Using containing equation⇥ ˜ in ( the57) potential and the fact (10) that S = d ( )4⌥(S + S R2)R˙ E d 2 1 can⇧ be=˜ neglected.aµ when⇥ ˜ Using= 0, equation (57) and the fact that ⇤ dR 2 ˙ SE = ⌥ ⇤d ( )4⌥(S2 + S1R )R ⇧ =˜aµ when⇥ ˜ = 0, d R2 ˜ aµ˜ 2 2⌃ 2 2 2 =⌥ ⇤⇤32⌥ dR (S2 + S1R )E(R). (66) S = dh h(h µ a˜ ) (71) R2 1 aµ˜ R1 2⌃˜ µ = ⇤32⌥ ⌥ dR (S + S R2)E(R). (66) ⌥0 2 2 2 2 1 S1 = dh h(h µ a˜ ) ⇥ (71) • using the expressionR1 for E(R). µ ˜0 Using the expression⌥ for E(R) given in (47) and neglect- ⌥⌃ 4 3 2 = a˜ µ . ⇥ (72) ing S2 in comparison to S1R , the euclidean action of the ˜ Using the expression for E(R) given in (47) and neglect- ⌃ 48 3 ingbounceS in comparison solution can to beS R written2, the euclidean action of the = a˜ µ . (72) 2 1 The value of can8 be obtained from equation (46)by bounce solutionR can2 be written 2 5 4 2 Thenoting value that of thecan terms be obtained multiplying fromr equationare large (46 compared)by SE = A dR ( R 4⌥ R CR + E0R ) R2 2 R notingto the that terms the terms independent multiplying of r andr are the large term compared multiplying ⌥ 1 5 4 2 2 SE = A dR ( R 4⌥ R CR + E0R ) (67) 1/r .Since⇤ is small, we can write r = R and equation R to the terms independent of r and the term multiplying ⌥ 1 2 where A = ⇤32 ⌥S1. In deriving the above expression,(67) 1/r(46.Since) becomes⇤ is small, we can write r = R and equation wherethe constantA = ⇤32E⌥S0 1=. InE(R deriving1) was the subtracted above expression, from the ex- (46) becomes R+ 2 pression for E(R)in(47) so that the bounce has a finite 1 2 the constant E0 = E(R1) was subtracted from the ex- = dr (h⇥) + V (h) . (73) R+ 2 pressionaction. for PullingE(R)in( out47 a) factor so that of R thefrom bounce the hassquare a finite root in R 1 2 2 = ⌥ dr2 (⇤h⇥) + V (h) . ⌅ (73) action.the integrand, Pulling out we a have factor of R from the square root in R 2 ⌥ 2 the integrand, we have Substituting for h⇥ from⇤ equation (55),⌅ becomes R2 Substituting for h from equation (55), becomes ⇥ R+ SE = A dR⇤R⇤ J (68) 2 R2 2 R1 ⇤ ⇤ = R+ dr (h⇥) (74) SE = A ⌥ dR R J (68) 2 R 2 2 4 R31 = ⌥ dr (h⇥) (74) where J = R 4⌥ ⌥R C + E R. The function J has 0 R 2 4 3 ⌥= dh (h⇥) (75) wherea doubleJ = rootR at4⌥ R R= R1C, a+ positiveE0R. The root function at R =JRhas2, and 0 a doublea negative root root at R at= RR1=, aR positive3. Since root we are at R working= R2, and with ⌅ = ⌥dh (h⇥) (75) 0 a negativesmall and root =4 at ⌥⌅R /=3,R we3. can Since neglect we are the working term containing with ⌅ ⌥= dh 2V (h) (76) 2 1/3 small while and solving=4⌥⌅/dE/dR3, we can= 0 neglect and obtain the termR1 containing(4e ) . = ⌥dh0 2V (h) (76) ⇥ 2 1/3 whileTo find solvingR3 wedE/dR also neglect= 0 and the obtain termR containing(4e ) , and. =0 S1. (77) 1 ⇥ ⌥ To find R3 we also neglect the term containing , and = S1. (77) 9 9 Using (77) and (72)in(70)yields VI. MONOPOLE DECAY IN A SUPERSYMMETRIC SU(5) GUT MODEL Using (77) and (72)in(70)yields VI. MONOPOLE DECAY IN A • which gives:144 ⌅⌃ 2 S4 R 5/2 R R 1 1 1 3 SUPERSYMMETRIC SU(5) GUT MODEL SE = 3 1 I , (78) 35 ⇥ 4 R2 5/R22 R2 The results of this work have direct relevance to a 144 ⌅⌃2 S1 R1 R1 R3 S = ⇤ 12 1 ⌅ ⇤ I ⌅, (78) E9⌃2 ⌅ µ 3 R 5/2 R R supersymmetric SU(5) model studied in [23]inwhich ˜23516 ⇥ R12 R12 R32 The results of this work have direct relevance to a = ⇤ a˜ 3 1 I , (79) supersymmetry symmetry breaking is sought directly 140 ⇥ ⇤ 12 R2 ⌅ 5⇤/R22 R2 ⌅ supersymmetric SU(5) model studied in [23]inwhich 9⌃2 ⌅ 2 16 µ R1 R1 R3 through O’Raifeartaigh type breaking. The Higgs sec- = ⇤˜ a˜ ⇤ 1 ⌅ ⇤ I ⌅, (79) supersymmetry symmetry breaking is sought directly 140 ⇥3 R R R tor, which contains two adjoint scalar superfields as the final value of the bounce action.2 From the2 values2 through O’Raifeartaigh type breaking. The Higgs1 sec- ⇤ ⌅ ⇤ ⌅ and and the superpotential, including leading non- of R1 and R2,wehave tor,2 which contains two adjoint scalar superfields as the final value of the bounce action. From the valuesrenormalizable terms, is of the form 1 and and the superpotential, including leading non- • whereof R1 andR R2,wehave1 1 ⇥ 2 1 renormalizable terms, is of the form = 2/3 1/3 (80) 2 1 3 2 2 R2 R e (41⇧) 31⇧ ⇥ W = Tr 2 µ 1 + ⇤ 1 + 1 + Tr( 1) 1 1 = (80) M M 1 2/316 1/13/3 ⇥ ⌥ ⇤ 2 1 3 22 ⌅ 2 =R2 e (4⇧) 3⇧ (81) W = Tr 2 ⇤µ 1 + ⇤ 1 + 1 + ⇧1 Tr( 1) 1 (⇤˜e)2/3 27 a˜16/3µ4 = ⇧1⇧2 µ ⇧1 +(7 1 +M 30 2) M (85) 1 16 1/3 ⇥ ⌥ ⌃⇤30 30M 2 ⌅ = ⇥ (81) ⇧ ⇤ ⌃⇧1 (⇤˜e)2/3 27 a˜16/3µ4 = ⇧1⇧2 µ ⇧1 +(7 1 + 30 2) (85) where the value of ⇧ has been expressed in terms of the where ⇧ and ⇧ are selected ⌃30 components of 30andM couplings appearing in the potential using⇥ equations (77) 1 2 ⇧ 1 ⌃ 2 where the value of ⇧ has been expressed in terms of therespectively, relevant to the symmetry breaking. Two and (72). From the expression given in (79), it is evident where ⇧1 and ⇧2 are selected components of 1 and 2 couplings appearing in the potential using equations (77)mass scales appear in the superpotential, µ and M,the that the bounce action S is zero when R = R as respectively, relevant to the symmetry breaking. Two and (72). From the expressionE given in (179), it is2 evidentlatter being a larger mass scale whose inverse powers de- expected. With ⇥ small, R /R is small, but it is inter- mass scales appear in the superpotential, µ and M,the that the bounce action1 S2 is zero when R = R astermine the magnitudes of the coe⇤cients of the non- esting to note that variations inE the couplings can reduce1 2 latter being a larger mass scale whose inverse powers de- expected. With ⇥ small, R /R is small, but it is inter-renormalizable terms. The scalar potential derived from the bounce action. For example,1 a reduction2 in the U(1) termine the magnitudes of the coe⇤cients of the non- esting to note that variations in the couplings can reducethis superpotential can be written as gauge coupling e has the e⇥ect of increasing the monopole renormalizable terms. The scalar potential derived from the bounce action. For example, a reduction in the U(1) mass and of reducing the bounce action. this superpotential2 can be written3 as3 2 gauge coupling e has the e⇥ect of increasing the monopole ⇤⇧1 7 1⇧1 2⇧1 We now compare our answer with the well known for- V = µ⇧1 + + mass and of reducing the bounce action. ⌃30 230 M 3M 3 2 mula of [13] relevant to homogeneous nucleation, i.e. tun- ⇤⇧1 7 1⇧1 2⇧1 We now compare our answer with the well known for- V⇤ = µ⇧12⇤⇧ (7+ + 30+ )⌅ 2 neling of the translation invariant false vacuum to the + ⇧ µ ⌃1 30+ 301 M 2 ⇧M2 . (86) mula of [13] relevant to homogeneous nucleation, i.e. tun- 2 ⇤ 1 ⌅ true vacuum. Denoting this bounce to be B0, ⌃30 2⇤⇧ 10(7M + 30 ) 2 neling of the translation invariant false vacuum to the ⇤ +⇤ ⇧ µ 1 + 1 ⌅⌅2 ⇧2 . (86) 2 ⌃ 10 M 1 true vacuum. Denoting2 this4 bounce to be B0, In [20], monopole solutions were30 shown to exist in this 27⌅ S1 ⇤ ⇤ ⌅⌅ B0 = (82) model and the classical instability of the vacuum struc- 2 ⇥3 2 4 In [20], monopole solutions were shown to exist in this 27⌅ S1 B0 = 2 12 (82)ture of this theory in the presence of such monopoles was 27⌅ ˜22 16⇥µ3 model and the classical instability of the vacuum struc- = ⇤ a˜ 3 . (83) discussed. 128 27⌅2 ⇥ µ12 ture of this theory in the presence of such monopoles was = ⇤˜2a˜16 . (83) Thindiscussed. walled monopoles can be obtained in this model Comparing this expression with128 our bounce⇥3 B S for under the condition ⇥ E Thin walled monopoles can be obtained in this model the monopole assisted tunneling given in (79), we see that under the condition Comparing this expression with our bounce B SE for ⇧1 ⌃30 the monopole assisted tunneling given in (79), we⇥ see that (87) 5/2 32⌃2 R1 R1 R3 µ ⌅⇧1 2⇤ ⌃30 B = B0 1 I , . (84) (87) 105 ⌅ R 5/R2 R µ ⌅ 2⇤ 32⌃2 2 R1 2 R21 R3 which is equivalent to the condition in Eqn. (13), and B = ⇤ B0 1 ⌅ ⇤ I ⌅, . (84) We see that unlike105 the⌅ homogeneous R2 case,R the2 R bounce2 hence the results of this paper could be applied directly ⇤ ⌅ ⇤ ⌅ which is equivalent to the condition in Eqn. (13), and can parametrically become indefinitely small and vanish there.hence In [20 the] the results region of ofthis parameter paper could space be studied applied did directly We see that unlike the homogeneous case, the bouncenot coincide with this condition, and thus the monopoles in the limit R1 R2. The interpretation of this limit is there. In [20] the region of parameter space studied did can parametrically⇧ become indefinitely small and vanish that the very presence of a monopole in this parameter werenot not coincide thin walled. with this The condition, monopoles and were thus classically the monopoles in the limit R1 R2. The interpretation of this limit isunstable when ⇥ M 4 was increased beyond a critical regimethat implies the very the presence unviability⇧ of of a monopole a state asymptotically in this parameter were not thin⇤ walled. The monopoles were classically approaching the vacuum deduced by a naive use of the value.unstable We can when recover⇥ thisM 4 behaviourwas increased from Eqn. beyond (79 a) critical as regime implies the unviability of a state asymptotically⇥ is increased, however⇤ it is important to note that our e⇥ectiveapproaching potantial. the If the vacuum parameters deduced in by the a e naive⇥ective use po- of the value. We can recover this behaviour from Eqn. (79) as tential explicitly depend on external variables such as approximation⇥ is increased, in this however paper it becomes is important invalid to note for large that our e⇥ective potantial. If the parameters in the e⇥ective po-enough ⇥. temperature, it may happen that the limit R1 R2 is approximation in this paper becomes invalid for large tential explicitly depend on external variables⇧ such as reachedtemperature, at a critical it value may happen of this external that the parameter. limit R InR is enough ⇥. this case, as the external parameter gets tuned to1 this⇧ 2 reached at a critical value of this external parameter. In VII. DISCUSSIONS AND CONCLUSIONS criticalthis value, case, the as monopoles the external will parameter become sites gets where tuned the to this true vacuumcritical value, is nucleated the monopoles without will any become delay and sites the where in- the VII. DISCUSSIONS AND CONCLUSIONS definitetrue growth vacuum of is such nucleated bubbles without will eventually any delay convert and the in- We have calculated the decay rate for so-called false the entiredefinite system growth to theof such true bubbles vacuum will without eventually the need convertmonopolesWe have in a simple calculated model the with decay a hierarchical rate for so-called struc- false for quantumthe entire tunneling. system to Such the a true phenomenon vacuum without may be the re- needturemonopoles of symmetry in breaking. a simple model The toy with model a hierarchical that we use struc- ferredfor to quantum as a roll-over tunneling. transition Such a[18 phenomenon] characterised may by be re-has ature breaking of symmetry of SU(2) breaking. to U(1) The which toy is model the false that vac- we use the relevantferred to critical as a roll-over value. transition [18] characterised byuum,has which a breaking in principle of SU happens(2) to atU(1) a higher which energy is the scale, false vac- the relevant critical value. uum, which in principle happens at a higher energy scale, 9 9 Using (77) and (72)in(70)yields VI. MONOPOLE DECAY IN A Using (77) and (72)in(70)yields VI.SUPERSYMMETRIC MONOPOLE DECAY SU(5) IN A GUT MODEL 4 5/2 SUPERSYMMETRIC SU(5) GUT MODEL 144 ⌅⌃2 S1 R1 R1 R3 4 5/2 S144E =⌅⌃2 S 3R1 1 R1 RI 3 , (78) 351 ⇥ R2 R2 R2 The results of this work have direct relevance to a SE = 3 1 I , (78) 35 ⇥ R⇤2 12 R⌅ 2 R5⇤2/2 ⌅ The resultssupersymmetric of this workSU have(5) model direct studied relevance in to [23 a]inwhich 9⌃2 ⌅ 2 16 µ R1 R1 R3 = ⇤⇤˜12 a˜ ⌅ 15⇤/2 ⌅I , (79)supersymmetric SU(5) model studied in [23]inwhich 9⌃2 ⌅ 2 16 µ 3R1 R1 R3 supersymmetry symmetry breaking is sought directly ˜ 140 ⇥ R2 R2 R2 = ⇤ a˜ 3 1 I , (79) supersymmetry symmetry breaking is sought directly 140 ⇥ R2⇤ R⌅2 R2⇤ ⌅ through O’Raifeartaigh type breaking. The Higgs sec- through O’Raifeartaigh type breaking. The Higgs sec- as the final value of⇤ the bounce⌅ action.⇤ From⌅ the values tor, which contains two adjoint scalar superfields 1 tor, which contains two adjoint scalar superfields 1 as the final value of the bounce action. From the values and 2 and the superpotential, including leading non- of R1 and R2,wehave and and the superpotential, including leading non- of R1 and R2,wehave 2 renormalizable terms, is of the form renormalizable terms, is of the form R1 1 1 ⇥ R1 1 = 12/3 ⇥ 1/3 (80) 2 1 3 2 2 = R22/3 e 1/3(4⇧) 3⇧ (80) W = Tr 2 µ2 1 + 1⇤ 31 + 2 1 +2 Tr( 1) 1 R2 e (4⇧) 3⇧ W = Tr 2 µ 1 + ⇤ 1 + 1 + MTr( 1)M 1 1 16 1/3 ⇥ ⌥ ⇤ M M 2 ⌅ 1= 16 1/3 ⇥ (81) ⌥ ⇤ ⇤ 2 ⌅ ⇧1 = (⇤˜e)2/3 27 a˜16/3µ4 (81) = ⇧1⇧2⇤ µ ⇧1 +(7 1 ⇧+1 30 2) (85) (⇤˜e)2/3 27 a˜16/3µ4 = ⇧1⇧2 µ ⇧1 +(7⌃30 1 + 30 2) (85)30M ⇥ ⌃30⇧ 30M ⌃ ⇧ ⌃ where the value of ⇧ has⇥ been expressed in terms of the where the value of ⇧ has been expressed in terms of the where ⇧ whereand ⇧⇧1areand selected⇧2 are components selected components of and of 1 and 2 couplingscouplings appearing appearing in the potential in the potential using equations using equations (77) (77) 1respectively,2 relevant to the symmetry1 breaking.2 Two and (72). From the expression given in (79), it is evidentrespectively, relevant to the symmetry breaking. Two and (72). From the expression given in (79), it is evident mass scalesmass appear scales in appear the superpotential, in the superpotential,µ and M,theµ and M,the that the bounce action SE is zero when R1 = R2 as that the bounce action SE is zero when R1 = R2 as latter beinglatter a larger being mass a larger scale mass whose scale inverse whose powers inverse de- powers de- expected. With ⇥ small, R1/R2 is small, but it is inter- expected. With ⇥ small, R1/R2 is small, but it is inter- terminetermine the magnitudes the magnitudes of the coe of⇤ thecients coe of⇤ thecients non- of the non- estingesting to note to that note variations that variations in the couplings in the couplings can reduce can reduce renormalizable terms. The scalar potential derived from the bounce action. For example, a reduction in the Urenormalizable(1) terms. The scalar potential derived from the bounce action. For example, a reduction in the U(1) this superpotentialthis superpotential can be written can be as written as gauge couplinggauge couplinge has thee has e⇥ect the of e increasing⇥ect of increasing the monopole the monopole 2 3 3 mass andmass of and reducing of reducing the bounce the bounce action. action. 2 ⇤⇧ 3 7 ⇧3 2 ⇧ 2 ⇤⇧1 7 1⇧11 2⇧11 1 2 1 We nowWe compare now compare our answer our answer with the with well the known well for- known for- V = µV⇧1 = µ⇧1+ ++ + Comparison with homogeneous ⌃30 30⌃M30 30MM M mula ofmula [13] relevantof [13] relevant to homogeneous to homogeneous nucleation, nucleation, i.e. tun- i.e. tun- ⇤ ⌅ ⇤ 2⇤⇧ (7 ⌅ + 302 ) 2 neling of the translation invariant false vacuum to the 2⇤⇧1 (7 1 1+ 30 2)1 2 2 2 neling of the translation invariant false vacuum to the + ⇧ µ+ ⇧2 µ+ + ⇧ . (86)⇧ . (86) nucleation 2 ⌃ 101 M 1 true vacuum.true vacuum. Denoting Denoting this bounce this bounce to be B to0, be B0, ⌃30 3010 M • The usual bounce action found by Coleman ⇤ ⇤ ⇤ ⇤ ⌅⌅ ⌅⌅ 27⌅2 S427⌅2 S4 In [20], monopoleIn [20], monopole solutions solutions were shown were to shown exist in to this exist in this and others is:B = B = 1 1 (82) (82) 0 02 ⇥3 2 ⇥3 model andmodel the classicaland the instabilityclassical instability of the vacuum of the struc- vacuum struc- 27⌅2 27⌅µ212 µ12 ture of thisture theory of this in theory the presence in the of presence such monopoles of such monopoles was was = =⇤˜2a˜16 ⇤˜2.a˜16 . (83) (83)discussed.discussed. 128 128⇥3 ⇥3 Thin walledThin monopoles walled monopoles can be obtained can be in obtained this model in this model ComparingComparing this expression this expression with our with bounce our bounceB SEBfor S underfor theunder condition the condition ⇥ E • thecomparing monopolethe monopole assisted with assisted tunnelingour expression tunneling given in given (79 we), inwe have: ( see79), that we ⇥ see that ⇧1 ⌃30⇧ ⌃30 1 (87) (87) ⌃ 5/2 5/2 µ ⌅ 2⇤ 32 2 32⌃2 R1 R1 R1 R3 R1 R3 µ ⌅ 2⇤ B = B =B0 1 B0 1 I , I . , (84). (84) 105 ⌅ 105 ⌅ R2 R R2 R2 R R ⇤ ⌅ ⇤2 ⌅ 2 2 which iswhich equivalent is equivalent to the condition to the condition in Eqn. ( in13), Eqn. and (13), and ⇤ ⌅ ⇤ ⌅ hence the results of this paper could be applied directly We seeWe that see unlike that unlike the homogeneous the homogeneous case, the case, bounce the bounce hence the results of this paper could be applied directly • canso parametricallywe see our action become indefinitelycan vanish! small and vanish there. Inthere. [20] the In region [20] the of region parameter of parameter space studied space did studied did can parametrically become indefinitely small and vanishnot coincide with this condition, and thus the monopoles in the limit R1 R2. The interpretation of this limit is not coincide with this condition, and thus the monopoles in the limit R1 R2. The interpretation of this limit is that the very presence⇧ ⇧ of a monopole in this parameter were not thin walled. The monopoles were classically that the very presence of a monopole in this parameter were not thin4 walled. The monopoles were classically regime implies the unviability of a state asymptotically unstableunstable when ⇥ whenM ⇥was increasedM 4 was increased beyond a beyond critical a critical regime implies the unviability of a state asymptoticallyvalue. We can recover⇤ this⇤ behaviour from Eqn. (79) as approachingapproaching the vacuum the vacuum deduced deduced by a naive by a use naive of the use of the value. We can recover this behaviour from Eqn. (79) as e⇥ective potantial. If the parameters in the e⇥ective po- ⇥ is increased, however it is important to note that our e⇥ective potantial. If the parameters in the e⇥ective po- ⇥ is increased, however it is important to note that our tential explicitly depend on external variables such as approximation in this paper becomes invalid for large tential explicitly depend on external variables such as approximation in this paper becomes invalid for large temperature, it may happen that the limit R R is enough ⇥. temperature, it may happen that the1 limit R2 R is enough ⇥. reached at a critical value of this external parameter.⇧ 1 In 2 reached at a critical value of this external parameter.⇧ In this case, as the external parameter gets tuned to this this case, as the external parameter gets tuned to thisVII. DISCUSSIONS AND CONCLUSIONS critical value, the monopoles will become sites where the VII. DISCUSSIONS AND CONCLUSIONS true vacuumcritical is value, nucleated the monopoles without any will delay become and sites the in- where the definitetrue growth vacuum of such is nucleated bubbles will without eventually any delay convert and the in-We have calculated the decay rate for so-called false the entiredefinite system growth to the of true such vacuum bubbles without will eventually the need convertmonopolesWe in a have simple calculated model with the a decay hierarchical rate for struc- so-called false for quantumthe entire tunneling. system Such to the a phenomenon true vacuum may without be re- the needture of symmetrymonopoles breaking. in a simple The model toy model with that a hierarchical we use struc- ferredfor to as quantum a roll-over tunneling. transition Such[18 a] phenomenon characterised may by behas re- a breakingture of of symmetrySU(2) to breaking.U(1) which The is the toy false model vac- that we use the relevantferred critical to as a value.roll-over transition [18] characteriseduum, by whichhas in a principle breaking happens of SU(2) at to a higherU(1) which energy is scale, the false vac- the relevant critical value. uum, which in principle happens at a higher energy scale, 10 and then a true vacuum which has no symmetry break- where ⇥ contains the determinantal and zero mode fac- ing. The symmetry broken false vacuum admits mag- tors, and I is defined in Eqn. (69). In the limit that netic monopoles. The false vacuum can decay via the R1 R2 the tunneling rate is unsuppressed while the usual creation of true vacuum bubbles [13], however we homogeneous⇤ tunneling rate for the nucleation of true find that this decay can be dramatically enhanced in the vacuum bubbles as found by Coleman [13] still remains presence of magnetic monopoles. Even though the false suppressed. Hence in this limit, the classical false vac- vacuum is classically stable, the magnetic monopoles can uum is classically stable, but subject to quantum insta- be unstable. At the point of instability, the monopoles bility through the nucleation of true vacuum bubbles, but are said to dissociate. This corresponds to an evolution the rate for such a decay can be quite small. However where the core of the monopole, which contains the true the existence of magnetic monopole defects render the vacuum, dilates indefinitely, [15, 16, 22]. However, be- false vacuum unstable, and in the limit of large monopole fore the monopoles become classically unstable, they can mass, the decay rate is unsuppressed. be rendered unstableDecay from quantum rate tunneling. We have computed the corresponding rate and find that as we ap- proach the regime of classical instability, the exponential • Thesuppression decay vanishes. rate per The unit tunneling volume amplitude is given behaves by: as 2 VIII. ACKNOWLEDGEMENTS Γ κ 16 2S1π ϵ ∼ exp − F(R1,R2,R3) V 2 105 32 ⇥ # 16 $2S1⇤ % ! " exp (R1,R2,R3) (88) We thank NSERC, Canada for financial support. The V ⇥ 2 ⇤105⇧ 3 F ⌅ visit of BK was made possible by a grant from CBIE, ⇥ • with Canada. The research of UAY is partly supported by a with grant from DST, India. The authors would like to thank 5/2 R. MacKenzie and P. Ramadevi for useful comments re- 7/2 R1 R1 R3 (R1,R2,R3)=R2 1 I , (89) garding this work. F R2 R2 R2 ⇥ ⇥ [1] S. R. Coleman, Subnucl. Ser. 13,297(1977). Nucl. Phys. 20,644(1975). [2] G. ’t Hooft, Nucl. Phys. B79,276(1974). [13] S. R. Coleman, Phys. Rev. D15,2929(1977). [3] A. M. Polyakov, JETP Lett. 20,194(1974). [14] S. R. Coleman and F. De Luccia, Phys. Rev. D21,3305 [4] A. M. Polyakov, Nucl. Phys. B120,429(1977). (1980). [5] R. Rajaraman, Solitons and Instantons. An Introduc- [15] P. J. Steinhardt, Nucl. Phys. B190,583(1981). tion to Solitons and Instantons in Quantum Field Theory [16] Y. Hosotani, Phys. Rev. D27,789(1983). (North-Holland, Amsterdam, 1982). [17] U. A. Yajnik, Phys. Rev. D34,1237(1986). [6] J. Preskill and A. Vilenkin, Phys. Rev. D47,2324(1993), [18] U. A. Yajnik and T. Padmanabhan, Phys. Rev. D35, hep-ph/9209210. 3100 (1987). [7] T. H. R. Skyrme, Nucl. Phys. 31,556(1962). [19] B. Kumar and U. A. Yajnik, Phys. Rev. D79,065001 [8] T. Gisiger and M. B. Paranjape, Phys. Rept. 306,109 (2009), 0807.3254. (1998), hep-th/9812148. [20] B. Kumar and U. Yajnik, Nucl. Phys. B831,162(2010), [9] M. Dine and A. E. Nelson, Phys. Rev. D48,1277(1993), 0908.3949. hep-ph/9303230. [21] S. R. Coleman, Subnucl. Ser. 15,805(1979). [10] K. A. Intriligator, N. Seiberg, and D. Shih, JHEP 04, [22] P. J. Steinhardt, Phys. Rev. D24,842(1981). 021 (2006), hep-th/0602239. [23] B. Bajc and A. Melfo, JHEP 04,062(2008),0801.4349. [11] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, [24] J. Terning (2003), hep-th/0306119. Phys. Rev. D68, 046005 (2003), hep-th/0301240. [25] See [24] for discussion of supersymmetric field theories. [12] I. Y. Kobzarev, L. B. Okun, and M. B. Voloshin, Sov. J. LEE et al. PHYSICAL REVIEW D 88, 105008 (2013) For sufficiently small , this energy functional is domi- Note that the Lagrangian is no longer renormalizable in natedLEE byet the al. first two terms. It is infinitely high for a small 3 1 dimensions; howeverPHYSICAL the understanding REVIEW D is88, that105008 it is an (2013) radius due to the magnetic energy, and will diminish to a effectiveþ theory obtained from a well defined renormalizable For sufficiently small , this energy functional is domi- Note that the Lagrangian is no longer renormalizable in local minimum when the linear wall energy begins to fundamental Lagrangian. The fields 0 and A" and the nated by the first two terms. It is infinitely high for a small 3 1 dimensions; however the understanding is that it is an become important. This will occur at a radius R0 well vacuumþ expectation value v have mass dimension 1, the beforeradius the due quadratic to the area magnetic energy, energy, due to andthe energy will diminish splitting to achargeeffectivee is dimensionless theory obtained and ! fromhas a mass well dimension defined renormalizable 2 since betweenlocal minimum the false vacuum when the and linear the true wall vacuum, energy becomes begins toit is thefundamental coupling constant Lagrangian. of the The sixth fields order scalar0 and potential.A" and the important,become for important. is sufficiently This will small. occur Clearly at a radius though,R for0 wellThevacuum potential expectation energy density value of thev have false mass vacuum dimension0 v 1, the largebefore enough the quadratic radius of area the thinenergy, wall due string to the configuration, energy splittingvanishes,charge whilee is that dimensionless of the true vacuum and ! has has massV 0 dimensionj !j v¼6 2. since thebetween energy the splitting false vacuum will be and the the most true important vacuum, term, becomesWe rescaleit is the analogous coupling constant to [2] of the sixth orderð Þ¼À scalar potential. andimportant, a thin-walled for vortexis sufficiently configuration small. of sufficiently Clearly though, large for The potential energy density of the false vacuum 0 v j j ¼6 radiuslarge will enough be unstable radius while of the expanding thin wall to string infinite configuration, radius. vanishes, while that of the true1= vacuum2 has V 0 2 1=2 !v . 0 v0 A" vA" e ! ve x x=ð vÞ¼À! However,the energy a vortex splitting of radius willR be0 will the be most classically important stable term, !We rescale analogous! to! [2] ! ð Þ andand only a thin-walled susceptible vortex to decay configuration via quantum of tunneling. sufficiently The large (4) amplituderadius will for besuch unstable tunneling, while in the expanding semiclassical to infinite approxi- radius. 1=2 2 1=2 0 v0 A" vA" e ! ve x x= v ! mation,However, has been a vortex calculated of radius in [2R].0 will be classically stableso that! all fields, constants! and the! spacetime coordinates! ð Þ andIn this only paper susceptible we consider to decay the generalization via quantum of tunneling. the model Thebecome dimensionless; then the Lagrangian density is still (4) toamplitude3 1 dimensions. for such tunneling, Here the vortexin the semiclassical can be continued approxi-given by Eq. (2) where now the potential is þ alongmation, the has third, been additional calculated dimension in [2]. as a string, often so that all fields, constants and the spacetime coordinates called a cosmic string. The interior of the string contains 2 2 2 In this paper we consider the generalization of the model becomeV dimensionless;0Ã0 0 then the0 Lagrangian1 : density(5) is still a largeto 3 magnetic1 dimensions. flux distributed Here the over vortex a region can of be the continued true given by Eq.ð (2Þ¼ð) wherej j nowÀ Þð thej potentialj À Þ is þ vacuum.along the It is third, separated additional by a thin dimension wall from as the a string, outside, oftenand there is an overall factor of 1= !v2 in the action. wherecalled the a scalar cosmic field string. is in The the false interior vacuum. of the The string analysis contains 2ð Þ 2 2 Initially, the cosmicV 0Ã0 string will0 be independent 0 1 of: z, the (5) ofa the large decay magnetic of two-dimensional flux distributed vortices over a cannot region directly of the truecoordinate along itsð length,Þ¼ð andj willj À correspondÞðj j À toÞ a tube of apply to the decay of the cosmic string, as the cosmic string vacuum. It is separated by a thin wall from the outside,radius R with a trapped magnetic flux in the2 true vacuum must maintain continuity along its length. Thus the radius and there is an overall factor of 1= !v in the action. where the scalar field is in the false vacuum. The analysisinside,Initially, separated the by cosmic a thin string wall from will beð the independent falseÞ vacuum of z, the ofof the the string decay at a of given two-dimensional position cannot vortices spontaneously cannot make directlyoutside. R will vary in Euclidean time ( and in z to yield the quantum tunneling transition to the larger isoenergetic coordinate along its length, and will correspond to a tube of apply to the decay of the cosmic string, as the cosmic stringan instantonradius R solution.with a trapped Thus we magnetic promote fluxR to in a the field trueR vacuum radius, called R1, without being continuously connected to ! must maintain continuity along its length. Thus the radiusR z;inside,( . Hence separated we will look by a for thin axially wall symmetric from the solutions false vacuum theof rest the of string the at string. a given The position whole string cannot could spontaneously in principle makeforð 0 Þand A in cylindrical coordinates r; ; z; ( . We use spontaneously tunnel to the fat string along its whole outside."R will vary in Euclidean timeð ( andÞ in z to yield the quantum tunneling transition to the larger isoenergeticthe following ansatz for a vortex of winding number n: length, but the probability of such a transition is strictly an instanton solution. Thus we promote R to a field R radius, called R , without being continuously connected to ! zero for an infinite1 string, and correspondingly small R z; ( . Hence we will look for axially symmetric solutions the rest of the string. The whole string could in principle ð 0Þ r; ; z; ( f r; R z; ( ein; for a closed string loop. The aim of this paper is to describe for 0 and A in cylindrical coordinates r; ; z; ( . We use spontaneously tunnel to the fat string along its whole ð " Þ¼ ð ð ÞÞ ð Þ(6) the tunneling transition to a state that corresponds to a the following ansatz forn " aij vortexr of winding number n: length, but the probability of such a transition is strictly A r; ; z; ( j a r; R z; ( ; spontaneously formed bulge in the putatively unstable i 2 zero for an infinite string, and correspondingly small ð Þ¼Àe r ð ð ÞÞ thin string. 0 r; ; z; ( f r; R z; ( ein; for a closed string loop. The aim of this paper is to describe ij ð Þ¼ ð ð ÞÞ (6) where " is the two-dimensional Levi-Civitaij symbol. This the tunneling transition to a state that corresponds to aansatz is somewhat simplistic, it isn clear" rj that if the radius of II. ENERGETICS AND DYNAMICS Ai r; ; z; ( 2 a r; R z; ( ; spontaneously formed bulge in the putatively unstablethe cosmic stringð swells outÞ¼À at somee r rangeð of ðz, theÞÞ mag- OF THE THIN, FALSE STRING thin string. netic flux will dilute and hence through the (Euclidean) ij A. Setup Maxwell’swhere " equationsis the two-dimensional some ‘‘electric’’ Levi-Civita fields will be symbol. gen- This Cosmic String Decay ansatz is somewhat simplistic, it is clear that if the radius of II. ENERGETICS AND DYNAMICS erated. In three-dimensional, source free, Euclidean elec- We• considerThe same the situation Abelian can Higgs be modelpossible (spontaneously for the cosmic string swells out at some range of z, the mag- OF THE THIN, FALSE STRING trodynamics, there is no distinct electric field; the Maxwell brokencosmic scalar electrodynamics)strings. with a modified scalar netic flux will dilute and hence through the (Euclidean) potential corresponding to our previous work [2] but now equations simply say that the three-dimensional magnetic A. Setup Maxwell’s equations some ‘‘electric’’ fields will be gen- generalized• Theories to 3 with1 dimensions. cosmic strings The Lagrangian require and density field is a divergence free and rotation free vector field that erated. In three-dimensional, source free, Euclidean elec- of theWe modelabelian consider has symmetryþ the the form Abelian to be Higgs spontaneous model (spontaneously broken. satisfies superconductor boundary conditions at the loca- broken scalar electrodynamics) with a modified scalartiontrodynamics, of the wall. It there is clear is no that distinct the electric correct field; form the of the Maxwell • The simplest model is the original Nielsen- equations simply say that the three-dimensional magnetic potential corresponding1 "# to our previous" work [2] but nowelectromagnetic fields will not simply be a diluted mag- L OlesenF "#modelF of Dthe"0 Abelian-Higgsà D 0 V 0 model,Ã0 ; (2) generalized¼À4 to 3 1þðdimensions.Þ ð TheÞÀ Lagrangianð Þ densityneticfield field is that a divergence always points free along and the rotation length free of the vector cosmic field that of thewith model a complex hasþ the form scalar field and an abelian stringsatisfies as with superconductor our ansatz; however boundary the correction conditions will at not the loca- where F"# @"A# @#A" and D"0 @" ieA" 0. givetion a major of the contribution, wall. It is and clear we that will the neglect correct it. Indeed, form of the gauge¼ field. À The Lagrangian density¼ð À is: Þ The potential1 is a sixth"# order polynomial" in 0 [1,5], the inducedelectromagnetic fields will fields always will be not smaller simply by be a a power diluted of mag- L F F D 0 D 0 V 0 0 ; (2) 2 written ¼À4 "# þð " ÞÃð ÞÀ ð Ã Þ 1=c neticwhen field the that usual always units are points used. along the length of the cosmic Thestring Euclidean as with action our ansatz; functional however for the the cosmic correction string will not 2 2 2 2 2 where FV"#0Ã0 @"A!# 0@#A"vand 0D"0 v @:" ieA(3)" 0.thengive has the a major form contribution, and we will neglect it. Indeed, ð ¼ Þ¼ ðjÀ j À Þðj j À ¼ðÞ À Þ The potential is a sixth order polynomial in 0 [1,5], the induced fields will always be smaller by a power of written 1=c2 when the usual units are used. The Euclidean action functional for the cosmic string V 0 0 ! 0 2 v2 0 2 v2 2: 105008-2(3) ð à ޼ ðj j À Þðj j À Þ then has the form 105008-2 LEE et al. PHYSICAL REVIEW D 88, 085031 (2013) f r 1;ar 1 as r : (10) n 1.Thismaybeaconsequenceoftheformofthepotential ð Þ ! ð Þ ! !1 chosen,¼ although we have yet to consider other potentials Conditions (9) are imposed for smoothness of the fields to see if thin-wall vortices with n 1 can be produced. at r 0, while Eqs. (10) are required for finiteness of However, with the potential (3), thin-wall¼ solutions do exist the energy.¼ More precisely, the behavior for small r can for sufficiently small and sufficiently large n [11–13]; one be found by linearizing the equations, which indicates such solution is displayed in Fig. 3(b). that f rn and a r2 as r 0. As r , we write The various contributions to the energy density as well f r $1 r and$a r 1! c r and!1 linearize in as the total energy density are shown for these solutions in andð Þ¼c . TheseÀ ð functionsÞ ð Þ¼ obey modifiedÀ ð Þ Bessel equations, Fig. 4; all of these vanish as r , as expected. Note that !1 and we find r r 1=2e 2p1 r and c r r1=2e p2er in both cases, the potential energy density is negative as LEE et al.À À À À PHYSICAL REVIEW D 88, 105008 (2013) as r [10].ð Þ$ ð Þ ! r 0; it then rises to a maximum and returns to zero, as ffiffiffiffiffiffiffi ffiffi ! Numerical!1 solutionsFor sufficiently for f smallr and, thisa r energyare displayed functional is in domi-expectedNote that given the Lagrangian the profile is of no longerf r and renormalizable the form in of the ð Þ Fig. 3(a) for nnatede by the1, first twoð Þ0: terms.1. The Itð isÞ vortex infinitely solution high for is a smallpotential3 1 dimensions; (see Fig. 2 however). The the totalunderstanding energy of is that the it thick-wall is an radius due to the magnetic energy, and will diminish to a effectiveþ theory obtained from a well defined renormalizable classically stable.¼ The¼ asymptotic¼ behavior (as r 0 and vortex is 5.38 in dimensionless units, which will be com- local minimum when the linear wall energy! begins to fundamental Lagrangian. The fields 0 and A" and the r ) of thebecome profile important. functions Thisf willr and occura atr ais radius as ex-R0 wellparedvacuum with that expectation of an ansatz value v wehave will mass use dimension in the next 1, the section. pected.!1 The solutionbefore the has quadratic a thick-wall area energy,ð Þ profile, due toð theÞ unlike energy the splittingThatcharge of thee is thin-wall dimensionless vortex and for! has the mass parameters dimension of 2 since Fig. 3(b) case of usualbetween vacuum the bubbles false vacuum studied and in the Ref. true vacuum, [7] which becomesis 92.5it is in the dimensionless coupling constant units. of the sixth order scalar potential. have a thin wallimportant, in the for limit is that sufficiently the vacuum small. Clearly degeneracy though, for LetThe us potential finish this energy section density with of the a few false comments vacuum 0 regardingv large enough radius of the thin wall string configuration, vanishes, while that of the true vacuum has V 0 j !j v¼6. splitting, controlled by the value of , is very small. But it vortices with n 1. As varies, the potentialð Þ¼À changes in a the energy splitting will be the most important term, We rescale analogous¼ to [2] should be notedand that a thin-walled here we vortex are configuration looking for of classically sufficiently largeway that has a dramatic effect on the vortices; this effect stable solitonradius solutions will be and unstable not whileinstanton-type expanding to solutions infinite radius.can be described as a sort of phase1=2 transition in that2 1 below=2 a 0 v0 A" vA" e ! ve x x= v ! analogous toHowever, the thin-wall a vortex vacuum of radius R bubbles0 will be classically found in stablecertain! critical value! of (which! depends on! e),ð thick-wallÞ Ref. [7]. Nonetheless,and only susceptible it will to prove decay via useful quantum to tunneling. have a Thevortices exist, while above it no stable vortices are(4) found. thin-wall vortexamplitude solution for since such then tunneling, tunneling in the can semiclassical be analyzed approxi-(It appears that thin-wall vortices are not seen for this mation, has been calculated in [2]. so that all fields, constants and the spacetime coordinates without recourse to numerical simulation, as indeed was the potential for n 1.) This phase transition is not a complete In this paper we consider the generalization of the model become dimensionless;¼ then the Lagrangian density is still case in Ref. [7to]. We3 have1 dimensions. not found Here any the such vortex solutions can be with continuedsurprise:given by for Eq. (<2) where0, now0 thehas potential greater is potential energy alongþ the third, additional dimension as a string, often ¼ called a cosmic string. The interior of the string contains 2 2 2 Vortex (n= 1, e= 1.00, ε= 0.10) VortexV (n=0 Ã50,0 e= 1.00,0 ε= 0.005)0 1 : (5) (a)a large magnetic flux distributed over a region of the true(b) ð Þ¼ðj j À Þðj j À Þ vacuum. It is separated by a thin wall from the outside, and there is an overall factor of 1= !v2 in the action. 1 1 where the scalar field is in the false vacuum. The analysis Initially, the cosmic string will beð independentÞ of z, the of the decay of two-dimensional vortices cannot directly coordinate along its length, and will correspond to a tube of apply to the decay of the cosmic string, as the cosmic string f radius R with a trapped magnetic fluxf in the true vacuum must maintain continuity along its length.a Thus the radius inside, separated by a thin wall froma the false vacuum of 0.5 the string at a given position cannot spontaneouslyB make 0.5outside. R will vary in Euclidean timeB ( and in z to yield the quantum tunneling transition to the larger isoenergetic an instanton solution. Thus we promote R to a field R radius, called R1, without being continuously connected to R z; ( . Hence we will look for axially symmetric solutions! the rest of the string. The whole string could in principle forð 0 Þand A in cylindrical coordinates r; ; z; ( . We use spontaneously tunnel to the fat string along its whole " ð Þ 0 0the following ansatz for a vortex of winding number n: length, 0 but the 1 probability 2 of 3 such 4 a transition 5 is strictly6 0 10 20 30 40 zero for an infinite string, and correspondingly small• with the ansatz 0 r; ; z; ( f r; R z; ( ein; FIG. 3 (color online).for a closed Vortex string profile loop. The for aim (a) of thick-wall this paper and is to (b) describe thin-wall vortices. Displayedð Þ¼ areð the functionsð ÞÞ f r and a r (6)and the ij ð Þ ð Þ magnetic field Bther tunnelingna r =er transition. (a) Vortex to a state (n that1, e corresponds1:00, e to0 a:10) and (b) Vortex (n 50,ne " r1j:00, e 0:005). 0 A r; ; z; ( a r; R z; ( ; spontaneouslyð Þ¼ ð Þ formed bulge in the¼ putatively¼ unstable¼ ið Þ¼À¼ e ¼r2 ð ð¼ ÞÞ thin(a) string. Vortex (n= 1, e= 1.00, ε= 0.10) (b) Vortex (n= 50, e= 1.00, ε= 0.005) where "ij is the two-dimensional Levi-Civita symbol. This ansatz is somewhat simplistic, it is clear that if the radius of II. ENERGETICS AND DYNAMICS 0.3the cosmic string swells out at some range of z, the mag- 0.4 OF THE THIN, FALSEρgrad STRING 0.02 ρgrad ρmag netic 0.01 flux will dilute and henceρmag through the (Euclidean) 0.2 0 A. Setup ρpot Maxwell’s equations some ‘‘electric’’ρpot fields will be gen- -0.01 0.2 ρtot erated. In three-dimensional, sourceρtot free, Euclidean elec- We consider the Abelian Higgs model (spontaneously 0.1 broken scalar electrodynamics) with a modified scalar trodynamics, there is no distinct electric field; the Maxwell 0 equations simply say that the three-dimensional magnetic potential corresponding to our previous work [2] but now 0 generalized to 3 1 dimensions. The Lagrangian density field is a divergence free and rotation free vector field that þ satisfies superconductor boundary conditions at the loca- of-0.2 the model has the form -0.1 0 1 2 3 4 5 6 tion0 of the 10 wall. It 20 is clear that 30 the correct 40 form of the 1 r electromagnetic fieldsr will not simply be a diluted mag- L F F"# D 0 D"0 V 0 0 ; (2) ¼À4 "# þð " ÞÃð ÞÀ ð Ã Þ netic field that always points along the length of the cosmic FIG. 4 (color online). Scalar field gradient energy density grad, magneticstring field as with energy our density ansatz; howevermag, potential the correction energy will density not pot, where F"# @"A# @#A" and D"0 @" ieA" 0. give a major contribution, and we will neglect it. Indeed, and total energy density ¼tot for (a)À thick-wall and¼ð (b) thin-wallÀ Þ vortices. (a) Vortex (n 1, e 1:00, e 0:10) and (b) Vortex (n 50, e 1:The00, e potential0:005 is). a sixth order polynomial in 0 [1,5], the induced fields will¼ always¼ be smaller¼ by a power of ¼ ¼ written¼ 1=c2 when the usual units are used. The Euclidean action functional for the cosmic string V 0 0 ! 0 2 v2 0 2 v2 2: (3) ð à ޼ ðj j À Þðj j À Þ then has the form 085031-4 105008-2 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) 1 4 1 1 Substituting Eqs. (5) and (6) into Eq. (7), we obtain BATTLESE A"; OF0 THE BULGE:2 d x DECAYF0 OFiF0i THEF THIN,i3Fi3 FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) ½ ¼!v 2 þ2 2 2 2 2 2 2 i 2% n a_ n a0 n @ a 1 Z X1 1 SSubstituting Eqs.dzd( (5)1 anddrr (6) into Eq. (7), we obtainð r Þ 1 4 1 E !v2 2e2r2 2e2r2 2e2r2 SE A";0 2 d x F0iF0i Fi3Fi3 ¼ 0 þ þ ½ ¼!v F03F03 2 FijFþij 2 @(0 à @(0 Z Z 2 2 2 2 2 2 þ2 iþ 4 þð Þ ð Þ 2% n a_ 2 n a0 n @ra Z X ij 2 2 1 2 n 2 2 X SE 2 f_ dzdf(0 @drrf 2 2 1 2a 2 f ð 2 2Þ 1 1 ¼ !v 0 r 2e rr2 þ 2e r þ 2e r F03F03 FijFij @(0 à @(0 þ þ þð Þ þ ð À Þ @ 0 à @ 0 D 0 à D 0 V 0Ã0 Z Z 2 þ2 z þ zij 4 þði Þi ð Þ n þð Þ ð XÞþ i ð Þ ð Þþ ð Þ f_ 2 f2 f 2 f@2 f 12 2 ; 1 a 2f2 (8) X 0 r 2 þþðþ À þðÞð ÀÞ Þþr ð À Þ @z0 à @z0 Di 0 à Di0 V 0Ã0 (7) þð Þ ð Þþ i ð Þ ð Þþ ð Þ where thef2 dot andf2 prime1 2 denote; differentiation with respect(8) X þð À Þð À Þ @a r;R to ( and z, respectively. Then a_ ð Þ R_ and a where i, j take values just over the two transverse direc-(7) ¼ð @R Þ 0 ¼ where@a r;R the dot and prime denote differentiation with respect tions and we have already incorporated that A0 A3 0. @Rð Þ R0, and likewise for f; hence the action becomes ¼ ¼ ð Þ @a r;R _ where i, j take values just over the two transverse direc- to ( and z, respectively. Then a_ @Rð Þ R and a0 @a r;R ¼ð Þ ¼ tions and we have already incorporated that A2 0 @a r;RA3 20. 2 @að r;RÞ R0, and2 likewise2 2 for f; hence the action becomes 2% n ð Þ R_ n @Rð Þ R n @ a @f r; R 2 1 ¼@R ¼ ð @R Þ 0 r _ SE 2 dzd( drr ðð 2 2Þ Þ ðð 2 2Þ Þ ð 2 2Þ ð Þ R ¼ !v 0 2e r þ 2e r þ 2e r þ @R Z Z 2 @a r;R 2 2 @a r;R 2 2 2 2% 2 n ð Þ R2_ n ð Þ R0 n @ a @f r; R 2 @f r; R 1 2@R n 2 2@R 2 2 r 2 _ SE 2 dzdð ( Þ R0 drr @ððrf 2 2Þ Þ 1 aððf 2 2Þf Þ fð 2 21Þ ð Þ R ¼ !v 0 2e r 2 þ 2e r þ 2e r þ @R þ @R þð Þ þ r ð À Þ þð À Þð À Þ Z Z 2 2 2 2 2 2 2 2@f% r; R n2 n@a r; R 2 2 @f r;2 R 2 2 n @ra ð Þ R01 @ f 1 a f f f _ 2 1 2 2 2 dz drr r 2 2 2 ð Þ ð Þ R R0 ð 2 2Þ @rf ¼þ!v @R 0 þð 2eÞ rþ r ð@RÀ Þ þ þð@RÀ Þð ð ÀþÞ Þþ 2e r þð Þ Z Z 2% 2 n2 @a r; R 2 @f r; R 2 n2 @ a 2 n 1 2 2 2 2 2 _ 2 2 r 2 2 dz1 adrrf f2 2 ðf Þ 1 : ð Þ R R0 ð 2 2Þ @rf (9) ¼ !vþ r2 ð À0 Þ þð2e rÀ Þð@R À Þ þ @R ð þ Þþ 2e r þð Þ Z Z n2 1 a 2f2 f2 f2 1 2 : (9) þ r2 ð À Þ þð À Þð À Þ We note the two-Euclidean-dimensional, rotationally invari- 2 2 2 2 R z; ( R z ( R & (13) ant form R_ R0 which appears in the kinetic term. ð Þ¼ ð þ Þ¼ ð Þ This allowsð usþ to makeÞ the O 2 symmetric ansatz for the with the imposed boundarypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi condition that R R . We note the two-Euclidean-dimensional,ð Þ rotationally invari- 2 2 ð1Þ¼ 0 instanton,2 and the2 easy continuation of the solution to Such a solutionR z; ( willR describez ( the transitionR & from a(13) string ant form R_ R0 which appears in the kinetic term. ð Þ¼ ð þ Þ¼ ð Þ Minkowskið time,þ toÞ a relativistically invariant O 1; 1 solu- of radius R0 at ( ,p toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a point in ( &0, say at z 0, This allows us to make the O 2 symmetric ansatzð forÞ the with the imposed boundary¼À1 condition that¼R R0. ¼ instanton,tion, once and the the tunneling easy continuation transitionð Þ has of been the completed. solution to whenSuch a a solution soliton-antisoliton will describe pair the starts transition to beð1 fromÞ¼ created. a string The In the thin wall limit, the Euclidean action can be configuration then develops a bulge which forms when the Minkowski time, to a relativistically invariant O 1; 1 solu- of radius R0 at ( , to a point in ( &0, say at z 0, tion,evaluated once the essentially tunneling analytically, transition has up been to corrections completed.ð Þ which whenpair a separates soliton-antisoliton to¼À a radius1 which pair starts has to to¼ be be again created.&0 because¼ The areIn the smaller thin by wall at least limit, one the power Euclidean of 1=R action. The method can be of configurationof O 2 invariance then develops and which a bulge is thewhich bounce forms point when of the the evaluation is identical to that in [2] (we shall not repeat the instantonð Þ along the z axis at ( 0. Finally the subsequent evaluated essentially analytically, up to corrections which pair separates to a radius which has¼ to be again &0 because aredetails), smaller and by weat least find one power of 1=R. The method of ofEuclideanO 2 invariance time evolution and which continues is the bounce in a manner point which of the is justð theÞ (Euclidean) time reversal of evolution leading up to evaluation1 is identical1 to that in [2] (we shall not repeat the instanton along the z axis at ( 0. Finally the subsequent • Swe find thed2 Euclideanx M R z;( actionR_ 2 R 2 E R z;( E R the bounce point configuration¼ until a simple cosmic string details),E ¼ ! andv2 we find2 ð ð ÞÞð þ 0 Þþ ð ð ÞÞÀ ð 0Þ Euclidean time evolution continues in a manner which is Z justof the radius (Euclidean)R0 is reestablished time reversal of for evolution( &0 leadingand all upz, to i.e. 1 1 (10) & & . The action functional is given by S d2x M R z;( R_ 2 R 2 E R z;( E R the bounce0 point configuration until a simple cosmic string E !v2 2 0 0 ¼ ð ð ÞÞð þ Þþ ð ð ÞÞÀ ð Þ of radius2%R is reestablished1 for@R &( 2 & and all z, i.e. where Z 0 0 (10) & SE & . The2 actiond&& functionalM R & is givenð Þ byE R & E R0 : 2 ¼0!v 2 ð ð ÞÞ @& þ ð ð ÞÞÀ ð Þ 2%n Z M R %R (11) 2% 1 @R & 2 where ð Þ¼ e2R2 þ (14) SE 2 d&& M R & ð Þ E R & E R0 : 2%n2 ¼!v 2 ð ð ÞÞ @& þ ð ð ÞÞÀ ð Þ M R n2È2 %R (11) The instantonZ equation of motion is E Rð Þ¼ e2R2 %þR %R2: (12) (14) ð Þ¼2%R2 þ À d dR 1 dR 2 &M R &M0 R &E0 R 0 (15) È is the total magneticn2È flux2 and R is the classically stable Thed instanton& ð equationÞ d& À 2 of motionð Þ d is& À ð Þ¼ E R %R 0 %R2: (12) thin tube stringð Þ¼ radius.2%R2 þ À withd the boundarydR condition1 thatdRR2 R , and we look &M R &M0 R &E0 R0 0 (15) È is the total magnetic flux and R is the classically stable ford& a solutionð Þ d& thatÀ has2 R ð RÞ dnear& ð1&ÀÞ¼0 ðwhereÞ¼R is the 0 1 ¼ 1 thin tube stringIII. INSTANTONS radius. AND THE BULGE large radius for which the string is approximately isoener- with the boundary condition that R R0, and we look getic with the string of radius R0.ð1 TheÞ¼ solution necessarily A. Tunneling instanton for a solution that has R R1 near & 0 where R1 is the ‘‘bounces’’ at ( 0 since @R ¼& =@( ( 0 R0 & III. INSTANTONS AND THE BULGE large radius for which¼ the string is approximatelyð Þ j ¼ ¼ isoener-ð Þ We look for an instanton solution that is O 2 symmetric; (=& ( 0 0. [The potential singularity at & 0 is ð Þ geticð withÞj ¼ the¼ string of radius R0. The solution necessarily¼ the appropriateA. ansatz Tunneling is instanton not there since a smooth configuration requires ‘‘bounces’’ at ( 0 since @R & =@( ( 0 R0 & ¼ ð Þ j ¼ ¼ ð Þ We look for an instanton solution that is O 2 symmetric; (=& ( 0 0. [The potential singularity at & 0 is the appropriate ansatz is ð Þ notð Þ therej ¼ ¼ since a smooth configuration requires¼ 105008-3 105008-3 LEE et al. PHYSICAL REVIEW D 88, 085031 (2013) R =2 vortices has also been shown explicitly and studied from þ 2 Ewall 4R drf0 various points of view in Refs. [11–14]. ¼ R =2 Z À We must reintroduce n into the action, giving R =2 þ 2 2 2 4R drf0 f 1 f ¼ R =2 ð À Þ ð À Þ Z À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S dt T E ; 1 ¼ ð À Þ 4R df f2 1 2 f2 : (27) Z ¼ ð À Þ ð À Þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the kinetic term is Z To leading order, we can put 0, giving Ewall R. In the exterior region (r>R¼=2), we assume¼f r n2a_ 2 1 _ 2 a r 1, and we find E þ 0. Summing theð threeÞ¼ T 2 drr f 2 2 (21) ext ¼ 0 þ 2e r ð Þ¼ ¼ Z contributions, we find and the energy of a static configuration is now 2n2 E R R R2: (28) ð Þ¼e2R2 þ À 2 2 2 2 1 2 n 1 a 2 n a0 E 2 drr f0 ð À2 Þ f 2 2 Finding the expected minimum of this function involves ¼ 0 þ r þ 2e r Z solving a fourth-order polynomial equation, which cannot 2 2 2 2=3 f f 1 : (22) be done exactly. It is useful to define R^ 2n=e À R, þð À Þð À Þ 2=3 ¼ð2=3 Þ E^ R^ 2n=e À E R = and ^ 2n=e . In terms ofð theseÞ¼ð new variables,Þ ð Þ ¼ð Þ Let us first determine the energy as a function of R of a static thin-wall configuration. We can divide the energy 1 E^ R^ R^ ^R^2; (29) integral (22) into three regions: ð Þ¼2R^2 þ À involving only one parameter ^, which we assume is small. E R Eint Ewall Eext: (23) ð Þ¼ þ þ This function is displayed in Fig. 8 for the parameters used in Fig. 3(b) (for which ^ 0:108). In the interior region (r The second and third terms of this equation can be dropped since r 1 in this region. Multiplying by f0, we can integrate the equation, giving 2 2 2 2 f0 f f 1 : (26) ^ ^ ¼ð À Þð À Þ FIG. 8. The rescaled energy E R with ^ 0:108. Numerical ð Þ ^ ¼ ^ values for the three parameters shown are E0 1:38, R0 1:09, Thus, and R^ 7:61. ¼ ¼ 1 ¼ 085031-8 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) 1 4 1 1 Substituting Eqs. (5) and (6) into Eq. (7), we obtain SE A";0 BATTLE2 d OFx THE BULGE:F0iF0i DECAYFi3F OFi3 THE THIN, FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) ½ ¼!v 2 þ2 2 2 2 2 2 2 i 2% n a_ n a0 n @ra Z X1 1 1 S Substitutingdzd( Eqs.1 drr (5) and (6) into Eq. (7), weð obtainÞ 1 1 4 E 2 2 2 2 2 2 2 SE A";0 2 d x F0iF0i Fi3Fi3 ¼ !v 0 2e r þ 2e r þ 2e r BATTLE OF THE BULGE: DECAY½F03F¼03 OF!v THEFij THIN,Fij 2 @ FALSE(0 Ãþ@2(0... PHYSICAL REVIEW2 2 D2 88,2 1050082 2 (2013) i Z2% Z 2n a_ n a0 n @ra þ2 þ ij Z4 Xþð Þ ð Þ S dzd( 1 drrn ð Þ X1 1 E _ 2 2 2 2 2 2 2 22 2 2 2 f¼ !vf0 @rf 0 22e1r þa2efr þ 2e r 1 1 1F03F03 FijFij @(0 à @(0Substituting Eqs.Z (5) andZ (6r) into Eq. (7), we obtain 4 þ2 þ 4 þð Þ ð Þ þ þ þð Þ þ ð 2À Þ SE A";0 d x @Fz0iFà 0@i z0 Fi3FDii30ij à Di0 V 0Ã0 2 2 2 n 2 2 2 X f_ f0 @ f 1 a f ½ ¼!v i þð2 Þ ð þÞþ2 i ð Þ ð Þþ ð Þ f2þ þf2 þð1 2r ;Þ2þ2r2 ð À2 Þ 2 2 (8)2 X 2% 1 n a_ n a0 n @ra Z X @z0 à @z0 Di 0 à Di0 SV 0Ã0 þðdzdÀ( Þð ÀdrrÞ þð Þ ð Þþ ð Þ ð Þþ(7)Eð Þ 2 2 2 2 2 2 2 2 ð 2 2Þ 1 1 Xi ¼ !v f 0 f 21 e ;r þ 2e r þ 2e r(8) F03F03 FijFij @(0 à @(0 where the dotþð andÀ primeÞð denoteÀ Þ differentiation with respect (7) Z Z 2 þ2 þ ij 4 þð Þ ð Þ to ( and z, respectively. Thenn a_ @a r;R R_ and a where i, j take values just over the two transverse direc- _ 2where the2 dot and prime2 denote differentiation@Rð2 Þ2 with respect0 X @a r;Rf f0 @rf 1 ¼ða @afr;RÞ ¼ to ( and z, respectively.2 Then a_ ð Þ R_ and a0 tions and wewhere have alreadyi, j take incorporated values just over that theA0 two transverseA3 0. direc-þ@Rð Þ Rþ0, andþð likewiseÞ forþf;r henceð À the actionÞ @R becomes ¼ ¼ ð Þ@a r;R ¼ð Þ ¼ @z0 à @z0 tions andD wei 0 haveà D alreadyi0 incorporatedV 0Ã0 that A0 A3 0. ð Þ R0, and likewise for f; hence the action becomes ¼ ¼ ð @R Þ þð Þ ð Þþ i ð Þ ð Þþ ð Þ 2 2 2 2 @a r;R 2 2 @a r;R f 2 2f 21 ; (8) X2% n ð Þ R_ n ð Þ R n @ a @f r; R 2 1 @R 2 @a r;R 2 þð@R 2 @a0 Àr;R Þð2 rÀ2 Þ 2 S dzd(2% drr ðð nÞ Þ ð Þ R_ðð n Þ ðÞ Þ R ð n Þ@ a ð@f r;Þ R R_ 2 E 2 1 (7)2 2 @R 2 2 @R 0 2 2 r _ ¼ !v SE 20 dzd( 2drre r ðð þ2 2Þ Þ 2e rðð 2 þ2Þ Þ2e r ð 2þ2Þ @R ð Þ R Z ¼ !vZ 0 2ewherer þ the dot2e r and primeþ 2e r denoteþ differentiation@R with respect Z2 Z 2 @f r; R @f r; R 2 n n2 2 2 2 2 2 @a r;R ð Þ R0 @Rrf @2f12 toa (1f anda 2ff2z, respectively.f2 f f12 1 2 Then a_ ð Þ R_ and a where i, j take values just overþ the two@R transverseðþðÞ 0 direc-Þ þ r r ð À 2Þ þð À Þð À Þ @R 0 þ @R þð Þ þ r ð À Þ þð À Þð À Þ ¼ð Þ ¼ 2 @a r;R 2 2 tions and we have already incorporated2% that2%A0 An3 @a0.nr;2 R @a2 r;ð R@fÞ 2Rr;0, R@f and2r; R likewise2 forn f@;n2 hencea@ a 2 the action becomes 1 1 @R TUNNELING_ 2 _ 2 DECAY2 2 OFr FALSEr VORTICES2 2 PHYSICAL REVIEW D 88, 085031 (2013) 2 dz 2drrdz¼ 2drr2¼ ð 2 2 Þ ð ð Þ Þð Þ ð RÞ RR0 R0 ð 2 ð2Þ2 2Þ @rf@rf ¼ !v ¼ !0v 20e r 2@Re r @Rþ @Rþ @Rð þð þÞþÞþ2e r2e rþðþð Þ Þ Z Z Z Z ^ ^ 2 n2 We can calculate R0 and E0 perturbatively in ^ 1; We are now in a position to write an expression for n 22 2@a1 r;Ra22f2 2 f22 2 @a2f2 r;R1 2 : 2 2 2 (9) 2% 1 an f2 ð fÞ R_ f n 1 ð: Þ R0 to lowestn @ ordera we@f canr; simply R 2 drop the third term(9) of the bounce action, following the procedure used to obtain þ1r2 ð À þÞ r ðþð@RÀ Þ À þðÞð ÀÀ ÞðÞ @RÀ Þ r _ SE 2 dzd( drr ðð 2 2Þ Þ ðð 2 2Þ Þ Eq. (29ð),2 and2Þ we find R^ð 1Þ, RE^ 3=2,withcorrections Eq. (20). The result is ¼ !v 0 2e r þ 2e r þ 2e r þ @R0 0 ¼ ¼ ^ Z We noteZ the two-Euclidean-dimensional, rotationally invari- (positive and negative, respectively) of order .The 2 2 R z; ( R z2 (2 R & (13) R 2 2 We note@f the two-Euclidean-dimensional,r; R _ 2 2 n rotationally invari- larger radius R^ does not2 exist2 for ^ 0, but it can be thin 1 n R 2n 2 ant form R R0 which2 appears in the2 kinetic2 term.2 2 R z;2( ð1 RÞ¼zð (þ Þ¼R &ð Þ (13) S 2 dR R R E ð_ 2 Þ R2 0 ð þ @ Þf 1 a f f f 1 ¼ B 2 2 2 2 0 ant form R ThisR allows0 which us to appearsmaker the inO 22 thesymmetric kinetic ansatzterm. for the writtenwith the as imposedð a LaurentÞ¼ boundaryð series;pþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi condition itÞ¼ is R^ thatð ÞR1=^ withR a. (nega- ¼ R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie R þ 2 e R þ À À þ ð@Rþ Þ þð Þ þ rð Þð À Þ þð À Þð À Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ð1Þ¼ 0 Z 0 This allows usinstanton, to make and the theO 2 easysymmetric continuation ansatz of the for thesolutionwith to tive) theSuch imposed correction a solution boundary of will order describe condition 1. These the transition that correctionsR fromR a are0 string. easily 2 2 2 ^ 3 ð Þ 2 2 ð1Þ¼ 2n 4=3 R1 dR^ 1 R^ 1 instanton,2% andMinkowski the easy time, continuation ton a relativistically@a ofr; the R invariant solutionO@f to1; 1r;solu- R Suchcalculatedof a radius solutionR0 andat will( are describen in@, goodtora a the point agreement transition in ( &0 from, with say at a the stringz values0, 2 3 4 1 ð Þ _ 2 2 ¼À1 ¼ 2 ¼ 2 E^0R^ R^ ^R^ ; Minkowski2 time,dztion, to once adrr relativistically the tunneling2 2 transition invariantð hasOÞ been1; 1 completed.solu-ð ofÞ radiuswhenRR aat soliton-antisolitonR( 0 , toð a2 point2 pairÞ in starts(@rf to& be, say created. at z The0, ¼ e ^ ^2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ 2 2À þ À ¼ !v 2e r @R þ @R givenð 0 inþ Fig. Þþ8. In terms2e r of theþð originalÞ0 variables, we find R0 R tion, once the tunnelingIn0 the thin transition wall limit, has been the Euclidean completed.ð Þ action canwhen be• with aconfiguration soliton-antisolitonparameters:¼À then1 develops pair a starts bulge which to¼ be forms created. when¼ The the Z 2Z Z pair separates to a radius which has to be again & because (33) In then thinevaluated wall limit, essentially the Euclidean analytically, action up to corrections can be whichconfiguration2n then2=3 develops a bulge3 2 whichn 2=3 forms when0 1 the are smaller2 2 by at least2 one power2 of 12=R. The method of Rof O 2 invariance;E and which is the bounce;R point of the; (30) evaluated2 essentially1 a f analytically,f up tof corrections1 : which pair separates0 ð Þ to a radius which0 has to be again & 1because (9) þ r ð ÀevaluationÞ isþð identicalÀ to thatÞð inÀ [2] (weÞ shall not repeat the instanton¼ e along the z axis¼ at 2( 0e. Finally the subsequent0 ¼ where we have gone to the hatted variables defined earlier. are smaller by at least one power of 1=R. The method of of O 2 invariance and which is¼ the bounce point of the ^ ^ ^ details), and we find ð EuclideanÞ time evolution continues in a manner which is Note that the last factor is the energy function E R E0 evaluation is identical to that in [2] (we shall not repeat the instantonwithjust along the corrections (Euclidean) the z axis smaller time at ( reversal by0 a. Finally of factor evolution the^ leading subsequent2n=e up2=3 to. R 2 ð ÞÀ 1 1 • the O(2) symmetric ansatz¼ is simply: 0 multiplied by R^ . This function, a quartic polynomial, has details), and weS find d2x M R z;( R_ 2 R 2 E R z;( E EuclideanR andtheE bounce timeare evolution point in good configuration agreement continues until in with a a simple manner the¼ð cosmic size which and stringÞ energy is E 2 0 0 0 four real roots, two of which are R^ and one of which is R^ . We note the two-Euclidean-dimensional,¼ !v 2 rotationallyð ð ÞÞð þ invari-Þþ ð ð ÞÞÀ ðjustÞ theof (Euclidean) radius R is time reestablished reversal2 of for2 evolution( & leadingand all z up, i.e. to 0 1 2 2 1 1 Z of theR thin-wallz; ( 0 R vortexz found( numericallyR0& earlier(13) [see ^ ant form R_ R which2 appears in the2 kinetic2 term. (10) & & . The action functional is given by The final root, which we will call R2, can be determined as a S 0 d x M R z;( R_ R0 E R z;( E R the bounceFigs. 3(b) pointð0 andÞ¼ configuration4(b)ð]. þ until aÞ¼ simpleð cosmicÞ string ð þ E ¼Þ !v2 2 ð ð ÞÞð þ Þþ ð ð ÞÞÀ ð 0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ^ ^ This allows us to make the O 2 symmetric ansatz for the with theof radius imposedR is boundary reestablished condition for ( 2 that& andR all z, i.e.R . power series in ; we find R2 :5 O . Thus, we can Z where The2 form0% of E 1R indicates@R that& bounce0 solutions exist.0 ¼À þ ð Þ ð Þ (10) & &S.E The action2 d&& functionalð MÞ R & is givenð Þ byE R &ð1Þ¼E R0 : write instanton, and the easy continuation of the solution2 to Suchwith aIn solution 0action order¼!v toand will compute equations describe2 ð theirð ÞÞ theof Euclidean@ motion:& transitionþ action,ð ð ÞÞÀ from weð a needÞ string to 2%n Z M R %R (11) determine the kinetic term (21).2 Once again we can con- Minkowski time,where to a relativistically invariant O 12; 12 solu- of radius R20%at ( 1 , to a point@R & in ( &0, say at(14)z 0, 4=3 ^ ^ ð Þ¼e R þ ^ 2n R1 dR ð Þ SE sider2 threed&&¼À regionsM1 R & (interior,ð Þ wall,E¼ andR & exterior).E R0 In:¼ the thin ^ ^ tion, once the tunneling transition has been2 completed. when a¼ soliton-antisoliton!v 2 ð ð ÞÞ pair@& startsþ toð beð ÞÞÀ created.ð Þ The SB 2 2 R R0 2%n 2 2 The instanton equation of motion is 2 2_ 3 ¼ sffiffiffi2 e R^ R^ ð À Þ n È 2 interior,Z f 0, while a r=R t , so a_ 2r R=R , 0 In the thin wall limit, theM EuclideanR E R2 2 action%R % canR % beR : (11)configuration(12) then¼ develops a bulge¼ð whichð ÞÞ forms¼À when the Z ð Þ¼ ðe Þ¼R 2þ%R2 þ À andd we find dR 1 dR 2 (14) &M R &M0 R &E0 R 0 (15) 3 1 evaluated essentially analytically,È is the total up magneticto corrections flux and R whichis the classicallypair stable separatesd& to a radiusð Þ d& À which2 ð hasÞ d& to beÀ againð Þ¼&0 because R^ R^1 R^ R^ R^2 : (34) 2 2 0 The instanton equation of motion is 2 _ 2  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ 2 ð À Þð À Þ are smaller by at least onethin power tube string ofn È1=R radius.. The method2 of of O 2 invariance and which is then R bounce point of the E R %R %R : (12) with the boundary conditionT that R : R0, and we look (31) ð Þ¼2%R2 þ À ð Þd dR 1 int ¼dRe2 2 ðR12Þ¼ evaluation is identical to that in [2] (we shall not repeat the instanton alongfor&M a solutionR the z thataxis has&M atR(R R1 0near. Finally& &E0 whereR the subsequentR01 is(15) the For small ^, we can make the approximations R^ 1, d& d& 2 0 d& ¼ 0 0 È is the total magneticIII. flux INSTANTONS and R0 is the AND classically THE BULGE stable largeð radiusÞ forÀ which theð stringÞ¼ is approximatelyÀ ð Þ¼ isoener- ^ ^ ^ ¼ details), and we find EuclideanInside time the evolution wall, we can continues assume that in af r; manner t is a function which of is R1 1=^, R2 :5, and E0 3=2, giving thin tube string radius. getic with the string of radius R0. The solutionð Þ necessarily ¼ ¼À ¼ A. Tunneling instanton withr the boundaryR t , that is,condition the time that dependenceR R of0,f andr; t weis look due only just the (Euclidean)‘‘bounces’’ time at ( reversal0 since of@R evolution& =@( ( 0 leadingR0 & up to 1 1 À ð Þ ¼ ð1Þ¼ð Þ j ¼ð ¼Þ ð Þ 2 We_ 2 look for2 an instanton solution that is O 2 symmetric;for ato solution( translation=& that0 has by. [TheRRt , potential theR1 near position& singularity of0 thewhere wall. at R&1 Thenis0 theis ^ 2n 4=3 1=^ dR^ SE d x M R z;( R R0 E R z;( E R0 the bounce point( configuration0 until a¼ simple cosmic string thin ^ 2 ð Þ large radiusð Þj for¼ ¼ which theð Þ string is approximately isoener-¼ SB 2 2 R 1 ¼ !v 2 ð ð III.ÞÞð INSTANTONSthe appropriateþ Þþ ansatz ANDð isð THEÞÞÀ BULGEð Þ not there since a smooth configuration requires ¼ sffiffiffi2 e 1 R^ ð À Þ of radius R0 is reestablished@f for ( &0 and all z, i.e. Z getic with the string of radius R0. The solution_ necessarily Z A. Tunneling instanton (10) f0 r Rt ; & &‘‘bounces’’0. The action at ( functional0 since@t ¼À is@R givenð&Þ =@ð Þ by( ( 0 R0 & 3 1 1 1 ¼ ð Þ j ¼ ¼ ð Þ R^ R^ R^ : (35) We look for an instanton solution that is O 2 symmetric; (105008-3=& ( 0 0. [The potential singularity at & 0 is  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ 2 ^ À þ 2 ð Þ ð2% andÞj ¼ the¼ first1 term of Eq.@R (21)& is 2 ¼ where the appropriate ansatz is S not thered&& sinceM R a& smooth configurationE R & requiresE R : E 2 ð Þ 0 ^ 2 ¼!v 2 ð ð RÞÞ=2 @& þRð ð ÞÞÀ ð Þ Finally, since the integral is dominated by R 1, we can 2%n þ 2 2 2 Z 2 drf0 R_ R_ ; drop the factors 1=2 (an approximation for which the M R 2 2 %R (11) R =2 ¼ 2 ð Þ¼ e R þ À (14) validity can easily be verified), after which the integral 105008-3 Z where we have used the fact that the integral has already can be evaluated exactly, giving n2È2 The instanton equation of motion is E R %R %R2: (12) been evaluated in Eq. (27). As was the case with the 2 d evaluationdR of the1 energy, thedR second2 term of Eq. (21) is ^ 2n 4=3 1=^ 1 ð Þ¼2%R þ À Sthin 2 dR^ R^ 1 R^ &MsmallerR than Eq.&M (310)R by a factor (&=RE0),R so it0 can(15) be B d& ð Þ d& À 2 ð Þ d& À ð Þ¼ ¼ sffiffiffi2 e 1 ð À Þsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^ À È is the total magnetic flux and R0 is the classically stable dropped, and Z thin tube string radius. ^ 2n 4=3 4 1 5=2 4p2 1 with the boundary condition that RR R0, and we look 2 1 ; (36) T ð1RÞ¼_ 2: ¼ sffiffiffi2 e 15 ^ À ’ 15ffiffiffi 2 for a solution that has R Rwall1 near¼ 2& 0 where R1 is the large radius for which the string is approximately¼ isoener- III. INSTANTONS AND THE BULGE Outside the wall both f and a are constant, and the con- where, in the last step, we have made the approximation ^ ^ getic withtribution the string to Eq. of ( radius21) vanishes,R0. The so the solution kinetic term necessarily is 1= 1 1= and we have also returned to the original A. Tunneling instanton ðvariables.ÞÀ Interestingly,’ the action is independent of n [aside ‘‘bounces’’ at ( 0 since @R & =@( ( 0 R0 & ¼ ðnÞ2 jR¼ ¼ ð Þ from the fact that Eq. (36) was derived for thin-wall vorti- We look for an instanton solution that is O 2 symmetric; (=& ( 0 0. [TheT T potentialT singularity atR_ 2:& 0(32)is ð Þ ð Þj ¼ ¼ int wall 2 2 2 ¼ ces, an approximation that is valid only for n 1]. the appropriate ansatz is not there since¼ a smoothþ ¼ configuratione R þ requires thin Recall that SB is an upper bound to the bounce action; We can now write the Euclidean action: thus, it gives a lower bound on the decay rate for the vortex, which is 2 2 n R 2 2n 2 SE dt R_ R R E0 : thin thin Sthin 105008-3 ¼ e2R2 þ 2 þ e2R2 þ À À À A eÀ B : (37) Z ¼ 085031-9 • The solution can only be obtained LEE et al. numerically andPHYSICAL looks REVIEW like: D 88, 105008 (2013) Energy(R) R 12000 12 000 10 000 10000 8000 8000 6000 6000 4000 R0 1292 2000 4000 R0 1292.18 R 2000 4000 6000 8000 0 0 5000 10 000 15 000 20 000 25 000 FIG. 1 (color online). The energy as a function of R, for n 100, e 0:005 and 0:0001. FIG. 2 (color online). The radius as a function of . ¼ ¼ ¼ depiction of the bounce point is given in Fig. 3. One should R0 0 0.] The equation of motion is better cast as an ð Þj ¼ ¼ imagine the radius R z along the cosmic string to be R to essentially conservative dynamical system with a ‘‘time’’ ð Þ 0 dependent mass and the potential given by the inversion of the left, then bulging out to the large radius as described by the energy function as pictured in Fig. 1, but in the pres- the mirror image of the function in Fig. 2 and then return- ence of a ‘‘time’’ dependent friction where plays the role ing to R0 according to the function in Fig. 2. This radius of time: function has argument z2 2. Due to the Lorentz ¼ þ invariance of the originalpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi action, the subsequent d dR 1 dR 2 1 dR Minkowski time evolution is given by R M R M0 R E0 R M R : ð Þ ! d ð ÞdÀ2 ð Þd À ð Þ¼À ð Þd R z2 t2 , which is of course only valid for z2 t2 ð À 2Þ 2 2 À ! (16) 0.p Fixedffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z t describes a spacelike hyperbola that asymptotes¼ to theÀ light cone. The value of the function As the equation is time dependent, there is no analytic trick R therefore remains constant along this hyperbola. This to evaluating the bounce configuration and the correspond- meansð Þ that the point at which the string has attained the ing action. We are, however, confident in the existence of a large radius moves away from z 0 to z at essen- solution which starts with a given R R at 0 and tially the speed of light. The other% side of!1 course moves % 1 ¼ achieves R R0 for > 0, by showing the existence of towards z . Thus the soliton-antisoliton pair sepa- an initial condition¼ that gives an overshoot and another rates quickly! À moving1 at essentially the speed of light, initial condition that gives an undershoot, in the same leaving behind a fat cosmic string, which is subsequently, manner of proof as in [6]. If we start at the origin at classically unstable to expand and fill all space. 0 high enough on the right side of the inverted energy The rate at which the classical fat string expands de- functional¼ pictured in Fig. 1, the equation of motion (16) pends on the actual value of . Once the string radius is will cause the radius R to slide down the potential and then large enough, its boundary wall is completely analogous to roll up the hill to R R . If we start too far up to the right, a domain wall that separates a true vacuum from a false ¼ 0 we will roll over the maximum at R R0 while if we do vacuum. The true vacuum exerts a constant pressure on the not start high enough we will never make¼ it to the top of the wall, and it accelerates into the region of false vacuum. hill at R R0. The rhs of (16) acts as a time dependent Obviously, if there is nothing to retard its expansion, it will friction, which¼ becomes negligible as , and once it accelerate to move at a velocity that eventually approaches is negligible, the motion is effectively!1 conservative. We the speed of light. The only effects retarding the velocity resort to numerical studies and we find with little difficulty increase are the inertia and possible radiation. Radiation that if we start at R 11506:4096, for n 100, e 0:005 should be negligible as there are no massless fields in the and 0:0001 we% generate the profile¼ function¼R in exterior and there are no accelerating charges. The accel- Fig. 2.¼ Actually, numerically integrating to 80; 000ð Þthe eration, a, is proportional to pressure divided by the mass function falls back to the minimum of the inverted% energy per unit area. The pressure is simply the energy density functional Eq. (1). On the other hand, we increase the difference, p . The mass per unit area can be obtained starting point by 0.0001; the numerical solution overshoots from Eq. (11).¼ Here the contribution to the mass per unit the maximum at R R0. Hence we have numerically length from the wall is simply R. Thus the mass per unit implemented the overshoot/undershoot¼ criterion of [6]. area, , is obtained from R L 2R L for a The cosmic string emerges with a bulge described by the given length L, which gives Â1=2¼. Then we have function numerically evaluated and represented in Fig. 2 ¼ which corresponds to R z; 0 . A three-dimensional a = 2: (17) ð ¼ Þ % ¼ 105008-4 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) (b) 0 • The three dimensional view(a) then gives: 10 000 10 000 20 000 5000 30 000 0 10 000 5000 5000 10 000 10 000 0 5000 5000 0 5000 10 000 10 000 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... 10 000 PHYSICAL REVIEW D 88, 105008 (2013) 5000 0 5000 20 000 (b) 0 20 000 10 000 0 (a) 10 000 FIG. 310 (color 000 online). (a)20 000 Cosmic string profile at the bounce point. (b) Cut away of the cosmic string profile at bounce point. 5000 30 000 0 10 000 5000 Thus it is clear that this acceleration can be arbitrarily5000 10 000 c:s: S0 S À NL À A ð Þ eÀ 0ð Þ (19) 10 000 small, for small , and it is reasonable to imagine that 2 2 0 Âð Þ¼ L ¼ 2 L 5000 once the tunneling transition has occurred the fat cosmic 5000 0 string will exist and be identifiable for a long time. or the decay rate per unit volume will be 5000 10 000 c:s: S0 S0 10 000 A ð Þ e 10 000 5000 À 2 À ð Þ B. Tunneling0 5000 amplitude ð Þ : (20) 20 000 0 20 000 10 000 L2 ¼ L2 It is difficult to say too much about the tunneling am- FIG. 3 (color online). (a) Cosmic string profile at the bounce point. (b) Cut away of the cosmic string profile at bounce point. plitude or the decay rate per unit volume analytically in the A comparable calculation with pointlike defects [2] would parameters of the model. The numerical solution we have give a decay rate per unit volume of the form obtained for some rather uninspired choices of the parame- point like point like point like S0 3=2 S Thus it is clear that this acceleration can be arbitrarily S0 point like A ð Þ eÀ 0 ð Þ ters gives rise to the profile of the instantonc:s: given inS0 Fig. 2. À 2 small, for small , and it is reasonable to imagine that À NL À 2 A ð Þ eÀ ð Þ 2 (19) (21) Âð Þ¼ L ¼ 2 L 3 3 once the tunneling transition has occurred the fat cosmicThis numerical solution could be then inserted into the L ¼ L Euclidean action to determine its numerical value; call it string will exist and be identifiable for a long time. or the decay rate per unit volume will be and the corresponding decay rate from vacuum bubbles S0 . It seems difficult to extract any analytical depen- c:s: S0 S (without topological defects) [6] would be denceð Þ on ; however it isÀ reasonableA ð Þ toeÀ expect0ð Þ that as B. Tunneling amplitude ð 2 Þ : (20) 2 2 vac:bubble 2 0 the tunneling barrier,L as¼ can beL seen in Fig. 1, will vac:bubble vac:bubble S0 Svac:bubble It is difficult to say too much about the tunneling am-! À A ð Þ eÀ 0 ð Þ: (22) get biggerA comparable and bigger andcalculation hence the with tunneling pointlike amplitude defects [2] would ¼ 2 plitude or the decay rate per unit volume analytically inwill the vanish. On the other hand, there should exist a limiting parameters of the model. The numerical solution we have give a decay rate per unit volume of the form Since the length scale L is expected to be microscopic, we value (call it c), where the tunneling barrier disappears at obtained for some rather uninspired choices of the parame- point like point like point like S0 3=2 S would then find that the number of defects in a macro- the so-called dissociationA point [7], suchð Þ thate as 0 c, Àpoint like 2 À !ð Þ ters gives rise to the profile of the instanton given in Fig.the2. action of the instanton will vanish, analogous to what (21)scopic volume (i.e. universe) could be incredibly large, 3 3 This numerical solution could be then inserted intowas the found in [2].L In general¼ the decay rateL per unit length suggesting that the decay rate from topological defects Euclidean action to determine its numerical value; call it of the cosmicand the string corresponding will be of the decay form rate from vacuum bubbleswould dominate over the decay rate obtained from simple S . It seems difficult to extract any analytical depen- vacuum bubbles a` la Coleman [6]. Of course the details do 0ð Þ (without topological defects) [6] would be dence on ; however it is reasonable to expect that as S0 depend on the actual values of the Euclidean action and the À Ac:s: e S0 ; (18) ð Þ vacÀ :bubbleð Þ 2 0 the tunneling barrier, as can be seen in Fig. 1, will vac:bubble¼ vac:bubble2 S0 Svac:bubble determinantal factor that is obtained in each case. ! À A ð Þ eÀ 0 ð Þ: (22) get bigger and bigger and hence the tunneling amplitude c:s: ¼ 2 will vanish. On the other hand, there should exist a limitingwhere A is the determinantal factor excluding the zero S IV. CONCLUSION Since0ð theÞ length scale L is expected to be microscopic, we value (call it c), where the tunneling barrier disappearsmodes at and 2 is the correction obtained after taking into wouldð thenÞ find that the number of defects in a macro- the so-called dissociation point [7], such that as accountc, the two zero modes of the bulge instanton. These There are many instances where the vacuum can be the action of the instanton will vanish, analogous to! whatcorrespondscopic to invariance volume (i.e. under universe) Euclidean could time be translation incredibly large,metastable. The symmetry broken vacuum can be meta- was found in [2]. In general the decay rate per unit lengthand spatialsuggesting translation that along the decay the ratecosmic from string topological [6]. In defectsstable. Such solutions for the vacuum can be important for of the cosmic string will be of the form general,would there dominatewill be a over length the decayL of cosmic rate obtained string from per simplecosmology and for the case of supersymmetry breaking; see volume vacuumL3. For bubbles a seconda` la orderColeman phase [6]. transition Of course to the the details[ do9] and the many references therein. In string cosmology, the c:s: S0 S depend on the actual values of the Euclidean action and the À A ð Þ eÀ 0ð Þ; metastable(18) vacuum, L is the correlation length at the inflationary scenario that has been obtained in [10] also ¼ 2 determinantal factor that is obtained in each case.1 temperature of the transition which satisfies LÀ gives rise to a vacuum that is metastable, and it must c:s: 2 % where A is the determinantal factor excluding the zerov Tc [8]. For first order transitions, it is not clear what necessarily be long-lived to have cosmological relevance. S IV. CONCLUSION modes and 0ð Þ is the correction obtained after takingthe into density of cosmic strings will be. We will keep L as a In a condensed matter context symmetry breaking ground ð 2 Þ account the two zero modes of the bulge instanton. Theseparameter butThere we do are expect many that instances it is microscopic. where the Then vacuum in a canstates be are also of great importance. For example, there are correspond to invariance under Euclidean time translationlarge volumemetastable. , we The will have symmetry a total broken length vacuumNL of cosmic can be meta-two types of superconductors [11]. The cosmic string is string, where N =L3. Thus the decay rate for the called a vortex line solution in this context, and it is relevant and spatial translation along the cosmic string [6]. In stable. Such¼ solutions for the vacuum can be important for general, there will be a length L of cosmic stringvolume per cosmologywill be and for the case of supersymmetry breaking; seeto type II superconductors. The vortex line contains an volume L3. For a second order phase transition to the [9] and the many references therein. In string cosmology, the metastable vacuum, L is the correlation length at the inflationary scenario that has been obtained in [10] also 1 temperature of the transition which satisfies LÀ gives rise to a vacuum that is metastable, and it must 2 % 105008-5 v Tc [8]. For first order transitions, it is not clear what necessarily be long-lived to have cosmological relevance. the density of cosmic strings will be. We will keep L as a In a condensed matter context symmetry breaking ground parameter but we do expect that it is microscopic. Then in a states are also of great importance. For example, there are large volume , we will have a total length NL of cosmic two types of superconductors [11]. The cosmic string is string, where N =L3. Thus the decay rate for the called a vortex line solution in this context, and it is relevant volume will be¼ to type II superconductors. The vortex line contains an 105008-5 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) (b) 0 (a) 10 000 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... 10 000 PHYSICAL20 000 REVIEW D 88, 105008 (2013) BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... PHYSICAL REVIEW D 88, 105008 (2013) 5000 30 000(b) 0 0 (b)10 000 10 000 0 (a) 5000 10 000 10(a) 000 20 000 5000 10 000 5000 30 000 10 000 20 000 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... 10 000 PHYSICAL REVIEW D 88, 105008 (2013) 5000 30 000 0 BATTLE OF THE BULGE: DECAY OF THE THIN, FALSE ... PHYSICAL REVIEW0 D 88, 105008 (2013) 10 000 5000 (b) 5000 0 10 000 0 (b) 5000 5000 0 0 10 000 5000 (a) 10 000 (a) 10 000 5000 10 000 10 000 5000 20 000 10 000 0 10 000 10 000 20 000 5000 5000 30 000 10 000 10 000 10 000 5000 30 000 5000 0 0 5000 5000 0 20 000 0 10 00020 000 10 000 0 0 5000 10 000 5000 5000 5000 5000 0 5000 10 000 10 000 5000 10 000 10 000 FIG. 310 000 (color online). (a) Cosmic string profile at the bounce point.5000 (b) Cut away of the cosmic string profile at bounce point. 10 000 10 000 5000 0 0 5000 10 000 20 000 0 20 000 0 10 000 10 000 5000 10 000 5000 5000 5000 0 5000 0 20 000 0 5000 20 000 10 000 0 FIG. 3 (color online). (a) Cosmic string profile at the bounce point. (b) Cut away of the cosmic string profile at bounce point. 5000 10 000 5000 Thus it is clear that this acceleration can be arbitrarily 10 000 10 000 c:s: S0 S 10 000 FIG. 3 (color5000 0 online).10 000 (a) Cosmic string profile at the bounce point. (b) Cut away of the cosmic0 string profile at bounce point. 10 000 5000 5000 À NL À A ð Þ eÀ ð Þ (19) 20 000 0 small,20 000 for small , and it is reasonable10 000 0 5000 to imagine that 2 2 20 000 0 20 000 Âð Þ¼ L ¼ 2 L 10 000 Thus itonce is clear the that tunneling this acceleration transition has can occurred be arbitrarily the fat cosmic FIG. 3 (color online). (a) Cosmic string profile at the bounce point. (b) Cut away of the cosmic string profile at bounce point. c:s: S0 S FIG. 3 (color online). (a) Cosmic stringstring profile will at exist the bounce and be point. identifiable (b) Cut away for of a long the cosmic time. string profileÀ orNL at bounce the decayÀ point. rateA per unitð volumeÞ eÀ 0ð willÞ be (19) small, for small , andThus it is it reasonableis clear that to this imagine acceleration that can beÂð arbitrarilyÞ¼ L2 ¼ 2 L2 S c:s: S0 S once the tunneling transition has occurred the fat cosmic À NLc:s: 0 À S0 A e 0 (19) small, for small , and it is reasonable to imagine that À A 2ð Þ eÀ 2ð Þ ð Þ À ð Þ 2 string will exist and be identifiableB. Tunneling for a amplitude long time. or the decay rate per unit volumeÂð will beðÞ¼Þ L :¼ 2 (20) L Thus it is clear that this acceleration can be arbitrarily S0 2 2 Thus it is clear that this acceleration can be arbitrarilyonce the tunneling transitionc:s: has occurredS0 S the fat cosmic L ¼ L À NL À A ð Þ eÀc:s:ð Þ 0 (19)S0 small, for small , and it is reasonable to imagine that À 2NL À A ð2 Þ e (19)c:s: S0 S0 small, for small , and it is reasonableIt is to difficult imagine to thatÂð say tooÞ¼ muchL about¼ the tunneling22 am-L À ð Þ 2 À Aor theð Þ decayeÀ ð rateÞ per unit volume will be once the tunneling transition has occurred the fat cosmic string will exist andÂð beÞ¼ identifiableL ¼ for a long2 time.A comparableL calculation2 with pointlike defects [2] would plitude orB. the Tunneling decay rate amplitude per unit volume analytically in the ð 2Þ : (20) once the tunneling transition has occurred the fator cosmic the decay rate per unit volume will be 2 ¼ L S string will exist and be identifiable for a long time. or the decay rate per unit volume will be give a decayL rate per unit volume of thec form:s: 0 S0 string will exist and be identifiableIt is difficultparameters for a long to time. say of the too model. much about The numerical the tunneling solution am- we have À A 2ð Þ eÀ ð Þ B. TunnelingS amplitude ð Þ : (20) c:s: 0 S0 A comparable calculation with pointlikepoint defects like 2 [2] would 2 plitude orobtained the decay for rate some per rather unit volume uninspiredÀ analyticallyA choicesð Þ ine ofÀ the theðc:Þs: parame-S0 S0 point like 2 A ð Þ e point like S0 L 3¼=2 S L B. Tunneling amplitude ð À Þ : 2 À ð Þ (20) A ð Þ e 0 B. Tunnelingparameters amplitudeters of gives the risemodel. toIt the The is profile difficult numerical of to the2 solution¼ say instanton too weL much2 given have about inð Fig.give theÞ 2 a. tunneling decay: rateÀ am-point per like unit(20) volume of the2 form À ð Þ L L2 ¼ L2 (21) It is difficult to say too much about the tunneling am- Apoint comparable like calculation with pointlike defects [2] would obtained for some ratherplitude uninspired or the choices decay rate of the per parame- unit volume analytically in the3 S 3=2 point3 like It is difficult to say too much aboutThis the numerical tunnelingA comparable solution am- could calculation be then with inserted pointlike intodefects the [2] would Lpoint like¼ 0 S L plitude or the decay rate per unit volume analytically in the A comparable calculation with pointlikepoint defects like [A2] wouldgive a decayð Þ e rateÀ 0 perð unitÞ volume of the form plitude or the decay rate perters unit gives volumeEuclidean rise analytically to the action profilegiveparameters in a tothe of decay determine the instantonrate of the per its model. unit given numerical volume The in Fig. numerical of value;2 the. form call solution itÀ we have 2 parameters of the model. The numerical solution we have give a decay rate per unit volume of the formand the corresponding3 decay rate from vacuum(21) bubbles parameters of the model. TheThis numerical numericalS solution. It solution seems weobtained have difficult could for be to some then extract rather inserted any uninspired analyticalinto the choices depen- ofL the3 parame-¼ L Spoint like 3=2 point like 0 point like point like point like 0 S obtained for some rather uninspired choices of the parame- S 3=2 point like (without topological defects)point like [6]A would be ð Þ e 0 obtained for some rather uninspired choicesð Þ of the parame- point like 0 S S0 3=2 point like À ð Þ Euclideandence action on to; determineters however givespoint likeits rise it numerical is toA reasonable the profile value; to ofpointð call expect theÞ like it instantoneÀ0 thatð asÞ givenS in Fig. 2. À 2 ters gives rise to the profile of the instanton given in Fig. 2. À point like 2A andð Þ theeÀ corresponding0 ð Þ decay rate from vacuum bubbles (21) ters gives rise to the profileS of the . instanton It seems given difficult in Fig. to2 extract. anyÀ analytical depen- 2 (21) 3 vac:bubble 3 0 0 the tunnelingThis numerical barrier,3 as solution can be seen could3 in be Fig. then1, will inserted into the (21) S ¼ 2 vac:bubbleL This numerical solution could be then insertedð Þ into the L ¼ 3 L (without3 topologicalvac:bubble defects)vac [:bubble6] wouldL 0 be S This numerical solution coulddence be on then!; inserted however into it is the reasonable toL expect¼ that as L À A ð Þ eÀ 0 ð Þ: (22) Euclidean action to determine its numerical value;get call bigger it andEuclidean bigger and action hence to the determine tunneling its amplitude numerical value; call it ¼ 2 Euclidean action to determine its numerical value;and call the it corresponding decay rate from vacuum bubbles andvac:bubble the corresponding 2 decay rate from vacuum bubbles S . It seems difficult to extract any analytical0 thewill depen- tunneling vanish. On barrier,S the . other It as seems canand hand, be the difficult seen there corresponding in should Fig. to extract1 exist, will decay a any limiting rate analyticalvac from:bubble vacuum depen-vac bubbles:bubble S0 Svac:bubble 0 S . It seems difficult to extract! any analytical depen-0 À A ð Þ e 0 : (22) ð Þ 0 get bigger and bigger(without andð Þ hence topological(without the tunneling defects) topological amplitude [6] would defects) be [6] would beSince the length(without scale L topologicalis expectedÀ defects) toð Þ be microscopic, [6] would we be dence on ; howeverð Þ it is reasonable to expectvalue that (call as it dencec), where on the; however tunneling it barrier is reasonable disappears to at expect¼ that as 2 dence on ; however it is reasonable to expect that as vac:bubble 0 the tunneling barrier, as can be seenwill in vanish. Fig. 1, On will the other hand, there should existS a limiting 2vac:bubblevac:bubble would then find that the number of defectsvac:bubble in a macro- 0 the tunneling barrier, as canthe be seen so-called in Fig. dissociation1,vac will0:bubblethe tunneling pointvac:bubble [7 barrier,], such0 that as can as beS seenc, in2 Fig. 1, will S 2 vac:bubble ! À Avac:bubble vac:bubbleð Þ S0eÀ Since0 ð theÞ: length(22)Svac:bubble scale L is expectedvac:bubble to be microscopic,vac:bubble 0 we S get bigger and! bigger and hence the tunnelingvalue (call amplitude it c), where! the tunnelingÀ barrier disappearsA 2 at ! ð Þ eÀscopic0 ð volumeÞ: (22) (i.e.À universe)A could be incrediblyð Þ large,eÀ 0 ð Þ: (22) get bigger and bigger and hence thethe tunneling action of amplitudeget the bigger instanton¼ and will bigger vanish,¼ and analogous hence the to2 tunnelingwould what then amplitude find that the number of defects¼ in a macro-2 will vanish. On the other hand, there shouldthe exist so-called a limiting dissociation point [7], such that as c, suggesting that the decay rate from topological defects will vanish. On the other hand, therewas should found exist in aSincewill limiting [2]. vanish. In the general length On scale the the decay otherL is expected hand, rate! per there to unit be should microscopic, lengthscopic exist volume a we limiting (i.e. universe) could be incredibly large, value (call it c), where the tunneling barrierthe action disappears of the at instanton will vanish,Since the analogous length scale to whatL is expected to bewould microscopic, dominate we over the decay rate obtained from simple value (call it c), where the tunnelingof barrier the cosmic disappearswouldvalue string at (call then will find it be of that), where the the form number the tunneling of defects barriersuggesting in a disappears macro- that the at decaySince rate the from length topological scale L defectsis expected to be microscopic, we the so-calledthe dissociation so-called point dissociation [7], such pointwas that found [7 as], such in [2 thatc]., In as general , the decaywouldc rate then per find unit that length the number of defectsvacuum in a bubbles macro- a` la Coleman [6]. Of course the details do the action of the instanton will vanish, analogous to! what scopicthe so-calledc volume dissociation (i.e. universe) point could [7 be], incredibly suchwould that dominate large, as over , thewould decay rate then obtained find that from the simple number of defects in a macro- of the cosmic string will! be of thescopic formS0 volume (i.e. universe) could be incrediblyc large, the action of the instanton will vanish, analogoussuggesting to what thatc:s: the decayS0 rate from topological defectsdepend! on the actual values of the Euclidean action and the was found in [2]. In general the decay rate per unit length the actionÀ A ofsuggesting theð instantonÞ eÀ thatð theÞ; will decay vanish, ratevacuum analogous(18) from topological bubbles to whata` la defectsColemanscopic [6]. volume Of course (i.e. the universe) details do could be incredibly large, was found in [2]. In general the decay rate per unitwould lengthS dominate¼ over2 the decay rate obtained from simpledeterminantal factorsuggesting that is that obtained the decay in each rate case. from topological defects of the cosmic string will be of the form wasc:s: found0 inwould [S20]. dominate In general over the the decay decaydepend rate rate per obtained on unit the length fromactual simple values of the Euclidean action and the of the cosmic string will be of the form À vacuumA bubblesð Þ eÀ a` ðlaÞ;Coleman [6].(18) Of course the details do where Ac:s: ¼isof the the determinantal cosmic2 vacuum string bubbles factor will be excludinga` la ofColeman the form the [determinantal6 zero]. Of course the factor details that dowould is obtained dominate in each over case. the decay rate obtained from simple c:s: S0 S depend on the actual values of the Euclidean action and the 0 S0 S IV. CONCLUSION À A ð Þ eÀ c:sð: Þ; cS:s0: (18) 0 depend on the actual values of the Euclidean action and thevacuum bubbles a` la Coleman [6]. Of course the details do ¼ 2À A whereð Þ eAÀmodesð isÞ; the and determinantaldeterminantal2ð Þ (18)is the correction factor factor excluding that obtained is obtained the after zero in taking each case. into ¼ 2 ð Þ determinantal factorS0 that is obtained in each case. S c:s: S0 IV. CONCLUSIONdepend on the actual values of the Euclidean action and the c:s: modes andaccount0ð Þ theis the two correction zero modes obtained ofÀ the after bulgeA taking instanton.ð intoÞ eÀ Theseð Þ; There(18) are many instances where the vacuum can be where A is the determinantalc:s: factor excluding the2 zero ¼ 2 determinantal factor that is obtained in each case. S where A is the determinantal factorcorrespondð excludingÞ to the invariance zero underIV. Euclidean CONCLUSION time translation metastable. The symmetry broken vacuum can be meta- BATTLE OF THE BULGE:modes DECAY and OF THE0ð Þ THIN,is the FALSE correctionS ... obtainedaccount after takingPHYSICAL the two into zero REVIEW modes D 88, of105008 the bulge (2013) instanton.IV. These CONCLUSIONThere are many instances where the vacuum can be ð 2modes Þ and 0ð Þ is the correction obtainedand spatial after taking translation into c:s: along the cosmic string [6]. In stable. Such solutions for the vacuum can be important for account the two zero modesð 2 ofÞ the bulgecorrespond instanton. These to(b) invariancewhereThere underA are Euclideanis many the instances determinantal time translation where factor the vacuummetastable. excluding can be theThe zero symmetry broken vacuum can be meta- account the two zero modes of the bulgegeneral, instanton. there0 These will be aS lengthThere areL of many cosmic instances string where per thecosmology vacuum can and be for the case of supersymmetryIV. CONCLUSION breaking; see correspond(a) to invariance under Euclideanand time spatial translation10 000 translationmetastable.modes along and the The0 cosmicð symmetryÞ is the string correction broken [6]. vacuum In obtainedstable. can after be Such meta- taking solutions into for the vacuum can be important for correspond to invariance10 000 under Euclidean20 000 time3 translation metastable.2 The symmetry broken vacuum can be meta- Rategeneral,30 000 ofvolume there vacuum willL . be For decay a alength secondðL of orderÞ cosmic phase string transition per tocosmology the [9 and] and for the the many case of references supersymmetry therein. breaking; In string see cosmology, the and spatial translation along the5000 cosmic string [6]. In stable.account Such the solutions two zero for modes the vacuum of the can bulge be important instanton. for These There are many instances where the vacuum can be and spatial translation along the cosmic3 string [6]. In stable. Such solutions for the vacuum can be important for general, there will be a length•0 LWithoutof cosmicvolume an analytic stringmetastableL . For perexpression a second vacuum,cosmology for10 order000 theL and is phase for the the correlation transition case of supersymmetry to length the at[9 breaking;] the and theinflationary many see references scenario therein. that In hasstring been cosmology, obtained the in [10] also 3 general, there will5000 be a length L of cosmic stringcorrespond per cosmology to invariance and for under the case Euclidean of supersymmetry1 time translation breaking; seemetastable. The symmetry broken vacuum can be meta- temperature of the5000 transition which satisfies L gives rise to a vacuum that is metastable, and it must volume L . For a second3 order10action 000 phase metastablewe transition cannot tosay vacuum, the much[ 9Labout] andis the thethe many decay correlation references length therein. at In the stringinflationary cosmology,À the scenario that has been obtained in [10] also volume L . For a second order phase2 transitionand to the spatial[9 translation] and the many along references1 the therein. cosmic% In string string cosmology, [6]. In thestable. Such solutions for the vacuum can be important for 10 000 metastable vacuum, L is the correlation length at the inflationary0 scenario that has been obtained in [10] also metastable vacuum,rate.L is temperature the correlationv Tc of[ length8]. the For transition at first the orderinflationary which transitions, satisfies scenario it isLÀ not that clear has beengives what obtained risenecessarily to ina vacuum [10] bealso long-lived that is metastable, to have cosmological and it must relevance. 5000 temperature of the transition which satisfies2 L 1 givesgeneral, rise there to a vacuum will be that a length is metastable,%L of cosmic and it must string per cosmology and for the case of supersymmetry breaking; see temperature of the transitionv Tc which[the8]. For densityÀ satisfies first of orderL cosmic1 transitions,5000 strings3gives will it rise is be. not to a We clear vacuum will what keep thatL isnecessarilyas metastable, a beIn and long-lived a condensed it must to have matter cosmological context symmetry relevance. breaking ground 0 v2T [8]. For first order transitions,• However, it is not in clearprinciple what% for anecessarily volumevolumeÀ L be . there long-lived For a second to have order cosmological phase transition relevance. to the [9] and the many references therein. In string cosmology, the c 2 parameter but we do%10 000 expect that it is microscopic. Then in a states are also of great importance. For example, there are 5000 v Tc [8]. For first order transitions,the density it of is cosmic not clear strings what willnecessarily be. We will be keep long-livedL as toa have cosmologicalIn a condensed relevance. matter context symmetry breaking ground 10 000 10 000 the density of cosmic strings willwill be. Webe a will length5000 keep L asof acosmicmetastableIn string. a condensed vacuum, matter contextL is the symmetry correlation breaking length ground at the inflationary scenario that has been obtained in [10] also the density of cosmic stringsparameter will0 be.5000large We but volume will we do keep expect L,as we that a will it is haveIn microscopic. a a condensed total length Then matterNL in context aof cosmicstates symmetry are alsotwo breaking of types great ground of importance. superconductors For example, [11]. The there cosmic are string is 20 000parameter but0 we do expect20 000 that it is microscopic. Then10 000 in a states are also of great importance. For example, there are 1 parameter but we do• expectFor a largelarge that it volume volume isstring, microscopic. where,, wewith willThentemperature N have in a a =L total states 3 . of Thus length are the also theNL transition of decayof great cosmic importance. rate which fortwo the satisfies For types example,called ofL superconductorsÀ a there vortex aregives line solution [11 rise]. The to in a this cosmic vacuum context, string that and is is it is metastable, relevant and it must large volume , we will have a total length NL of cosmic two2 types of superconductors [11]. The cosmic string is % FIG. 3 (color online). (a) Cosmic string profilelarge at volume the bounce , point. wethere will (b) have Cut willstring, away a be total of a wherevolume thelength length cosmicN NL string willofof =L profilecosmicv be3T¼.c atThus[ 8 bouncestring].two For the point.in types first decay order of rate superconductors transitions, for the called it [11 is]. not The a vortex clearto cosmic type line what string solutionII superconductors. isnecessarily in this context, be The long-lived and vortex it is relevant to line have contains cosmological an relevance. string, where N =L3. Thus the decay rate for the called a vortex line solution in this context, and it is relevant string, where N =L3.volume Thus the decaywill be¼ ratethe for density the called of cosmic a vortex strings line solution will be. in this Weto context, type will keep II and superconductors.L it isas relevant a In The a condensed vortex line matter contains context an symmetry breaking ground volume will be¼ the volume and decay rate:to type II superconductors. The vortex line contains an volume will be¼ parameter butto type we do II expect superconductors. that it is microscopic. The vortex line Then contains in a anstates are also of great importance. For example, there are Thus it is clear that this acceleration can be arbitrarily c:s: S0 S À NL À A largeð Þ e volumeÀ 0ð Þ ,(19) we will have a total length NL of cosmic two types of superconductors [11]. The cosmic string is small, for small , and it is reasonable to imagine that Âð Þ¼ L2 ¼ 2 L2 once the tunneling transition has occurred the fat cosmic string, where N =L3. Thus the decay105008-5 rate for the called a vortex line solution in this context, and it is relevant or the decay rate per unit volume105008-5 will be ¼ 105008-5 string will exist and be identifiable for a long time. volume105008-5 will be to type II superconductors. The vortex line contains an c:s: S0 S À A ð Þ eÀ 0ð Þ B. Tunneling amplitude ð 2 Þ : (20) L2 ¼ L2 It is difficult to say too much about the tunneling am- A comparable calculation with pointlike defects [2] would plitude or the decay rate per unit volume analytically in the 105008-5 parameters of the model. The numerical solution we have give a decay rate per unit volume of the form obtained for some rather uninspired choices of the parame- point like point like point like S0 3=2 S A ð Þ e 0 Àpoint like 2 À ð Þ ters gives rise to the profile of the instanton given in Fig. 2. (21) This numerical solution could be then inserted into the L3 ¼ L3 Euclidean action to determine its numerical value; call it and the corresponding decay rate from vacuum bubbles S . It seems difficult to extract any analytical depen- 0 (without topological defects) [6] would be denceð Þ on ; however it is reasonable to expect that as vac:bubble 2 0 the tunneling barrier, as can be seen in Fig. 1, will vac:bubble vac:bubble S0 Svac:bubble ! À A ð Þ e 0 : (22) get bigger and bigger and hence the tunneling amplitude ¼ 2 À ð Þ will vanish. On the other hand, there should exist a limiting Since the length scale L is expected to be microscopic, we value (call it c), where the tunneling barrier disappears at would then find that the number of defects in a macro- the so-called dissociation point [7], such that as c, the action of the instanton will vanish, analogous to! what scopic volume (i.e. universe) could be incredibly large, was found in [2]. In general the decay rate per unit length suggesting that the decay rate from topological defects of the cosmic string will be of the form would dominate over the decay rate obtained from simple vacuum bubbles a` la Coleman [6]. Of course the details do c:s: S0 S depend on the actual values of the Euclidean action and the À A ð Þ eÀ 0ð Þ; (18) ¼ 2 determinantal factor that is obtained in each case. where Ac:s: is the determinantal factor excluding the zero S IV. CONCLUSION modes and 0ð Þ is the correction obtained after taking into ð 2 Þ account the two zero modes of the bulge instanton. These There are many instances where the vacuum can be correspond to invariance under Euclidean time translation metastable. The symmetry broken vacuum can be meta- and spatial translation along the cosmic string [6]. In stable. Such solutions for the vacuum can be important for general, there will be a length L of cosmic string per cosmology and for the case of supersymmetry breaking; see volume L3. For a second order phase transition to the [9] and the many references therein. In string cosmology, the metastable vacuum, L is the correlation length at the inflationary scenario that has been obtained in [10] also 1 temperature of the transition which satisfies LÀ gives rise to a vacuum that is metastable, and it must 2 % v Tc [8]. For first order transitions, it is not clear what necessarily be long-lived to have cosmological relevance. the density of cosmic strings will be. We will keep L as a In a condensed matter context symmetry breaking ground parameter but we do expect that it is microscopic. Then in a states are also of great importance. For example, there are large volume , we will have a total length NL of cosmic two types of superconductors [11]. The cosmic string is string, where N =L3. Thus the decay rate for the called a vortex line solution in this context, and it is relevant volume will be¼ to type II superconductors. The vortex line contains an 105008-5 Domains Walls • False domain walls can exist in interesting models, where the true vacuum is trapped inside a domain wall, with false vacuum outside. • It cannot be classically stable for the same reasons as monopoles or cosmic strings, since the wall region is essentially a point. It has no area or length. I. INTRODUCTION A theory of scalar fields with spontaneous symmetry breaking can break the symmetry to discrete, distinct vacua. Typically, one of the vacua is the least energetic, usually normalized to zero energy, while the others are false vacua that are inherently unstable. It can further occur that the true and false vacua are themselves degenerate, there are multiple, discrete copies of the true vacua and the false vacua. Then field configurations that interpolate between different vacua, true or false, give rise to domain walls, which themselves are stable or metastable. In this paper we consider domain walls where regions of true vacuum are trapped, albeit in a metastable fashion, within the false vacuum. We further consider the possibility that the domain wall itself entraps several quanta of a different quantum field, , which we call a field of sheep, and these sheep are herded into staying inside the domain wall, by a field we call the shepherding field, . The shepherding field is unstable to quantum tunnelling to its true vacuum, which releases the sheep and allows them to spread out to infinity. We compute the amplitude for such a decay in a specific model analytically, aided by numerical calculations, within a well defined approximation scheme. II. THE MODEL• However it is easy to conceive of models where the dynamics allows trapped regions Consider the modelof definedtrue vacuum. by the Lagrangian Consider density the model: 1 µ µ = (@µ @ + @µ @ ) V ( , ) (1) L 2 with V ( , )=V ( )+V ( )+V ( , ) V0 (2) where 2 2 2 V ( )=↵( + a) (( a) + ✏ )) (3) 2 2 V ( )= (sin (⇡ )+✏ sin (⇡ /2)) (4) while 2 2 2 ( a) (( + a) + ✏ )) V ( , )= (5) (V ( ) V (1/2)2 + 2 is considered to be very small compared to ↵ and is considered small compared to V (1/2). We impose that V ( , ) 0,clearlythefirstthreetermsarepositivesemi-definite,andthen V0 is chosen exactly such that V ( , ) vanishes at its global minimum. 2 • The potentials look like: 1.0 0.8 0.6 0.4 0.2 -4 -2 2 4 FIG. 1. (Color online) The potential V for the field . FIG. 2. (Color online) The potential V for the field with a =1. It is easy to see that the global minimum of V occurs at = a while a local minimum occurs near = a. These values are mildly perturbed by V ( , ) but for small values of , the changes can be made arbitrarily small. The minimum of V ( ) occurs at even integers =2k,foranyintegerk, with a slightly higher local minimum occurring at odd integers, =2k +1,andthelocalmaximaoccurateachhalfoddinteger.Again,these values are mildly perturbed by V ( , ) but for small values of ,thechangescanbemade arbitrarily small. Obviously for =0,thetruevacuumcorrespondsto = a, =2k { } while = a, =2k +1 or = a, =2k or 2k +1 correspond to the false vacuum. { } { } Of course with non-zero, the values of the fields will be slightly modified, however the structure of the false and true vacua will remain unchanged. In the ensuing discussion we will identify the vacua by the values that the fields take when =0, but always with the understanding that the actual values that the fields take are slightly perturbed. 3 FIG.• and 1. (Color online) The potential V for the field . 5 4 3 2 1 -1.5 -1.0 -0.5 0.5 1.0 1.5 FIG. 2. (Color online) The potential V for the field with a =1. It is easy to see that the global minimum of V occurs at = a while a local minimum occurs near = a. These values are mildly perturbed by V ( , ) but for small values of , the changes can be made arbitrarily small. The minimum of V ( ) occurs at even integers =2k,foranyintegerk, with a slightly higher local minimum occurring at odd integers, =2k +1,andthelocalmaximaoccurateachhalfoddinteger.Again,these values are mildly perturbed by V ( , ) but for small values of ,thechangescanbemade arbitrarily small. Obviously for =0,thetruevacuumcorrespondsto = a, =2k { } while = a, =2k +1 or = a, =2k or 2k +1 correspond to the false vacuum. { } { } Of course with non-zero, the values of the fields will be slightly modified, however the structure of the false and true vacua will remain unchanged. In the ensuing discussion we will identify the vacua by the values that the fields take when =0, but always with the understanding that the actual values that the fields take are slightly perturbed. 3 FIG. 1. (Color online) The potential V for the field . FIG. 1. (Color online) The potential V for the field . FIG. 1. (Color online) The potential V for the field . attract each other in pairs. This is evident as the energy is minimized by reducing the region over which the field configuration is in the false vacuum. Thus the net soliton passes from even integer to even integer. But then these full solitons further minimize their energy by separating from each other, as this minimizes any gradient energies that occur due to their proximity, hence they have a repulsive interaction. The half solitons on either end are also FIG. 2. (Color online) The potential V for the field with a =1. simply repelled by the adjacent full solitons. FIG. 2. (Color online) The potential V for the field withThusa =1 in. the example shown in Figure (4), the half soliton from -3 to -2 is repelled by the It is easyfull to solitons see that from -2the to global0, which alsominimum repels the of subsequentV occurs full soliton at from= 0a towhile 2 which a in local minimum FIG. 2. (Color online) The potential V for the field with a =1. It is easy to see that the global minimum of V occurs at = turna while repels a local the final minimum half soliton from 2 to 3. The solitons have the false vacuum a occurs near = a. These values are mildly perturbed by V ( , ) but⇡ for small values occurs near = a. These values are mildly perturbed by V ( in, between) but for them, small hence, values they have an attractive interaction between them. The sheep want It is easy to see that the global minimum of V occurs at = a while a local minimum of , the changes can be made arbitrarily small. The minimum of V ( ) occurs at even of , the changes can be made arbitrarily small. The minimumto of separateV ( ) occurs as far as at possible even from one another, but they cannot do so without pushing the occurs near = a. These values are mildly perturbed by V ( , ) but for small values integers =2k,foranyintegerk, with a slightly higherintegers local minimum=2shepherdk,foranyinteger occurring solitons further at odd and furtherk, with apart. a slightlyThe energy, higher in doing local so, increases minimum linearly with occurring at odd of , the changes can be made arbitrarily small. The minimum of V ( ) occurs at even the separation of the shepherd solitons, while the energy is at best decreased to a constant integers, =2k +1,andthelocalmaximaoccurateachhalfoddinteger.Again,theseintegersintegers, =2 k,foranyinteger=2k +1,andthelocalmaximaoccurateachhalfoddinteger.Again,thesek, with a slightly higher local minimum occurring at odd value as the separation between the sheep becomes large. Hence, energetically, the shepherd values are mildly perturbed by V ( , ) but for small values =2 of k,thechangescanbemade+1 integers,values• The are free mildly potentials,andthelocalmaximaoccurateachhalfoddinteger.Again,these perturbed have one by setV of true( , ) but for small values of ,thechangescanbemade solitons force the sheep solitons to be herded together at a finite size where the pressure arbitrarily small. Obviously for =0,thetruevacuumcorrespondstominima: = Va,( ,=2 ) k values are mildly perturbed{ by but} for small values of ,thechangescanbemade arbitrarilyfrom small. the sheep Obviously is balanced for against =0 the,thetruevacuumcorrespondsto vacuum pressure keeping the shepherds together. An= a, =2k while = a, =2k +1 or = a, =2k orarbitrarily2k +1• andcorrespond small. three Obviously sets to of the false for false minima:=0 vacuum.,thetruevacuumcorrespondsto = a, =2k { } { } { } { } even more extreme example is given in Figure (6)andthecorrespondingzoomsinFigures Of course with non-zero, the values of the fieldswhilewhile will be= slightly a,= =2 modified,a,k +1=2or howeverk +1= a, theor=2k or=2ka,+1 =2correspondk or 2 tok the+1 falsecorrespond vacuum. to the false vacuum. { { } { } { } } (7). structure of the false and true vacua will remainOf unchanged.Of course• courseAt withnon In withzero thenon-zero, ensuing butnon-zero, the discussion small, values the of the we thefield values fields values will of be the slightly fields modified, will behowever slightly the modified, however the structurewill of thechange falseHowever, slightly, and true the vacuabut herded the will structure sheep remain are unchanged. actuallyof the in In a metastable the ensuing configuration. discussion we Quantum tun- will identify the vacua by the values that the fieldsstructure take when of=0 the, but false always and with true the vacua will remain unchanged. In the ensuing discussion we vacua willnelling be unchanged. processes can permit the field to make a quantum transition to its true vacuum at understanding that the actual values that the fieldswill take identify are slightly the vacua perturbed. by the values that the fields take when =0, but always with the understandingwill• Consider identify thatall solitons the points the actual vacua where in values the the by that the fieldfield the values is that near fields its take that true are vacuum the slightly fields (and perturbed. specifically take when away from=0 values, but where always with the 3 understandinginterpolateV ( )=from thatV (1 -1/ the2) to). 1. Such actual These transitions values are just truly that kink liberate the the fields sheep, take which are can now slightly move apart perturbed. freely. solitons passing from false vacuum3 to false vacuum. Since they must pass through 0, 3 they areIII. unstable ANALYTICAL to separating CALCULATION at that point. OF THE DECAY RATE IN THE FREE KINK APPROXIMATION The soliton is metastable, it will decay via quantum tunnelling. The tunnelling transition is mediated by an instanton, which is called a bounce. The bounce corresponds to a tra- jectory in Euclidean time, t i⌧. This changes the sign of the kinetic energy, T T , ! ! and therefore bounce configuration can be thought of as motion in the potential V where we include the spatial gradient terms as part of the potential. The Euclidean action can be written as SE = T + V and the conserved Euclidean energy isE = T V . The bounce 5 attract eachattract other in each pairs. other This in is pairs. evident This as is the evident energy as is the minimized energy is by minimized reducing bythe reducing region the region over which the field configuration is in the false vacuum. Thus the net soliton passes from over which the field configuration is in the false vacuum. Thus the net soliton passes from even integer to even integer. But then these full solitons further minimize their energy by even integer to even integer. But then these full solitons further minimize their energy by separating from each other, as this minimizes any gradient energies that occur due to their separating from each other, as this minimizes any gradient energies that occur due to their proximity, hence they have a repulsive interaction. The half solitons on either end are also proximity, hence they have a repulsive interaction. The half solitons on either end are also simply repelled by the adjacent full solitons. simply repelled by the adjacent full solitons. Thus in the example shown in Figure (4), the half soliton from -3 to -2 is repelled by the Thus in the example shown in Figure (4), the half soliton from -3 to -2 is repelled by the full solitons from -2 to 0, which also repels the subsequent full soliton from 0 to 2 which in full solitons from -2 to 0, which also repels the subsequent full soliton from 0 to 2 which in turn repels the final half soliton from 2 to 3. The solitons have the false vacuum a turn repels the final half soliton from 2 to 3. The solitons have the false⇡ vacuum a in between them, hence, they have an attractive interaction between them. The sheep want ⇡ in between them, hence, they have an attractive interaction between them. The sheep want to separate as far as possible from one another, but they cannot do so without pushing the to separate as far as possible from one another, but they cannot do so without pushing the shepherd solitons further and further apart. The energy, in doing so, increases linearly with shepherd solitons further and further apart. The energy, in doing so, increases linearly with the separation of the shepherd solitons, while the energy is at best decreased to a constant the separation of the shepherd solitons, while the energy is at best decreased to a constant value as the separation between the sheep becomes large. Hence, energetically, the shepherd value as the separation between the sheep becomes large. Hence, energetically, the shepherd solitons force the sheep solitons to be herded together at a finite size where the pressure solitons force the sheep solitons to be herded together at a finite size where the pressure from the sheep is balanced against the vacuum pressure keeping the shepherds together. An from the sheep is balanced against the vacuum pressure keeping the shepherds together. An even more extreme example is given in Figure (6)andthecorrespondingzoomsinFigures (7). even more extreme example is given in Figure (6)andthecorrespondingzoomsinFigures (7). However, the herded sheep are actually in a metastable configuration. Quantum tun- • But nownelling we processes add a However,soliton can permit in the the the herded field field, sheep to make are actually a quantum in a transition metastable to its configuration. true vacuum Quantum at tun- that startsall points in the where nellingtrue the vacuum processes field is at can near -a permit itspasses true the vacuumto field (and to make specifically a quantum away transition from values to its where true vacuum at the falseV ( vacuum)=V (1/all2) at). points a Such before where transitions the the trulyfield solitons, is liberate near its the true sheep, vacuum which (and can specifically now move apart away freely. from values where and returns to theV (true )= vacuumV (1/2)). Suchafter, transitions at -a. truly liberate the sheep, which can now move apart freely. III. ANALYTICAL CALCULATION OF THE DECAY RATE IN THE FREE KINK APPROXIMATIONIII. ANALYTICAL CALCULATION OF THE DECAY RATE IN THE FREE KINK APPROXIMATION The soliton is metastable, it will decay via quantum tunnelling. The tunnelling transition is mediated byThe an instanton, soliton is metastable, which is called it will a decay bounce. via The quantum bounce tunnelling. corresponds The to tunnelling a tra- transition jectory in Euclideanis mediated time, byt an instanton,i⌧. This which changes is the called sign a of bounce. the kinetic The energy, bounceT correspondsT , to a tra- ! ! and thereforejectory bounce in configuration Euclidean time, cant be thoughti⌧. This of as changes motion the in sign the potential of the kineticV where energy, T T , ! ! we include theand spatial therefore gradient bounce terms configuration as part of can the be potential. thought of The as motion Euclidean in the action potential can V where FIG. 3. (Color online) The metastable domain wall configuration, passing from 3 3 and be written asweSE include= T + theV and spatial the gradient conserved ! terms Euclidean as part energy of the isE potential.= T V The. The Euclidean bounce action can passing from 1 1 1. ! ! be written as SE = T + V and the conserved Euclidean energy isE = T V . The bounce 5 begins at the metastable configuration, arrives at the configuration which will materialize upon the event of the quantum transition, and then returns to the initial configuration. 5 Therefore, the bounce will began at the metastable soliton as in Figure (4)followedbyan evolution during which the field which is equal to a in the region in between the two half solitons sinks down until it finally makes it to = a corresponding to a configuration of two free half solitons as depicted in Figure (6). The two half solitons must be created with some kinetic energy, since during the evolution the Euclidean “energy”, T V is conserved, and V is more negative at the configuration. In terms of the Minkowski dynamics, this configuration corresponds to the top of the barrier, and hence has much more energy. Then in the Euclidean dynamics, it has energy E,hencetherehastobecompensatingkinetic energy. The solitons then separate and rise up the potential until the stop, and bounce back. When they reach the bounce point, that is the configuration at which they will materialize when the quantum tunnelling transition occurs. 6 I. INTRODUCTION A theory of scalar fieldsattract with spontaneous each other in symmetry pairs. This breaking is evident can breakas the the energy symmetry is minimized to by reducing the region discrete, distinct vacua. Typically,over which one the of fieldthe vacua configuration is the least is in energetic, the false usually vacuum. normalized Thus the net soliton passes from to zeroattract energy, each while other the in otherseven pairs. integer are This false is to evident vacua even integer. that as the are energy But inherently then is minimized these unstable. full by solitons It reducing can further further the region minimize their energy by attract each other in pairs. This is evident as the energy is minimized by reducing the region occurover that which the true the and field false configurationseparating vacua are from is themselves in each the other, false degenerate, vacuum. as this minimizes Thus there the are netany multiple, gradientsoliton discrete passes energies from that occur due to their over which the field configuration is in the false vacuum. Thus the net soliton passes from copieseven of the integer true to vacua evenproximity, and integer. the false But hence then vacua. they these haveThen full a field solitons repulsive configurations further interaction. minimize that The interpolate their half solitonsenergy by on either end are also even integer to even integer. But then these full solitons further minimize their energy by betweenseparating different from vacua, each truesimply other, or false, repelled as this give minimizes by rise the to adjacent domain any gradient walls, full solitons. which energies themselves that occur are stable due to their separating from each other, as this minimizes any gradient energies that occur due to their or metastable.proximity, In hence this they paper have we a consider repulsive domain interaction. walls where The half regions solitons of true on either vacuum end are are also proximity, hence they haveThus a repulsive in the example interaction. shown The in Figure half solitons (4), the on half either soliton end arefrom also -3 to -2 is repelled by the trapped,simply albeit repelled in a metastable by the adjacent fashion, full within solitons. the false vacuum. We further consider the simply repelled by thefull adjacent solitons full from solitons. -2 to 0, which also repels the subsequent full soliton from 0 to 2 which in possibility that the domain wall itself entraps several quanta of a different quantum field, , Thus in the exampleturn shown repels in the Figure final ( half4), the soliton half from soliton 2 fromto 3. -3 The to -2solitons is repelled have by the the false vacuum a Thus in the example shown in Figure (4), the half soliton from -3 to -2 is repelled by the ⇡ whichfull we solitonscall a field from of sheep, -2 to 0, and which these also sheep repels are herded the subsequent into staying full inside soliton the from domain 0 to wall, 2 which in fullattract solitons each from other -2 to inin 0, pairs. between which This also them, is repels evident hence, the as subsequentthey the have energy an full isattractive minimizedsoliton from interaction by 0 reducing to 2 whichbetween the in region them. The sheep want by a fieldturn we repels call the the final shepherding half soliton field, from. 2 The to 3. shepherding The solitons field is have unstable the false to quantum vacuum a turnover repels which the the final field halfto configuration separatesoliton from as far is 2 in to as the 3. possible The false vacuum. fromsolitons one Thusanother,have the the but false net they vacuumsoliton cannot passes do⇡a so from without pushing the tunnelling to its true vacuum, which releases the sheep and allows them to spread out to⇡ in between them, hence, they have an attractive interaction between them. The sheep want in betweeneven integer them, to hence, evenshepherd integer. they have solitons But an attractive then further these and interaction full further solitons between apart. further The them. minimize energy, The in sheep their doing want energy so, increases by linearly with infinity. We compute the amplitude for such a decay in a specific model analytically, aided totoseparating separate separate as from as far far as each as possiblethe possible other, separation from fromas onethis one of another, minimizes the another, shepherd but but any they solitons, they gradient cannot cannot while energies do do so the so without energy without that occur pushing is pushing at due best the to decreasedthe their to a constant by numerical calculations, within a well defined approximation scheme. shepherdshepherdproximity, solitons solitons hence further further theyvalue and have and as further the a further repulsive separation apart. apart. interaction. The between The energy, energy, the The in sheep in doing half doing becomes so, solitons so, increases increases large. on either linearly Hence, linearly end with energetically, with are also the shepherd thethesimply separation separation repelled of of the by the shepherd thesolitons shepherd adjacent force solitons, solitons, full the whilesolitons. sheep while the solitons the energy energy to is be at is herdedat best best decreased together decreased to at to a a constant a finite constant size where the pressure II. THE MODEL valuevalueThus as as the the in separation the separation examplefrom between between shown the sheep the the in sheep Figure is sheep balanced becomes becomes (4), the against large. large. half Hence, the soliton Hence, vacuum energetically, from energetically, pressure -3 to -2 the keeping is the repelled shepherd shepherd the shepherds by the together. An Considersolitonssolitonsfull solitons force the force model the from the sheep defined sheep -2even to solitons 0, solitons bymore which the to extreme Lagrangian to bealso be herded repels herded example density together the together is subsequent given at at a in finite a Figure finite full size soliton ( size6)andthecorrespondingzoomsinFigures where where from the the 0 pressure to pressure 2 which in from the sheep is balanced against the vacuum pressure keeping the shepherds together. An fromturn the repels sheep the is balanced final(7 half). against1 soliton theµ from vacuum 2 toµ pressure 3. The keepingsolitons the have shepherds the false together. vacuum An a = (@µ @ + @µ @ ) V ( , ) (1) ⇡ evenevenin more between more extreme extreme them, example hence, exampleLHowever, they is2 given is givenhave the in an in Figureherded Figureattractive ( sheep6)andthecorrespondingzoomsinFigures (6 )andthecorrespondingzoomsinFigures interaction are actually between in a metastable them. The configuration. sheep want Quantum tun- with(7).(to7). separate• asNow far asnelling the possible coupling processes from oneterm can another, permit plays the a but crucial theyfield tocannot role. make do a quantum so without transition pushing to the its true vacuum at V ( , )=V ( )+V ( )+V ( , ) V0 (2) shepherdHowever,However, solitons the theIt herded does herded furtherall sheep pointsnot sheep and allow are where further are actually actuallythe the apart. in field insolitons a The metastable a is metastable near energy, to its pass true configuration. in configuration. doing vacuum so, (and increases Quantum specifically Quantum linearly tun- tun- away with from values where wherenellingnellingthe separation processes processesthrough can of can the permitV permit( shepherd )=the the V the (1 /field solitons,2)solitons.field). to Such to make while make transitions a quantum thea quantum energy truly transition transition liberateis at best to the to its decreased sheep, its true true vacuum which vacuum to a can constant at atnow move apart freely. 2 2 2 allallvalue points points as where the where• separation theFor the field field V equal between is( near is)= near to↵ its( the 1/2, its true+ sheep truea )the vacuum(( vacuum becomesdenominatora (and) (and+ ✏ large. specifically)) specifically Hence,can be away energetically,away from from values values the(3) where where shepherd V (Vsolitons )=( )=V (1V force/(12)/).2)made the). Such Such sheep transitions toV transitions (almost solitons)= truly(sin vanish truly to2( liberate⇡ be liberate)+ herded and✏ thesin this the2 sheep,together(⇡ sheep,is/ 2)) which which at a can finite can now now size move move where apart apart the freely. freely. pressure III. ANALYTICAL CALCULATION OF THE DECAY RATE(4) IN THE FREE KINK from the sheepenergetically is balanced against forbidden. the vacuum pressure keeping the shepherds together. An while APPROXIMATION even more extreme example is given( ina) Figure2(( + a (6)2)andthecorrespondingzoomsinFigures+ ✏2 )) III. ANALYTICAL CALCULATION OF THE DECAY RATE IN THE FREE KINK III. ANALYTICALV CALCULATION ( , )= OF THE DECAY2 2 RATE IN THE FREE(5) KINK (7). (V ( ) V (1/2) + APPROXIMATION The soliton is metastable, it will decay via quantum tunnelling. The tunnelling transition is consideredAPPROXIMATION to be very small compared to ↵ and is considered small compared to V (1/2). However,• theUnless herded the sheep numerator are actually is zero in aor metastable very small, configuration. Quantum tun- is mediated by an instanton, which is called a bounce. The bounce corresponds to a tra- We impose that V ( , ) 0,clearlythefirstthreetermsarepositivesemi-definite,andthen nelling processeswhich can permitrequires the field equal to maketo a or a quantum -a but with transition to its true vacuum at TheThe soliton soliton is metastable, is metastable,jectory it in will it Euclidean will decay decay via time, via quantum quantumt tunnelling.i⌧ tunnelling.. This changes The The tunnelling tunnelling the sign transition of transition the kinetic energy, T T , V0 is chosen exactly such that V ( , ) vanishes at its global! minimum. ! is mediatedall points by where ana small instanton, the fieldenergy which is near penalty. is its called true a vacuum bounce. (and The specifically bounce corresponds away from to values a tra- where is mediated by an instanton,and therefore which bounce is called configuration a bounce. can The be bounce thought corresponds of as motion to in a tra- the potential V where V ( )=V (1/2)). Such transitions truly liberate the sheep, which can now move apart freely. jectoryjectory in in Euclidean Euclidean time, time,t t i⌧.i⌧ This. This2 changes changes the the sign sign of of the the kinetic kinetic energy, energy,T T T ,T , we include! ! the spatial gradient terms as part of the potential.! ! The Euclidean action can and therefore bounce configuration can be thought of as motion in the potential V where and therefore bounce configurationS can= beT + thoughtV of as motion in the potential V whereE = T V be written as E and the conserved Euclidean energy is . The bounce we include the spatial gradient terms as part of the potential. The Euclidean action can weIII. include ANALYTICAL the spatial gradient CALCULATION terms as part OF of THE the DECAY potential. RATE The Euclidean IN THE FREEaction can KINK be written as SE = T + V and the conserved Euclidean energy isE5 = T V . The bounce beAPPROXIMATION written as SE = T + V and the conserved Euclidean energy isE = T V . The bounce 5 The soliton is metastable, it will decay via5 quantum tunnelling. The tunnelling transition is mediated by an instanton, which is called a bounce. The bounce corresponds to a tra- jectory in Euclidean time, t i⌧. This changes the sign of the kinetic energy, T T , ! ! and therefore bounce configuration can be thought of as motion in the potential V where we include the spatial gradient terms as part of the potential. The Euclidean action can be written as SE = T + V and the conserved Euclidean energy isE = T V . The bounce 5 • A soliton with more “sheep” is pictured here: FIG. 4. (Color online) The metastable domain wall configuration, passing from 3 3 and ! passing from 1 1 1. ! ! The Euclidean Lagrangian is given by 1 2 1 2 1 2 1 2 LE = dx ˙ + ˙ + 0 + 0 + V ( , ) . (6) 2 2 2 2 Z ✓ ◆ The fields depend on time only through their dependence on the positions of the solitons. Indeed, in a first approximation, the fields depend independently on the positions of the solitons. This means we can approximate (x, x1,x2)= 1(x x1)+ 2(x x2) while (x, x1,x2)= 1(x x1)+ 2(x x2). This is approximation is certainly valid when the solitons are separated from one another. It is not valid when they are very close, and we will have to resort to numerical calculations to find the action there, but we leave that for later. Hence, with a slight abuse of notation in the arguments of the fields, for example for ˙ we get ˙ = @x1 (x, x1,x2)˙x1 + @x2 2(x, x1,x2)˙x2 = @ (x x )˙x + @ (x x )˙x x1 1 1 1 x2 2 2 2 = @x 1(x x1)˙x1 + @x 2(x x2)˙x2. (7) 7 1sheep Energy density(Sorry, I shouldand profile call him 2 sheep) 1.5 Etotal 1.0 0.5 1.0 E -10 -5 5 10 0.5 E -0.5 R1 -10 -5 5 10 -1.0 V M1 = Etotaldx =4.72273 Z R =2.1775 ( R ) 0 1 ± 1 ⌘ 5,10,15,20,25 sheep 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 -10 -5 5 10 -15 -10 -5 5 10 15 -20 -10 10 20 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 -20 -10 10 20 -20 -10 10 20 Silence of the lambs: the decay of the soliton • The solitons can decay through tunnelling. Starting from: (b) (a) (c) (d) FIG. 7. (Color online) (a) Metastable soliton configuration. (b) Solitons separated by a microscopic amount. (c) Solitons separated by a macroscopic amount. (d) Solitons separated by a macroscopic amount. corresponds to the part of the potential starting at the top of the barrier and descending with the very slightly sloped straight line. The bounce point is achieved at xB =2m /↵✏ , which obviously behaves as o(1/✏ ). The trajectory from the metastable state to the top of the barrier, in the parametrized set of configurations, achieves the top of the barrier with a variation of x which is of the order of the size of the solitons, a size which is o(1) when compared with o(1/✏ ). Using and analysis identical to that preceding Eqn. (26), the contribution to the action is given by xT 3/2 SE = dx 2m x m +2m = 2m m +2m xT o(1). (29) ⇠ Z0 p p p p Thus it is perfectly reasonable to neglect this contribution to the Euclidean action. 14 • Followed by a transition: (b) (a) (c) (d) FIG. 7. (Color online) (a) Metastable soliton configuration. (b) Solitons separated by a microscopic amount. (c) Solitons separated by a macroscopic amount. (d) Solitons separated by a macroscopic amount. corresponds to the part of the potential starting at the top of the barrier and descending with the very slightly sloped straight line. The bounce point is achieved at xB =2m /↵✏ , which obviously behaves as o(1/✏ ). The trajectory from the metastable state to the top of the barrier, in the parametrized set of configurations, achieves the top of the barrier with a variation of x which is of the order of the size of the solitons, a size which is o(1) when compared with o(1/✏ ). Using and analysis identical to that preceding Eqn. (26), the contribution to the action is given by xT 3/2 SE = dx 2m x m +2m = 2m m +2m xT o(1). (29) ⇠ Z0 p p p p Thus it is perfectly reasonable to neglect this contribution to the Euclidean action. 14 (b) (a) • Separation: (c) (d) FIG. 7. (Color online) (a) Metastable soliton configuration. (b) Solitons separated by a microscopic amount. (c) Solitons separated by a macroscopic amount. (d) Solitons separated by a macroscopic amount. corresponds to the part of the potential starting at the top of the barrier and descending with the very slightly sloped straight line. The bounce point is achieved at xB =2m /↵✏ , which obviously behaves as o(1/✏ ). The trajectory from the metastable state to the top of the barrier, in the parametrized set of configurations, achieves the top of the barrier with a variation of x which is of the order of the size of the solitons, a size which is o(1) when compared with o(1/✏ ). Using and analysis identical to that preceding Eqn. (26), the contribution to the action is given by xT 3/2 SE = dx 2m x m +2m = 2m m +2m xT o(1). (29) ⇠ Z0 p p p p Thus it is perfectly reasonable to neglect this contribution to the Euclidean action. 14 (b) (a) • Till the bounce point: (c) (d) FIG. 7. (Color online) (a) Metastable soliton configuration. (b) Solitons separated by a microscopic amount. (c) Solitons separated by a macroscopic amount. (d) Solitons separated by a macroscopic amount. corresponds to the part of the potential starting at the top of the barrier and descending with the very slightly sloped straight line. The bounce point is achieved at xB =2m /↵✏ , which obviously behaves as o(1/✏ ). The trajectory from the metastable state to the top of the barrier, in the parametrized set of configurations, achieves the top of the barrier with a variation of x which is of the order of the size of the solitons, a size which is o(1) when compared with o(1/✏ ). Using and analysis identical to that preceding Eqn. (26), the contribution to the action is given by xT 3/2 SE = dx 2m x m +2m = 2m m +2m xT o(1). (29) ⇠ Z0 p p p p Thus it is perfectly reasonable to neglect this contribution to the Euclidean action. 14 • The potential can be analytically computed in the free kink approximation, once the free solitons have formed. It is just a linear function of the separation. FIG. 8. (Color online) Potential for the parametrization of the bounce trajectory. IV. DISCUSSION We have computed the decay of a domain wall which traps the true vacuum inside the false vacuum. On first glance, one would think that such a domain wall would be classically unstable to dissociating. There is furthermore no analogous surface energy, that occurs in two [2, 3]andthree[4] spatial dimensional examples, which could trap the true vacuum inside, at least classically. However, in this paper we find that we can build on the analogy with the higher dimensional models. In those models, the true vacuum was separated from the false vacuum by a thin wall. The thin wall was achieved by have several topological quanta of one of the fields trapped inside the wall. The pressure of these quanta made the wall thin. 15 shepherd soliton, hence, looking at the potential Eqn. (4) we see that the contribution will be x2 dxV ( , )= ↵✏ x2 x1 . (19) | | Zx1 Then the energy for the configuration at the top of the barrier will be 1 E E = (m +2m )(x ˙ 2 +˙x2)+2(m +2m ) 2(m + m ) ↵✏ x x T 0 2 1 2 | 2 1| 1 = (m +2m )(x ˙ 2 +˙x2)+2m ↵✏ x x 2 1 2 | 2 1| = T + V (20) The Minkowski action is SM = dt (T V ) while the energy is E = T + V .Analytically continuing to Euclidean time, TR T and SM d⌧ (T + V )= SE. Then the ! ! Euclidean action is SE = d⌧ (T + V ) and easily read offRas R 1 2 2 SE = d⌧ (m +2m )(x ˙ +˙x )+2m ↵✏ x2 x1 (21) 2 1 2 | | Z ✓ ◆ where the overdot now means the derivative with respect to ⌧. The Euclidean trajectory that we can analytically compute, starts at the configuration at the top of the barrier, the solitons separate until they reach the bounce point, and then the solitons come back together and coalesce. We define center of mass and relative coordinates X =(x1 + x2)/2 2 2 2 1 2 and x = x2 x1,thenx˙ +˙x =2X˙ + x˙ .Wetakethecenterofmasscoordinatetobe 1 2 2 constant, X = X˙ =0so that x˙ 2 +˙x2 = 1 x˙ 2. The Euclidean “energy”, T V ,isconserved 1 2 2 and normalized to zero 1 m 2 0=T V = ( + m )˙x 2m + ↵✏ x (22) 2 2 where we restrict x>0. The solitons separate until x˙ =0,henceatthatpointx = xB which is given by 2m xB = . (23) • The action from the top↵✏ of the barrier is Then the contributionsimply to the given action by: from this part of the trajectory will be 2 evidently the integrals are equal. For ,thepotentialisV = ↵ sin (⇡ ) which gives 1 m 2 SE = d⌧ ( + m )˙x +2m ↵✏ x . (24) xf xf 0 0 m 1 2 2 2 ↵ ↵ cos(| |⇡ ) p2↵ = dx 0 = ZdxV ( )= d ( sin(⇡ )) = = . 2 xi 2 xi 2 1 2 2 ⇡ 2 1 ⇡ evidentlyZ the integralsZ are equal.r ForZ ,thepotentialisVr = ↵ sin (⇡ ) which gives evidently the integrals• are equal. For ,thepotentialisV = ↵ sin (⇡ ) which gives The energy conditionwhere Eqn. ( 22)implies (16) x x 0 0 xf m xf f 1 f 0 ↵ 0 p↵ cos( ⇡ ) p2↵ m The minus1 sign2 in front of2 the sin(↵ ⇡ ) comes from the square↵ cos(⇡ root) since for2↵ the first kink = dx 0 == dxVdx ( 0)== dxVd (( )=sin(⇡ )) = d ( sin(⇡ )) == . = . 2 2 2 2 2 2 2 ⇡ ⇡2 ⇡ ⇡ xi xixi xi 1 (2m ↵✏1 x) 2 1 2 2 1 Z [ 1, 0] whereZZsin(⇡ ) 0.ForthefieldrZ Z wer haveZ rV = ( a )rwhich gives x˙ =2 . (16) (16) (25) 2 m +2m xf xf s a The minus signThe inm front minus of sign the sin( in1 front⇡ 2 ) comes of the fromsin(⇡ the) squarecomes root from since the2 square for2 the rootfirst2 kink since3 for the first kink = dx 0 = dxV ( )= d (a )= 2 a (17) 2 2 2 2 2 2 32 2 2 [ 1, 0] where sin([ ⇡ 1, 0]) xiwhere0.Forthefieldsin(⇡ )xi 0.Forthefieldwe have V r= ( wea havea ) Vwhich = gives( a ) which gives 2 2 Z Z Z p 11 2 2 We notem the signxf change1 inx thefxf intermediate xf equationa as ( a2a ) is negative for [ a, a] m2 1 2 2 2 32 2 2 2 3 = dx 0 == dxVdx ( 0)== dxVd (( a)= )= d2 (aa )=(17) 2 a (17) 2 x x 2 2 x x 2 2 a x2 3x 3 and the limitsi i and f areixi assumed tor bexi both close tor 1 or a 2 as required. Z ZZ Z Z Z p 2 2 p We note theWe sign willWe change evaluate note the in the sign the intermediate changeenergy, in first the equation in intermediate the as metastable( equationa ) is negativeconfiguration as ( 2 for a2) andis[ negativea, second a] for in the [ a, a] 2 2 and theconfiguration limits xandi and the atxf limits theare top assumedxi ofand the toxf potential beare both assumed close barrier, to tox be1 soor both thatx2 as close eventually required. to x1 or wex2 canas required. normalize the WeEuclidean will evaluate actionWe the will properly energy, evaluate firstand the then in theenergy, evaluate metastable first it. in For configuration the the metastable metastable and configuration configuration, second in the and the energysecond in the configurationessentially atconfiguration the obtains top of contributions the at potential the top from of barrier, the the potential so locations that eventually barrier, of the so kink we that can solitons, eventually normalize two thewe for can and normalize two the Euclideanfor action,henceEuclidean properly actionand then properly evaluate and it. then For the evaluate metastable it. For configuration, the metastable the energy configuration, the energy essentially obtains contributions from the locations of the kink solitons, two for and two essentially1 obtains contributions2 2 2 from2 the locations of the kink solitons, two for and two E0 = dx 10 + 20 + 10 + 20 + dxV ( , ) for ,hence x x 2 ⇡ 1 for ,henceZ Z x x2 ⇡ 1 2 2 2 2 E0 = =(dx m 10 + m201 +)+ 10 + 20 dx+(V2( )+dxV2V (( ),2+ ) dx(V ( )+V ( )) E = dx 0 + 0 x+x 0 + 0 + dxV ( , ) 2 0 x x1 2 ⇡ 1 1 2 x x2 Z 2 Z ⇡ Z x x2 Z ⇡ x x1 ⇡ x⇡x =2( m + m Z) Z ⇡ 2 (18) =(m + m )+ dx(V ( )+V ( )+ dx( V ( )+V ( )) =(x mx1 + m )+ dxx(Vx(2 )+V ( )+ dx(V ( )+V ( )) Z ⇡ E x x1 Z ⇡ x x2 using Eqns.=2( (m16 ,+17m). We) will subtract Z0 from⇡ the potential so thatZ ⇡ we normalize(18) the initial =2(m + m ) (18) configuration to vanishing energy. using Eqns. (16 ,17). We will subtract E0 from the potential so that we normalize the initial For theusing configuration Eqns. (16 , at17 the). We top will of subtractthe potentialE0 from barrier the wepotential must take so that into we account normalize of the the initial configuration to vanishing energy. followingconfiguration differences: the to vanishing solitons must energy. have kinetic energy in order to conserve energy, we For the configuration at the top of the potential barrier we must take into account of the For the configuration at the top of the potential barrier we must take into account of the have two additional kinks in the field and if the solitons separate by a distance x2 x1 following differences: the solitons must have kinetic energy in order to conserve energy, we | | x the energyfollowing increases diff (iterences: actually the decreases solitons as must this have contribution kinetic energy is negative) in order by to conserve2 dxV ( , energy,). we have two additional kinks in the field and if the solitons separate by a distance x2 xx11 | | have two additional kinks in the field and if the solitons separatex by a distance x2 x1 the energyFor the increases metastable (it actually configuration decreases aswe this did contribution not consider is that negative) the solitons by 2 dxV could( ,R separate,). it is| | x1 x the energy increases (it actually decreases as this contribution is negative) by 2 dxV ( , ). For theassumed metastable that configuration they are herded we did together not consider by the that shepherd the solitons field. could If theR separate, solitons it at is the topx1 of For the metastable configuration we did not consider that the solitons couldR separate, it is assumedthe that potential they are barrier herded separate, together they by the leave shepherd behind field. the true If the vacuum. solitons Sinceat the we top will of normalize assumed that they are herded together by the shepherd field. If the solitons at the top of the potentialthe vacuum barrier energy separate, so that they the leave false behind vacuum the true outside vacuum. the Since shepherd we will soliton normalize has zero energy the potential barrier separate, they leave behind the true vacuum. Since we will normalize the vacuumdensity, energy the true so that vacuum the false between vacuum will outside have the negative shepherd energy, soliton so has this zero contribution energy in fact the vacuum energy so that the false vacuum outside the shepherd soliton has zero energy density,decreases the true the vacuum energy. between The willfield have is in negative its true energy, vacuum so thisoutside contribution the shepherd in fact soliton, thus density, the true vacuum between will have negative energy, so this contribution in fact decreasesthere the is energy. no energy The density field di isff inerence its true for vacuum it, but the outside field the is shepherd in its false soliton, vacuum thus outside the there is no energydecreases density the difference energy. for The it, butfield the isfield in its is truein its vacuum false vacuum outside outside the shepherd the soliton, thus there is no energy density difference10 for it, but the field is in its false vacuum outside the 10 10 (b) (a) (c) (d) FIG. 7. (Color online) (a) Metastable soliton configuration. (b) Solitons separated by a microscopic amount. (c) Solitons separated by a macroscopic amount. (d) Solitons separated by a macroscopic amount. Replacing, in the usual way as in Eqns. (11-15)forx˙ in the first term of Eqn. (24)wefind corresponds to the part of the potential starting at the top of the barrier and descending 1 m (2m ↵✏ x) 1 SE = d⌧ ( + m )2 x˙ + 2m ↵✏ x m +2m x˙ with the very slightly sloped straight line. The bounce point is achieved at xB =2m2 /2↵✏ , m +2m 2 Z s p p which obviously behaves as o(1/✏ ). The trajectory from the metastable state= tod⌧ the2m top ↵✏ x m +2m x˙ x Z x 3/2 B of the barrier, in the parametrized set of configurations, achieves the top of theB p barrier p 2 m +2m (2m ↵✏ x) • Which isReplacing, trivially= inevaluated: thedx usual2m way↵✏ asx in Eqns.m +2 (m11-15= )for x˙ in the first term of Eqn. (24)wefind x o(1) 3↵✏ with a variation of which is of the order of the size of the solitons, a size whichxT is p Z xT 0 p 3/2p ⇡ when compared with o(1/✏ ). Using and analysis identical to that preceding Eqn.2 m (26 1 +2),m the m (2m ) (2m ↵✏ x) 1 SE = d⌧ ( + m )2 x˙ + 2m ↵✏ x m +2m x˙ (26) 2 32↵✏ m +2m 2 contribution to the action is given by Zp s • It is obvious that the contribution to the p p where xT=is thed⌧ value2m of x↵✏at thex m top +2 of them x barrier,˙ which is of the order of the size of the action from the “creation”Z part of the xB xT solitons, butx weB can approximate it as zero).This is the contribution to the action3/2 of the 3/2 p p 2 m +2m (2m ↵✏ x) SE = dx 2m x m +2m = 2m mtrajectory +2m xT is o = (1) . anddx therefore(29)2m ↵✏ it x is m +2m = part of⇠ the instanton from the top of the barrier to the bounce point.3↵✏ 0 xT p Z negligible. Z xT 0 p p p p p 3/2p ⇡ 2 m +2m (2m ) = (26) 3↵✏ Thus it is perfectly reasonable to neglect this contribution to the EuclideanA. Contribution action.p of the interaction potential where xT is the value of x at the top of the barrier, which is of the order of the size of the 14 solitons,In this but subsection we can we approximate explain why it the as zero).This interaction is potential the contributionV does not to contribute the action sig- of the partnificantly of the to instanton the energy from and the the top action. of the If barrier we imagine to the the bounce sheep point. solitons separating, then between them the field will be in its true vacuum but the field will be in its false vacuum. Thus as they separate, the total energy will grow linearly. thus the shepherd solitons give A. Contribution of the interaction potential acompressionthatholdsthesheeptogether.Howeverastheshepherdsolitonscompress the sheep more and more, there will be a pressure increase, and the system will come to In this subsection we explain why the interaction potential V does not contribute sig- an equilibrium. There are many contributions to the pressure forces, the kinks becoming nificantly to the energy and the action. If we imagine the sheep solitons separating, then steeper than their natural slope will add energy and hence pressure. But there will be a between them the field will be in its true vacuum but the field will be in its false vacuum. contribution coming from the interaction term. Effectively the shepherd solitons would like Thus as they separate, the total energy will grow linearly. thus the shepherd solitons give to approach each other and annihilate completely in order to minimize their energy. The acompressionthatholdsthesheeptogether.Howeverastheshepherdsolitonscompress sheep solitons get in the way. The energy diminishes linearly as the separation of the shep- the sheep more and more, there will be a pressure increase, and the system will come to herd solitons goes to zero, with coefficient ✏ .Ifoneforcestheshepherdsolitonscloser an equilibrium. There are many contributions to the pressure forces, the kinks becoming and closer, the value of the field at the point where 1/2 starts to move significantly steeper than their natural slope will add energy and hence⇡ pressure. But there will be a away from a, the values for which the numerator of the interaction term vanishes. This contribution± coming from the interaction term. Effectively the shepherd solitons would like will make the energy increase, to first approximation linearly, however with much greater to approach each other and annihilate completely in order to minimize their energy. The coefficient ⇠ > ✏ . Thus the dynamics can be simply modelled as given by a potential of sheep solitons get in the way. The energy diminishes linearly as the separation of the shep- the form herd solitons goes to zero, with coefficient ✏ .Ifoneforcestheshepherdsolitonscloser Veff. = ✏ x ⇠(x x0), (27) and closer, the value of the field at the point where 1/2 starts to move significantly ⇡ away from a, the values for which the numerator12 of the interaction term vanishes. This ± will make the energy increase, to first approximation linearly, however with much greater coefficient ⇠ > ✏ . Thus the dynamics can be simply modelled as given by a potential of the form Veff. = ✏ x ⇠(x x0), (27) 12 • The evolution of a large soliton would proceed why multiple, sequential tunnelling trajectories. However, after the first “cut”, the two lumps would immediately separate, converting false vacuum to true vacuum. FIG. 5. (Color online) The metastable domain wall configuration with 31 sheep, passing from 15 15 and passing from 1 1 1. ! ! ! @x 1(x x1)= 0 is a function that is sharply peaked around x1 and @x 2(x x2)= 0 1 2 thus the product of these functions essentially vanishes. This gives 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 2 L = dx 0 x˙ + 0 x˙ + 0 x˙ + 0 x˙ + 0 + 0 + 0 + 0 + V ( , ) . E 2 1 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 Z ✓ ◆ (8) Furthermore, the nature of the solitons at x1 and x2 in each field are essentially identical, they are simply the “kink” type solitons of an (almost) doubly (or multiply but only adjacent minima are considered here) degenerate potential. Therefore we will compute the contribu- tion of the kinks in the limit that the potential is exactly degenerate, and we note that kink or anti-kink obviously give an equal contribution. Thus we write 1 2 1 2 m 1 2 1 2 m dx 0 = dx 0 and dx 0 = dx 0 . (9) 2 1 2 2 ⌘ 2 2 1 2 2 ⌘ 2 Z Z Z Z m and m should not be confused with the actual masses of the perturbative excitations of each respective field. The contribution of the potential to the action will also be easily deconstructed. There will be a contribution from around the region of each soliton and there 8 Conclusions • We have computed the induced decay rate of the vacuum due to “false” topological solitons in the false vacuum. • The rate can be unsuppressed for heavy monopoles. • Cosmic strings should induce vacuum decay in a similar manner. • With discrete multiple vacua, false domain walls could also form. • Whether these additional decay channels significantly affect the vacuum stability depends on the specific values of various parameters.