The Battle of the Bulge: Decay of the Thin, False Cosmic String
Bum-Hoon Lee, Wonwoo Lee, Richard MacKenzie, M. B. Paranjape, U. A. Yajnik, Dong-han Yeom
UUdeM-GPP-TH-13-226, CQUeST-2013-0622, YITP-13-82
The Battle of the Bulge: Decay of the Thin, False Cosmic String
Bum-Hoon Lee a,b , ∗ Wonwoo Lee b , † Richard MacKenzie c , ‡ M. B. Paranjape c , § U. A. Yajnik d , ¶ and Dong-han Yeom b,e ∗∗ a Department of Physics and BK21 Division,Sogang University, Seoul 121-742, Korea b Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea c Groupe de Physique des Particules,Département de physique,Université de Montréal,C. P. 6128, Succursale Centre-ville,Montreal, Québec, Canada, H3C 3J7 d Department of Physics, Indian Institute of Technology Bombay, Mumbai, India and e Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan a r X i v : . [ h e p - t h ] O c t bstract We consider the decay of cosmic strings that are trapped in the false vacuum in a theory of scalarelectrodynamics in 3+1 dimensions. This paper is the 3+1 dimensional generalization of the 2+1dimensional decay of false vortices which we have recently completed [2]. We restrict our analysisto the case of thin-walled cosmic strings which occur when large magnetic flux trapped inside thestring. Thus the string looks like a tube of fixed radius, at which it is classically stable. The core ofthe string contains magnetic flux in the true vacuum, while outside the string, separated by a thinwall, is the false vacuum. The string decays by tunnelling to a configuration which is representedby a bulge, where the region of true vacuum within, is ostensibly enlarged. The bulge can bedescribed as the meeting, of a kink soliton anti-soliton pair, along the length of the string. It canbe described as a bulge appearing in the initial string, starting from the string of small, classicallystable radius, expanding to a fat string of large, classically unstable (to expansion) radius and thenreturning back to the string of small radius along its length. This configuration is the bounce pointof a corresponding O (2) symmetric instanton, which we can determine numerically. Once the bulgeappears it explodes in real time. The kink soliton anti-soliton pair recede from each other alongthe length of the string with a velocity that quickly approaches the speed of light, leaving behind afat tube. At the same time the radius of the fat tube that is being formed, expands (transversely)as it is no longer classically stable, converting false vacuum to the true vacuum with ever dilutingmagnetic field within. The rate of this expansion is determined by the energy difference betweenthe true vacuum and the false vacuum. Our analysis could be applied to a network, of cosmicstrings formed in the very early universe or vortex lines in a superheated superconductor. PACS numbers: 11.27.+d, 98.80.Cq, 11.15.Ex, 11.15.Kc ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] . INTRODUCTION We continue our study of the decay of the false vacuum precipitated by the existence oftopological defects in that vacuum [1, 2]. Here we consider the case of cosmic strings in aspontaneously broken U (1) gauge theory, a generalized Abelian Higgs model. The potentialfor the complex scalar field has a local minimum at a nonzero value and the true minimum isat vanishing scalar field. We assume the energy density splitting between the false vacuumand the true vacuum is very small. The spontaneously broken vacuum is the false vacuum.In a scenario where from a high temperature phase, the theory passes through an in-termediate phase of spontaneous symmetry breaking, finally arriving at a true vacuum ofunbroken symmetry, it is generic that there will be topologically defects. The phase of thecomplex scalar field can wrap an integer number of times around a given line in 3 dimen-sional space. The line can be infinite or form a closed loop. Homologous to the given linethere must exist a line of zeros of the scalar field. Where the scalar field vanishes correspondsto the true vacuum. The corresponding minimum energy configuration is called a cosmicstring, alternatively a Nielsen-Olesen string [3] or a vortex string [4]. In the scenario thatwe have described, the true vacuum lies at the regions of vanishing scalar field, thus theinterior of the cosmic string is in the true vacuum while the exterior is in the false vacuum.In our recent work [2], we considered the decay of vortices in the strictly two spatialdimensional context. There, the vortex was classically stable at a given radius R . Throughquantum tunelling, the vortex could evolve to a larger vortex of radius R , which was nolonger classically stable. Dynamically the interior of the vortex was at the true vacuum,thus energetically lower by the energy density splitting multiplied by the area of the vortex.The gain in energy behaved as ∼ R , while the magnetic field energy behaved like ∼ /R and the energy in the wall behaved like ∼ R . Thus the energy functional had the form E = α/R + βR − (cid:15)R . (1)For sufficiently small (cid:15) , this energy functional is dominated by the first two terms. It isinfinitely high for a small radius due to the magnetic energy, and will diminish to a localminimum when the linear wall energy begins to become important. This will occur at aradius R , well before the quadratic area energy, due to the energy splitting between thefalse vacuum and the true vacuum becomes important, for (cid:15) is sufficiently small. Clearlythough, for large enough radius of the thin wall string configuration, the energy splitting3ill be the most important term, and a thin walled vortex configuration of sufficiently largeradius will be unstable to expanding to infinite radius. However, a vortex of radius R willbe classically stable and only susceptible to decay via quantum tunnelling. The amplitudefor such tunnelling, in the semi-classical approximation, has been calculated in [2].In this paper we consider the generalization of the model to 3+1 dimensions. Here thevortex can be continued along the third, additional dimension as a string, often called acosmic string. The interior of the string contains a large magnetic flux distributed over aregion of the true vacuum. It is separated from the region outside by a thin wall, where thescalar field is in the false vacuum. The analysis of the decay of two dimensional vorticescannot directly apply to the decay of the cosmic string, as the cosmic string must maintaincontinuity along its length. Thus the radius of the string at a given position cannot sponta-neously make the quantum tunnelling transition to the larger iso-energetic radius, called R ,without being continuously connected to the rest of the string. The whole string could inprinciple spontaneously tunnel to the fat string along its whole length, but the probabilityof such a transition is strictly zero for an infinite string, and correspondingly small for aclosed string loop. The aim of this paper is to describe the tunnelling transition to a statethat corresponds to a spontaneously formed bulge in the putatively unstable thin string. II. ENERGETICS AND DYNAMICS OF THE THIN, FALSE STRINGA. Set-up
We consider the abelian Higgs model (spontaneously-broken scalar electrodynamics) witha modified scalar potential corresponding to our previous work [2] but now generalized to3+1 dimensions. The Lagrangian density of the model has the form L = − F µν F µν + ( D µ φ ) ∗ ( D µ φ ) − V ( φ ∗ φ ) , (2)where F µν = ∂ µ A ν − ∂ ν A µ and D µ φ = ( ∂ µ − ieA µ ) φ . The potential is a sixth-order polynomialin φ [1, 5], written V ( φ ∗ φ ) = λ ( | φ | − (cid:15)v )( | φ | − v ) . (3)Note that the Lagrangian is no longer renormalizable in 3+1 dimensions, however the un-derstanding is that it is an effective theory obtained from a well defined renormalizable4undamental Lagrangian. The fields φ and A µ , the vacuum expectation value v have massdimension 1, the charge e is dimensionless and λ has mass dimension 2 since it is the cou-pling constant of the sixth order scalar potential. The potential energy density of the falsevacuum | φ | = v vanishes, while that of the true vacuum has V (0) = − λv (cid:15) . We rescaleanalogous to [2] φ → vφ A µ → vA µ e → λ / ve x → x/ ( v λ / ) (4)so that all fields, constants and the spacetime coordinates become dimensionless, then theLagrangian density is still given by Eqn. (2) where now the potential is V ( φ ∗ φ ) = ( | φ | − (cid:15) )( | φ | − . (5)and there is an overall factor of / ( λv ) in the action.Initially, the cosmic string will be independent of z the coordinate along its length andwill correspond to a tube of radius R with a trapped magnetic flux in the true vacuuminside, separated by a thin wall from the false vacuum outside. R will vary in Euclideantime τ and in z to yield an instanton solution. Thus we promote R to a field R → R ( z, τ ) .Hence we will look for axially-symmetric solutions for φ and A µ in cylindrical coordinates ( r , θ , z , τ ) . We use the following ansatz for a vortex of winding number n : φ ( r, θ, z, τ ) = f ( r, R ( z, τ )) e inθ , A i ( r, θ, z, τ ) = − ne ε ij r j r a ( r, R ( z, τ )) , (6)where ε ij is the two-dimensional Levi-Civita symbol. This ansatz is somewhat simplistic, itis clear that if the radius of the cosmic string swells out at some range of z , the magneticflux will dilute and hence through the (Euclidean) Maxwell’s equations some “electric” fieldswill be generated. In 3 dimensional, source free, Euclidean electrodynamics, there is nodistinct electric field, the Maxwell equations simply say that the 3 dimensional magneticfield is divergence free and rotation free vector field that satisfies superconductor boundaryconditions at the location of the wall. It is clear that the correct form of the electromagneticfields will not simply be a diluted magnetic field that always points along the length of thecosmic string as with our ansatz, however the correction will not give a major contribution,and we will neglect it. Indeed, the induced fields will always be smaller by a power of /c when the usual units are used. 5he Euclidean action functional for the cosmic string then has the form S E [ A µ , φ ] = 1 λv (cid:90) d x (cid:34)(cid:88) i (cid:18) F i F i + 12 F i F i (cid:19) + 12 F F + (cid:88) ij F ij F ij + ( ∂ τ φ ) ∗ ( ∂ τ φ ) + ( ∂ z φ ) ∗ ( ∂ z φ ) + (cid:88) i D i ( φ ) ∗ ( D i φ ) + V ( φ ∗ φ ) (cid:35) (7)where i, j take values just over the two transverse directions and we have already incorpo-rated that A = A = 0 .Substituting Eqns. (5,6) into Eqn. (7), we obtain S E = 2 πλv (cid:90) dzdτ (cid:90) ∞ dr r (cid:20) n ˙ a e r + n a (cid:48) e r + n ( ∂ r a ) e r + ˙ f + f (cid:48) + ( ∂ r f ) + n r (1 − a ) f + ( f − (cid:15) )( f − (cid:21) , (8)where the dot and prime denote differentiation with respect to τ and z , respectively. Then ˙ a = (cid:16) ∂a ( r,R ) ∂R (cid:17) ˙ R and a (cid:48) = (cid:16) ∂a ( r,R ) ∂R (cid:17) R (cid:48) , and likewise for f , hence the action becomes S E = 2 πλv (cid:90) dzdτ (cid:90) ∞ dr r n (cid:16)(cid:16) ∂a ( r,R ) ∂R (cid:17) ˙ R (cid:17) e r + n (cid:16)(cid:16) ∂a ( r,R ) ∂R (cid:17) R (cid:48) (cid:17) e r + n ( ∂ r a ) e r + (cid:18) ∂f ( r, R ) ∂R ˙ R (cid:19) + (cid:18) ∂f ( r, R ) ∂R R (cid:48) (cid:19) + ( ∂ r f ) + n r (1 − a ) f + ( f − (cid:15) )( f − (cid:35) = 2 πλv (cid:90) dz (cid:90) ∞ dr r (cid:34)(cid:32) n e r (cid:18) ∂a ( r, R ) ∂R (cid:19) + (cid:18) ∂f ( r, R ) ∂R (cid:19) (cid:33) ( ˙ R + R (cid:48) )+ n ( ∂ r a ) e r + ( ∂ r f ) + n r (1 − a ) f + ( f − (cid:15) )( f − (cid:21) . (9)We note the two Euclidean dimensional, rotationally invariant form ( ˙ R + R (cid:48) ) which appearsin the kinetic term. This allows to make the O (2) symmetric ansatz for the instanton, andthe easy continuation of the solution to Minkowski time, to a relativistically invariant O (1 , solution, once the tunnelling transition has been completed.In the thin wall limit, the Euclidean action can be evaluated essentially analytically, upto corrections which are smaller by at least one power of /R . The method of evaluation isidentical to that in [2], we shall not repeat the details, we find S E = 1 λv (cid:90) d x M ( R ( z, τ ))( ˙ R + R (cid:48) ) + E ( R ( z, τ )) − E ( R ) (10)6here M ( R ) = (cid:20) πn e R + πR (cid:21) (11) E ( R ) = n Φ πR + πR − (cid:15)πR (12)and R is the classically stable thin tube string radius. R (cid:187) (cid:72) R (cid:76) FIG. 1. (color online) The energy as a function of R , for n = 100 , e = . and (cid:15) = . . III. INSTANTONS AND THE BULGEA. Tunnelling instanton
We look for an instanton solution that is O (2) symmetric, the appropriate ansatz is R ( z, τ ) = R ( √ z + τ ) = R ( ρ ) (13)with the imposed boundary condition that R ( ∞ ) = R .Such a solution will describe the transition from a string of radius R at τ = −∞ , to apoint in τ = ρ say at z = 0 when a soliton anti-soliton pair is started to be created. Theconfiguration then develops a bulge which forms when the pair separates to a radius which7as to be again ρ because of O (2) invariance and which is the bounce point of the instantonalong the z axis at τ = 0 . Finally the subsequent Euclidean time evolution continues in amanner which is just the (Euclidean) time reversal of evolution leading up to the bouncepoint configuration until a simple cosmic string of radius R is re-established for τ ≥ ρ andall z , i.e. ρ ≥ ρ . The action functional is given by S E = 2 πλv (cid:90) dρ ρ (cid:34) M ( R ( ρ )) (cid:18) ∂R ( ρ ) ∂ρ (cid:19) + E ( R ( ρ )) − E ( R ) (cid:35) . (14)The instanton equation of motion is ddρ (cid:18) ρM ( R ) dRdρ (cid:19) − ρM (cid:48) ( R ) (cid:18) dRdρ (cid:19) − ρE (cid:48) ( R ) = 0 (15)with the boundary condition that R ( ∞ ) = R , and we look for a solution that has R ≈ R near ρ = 0 . The solution necessarily “bounces” at τ = 0 since ∂R ( ρ ) /∂τ | τ =0 = R (cid:48) ( ρ )( τ /ρ ) | τ =0 = 0 . (The potential singularity at ρ = 0 is not there since a smooth con-figuration requires R (cid:48) ( ρ ) | ρ =0 = 0 .) The equation of motion is better cast as an essentiallyconservative dynamical system with a “time” dependent mass and the potential given bythe inversion of the energy function as pictured in Fig. (1), but in the presence of a “time”dependent friction where ρ plays the role of time: ddρ (cid:18) M ( R ) dRdρ (cid:19) − M (cid:48) ( R ) (cid:18) dRdρ (cid:19) − E (cid:48) ( R ) = − ρ (cid:18) M ( R ) dRdρ (cid:19) . (16)As the equation is “time” dependent, there is no analytic trick to evaluating the bounceconfiguration and the corresponding action. We are, however, confident in the existence ofa solution which starts with a given R ≈ R at ρ = 0 and achieves R = R for ρ > ρ ,by showing the existence of an initial condition that gives an overshoot and another initialcondition that gives an undershoot, in the same manner of proof as in [6]. If we start at theorigin at ρ = 0 high enough on the right side of the inverted energy functional pictured in Fig.(1), the equation of motion (16) will cause the radius R , to slide down the potential and thenroll up the hill to R = R . If we start too far up to the right, we will roll over the maximumat R = R while if we do not start high enough we will never make it to the top of the hillat R = R . The RHS of (16) acts as a “time” dependent friction, which becomes negligibleas ρ → ∞ , and once it is negligible, the motion is effectively conservative. We resort tonumerical studies and we find with little difficulty, that if we start at R ≈ . , for n = 100 , e = . and (cid:15) = . we generate the profile function R ( ρ ) in Figure (2). Actually,8 (cid:61) Ρ (cid:72) Ρ (cid:76) FIG. 2. (color online) The radius as a function of ρ . numerically integrating to ρ ≈ , the function falls back to the minimum of the invertedenergy functional Eqn. (1). On the other hand, we increase the starting point by . ,the numerical solution overshoots the maximum at R = R . Hence we have numericallyimplemented the overshoot/undershoot criterion of [6].The cosmic string emerges with a bulge described by the function numerically evaluatedand represented in Figure (2) which corresponds to R ( z, τ = 0) . A 3 dimensional depictionof the bounce point is given in Figure (3). One should imagine the radius R ( z ) along thecosmic string to be R to the left, then bulging out to the the large radius as describedby the mirror image of the function in Figure (2) and then returning to R according tothe function in Figure (2). This radius function has argument ρ = √ z + τ . Due to theLorentz invariance of the original action, the subsequent Minkowski time evolution is givenby R ( ρ ) → R ( √ z − t ) , which is of course only valid for z − t ≥ . Fixed ρ = z − t describes a space-like hyperbola that asymptotes to the light cone. The value of the function R ( ρ ) therefore remains constant along this hyperbola. This means that the point at whichthe string has attained the large radius moves away from z ≈ to z → ∞ at essentiallythe speed of light. The other side of course moves towards z → −∞ . Thus the soliton9 a (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) b (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) FIG. 3. (Color online) (a) Cosmic string profile at the bounce point. (b) Cut away of the cosmicstring profile at bounce point. anti-soliton pair separates quickly moving at essentially the speed of light, leaving behind afat cosmic string, which is subsequently, classically unstable to expand and fill all space.
B. Tunnelling amplitude
It is difficult to say too much about the tunnelling amplitude or the decay rate per unitvolume analytically in the parameters of the model. The numerical solution we have obtainedfor some rather uninspired choices of the parameters gives rise to the profile of the instantongiven in Fig. (2). This numerical solution could be then inserted into the Euclidean actionto determine its numerical value, call it S ( (cid:15) ) . It seems difficult to extract any analyticaldependence on (cid:15) , however it is reasonable to expect that as (cid:15) → the tunnelling barrier,as can be seen in Fig. (1), will get bigger and bigger and hence the tunnelling amplitudewill vanish. On the other hand, there should exist a limiting value, call it (cid:15) c , where thetunnelling barrier disappears at the so-called dissociation point [7], such that as (cid:15) → (cid:15) c , theaction of the instanton will vanish, analogous to what was found in [2]. In general the decayrate per unit length of the cosmic string will be of the form Γ = A c . s . (cid:18) S ( (cid:15) )2 π (cid:19) e − S ( (cid:15) ) . (17)where A c . s . is the determinantal factor excluding the zero modes and (cid:16) S ( (cid:15) )2 π (cid:17) is the correc-tion obtained after taking into account the two zero modes of the bulge instanton. Thesecorrespond to invariance under Euclidean time translation and spatial translation along thecosmic string [6]. In general, there will be a length L of cosmic string per volume L . For a10econd order phase transition to the metastable vacuum, L is the correlation length at thetemperature of the transition which satisfies L − ≈ λv T c [8]. For first order transitions, itis not clear what the density of cosmic strings will be. We will keep L as a parameter butwe do expect that it is microscopic. Then in a large volume Ω , we will have a total length N L of cosmic string, where N = Ω /L . Thus the decay rate for the volume Ω will be Γ × ( N L ) = Γ (cid:18) Ω L (cid:19) = A c . s . (cid:18) S ( (cid:15) )2 π (cid:19) e − S ( (cid:15) ) Ω L (18)or the decay rate per unit volume will be Γ L = A c . s . (cid:16) S ( (cid:15) )2 π (cid:17) e − S ( (cid:15) ) L . (19)A comparable calculation with point-like defects [2] would give a decay rate per unit volumeof the form Γ point like L = A point like (cid:16) S point like0 ( (cid:15) )2 π (cid:17) / e − S point like0 ( (cid:15) ) L (20)and the corresponding decay rate from vacuum bubbles (without topological defects) [6]would be Γ vac . bubble = A vac . bubble (cid:18) S vac . bubble0 ( (cid:15) )2 π (cid:19) e − S vac . bubble0 ( (cid:15) ) . (21)Since the length scale L is expected to be microscopic, we would then find that the numberof defects in a macroscopic volume ( i.e. universe) could be incredibly large, suggesting thatthe decay rate from topological defects would dominate over the decay rate obtained fromsimple vacuum bubbles à la Coleman [6]. Of course the details do depend on the actualvalues of the Euclidean action and the determinantal factor that is obtained in each case. IV. CONCLUSION
There are many instances where the vacuum can be meta-stable. The symmetry brokenvacuum can be metastable. Such solutions for the vacuum can be important for cosmologyand for the case of supersymmetry breaking see [9] and the many references therein. Instring cosmology, the inflationary scenario that has been obtained in[10], also gives rise toa vacuum that is meta-stable , and it must necessarily be long-lived to have cosmologicalrelevance. 11n a condensed matter context symmetry breaking ground states are also of great im-portance. For example, there are two types of superconductors [11]. The cosmic string iscalled a vortex line solution in this context, and it is relevant to type II superconductors.The vortex line contains an unbroken symmetry region that carries a net magnetic flux,surrounded by a region of broken symmetry. If the temperature is raised, the true vacuumbecomes the unbroken vacuum, and it is possible that the system exists in a superheatedstate where the false vacuum is meta-stable [12]. This technique has actually been used toconstruct detectors for particle physics [13]. Our analysis might even describe the decay ofvortex lines in superfluid liquid 3Helium [14].The decay of all of these metastable states could be described through the tunnellingtransition mediated by instantons in the manner that we have computed in this article. Forappropriate limiting values of the parameters, for example when (cid:15) → (cid:15) c , the suppressionof tunnelling is absent, and the existence of vortex lines or cosmic strings could cause thedecay of the meta-stable vacuum without bound. Experimental observation of this situationwould be interesting. ACKNOWLEDGEMENTS
This work was financially supported in part by the National Research Foundation ofKorea grant funded by the Ministry of Education, Science and Technology through theCenter for Quantum Spacetime (CQUeST) of Sogang University (2005-0049409), by theNatural Science and Engineering Council of Canada, by the Department of Science andTechnology, India, by the Coopération Québec-Maharashtra (Inde) program of the Ministèredes relations internationales du Québec and by the Direction de relations internationales del’Université de Montréal. WL was supported by the Basic Science Research Program throughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (2012R1A1A2043908). DY is supported by the JSPS Grant-in-Aidfor Scientific Research (A) No. 21244033. RM thanks McGill University for hospitality whilethis work was in progress. 12
1] B. Kumar, M. B. Paranjape, and U. A. Yajnik, Phys. Rev. D , 025022 (2010).[2] B-H. Lee, Wonwoo Lee, Dong-han Yeom, R. MacKenzie, M. B. Paranjape, and U. A. Yajnik,to be published.[3] H. B. Nielsen and P. Olesen, Nucl. Phys. B61 , 45 (1973).[4] A. Abrikozov, JETP(Sov.Phys.), , 1173 (1957).[5] P. J. Steinhardt, Phys. Rev. D , 842 (1981).[6] S. R. Coleman, Phys. Rev. D15, 2929 (1977); C. A. Callan and S. Coleman, Phys. Rev D16,1762 (1977).[7] U. A. Yajnik, Phys. Rev. D34, 1237 (1986); B. Kumar and U. A. Yajnik, Phys. Rev. D ,065001 (2009) [arXiv:0807.3254 [hep-th]]; B. Kumar and U. Yajnik, Nucl. Phys. B , 162(2010) [arXiv:0908.3949 [hep-th]].[8] T. W. B. Kibble, J. Phys. A , 1387 (1976); T. W. B. Kibble, Phys. Rep. 67, 183 (1980); W.H.Zurek, Nature (London) 317, 505 (1985); W.H. Zurek, Acta Phys. Pol. B 24, 1301 (1993).[9] S. A. Abel, C. -S. Chu, J. Jaeckel and V. V. Khoze, JHEP , 089 (2007) [hep-th/0610334];Willy Fischler, Vadim Kaplunovsky, Lorenzo Mannelli, Marcus Torres and Chethan Krishnan,JHEP03(2007)107.[10] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, Phys. Rev. D68, 046005 (2003), hep-th/0301240.[11] N. W. Ashcroft and N. D. Mermin, Solid state physics, (Harcourt College Publishers, 1976).[12] A. J. Dolgert, S. J. Di Bartolo, and A. T. Dorsey, Phys. Rev. B 53, 5650, (1996).[13] Superconductive Particle Detectors, edited by A. Barone (World Scientific, Singapore, 1987);K. Pretzl, J. Low Temp. Phys. 93, 439 (1993).[14] A. J. Leggett, “A Theoretical Description of the New Phases of Liquid 3He”, Rev. Mod. Phys.47, 331, 1975., 089 (2007) [hep-th/0610334];Willy Fischler, Vadim Kaplunovsky, Lorenzo Mannelli, Marcus Torres and Chethan Krishnan,JHEP03(2007)107.[10] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, Phys. Rev. D68, 046005 (2003), hep-th/0301240.[11] N. W. Ashcroft and N. D. Mermin, Solid state physics, (Harcourt College Publishers, 1976).[12] A. J. Dolgert, S. J. Di Bartolo, and A. T. Dorsey, Phys. Rev. B 53, 5650, (1996).[13] Superconductive Particle Detectors, edited by A. Barone (World Scientific, Singapore, 1987);K. Pretzl, J. Low Temp. Phys. 93, 439 (1993).[14] A. J. Leggett, “A Theoretical Description of the New Phases of Liquid 3He”, Rev. Mod. Phys.47, 331, 1975.