We show that the annihilation of vortex-antivortex pairs can lead to very long-lived oscillon states in 2d Abelian Higgs models. The emergence of oscillons is controlled by the ratio of scalar and vector field masses, β=(ms/mv)2 and can be described as a phase transition in field configuration space with critical value βc≃0.13(6)±2: only models with βO(β)∼|β−βc|o, where O is an order parameter indicating the presence of oscillons and o=0.2(2)±2 is the critical exponent

Gleiser, M.

Thorarinson, J.

English

Dartmouth Digital Commons (Dartmouth College)

Dartmouth College Dartmouth Digital Commons Open Dartmouth: Published works by Dartmouth faculty Faculty Work 8-15-2007 Phase Transition in U(1) Configuration Space: Oscillons as Remnants of Vortex-Antivortex Annihilation M. Gleiser Dartmouth College J. Thorarinson Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa  Part of the Physical Sciences and Mathematics Commons Dartmouth Digital Commons Citation Gleiser, M. and Thorarinson, J., "Phase Transition in U(1) Configuration Space: Oscillons as Remnants of Vortex-Antivortex Annihilation" (2007). Open Dartmouth: Published works by Dartmouth faculty. 3034. https://digitalcommons.dartmouth.edu/facoa/3034 This Article is brought to you for free and open access by the Faculty Work at Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: Published works by Dartmouth faculty by an authorized administrator of Dartmouth Digital Commons. For more information, please contact dartmouthdigitalcommons@groups.dartmouth.edu. Phase transition in U(1) configuration space: Oscillons as remnants of vortex-antivortexannihilationM. Gleiser* and J. Thorarinson†Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA(Received 31 January 2007; published 15 August 2007)We show that the annihilation of vortex-antivortex pairs can lead to very long-lived oscillon states in 2dAbelian Higgs models. The emergence of oscillons is controlled by the ratio of scalar and vector fieldmasses,   ms=mv2 and can be described as a phase transition in field configuration space with criticalvalue c ’ 0:136  2: only models with <c lead to oscillonlike remnants. The critical behavior ofthe system obeys a power law O  j cjo, where O is an order parameter indicating the presenceof oscillons and o  0:22  2 is the critical exponent.DOI: 10.1103/PhysRevD.76.041701 PACS numbers: 11.27.+d, 11.15.Ex, 98.80.CqI. INTRODUCTIONIn relativistic and nonrelativistic field theories, modelsthat exhibit spontaneous symmetry breaking support aplethora of topologically stable field configurations ordefects [1], static solutions of the nonlinear equations ofmotion with properties determined by the topology of thevacuum manifold. One of the simplest examples is the kinkin 1d 4 models, the result of a discrete symmetry break-ing: the vacuum has two disconnected points and the kinkinterpolates between them. Once formed, kinks cannot bedestroyed. Unless, of course, they meet an antikink. Low-momentum kink-antikink scattering leads to the formationof breathers, long-lived time-dependent bound states char-acterized by large-amplitude oscillations of the scalar field[2]. At larger velocities, breathers have been shown to formonly at certain windows or resonances [3]. Although 4breathers have not been seen to decay in numerical studies[3], they may decay via nonperturbative effects [4], radiat-ing their energy to spatial infinity.In 2d, the simplest stable topological defect with local-ized energy is the Nielsen-Olesen vortex found in AbelianHiggs models (AHMs) [1]. These vortices are of greatinterest in many areas of physics, e.g., as prototypes forstudying nonperturbative effects in the standard model andits extensions, or, in the nonrelativistic limit, as Abrikosovvortices in the Landau-Ginzburg theory of superconduc-tivity [5]. Vortex-like solutions may also have a variety ofcosmological roles, being formed during a phase transitionat the grand unified theory (GUT) scale or thereafter [1,6]or during reheating after inflation [7].Inspired by the existence of long-lived breathers in 1d4 models, in this work we investigate the annihilation ofbound configurations of vortices and antivortices in AHMs.Vortices and antivortices carry equal and opposite topo-logical charge. This allows for their annihilation. The onlyadjustable parameter in the model is   =2g2, the ratioof squared masses of the scalar and vector fields. We willshow that for  smaller than a critical value c, vortex-antivortex (henceforth vav) annihilation does give rise tohigher-dimensional breatherlike configurations known asoscillons [8], also characterized by long-lived oscillationsof the Higgs field magnitude. They were found in 2d [9], 3d[10], and higher-dimensional real scalar field models [11].Their properties have been studied in great detail [12].Long-lived oscillatory solutions have also been found inthe decay of sphalerons in the 1d AHM [13]. Recently,oscillons were found in 3d gauged SU(2) models [14] andgauged SU2  U1 models [15] when the Higgs mass istwice the W boson mass.In the above references, the procedure was to solve therelevant equations of motion assuming or not sphericalsymmetry and using an initial profile that approximatesthe oscillon solution. So long as certain conditions aresatisfied [11], the system relaxes into oscillons as it evolvesin time. Here, we show that oscillons may also emergedynamically, as remnants of vav annihilation. (Previousattempts to find oscillons in vortex-vortex scattering wereunsuccessful [16].) Our finding complements recent resultswhere oscillons were shown to form after a rapid quench inreal scalar field models with symmetric [17] and asymmet-ric [18] potentials. The fact that oscillons may emergedynamically raises their stakes considerably: they canplay a crucial role during spontaneous symmetry breakingin a variety of physical systems and situations, from theearly universe to high energy collisions, and possibly insuperfluids and superconductors.II. THE MODELThe Abelian Higgs Lagrangian density is L DyD14FF 4y 22; (1)where D  @  igA. The scalar and vector massesare, respectively, ms p and mv 2pg. Their ratiodefines the parameter   ms=mv2  =2g2. In the non-relativistic limit,   1 defines the boundary between*gleiser@dartmouth.edu†thorvaldur@dartmouth.eduPHYSICAL REVIEW D 76, 041701(R) (2007)RAPID COMMUNICATIONS1550-7998=2007=76(4)=041701(5) 041701-1 © 2007 The American Physical SocietyType I (< 1) and Type II (> 1) superconductors. Withthe scaling A ! 1A, ! 1 and x! gx, theenergy is E  2RM d2xH , where the Hamiltonian den-sity in the temporal gauge A0  0 is H 12E2 	 B2 	 @ty@t	Diy Di	2y 12: (2)From Eq. (1), one derives the equations of motion D D  2jj2  2;@F  j  2g ImfyDg:(3)Nielsen-Olesen vortices are solutions satisfying r einfr and Ar  n=grr, with boundary condi-tions at spatial infinity [r  x2 	 y21=2 ! 1]fr; r ! 1. n is the winding number. At the vortexcenter, f0  0  0. Vortices have a quantized mag-netic flux B  2	n=g. For a vav pair, the net flux is zero.Quantities are measured in units of   1.III. LATTICE IMPLEMENTATIONWe work on the temporal gauge A0  0 and follow theHamiltonian implementation for lattice gauge theories de-scribed in Refs. [16,19]. The discrete Hamiltonian is H XxH 	x ;x x2; (4)where the density is H  	yx	x 	 ~Dyx  ~Dx 	12~Ex  ~Ex 	12x2XijAix	j Aix  Ajx	i 	 Ajx2 	 Vyxx; (5)and the lattice covariant derivative is D eigxAx x	 xx1. The Hamiltonian density isinvariant under that lattice gauge transformations x ! eixx; 	x ! eix	x;Ax ! Ax 	 x	  x=gx; Ex ! Ex :(6)The equations of motion are then @t	x  @xXxH 	x ;x ;@tx  @	xXxH 	x ;x (7)for each of the fields 	  f	;Eig and   f;Aig. Thetemporal evolution follows a second-order leapfrogscheme with t < x=2p. Provided that the initial condi-tions satisfy the lattice Gauss’s law constraint, 1xXiEixi  Eix  i	xyx  	yxx; (8)it will remain satisfied during the evolution. This can beimplemented by, e.g., setting all temporal derivatives tozero as the simulation starts. Energy is conserved toOt2. We used t  0:02 and x  0:2 and checkedthat the results are consistent for x  f0:1$ 0:3g.Larger values compromise the accuracy of the results.Since we use periodic boundary conditions, to preventenergy from the initial condition to come back to theconfiguration we use a lattice much larger then the regionoccupied by the vav pair. However, as oscillons are verylong-lived, we also use an adiabatic absorbing wall [9]implemented by adding gauge invariant x; y	 termsto the evolution equations, where x; y has support onlyon a band near the lattice boundary and slowly increasesfrom 0! 0:1.IV. VORTEX-ANTIVORTEX ANNIHILATION:RESULTSWe set the initial conditions by using an ansatz thatapproximates a vav pair separated by an initial distance0. This is then allowed to evolve in time under theequations of motion, Eq. (3), at high viscosity [we setx; y  1] until it settles into the vav configuration. Atlarge 0 and for   1 (the case of interest to us) wecomputed numerically the total energy of a vortex, findinga good fit for Ev  2	1=5; (9)which agrees near the region  1 with the result of [20].Also, the contribution in gauge fields to the vortex energy,Eg  E2 	 B2=2, can be fitted as Eg 	6p 1=4: (10)After the initial ansatz settles into a vav pair, we let themapproach until their centers reach the distance i  4m1s ,large enough to avoid any tail overlap [20]. At this point,we set the viscosity to zero and let the pair attract andscatter. We then repeat the experiment for different valuesof .In Fig. 1 we show a sequence of snapshots of an anni-hilation process for   0:08, where an oscillon is formed.We plot both the amplitude jj2 and the magnetic field Bz.One can see the outgoing wave front of radiation (center)as the vortices coalesce into the oscillon and its oscillatingdipole magnetic field (right).In Fig. 2 we plot the associated energy (H ) and Chern-Simons (nCS) densities, where nCS  4	1"AF.Note that oscillons have a distinctive signature (localizedoscillations of ncs, top right) and thus could have aninteresting role in theories with current anomalies of theform @JB / F ~F leading to baryogenesis, including dur-M. GLEISER AND J. THORARINSON PHYSICAL REVIEW D 76, 041701(R) (2007)RAPID COMMUNICATIONS041701-2ing postinflationary reheating [21], a possibility we areexploring.Vortex-antivortex annihilation can lead to two possibleoutcomes, depending on the value of : either the vav pairquickly radiates all its energy to spatial infinity, 	 	 ! 0, or it settles into an intermediate, long-livedoscillatory state before radiating its energy to spatial in-finity, 	 	  ! 	 ! 0. In order to investigate theprocess quantitatively, we measured the energy in a disk(Edisk) surrounding the vav pair. The results are shown inFigs. 3 and 4. In Fig. 3 we show the energies in a disk ofradius r  15 for different values of . In Fig. 4 we showthe total energy and the associated energy in gauge fieldsfor   0:052. The inset shows that there are two charac-teristic frequencies, approximately set by the scalar andgauge field masses,ms andmv. For  0:052, one obtainsexcellent agreement with mv  4:4ms.By inspecting Fig. 3, we identify an oscillon wheneverthe energy develops a plateau. To read off a value of theplateau energy we examine the energy vs lnt plot. Whenoscillons are present, the curve assumes a ‘‘hockey stick’’shape. We read off the oscillon’s energy (Eosc) as the valueof Edisk at the location of the elbow. [Arrow in inset inFIG. 2 (color online). Chern-Simons density (top) and energydensity (bottom) for the same snapshots of Fig. 1.0 200 400 600 800 1000time02468Energyβ=0.22β=0.02βc10 100 100010E=7.85E=3.06β=0.108FIG. 3 (color online). Energy within a disk of radius r  15surrounding the initial vav pair for various values of  on a 1002lattice. The range of  is indicated on the top left. There are tworegimes: for   c ’ 0:136, the pair quickly annihilates, radi-ating its energy to spatial infinity. For <c, the pair shedssome of its initial energy and settles into a long-lived intermedi-ate state, where almost no energy is radiated. This state is theU(1) Abelian Higgs oscillon. Inset: change in energy (E) radi-ated as a function of lnt for   0:108. Note the ‘‘hockey stick’’shape. The arrow pointing right denotes the oscillon’s energy,Eosc.0 1000 2000 3000 4000 5000time01234567Energy 960 965 970 975 980 985 99001234FIG. 4 (color online). Top curve: total energy for   0:052 ina disk of radius r  15 surrounding the initial vav pair. Lowercurve: energy in the gauge fields. Black line is the runningaverage of the gauge energy. The inset shows the total energyand the energy in gauge fields for a small time interval.FIG. 1 (color online). Snapshots of jj2 (top) and Bz (bottom)for a configuration with   0:08. Time proceeds from theinitial vav pair on the left to the oscillon on the right.PHASE TRANSITION IN U(1) CONFIGURATION SPACE: . . . PHYSICAL REVIEW D 76, 041701(R) (2007)RAPID COMMUNICATIONS041701-3Fig. 3.] We have observed that oscillon lifetimes are all inexcess of osc  1045. For example, for   0:098,osc  5:0 105. Note that with the rescaling defined afterEq. (1), time is in units of g1  mv=2p. So, it is set bythe couplings of the model and the symmetry-breakingscale. If this were the standard model, g1 would bethe inverse mass of theW boson and time would be in unitsof 80 GeV  1 8:2 1027 sec. We leave a detailedstudy of their extreme longevity for future work.We identify c  0:136  2 as the critical value sep-arating the two regimes, as it exhibits the critical slowingdown typical of continuous phase transitions. For > 0:2,well outside the critical region, no oscillons form: vav pairsradiate their initial energy to spatial infinity so that by t500, Edisk & 0:4 Eosc. [We neglect here some smallremnant energy Edisk < 0:1 0:2, that persists long be-yond the oscillon’s demise.]The quantity Eosc=Ev naturally lends itself as an effec-tive order parameter describing the transition from vavpairs to oscillons in field configuration space. In Fig. 5we plot Eosc=Ev vs . The curve is well fitted for Eosc=Ev  j cj0:222: (11)Near criticality the exponent 0.2 will have small correc-tions [22]. Given the long integration times needed toextract a more precise value for the energy of the oscillonsaround c (akin to critical slowing down, as can be seen inFig. 3), we did not attempt to find these corrections here.From the theory of continuous phase transitions, the criti-cal point is related to infrared divergences. In fact, theeffective size of the oscillon increases near c: disks oflarger radii are needed to capture the full energy of theconfiguration. The associated decrease in the effectivescalar field mass near the core [ A2  1=2] is the resultof screening by the gauge field. Above criticality, the massat the core becomes positive, eventually matching theasymptotic vacuum as the vav annihilates.Our results show that the annihilation of a vav pair in the2d AHM can be described as a phase transition in configu-ration space, with complete annihilation corresponding tothe ‘‘symmetric’’ phase Eosc=Ev ! 0, and oscillons corre-sponding to long-lived ‘‘broken-symmetric’’ states, withthe associated order parameter attaining a nonzero value.It will be interesting to investigate if oscillons can existin superconductors, how their lifetimes vary with , and ifthey can be created in non-Abelian models via, for ex-ample, low-momentum monopole-antimonopolescattering.ACKNOWLEDGMENTSWe thank Noah Graham for stimulating discussions.[1] A. Vilenkin and E. P. S. Shellard, Cosmic Strings andOther Topological Defects (Cambridge University Press,Cambridge, England, 1994).[2] R. Rajamaran, Solitons and Instantons (North-Holland,Amsterdam, 1987).[3] D. K. Campbell, J. F. Schonfeld, and C. A. Wingate,Physica D (Amsterdam) 9, 1 (1983).[4] H. Segur and M. D. Kruskal, Phys. Rev. Lett. 58, 747(1987).[5] L. D. Landau and E. M. Lifshitz Statistical Physics(Pergamon, New York, 1980), 3rd ed., part 2.[6] T. W. B. Kibble, J. Phys. A 9, 1387 (1976); E. W. Kolb andM. S. Turner, The Early Universe (Addison-Wesley,Reading, MA, 1990);[7] A. Rajantie and E. J. Copeland, Phys. Rev. Lett. 85, 916(2000).[8] I. L. Bogolubsky and V. G. Makhankov, Pis’ma Zh. Eksp.Teor. Fiz. 24, 15 (1976) [JETP Lett. 24, 12 (1976)]; M.Gleiser, Phys. Rev. D 49, 2978 (1994).[9] M. Gleiser and A. Sornborger, Phys. Rev. E 62, 1368(2000); M. Hindmarsh and P. Salmi, Phys. Rev. D 74,105005 (2006).[10] E. J. Copeland, M. Gleiser, and H.-R. Müller, Phys. Rev. D52, 1920 (1995); E. Honda and M. Choptuik, Phys. Rev. D65, 084037 (2002).[11] M. Gleiser, Phys. Lett. B 600, 126 (2004).[12] P. M. Saffin and A. Tranberg, J. High Energy Phys. 01(2007) 030; G. Fodor, P. Forgács, P. Grandclément, and I.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.1800.20.40.60.81O(β)βFIG. 5 (color online). Phase diagram for the vav!oscillon transition. Plotted is the order parameter O Eosc=Ev as a function of . The diamonds are numerical results,while the continuous line is the fit of Eq. (11).M. GLEISER AND J. THORARINSON PHYSICAL REVIEW D 76, 041701(R) (2007)RAPID COMMUNICATIONS041701-4Rácz, Phys. Rev. D 74, 124003 (2006); A. Adib, M.Gleiser, and C. A. S. Almeida, Phys. Rev. D 66, 085011(2002).[13] C. Rebbi and R. Singleton, Phys. Rev. D 54, 1020 (1996);P. Arnold and L. McLerran, Phys. Rev. D 37, 1020 (1988).[14] E. Farhi, N. Graham, V. Khemani, R. Markov, and R.Rosales, Phys. Rev. D 72, 101701 (2005).[15] N. Graham, Phys. Rev. Lett. 98, 101801 (2007).[16] K. J. M. Moriarty, E. Myers, and C. Rebbi, Phys. Lett. B207, 411 (1988).[17] M. Gleiser and R. C. Howell, Phys. Rev. E 68, 065203(R)(2003).[18] M. Gleiser and R. C. Howell, Phys. Rev. Lett. 94, 151601(2005).[19] J. Smit, Introduction to Quantum Fields on a Lattice(Cambridge University Press, Cambridge, England, 2001).[20] L. Jacobs and C. Rebbi, Phys. Rev. B 19, 4486 (1979).[21] See, e.g., J. Garcı́a-Bellido, M. G. Pérez, and A. González-Arroyo, Phys. Rev. D 69, 023504 (2004); A. Tranberg, J.Smit, and M. Hindmarsh, J. High Energy Phys. 01 (2007)034; A. Tranberg and J. Smit, J. High Energy Phys. 08(2006) 012.[22] J. Zinn-Justin, Quantum Field Theory and CriticalPhenomena (Oxford University, New York, 1989).PHASE TRANSITION IN U(1) CONFIGURATION SPACE: . . . PHYSICAL REVIEW D 76, 041701(R) (2007)RAPID COMMUNICATIONS041701-5

Phase Transition in U(1) Configuration Space: Oscillons as Remnants of Vortex-Antivortex Annihilation

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Phase Transition in U(1) Configuration Space: Oscillons as Remnants of Vortex-Antivortex Annihilation

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