JHEP05(2016)074 Springer May 6, 2016 May 12, 2016 : April 25, 2016 : March 10, 2016 : : er ymmetric self-dual Revised Accepted c Published Received 10.1007/JHEP05(2016)074 doi: s, or, equivalently, when the ped by the self-dual vortices an Higgs model are equal. In S topological defects arise at anca, nca, Facultad de Ciencias, Ciencias, [email protected] Published for SISSA by , and J. Mateos Guilarte b [email protected] , W. Garcia Fuertes . 3 a 1602.09084 The Authors. c Spontaneous Symmetry Breaking, Topological States of Matt

In this paper the zero modes of fluctuation of cylindrically s , [email protected] Departamento de Fisica Fundamental, UniversidadPlaza de de Salama la Merced, E-37008 Salamanca, Spain E-mail: Departamento de Matematica Aplicada, UniversidadFacultad de de Ciencias Salam Agrarias yAv. Ambientales, Filiberto Villalobos 119, E-37008Departamento Salamanca, de Spain Fisica, UniversidadCalle de Oviedo, Calvo Facultad Sotelo de s/n, E-33007 Oviedo, Spain b c a Open Access Article funded by SCOAP Abstract: A. Alonso Izquierdo, vortices in Ginzburg-Landau superconductors Dissecting zero modes and bound states on BPS ArXiv ePrint: Keywords: vortices are analyzed andthe described critical in point full between detail. Typemasses II of These the and BP Higgs Type particle Iaddition, and superconductor the novel vector bound boson states inat of the their Higss Abeli core and are vector found bosons and trap investigated. JHEP05(2016)074 7 ], 1 4 6 17 27 27 29 11 14 5 31 10 17 ]. An ]. The 3 D 1 ndri- ] a system 4 ducting s es of fluc- moving around tric quantum me- mmetric vortices ty was refined in [ eminal paper [ ces at the transition point first-order operator uctors in Reference [ properties of these extended lny-Prasad-Sommerfield, [ internal and scattering re discovered by Abrikosov in tical value where the quotient ext of the Abelian Higgs model ofile near the center of the core – 1 – ], where their stringy nature was emphasized. Analytic 2 tuation of bound state type 5.2.1 Numerical procedure for finding5.2.2 BPS vortex excited BPS mod vortex bound state wave functions modes magnetic flux quanta chanics cally symmetric vortices 5.1 Particles on BPS topological vortices:5.2 translational, Portrait of bosonic bound states on BPS cylindrically sy 4.1 Analytical investigation4.2 of the algebraic Weakly kernel deformed of the cylindrically symmetric BPS vortices: 3.1 The first-order fluctuation operator: hidden supersymme systems important step forward in ourstructures knowledge of was the achieved mathematical byof Bogomolny, first-order who PDE identified such inbetween that Type the their II s and solutions Type are Iof superconductivity the the phases, scalar ANO the and vorti cri vector particle masses is one. These Bogomo same magnetic flux tubes reappearedin in the the paper relativistic of cont Nielsen andformulas Olesen are [ available in thisand Reference far for away the from 1-vortex pr the origin, although the behaviour at infini Vortex filaments carrying a single quantum ofthe magnetic realm flux we of the Ginzburg-Landau theory of Type II supercond 1 Introduction 6 Brief summary and outlook 5 Boson-vortex bound states. Internal fluctuation modes on BPS cyli 4 Zero mode fluctuations of BPS cylindrically symmetric vortice 3 Self-dual/BPS vortices and their fluctuations Contents 1 Introduction 2 Topological defects carrying quantized magnetic flux in supercon JHEP05(2016)074 | ]. n | 14 ], see , 8 13 d by E. ]. The self- 10 , 9 etric vortex zero ic operators. This ux is a topological ]. The BPS multivor- 6 the solutions with a low to extending the Atiyah- as the precision attained integer, is the space of ose in References [ ortex zero modes of fluctu- tices with more than three nce [ ions. Proof of the existence ifying analytically the zero this study is twofold: (1) It ly by Fourier analysis. The se, see e.g. [ egenerate vacua. (2) At the ed numerically. The angular nberg seminal paper [ modes of fluctuation around Z l interpolation methods have th or without boundaries, to rm the interpolation between mit of the Georgi-Glashow model r. They are thus free to move k-Burzlaff ansatz, we describe ecial interest. In this case the on in the moduli space of BPS nal series expansion, to obtain ∈ ial decay at infinity, all together linearly independent vortex zero n terms of elementary or special ial equation system which is solv- n | n bit is a two-dimensional sphere. “Self- | er governing vortices or monopoles, come ] two of us improved on the one-loop 17 , with n ty Yang-Mills equations. π e 2 3. In the first half of this paper we perform , ] and [ ]. Given the of rˆole the vortex zero modes in the – 2 – 16 = 2 12 n ]. The freedom in the locations of the centers preludes 7 ], better ansatzes for the analytical structure of the zero 15 plane [ ]–[ 7 2 R linearly independent zero modes of fluctuation, a fact prove | n | ]. In a later development, the self-dual cylindrically symm 11 ] with a shrewd generalization of the index theorem of ellipt vortices have very interesting features: (1) The magnetic fl 8 1 The primary aim of this work is to investigate the self-dual v Prasad and Sommerfield found magnetic monopoles at the BPS li 1 of this type of solitons was given by Jaffe and Taubes in Refere ation. The BPS vortex PDE equations admit multivortex solut unordered points in the Weinberg in [ analysis of the low-energy vortexvortex dynamics solutions, as geodesic see moti e.g.mode [ fluctuations of BPSThis vortices task were was proposed fully fornumber of this achieved magnetic purp in flux the quanta, papers e.g., just mentioned for been developed, either numerically orthe through full some multivortex functio solution,functions, which see is [ never expressiblemode i fluctuations were studiedalso in Chapter detail 3 starting in the with recent E. monograph Wei [ Singer index theorem usuallyopen observed in spaces compact where spaces, problems withdual wi the vortex continuous solutions spectrum with ari vortex cylindrical first-order symmetry equations reduce aroused to sp anable ordinary near different the origin and very far from the vortex core. Severa tex moduli space with magnetic flux equal to the existence of 2 computation, motivated by a physical problem, paved the way quantity related to the firstBPS homotopy limit, group these line of defects theand do circle zero not of modes interact d of with BPS-vortex one fluctuations anothe exist. where the Higgsdual”refers potential to disappears thefrom fact but two that different still dimensional the the reductions first-order of vacuum PDE the or self-duali systems, eith or self-dual, these two regimes by means ofdependence a shooting of procedure the implement zeroregularity mode of the wave wave function function is nearguaranteeing the fixed origin normalizability, analytical impose and exponent the existence of 2 a complete and detailedBPS analysis cylindrically of symmetric the vortices. structurethe Relying of on vortex the the zero zero Rubac modein profiles the knowledge with of themode the radial same profile BPS level near vortices the of themselves. core precision and close After to ident infinity we perfo quanta of magnetic flux. (2) Recently, in [ modes in concordance with theextends index the theorem. work The of interest several of authors on this subject to BPS vor JHEP05(2016)074 h isotope, ]. He 2 3 26 ]. Taking profit bound states in ising when scalar sions. 32 kink. Fluctuations – onductors, interest 4 linearly independent 30 | ], discovered that the in critical vortices be- λφ n lassical vortex field are | 24 ode wave functions such [ hase of the henomenon of fractionary though without complete translational (zero) modes ntext of the Abelian Higgs oodman and Hindmarsh in es the thout disturbing the defect . Quite recently, this prob- uation near a magnetic flux au phenomenological theory rtex in the framework of the vortex-fermion system obeys mprove the results obtained ion regularization procedure. ance in the mathematics and to BPS topological defects we ison with the well known BPS ector particles and the vortex ized are [ antum states in the middle of resting effects of these bound s of the meson bound states on nsions by controlling the inac- oned above the particles trapped ped at the vortex core by looking cts but also to instantons, see Refer- is the vortex magnetic charge. From a n plane. Besides these topological roots under- – 3 – 2 R ], or, in class A chiral superconductors, see [ ] to offer a quite detailed description of such bizarre bound 25 33 ]. This pioneer paper by de Gennes et al. prompted a long searc 27 ] in the mid nineties. Another papers where the of rˆole these 29 ] by diminishing the impact of zero modes in heat kernel expan ]. As a secondary goal, we shall study here the bound states ar ]. 22 – 23 28 Soon after the discovery of vortex filaments in Type II superc However, vortex zero modes exist and are influential not only Our method applies not only to conventional topological defe 18 2 physics of graphene, see e.g. [ the mass gap of thecharge. Dirac spectrum Thus, which, the inconsidered turn, context is induce in completely the different: which p external fermionic there field are zero does no modes not scalarlem fluctuate in and gained as v a in importance c the in Abelian condensed Higgs matter model physics, for inst lying their existence, the vortex-fermion zero modes add qu of the domain wallwhere defects a in meson this travels model together are with of the three kink center types: of 1) mass wi vortex zero modes. Our approach follows the pattern found in the framework of topological defectsof in the Cosmology is supersymmetric emphas quantum mechanicalwere structure able linked in the shortstates. letter We [ shall develop in thisBPS work vortices a and more we complete shall analysi discuss their properties by compar Abelian Higgs model, mutatis mutandisof in superconductivity. the Contrarily Ginzburg-Land to the boundby states the menti BPS vortices aremodel bosons bound rather states than of fermions. mesonsReference by [ In vortices the were co discovered by G aimed at unveiling thesuccess nature from the of analytic thisstates point type on of of the view. bound vortex core states, Nevertheless, have inte al beenand/or disclosed vector bosons in are a trapped superfluid at the p core of a self-dual vo aroused in the investigation of fermionicat bound the states trap one-particle spectrumline of background, the see Bogolyubov-de [ Gennes eq spectrum of the Dirac operator in a vortex background includ fermionic eigenfunctions of zero eigenvalue, where ence [ mathematical point of view thean existence of index zero theorem modes in on the a open space, the curacies induced by zero modesThe in new the method heat requires kernel/zetathat funct precise the information information about gatheredin in the [ this zero paper m is necessary to i shift calculations of kink masses and domain wall surface te see [ tween Type I and II superconductors. Jackiw et alli, see e.g. JHEP05(2016)074 ) x ( φ (2.1) )) x ( µ 2, and use , iA 1 , . − perconduct- and unveils the  µ 2 = 0 ∂ n 1) ), the vector poten- BPS vortices in the x − ( µ, ν ggs model is revisited or vortex fluctuations 2 ) = ( φ pe II superconductivity ∗ = 0, the energy of static x s devoted to describing ). We choose the metric iφ eigenvalue is unsettled a ( φ 0 tic flux supersymmetric quantum x ibes the minimal coupling ( trapped forming a meson- φ spects. where the gauge symmetry ( ation of the defect profile. bing the features of several with A µ 2 tic solutions of finite energy of the continuous spectrum )+ µ he paper. Section 4 offers a 8 r of the bound state in the , κ x D sons through the wall. A entified via diagonalization of ( ge 1) developed. Finally, in section 6 ν ons will be found numerically. ex solutions is discussed in this 1 inates, couplings and fields, the trum of cylindrically symmetric − mate the radial ODE by means symmetric BPS vortex carrying ch is usually referred to as the ∂ er PD operators is explained. In φ − this ODE into a linear system of φ , − µ 1 ) D ) = − x ∗ , x ( Minkowski space-time reads: ) ( ν φ φ 2 , µ A 1 µ D R ∂ ( 1 2 = diag(1 ) = + – 4 – x µν ( µν g µν F F µν F 1 4 )), the covariant derivative x − (  2 x 3 ,A ) d x ( Z 1 ,A ] = ) x ( 0 φ, A [ A S ) = ( x ( µ The paper is organized as follows: in section 2 the Abelian Hi We shall address types 1) and 2) of fluctuation concerning the ing systems A quanta of magnetic flux. In section 5 a similar picture descri comprehensive analysis of thegeneral BPS pattern vortex of zero zero mode modesn fluctuations of of magne a cylindrically density. Also, theHessian, second-order determining differential the small operator, fluctuationssection whi around and the its vort factorization assections the 4 product and of 5 the two first-ord generalBPS structure of vortices the is fluctuation developed spec forming the main contribution of t with the aimin of a fixing detailed our mannerleading notational to the the critical conventions. BPS system regime of between Section first-order Type PDE 3 governing I the i and sta Ty 2 Topological defects carrying quantized magnetic flux in su boson-vortex bound states also with lowwe magnetic draw charge, some is conclusions and speculate about some future pro is broken spontaneously. In termsaction of functional non-dimensional coord for this relativistic system in We start from the actionbetween of a U(1)-gauge the field Abelian and Higgs a model charged scalar that field descr in a phase tial the matrix of the linear system, after which the eigenfuncti with frequencies greater thanonly 0 differs but from lower the thanpriori. search the for threshold The zero shooting modesintermediate procedure region in is for the thus obtaining fact ineffectiveof and that the a we the form discretization shall of approxi facto thedifference radial equations. coordinate, The transforming bound state eigenvalues will be id Abelian Higgs, a much more difficult task. In fact, the search f tensor in Minkowski space in the form and the electromagnetic field tensor The main ingredients are one complex scalar field, the Einstein repeated index convention. In the temporal gau kink bound state thatThe produces existence of an this oscillatingmechanics second of in type the time kink of deform stability fluctuations problem. is 3) due Scattering to of the me profile. 2) kink internal modes of fluctuation where a meson is JHEP05(2016)074 m )] 2 . (2.5) (2.4) )] , x 2 1  circle at x , x 2 ( 2 1 ∗ , 1) , x ) , ( ) φ onfigurations ∗ −  1 ) = 1 ) φ φ inθ D r ∗ 2 e ] where the order ( . 2 )( φ 1 β D ( 2 are asymptotically ∂ Z 2 1 2 ) is purely vorticial. )( 8 n , x 2 -plane to the vacuum r x κ 2 inθ ∈ 1 2 C 6= 1 and a quantum of , ), one checks that the − x ) x , x = 0 (2.3) + ,A 2 n ( , i, j r 1 : ie 1 κ ( 2.3 φ φ x 1  ∞ 1 A − 2 ( β 2 1] (2.2) S , x | ) = − φ 2 2 D 2 1) ) − ∗ r 12 is the winding number of the x inθ 2 hus, all the field configurations ) − ) F − dx e 2 − φ ) n rmula (17) in [ 2 1 , x 2 superconducting material arising 2  φ ∂ 1 , x A ∗ topologically disconnected sectors: D x 1 , x φ ( ( 1 inθ + x ( Z 1 2 φ ( x − 2 1 )) = 1 ( 8 φ 2 κ ie + ∗ φ D dx 2 φ ) − , x 1 φ + 2 1 1 D )[ A φ ) x 2 ( i , x D = 0 and ( 2 = ( 1 ∗ 2 1 ∞ D , x ) x 1 ∞ S , x ∗ ∞ 1 ( φ | – 5 – S 1 ) I ,A ) with finite energy for | ∗ x 1 ) ) x φ ( 2 φ φ i . The fields in each sector ( π 2 D [ i 1 ], φ ∗ n 2 2.2 ( i 2 D 2 ,A φ C , x , 1 2 ( [ 1 2 1 κ 1 i = 2 1 Z x A 1 1 2 2 + − ∈ ( 12 1 n + 2 12 → ∞ ⊔ ) where it is evident that A ij F ( from the circle at infinity in the ,D 2 2 x F ) = ) = ) = F 2 1 2 x 2 2 2 = 1 2.4 ij d ] are the static fields satisfying the second-order PDE syste  6= 0, are endowed with a quantized magnetic flux: is an integer, as representatives of ( S 2 + 2 , x , x , x F = 1 1 1 1 n R and ( n 2 1 4 1 , ∞} x x x dx Z 1 ∞ x φ, A  ( ( ( 1 [ S iθ + 1 2 x | −→ π φ e 1 p 3 V ) 2 ) φ dx 1, are vortices given that the vector field ( A A and < d 2 ∗ r 1 r 2 1 = ] ( ± 2 1 φ ∂ ∂ S D inθ f x x Z r 1 2 2 = 1, the ansatz, see [ e ZZ = ∂ ∂ ) that comply with the asymptotic boundary conditions at the Φ = D φ, A n n = [ →∞ − − ] = ) = r 2.2 + responds to the Cooper pairs density. The search for static c ] = 2 ) ) ∞ 2 2 1 /V | φ ) , x φ φ, A D 1 , x , x [ : lim φ, A 1 = arctan 1 1 [ x E ( x x D ∞ φ, A V θ ( ( ( | φ ( 2 1 φ { A A 2 2 1 2 = ∂ ∂ Choosing, e.g., magnetic flux, have finite energy. In fact, choosing orbit determined by the phase ofin the scalar the field non-trivial at sectors, infinity. T configuration space of the static fields is the union of map infinity, i.e. when parameter requires us to look at the extrema of the functional: Solutions of ( The critical points of where field configurations becomes which, in a non-relativistic context,in is the the Ginzburg-Landau free theory energy of of superconductivity, a see fo Rotationally symmetrical solutions of ( C constrained by the formula ( JHEP05(2016)074 )  1 1 n − 2 he 2 ) ) > φ 2.7 φ φ (2.8) (2.7) 2 ∗ 2 ∗ κ > φ κ φ D ), ( ∗ − − φ [ 2.6 (1 1 (1 ∂ x 2 2 1 2 + d − 2 4 and electromag- profiles can only R 2 φ κ < κ > ∗ κ Z 4 1 φ φ if if + − ]: 12 8 2 4 12 r F κr 2 κ F − − , −  e e x r ]+ r 2 1 ) = 1, and the regularity 1 ) into a second-order ODE = } vortex solutions carrying √ ∓ d · ces piercing the material. n and appropriate fall-off at = 0 φ ly produce repulsive and at- · t: } ∞ 2  whole plane if the fields tend 1 2.2 DE system. The asymptotic ( R ) = 0 (2.6) H φ 2 2 der to investigate these critical dium. Values such that e same type. Thus, an effective H | 2 Z c β ic flux to either repel, if 1 D f c 1 (Type I superconductivity ma- ∗ φ f ) 2 1 D ) − 2 β ∗ − angular Abrikosov lattice, whereas φ − ± ) 1 2 1 iD −  φ κ < (1 2 2 D |    ± ( f D , determined by the φ 2 ( φ 2 − κ ]. Linearization of equations ( 2 + (1 κ 1 3 φ − ≃ iD ) = 1 and 2 →∞ D r + | dr φ dβ D ± ∞ ) 2 2 ∗ r ( 1 + r ) φ ) D ( f 1 2 – 6 – β φ ∗ 2 − 1 ) D  r | − φ 2 ] in the form, see Reference [ β D 1) 1 2 + ( (1 dr d { 2 D − i ( , f  φ, A − { [ φ r − 1) ∗ i − V ] we obtain φ 1) df dr − − ( re φ φ r 1 − 1 2 1 ∗ √ φ φ D · + ∗ ± = 1 between Type I and II superconductors no forces exist ( ∗ (0) = 0, convert the PDE system ( , measures the quotient between the penetration lengths of t φ 1 2 V 2 φ f ( 2 12 c β κ λ 2 e ± F dr 12 − d F 12 = φ [ 1 + . Because  ∗ F 2 2 ) x x 2 κ 2 φ  d 1 dx d are integration constants. The full Nielsen-Olesen vortex ≃ 2 x →∞ 1 2 2 R r D H R d Z ) [( c dx 2 1 Z 2 r 2 (0) = 0 and R ( ∂ 2 1 f ∓ Z = β 2 1 and x ] + 2 V ] = φ d c ∗ ] = ) φ φ, A 2 [ φ, A [ D V between the vortices, which thus, becomevortices very it special. is convenient In to or write in Type I superconductors the magnetic flux aggregates on3 sli Self-dual/BPS vortices andAt their the fluctuations transition point terials). In the first case, the vortices are arranged in a tri (Type II superconductors), or attract each other when quanta of magnetic flux.tractive Vector forces and of scalar Yukawa type between mesons chargedpotential respective objects arises of prompting th vortex solutions of unit magnet be identified by numerical methods. One might also search for to their vacuum values at infinity. The parameter around the scalar and vector fields vacuum values reveals tha system netic couplings as where scalar and electromagnetic fields in the superconducting me where together with the asymptotic conditions, V and at the same timeinfinity. guarantees There regular behavior is atform no the of available origi these analytical solutions, solution however, to is known, this see O [ ( conditions, up to a total derivative term that integrates to zero over the JHEP05(2016)074 , . is = ) n 2 κ φ , x 1 x uantum = ( ). It is clear ~x are no forces ) characterized 2.4 is the quantized the zeroes of the on a finite mass. 3.1 k topological bound: corresponding to a V , , and the vector meson, A )) with complex scalar field n λ v ) vortex internal modes of onditions ( ; . , where Φ √ ~x k ]. Given a positive integer 1) = 0 ( al quantity, there is no doubt theme of this paper, however, n energy per unit length of self-dual iational calculus terminology), 6 2 = hereas type I superconductors ontinuous spectrum. k nfunctions of the Hessian in the − es. We shall concentrate on two ,V Φ H ] = 0 ] = 0 (3.1) is the vacuum value of the scalar field. e PDE system ( φ ) 1 2 1) = 0 ∗ v m n φ φ 0 =1 φ ; 2 2 n k − ( is the total number of flux lines, see ~x A A 2 1 2 2 ( 0 P 1 φ n + + V ± , where = + 2 1 2 12 φ φ 2 1 v n | = ( 2 2 φ n ). Proof of the existence of vorticial solutions ∂ ∂ ( | [ V π 2 1 − – 7 – [ 2.2 ± vortex zero modes, those belonging to the kernel ± , i.e., and the vector potential 1 ] = ± k as: n φ ,F 2 12 n 1 V φ n φ, A F φ [ A 1 ,A Additionally, it may be checked that self-dual vortices = 0 V ) A − -plane. , n 3 φ 2 2 n . ; + 2 | x φ ~x 1 - = n = 1) of one or several quanta of magnetic flux, which thus ( 1 1 | φ 2 iD 1, as explained above. In the QFT context the parameter ∂ x 1 π 12 κ ∂ F ± < i ψ = 2 φ 2 R | ) has been developed in Reference [ 1 R κ )+ 12 π n e D 2 3.1 ; ~x xF ( 2 1 d ]. Behind this particular structure lies the fact that there Φ = e ψ 7 2 R R | ) = 1 2 n = 1 the energy of self-dual vortices saturates the Bogomolny ; , after the Higgs mechanism has taken place giving to the phot 2 ~x , read κ ( 2 parameters, the centers of the magnetic flux tubes located at e v ] = mechanics ψ iφ n Let us denote the scalar field The critical vortices are solutions of the first-order PDE’s = = Recovering the physical dimensions, the magnetic flux and the + 3 φ, A V [ V 1 φ of the second-order fluctuation operatorwhich (the arise Hessian because in of var fluctuation the corresponding freedom to of bound state motiondiscrete normalizable of spectrum eige the and centers. 3) scattering 2 eigenfunctions in the c self-dual vortex solution of vorticity magnetic flux of a vortex (the multiplicity) and V also Reference [ scalar field counted with multiplicity between self-dual vortices ( also solve the second-order PDEof system the ( PDE system ( by 2 that at there exists a moduli space of self-dual vortices solving th special modes of fluctuations: 1) the 2 3.1 The first-order fluctuation operator:Knowing that hidden the energy supersymmetricabout of q the self-dual stability vortices of isis these a the topological topologic analysis solitons. of field The fluctuations main around self-dual vortic vortices would be: move freely throughout the (metals) correspond to m characterize Type II superconductors (typically alloys) w which, written in terms of the real and imaginary parts of the Moreover, the BPS vortices are subjected to the asymptotic c φ the quotient between the masses of the Higgs particle, JHEP05(2016)074        (3.4) k V k ) (3.2) V solutions ~x 2 k ( 2 V L k 1) + ∂ ǫ ϕ und” gauge 1 1 ) together with − ψ ψ − 1 2 2 ) built around the | k ) + 3.1 is the second-order ~x D D ∂ n ψ ( | 2 2 k . ; 2 + V ~x ) ) = 0 (3.3) (3 ( 2 H iϕ 1 2 ~x 2 tem of four PDE equa- ( − = 0 = 0= 0 (3.5) = 0 ψ 1 . )) )+ 2 2 2 2 ϕ ~x tex fluctuations demanding elian Higgs model not only ~x t ( ϕ ϕ ϕ ϕ ) gate the spectral condition ∆ +        ( 2 ilated by the first-order PDE ) ) 1 2 ) =  ~x 1 1 2 2 1 s, and − ( ψ ψ ) n , a ϕ V V 2 V V rmine the dynamics of mesons ; ~x ) k ( ψ + + ~x 2 1 ~x V + − − + 2 ( ( 1 1 k ψ ψ 2 2 1 ) = − 1 2 ) are still solutions of the first-order 1 ϕ ϕ ϕ V ∂ ∂ ∂ ∂ a ~x ) 2 1 ) ( ( ~x 3.2 − − − − ~x ψ ψ k ǫ ( ϕ ( 2 V 1 1 − + 1) + ) BPS vortex fluctuation fields belong to 2 k 2 2 ϕ + ( + ( ) = 0, or, in components, are V ϕ , φ 2 2 ~x V ∂ 2 ) ψ ψ ~x 1 1 ( − ) )) + 1 a a ) ( 1 2 + ~x ψ 0 ϕ ϕ 2 1 ~x + 0 + 2 n ( ~x ψ ξ 2 | ) ) ( D D ξ ∂ ∂ )) and ; ( − 1 1 k 2 1 1 ∂ 2 2 2 ψ ~x ~x ψ ∂ ∂ D | V V – 8 – ( ( a − + − − − k ( 2 2 ǫ ϕ 1 1 (3 ) V + + 2 2 1 2 a a − 1 2 ~x 1 , a ,V ∂ 1 2 ψ 1 2 ( ) ) ) ∂ ψ ∂ ∂ 1 ) + ∂ ∂ − − ~x n ~x a ( n − ( ; − − 2 1 1 ; k ∆ + + (  ~x 1 2 ∂ ∂ a ~x a ( 2 ψ ψ − ( + ( 1 k is a label in either the discrete or the continuous spec- − − 1 a ∂ V 2 2 1 ) = | ψ 2 a λ        ψ 1 ~x 2 ψ ψ ( | ψ 2 ψ ) = = ξ + 2 0 ) = V D 1 − )) = ( D D n 2 a n 1 2 ; ∆ + 2 − ; a ~x ψ − 1 ( ~x , ϕ, φ ( 1 ), where ψ 2 k 2 | φ ~x 2 a operator is obtained by deforming the PDE system ( ( 1 ( ψ ψ | λ ,A ψ 1 ξ ) B 1 ) is satisfied if and only if the four-vector field D 0 n D 2 λ ). To discard pure gauge fluctuations, we select the “backgro D ; 2 ω 3.3 2 ∆ + ~x − ( 3.1 − 1 )–( A        ) = ( 3.1 ~x = ( λ + ξ + H vortex small fluctuation operator trum useful to enumerate the eigenfunctions and eigenvalue positive energy becausein these positive the perturbations differentH dete topological sectors. Thus, we shall investi Important information about thecomes particle from spectrum the vortex in zero the mode fluctuations Ab but also from vor of the PDE system: are zero modes of fluctuation if the perturbed fields ( BPS vortex fields as the gauge fixingtions condition ( on the fluctuation modes. The sys the background gauge. Thus, the zero mode equations ( the kernel of the first-order PDE operator, The self-dual vortex fluctuations ( which assembles all the BPSoperator: vortex field fluctuations is annih Note that this JHEP05(2016)074 are . (3.7) (3.6) lacks        ystem k − V ]. ]. In the k 8 H V 14 , k V ∞ n , k ∂ + belonging to the = 2 ] and [ + 1) + − 4 < -spectrum consists 2 k R i  | 13 + ∂ 2 2 ψ k ⊕ | H )) , i.e., the dimension of ) V ( M ~x 2 n 2 1 2 ( 2 R 2 + − ( , ϕ M † ux ranging in a dense set. One 2 ]. Analysis of the zero mode nd gauge) around the BPS L ∆ + . ν † ! Explicitly, we find 12 e index theorem dictates: + ( DD † − ∈ ! 2 4 ) D is one of the two SUSY partner k ies, which are explained in [ − DD − )) linearly independent normalizable ~x V 0 0 0 0 ( ~x 2 + k H n j = ) with ( V ξ 1

H ~x M + − ( ϕ k 0 2 ν = H + H V ξ . By inspection one sees that † k M + (

D ∂ n 2 † + 1) + is bounded: = + )) 0 0 is D ) belong in general to a rigged Hilbert space, 2 k | ~x k ~x ) Q  ( ( + ∂ ψ – 9 – † 2 2 | ~x ξ k ] within the framework of index theory in open ( ( L a H Q 8 V ξ 2 1 , ,Q k Tr + D + ( ! † † 2 0 ∆ + D →∞ )) 2-matrices, both the supercharges and the SUSY Hamiltonian 0 0 − QQ D ~x M -vortex solutions implies that the = ( × 2 1 n

| = = lim + a 5 ψ ( zero modes, see appendix B in [ | = H H h D as n x 0 0 0 ] and was further developed in References [ Q 2 8 , henceforth of D d ∆ + D 2 are superpartners in a supersymmetric quantum mechanical s − has 2 R Z ± 2 + | H = ψ H | ], E. Weinberg proved that there are 2 = dim Ker 2 0 0 2 0 0 0 ]. The fluctuation vectors k 10 D ) ~x 20 ∆ + ]–[ ( 9 ξ ind − k        = The stability of the BPS − The continuous part of the spectrum might cause some difficult Although written in the text as 2 8-matrices of partial differential operators. 4 5 H × operators obtained from which are isospectral in the positive part of the spectrum. observes that the second-order differential operator together with continuous spectrum eigenfunctions which is governed by the SUSY Hamiltonian: which means that of non-negative eigenvalues. Indeed in [ discrete spectrum, for which the norm eigenfunctions begun in [ spaces, see [ such that there exist square integrable eigenfunctions The Hamiltonians 8 vortices, see [ coming from linearizing the field equations (in the backgrou zero modes (all the potential wells are non-negative) and th BPS vortex zero modes in the topological sector of magnetic fl the algebraic kernel of built from the “supercharges” JHEP05(2016)074 ) are 4.2 The (4.1) (4.2) (4.6) 2 e 6 -vortices. and , ices n n = 1, i.e., the . In fact, it is d κ = 0 (4.5) = 0 = 0 (4.4) = 0 (4.3) , in cylindrical ) to be positive. . n θ θ 2 2 C a a )] ϕ ϕ → ∞ 3.1 ) ) r ) ) . ( r θ 2 n nθ nθ , where nθ nθ ) when f 2 f the vortex zero modes . r cos − 2 1 as ) 2.8 θ e r ) sin( a ) cos( ) sin( ( [1 ) cos( ly symmetric self-dual vor- earch for the eigenfunctions r → r r a ( n = 0 also fixes the behaviour r ( n ), the following ODE system ( ( + r ( enceforth less than twice) the ) e neighborhoods of the origin roach of geodesic dynamics in 2 n ) = lar and/or vector mesons. vortex zero modes came from r mmetric self-dual n n n r θ ilored to fit the asymptotic be- r f ows us to solve the system ( f f n β ( f 3.1 ( n n rest was focused in sectors of very − − + + β sin ) = β r r 1 r ) = r 1 a ( a a ϕ ϕ ) ) n ) ) r, θ and = ( dr nθ dβ nθ 1 and θ nθ ) are the self-dual vortex solutions. 2 nθ n r a = 0, such that the vector field is purely ) in polar coordinates: r ( → n rA n r d ) 3.5 β ) cos( ) sin( A r ) sin( ) cos( r r and ( r r ( ( , ( ( n n n θ ) = – 10 – n n f )] f f r f f ; r ( , in particular the bound states, i.e., the internal ) and ( sin n + − r + − + n and the signs in the first-order system ( f ( θ 2 1 inθ β r θ H n a e n ϕ ϕ a a f ) − ) ) − r 1 r r 1 r r ( θ ( ( ) generalized to the topological sector n + )[1 − n n r r ]. f r r r ( 2.5 cos 15 n n β n β ∂θ r ∂r ∂a f ∂a ) = ) demand that a r 1 r − − n ~x ( − = 2 2 2.3 ) into the first-order PDE system ( φ − θ 1 ∂r ∂θ θ a ) = ∂ϕ ∂ϕ ∂θ 4.1 r ∂a ∂r r 1 ( ∂a − r 1 n + 1 dr − df 1 ∂θ ∂ϕ ∂r r ∂ϕ 1 − − Investigation of zero mode fluctuations around cylindrical The first goal in this paper is to seek a complete description o Without loss of generality, we restrict 6 vorticial. Plugging ( In this ansatz we assume the radial gauge coordinates also in field space, reads: modes of fluctuation where the self-dual vortex captures sca 4 Zero mode fluctuationsThe of Nielsen-Olesen BPS cylindrically ansatz symmetric ( vort by interpolating the fieldand profiles of between infinity. their In shapes this in way th we construct the cylindrically sy emerges: asymptotic conditions ( in the strictly positive spectrum of not fully developed in the previouslow references magnetic because charge. inte The second task that we envisage is the s haviour. A shooting procedure implemented numerically all integration constants. Choice of these constants must be ta tices begins with rewriting the PDE system ( immediate to check that the asymptotic behaviour fits formul The solutions for the radial profiles of the solutions near the origin to be where we recall that the study of vortex scatteringtheir at moduli low space, energies see within e.g. the [ app last two references the motivation to describe in detail the critical value where the mass ofmass the of Higgs the boson vector is boson. equal to The (h requirement of regularity at JHEP05(2016)074 , )  + 4.6 H (4.7) (4.8) )–( 4.3 ] θ ) k der operator tuation planes, − ma we are only -inner product. n 4 , R ) = 0 (4.9) , ) satisfies the second- r ⊕        r ( ) ( ]        ] -fluctuations of the self- ) cos[( 2 n θ nk ] ] 4 r θ h R θ h ( θ 1) R ( 1) ) 2 1) 1) nk ), their orthogonal partners r ) veils the existence of a useful − ⊕ ) L ( g ) ) is a second zero mode of − − 2 the analytical behavior of the − ) k n acting on self-dual rotationally kθ evaluated on self-dual cylindri- t k 2 kθ kθ kθ ) k k ) = 0. − 1 D R r f ) = r − D ( ( − x n, k ~x, ϕ − sin( n 2 is a zero mode of the second-order ( sin( n cos( cos( − n 0 n ) r, θ nk L t ) + ) ) , r ξ ) ( ). ) ) ) r ) h ) ( r r r 2 r θ ( 2 r r ( ~x r ( ( ( ( ( ( a ). This discrete rotation also transforms n ( ′ nk n f n n 0 ′ nk ) sin[( , ϕ ′ nk ′ nk h ′ nk , ϕ f f f ξ h r ) cos[( h r →∞ ) suggests the ansatz h 1 h ) sin[( 4.6 ) cos[( ( r r a − r r − ( − )] ( ( , ϕ − 4.6 nk , is a symmetry of the ODE system ( r θ , nk r h ( nk nk θ h a n )–( , a h a h − – 11 – r zero modes ) with ( a               n β 4.3 ]; 2 1 1 n θ 4.5 ) = ( − − −→ − ) − k k ~x 6= 0 and lim k ( θ ) follows directly from the ) = ( − − k a ~x ~x ⊥ 0 n n − ( ( ξ r r (0) 0 ⊥ 0 n orthogonal zero mode ξ ) and ( -vortex solution of the form ) belongs to the kernel of ξ nk and ) is also a zero mode around the same vortex solution. The or- n ~x n h ~x ( θ ) = ) = 4.4 ( 0 a ξ ⊥ 0 ) sin[( ) + [1 + 2 1, and the zero mode radial form factor ξ ) and r . Then, → r ( ~x ( − + ) by ( r ( x n, k ~x, x n, k ~x, nk 0 a ( ( ′′ nk H ξ g 0 , ⊥ 4.3 0 ξ 1 ξ r h ϕ ) = there exist 2 , . . . , n − ). Thus, if 2 -rotation in the internal space of scalar and vector field fluc , ~x 2 π r, θ ( 1 → − ( , ⊥ r 0 let us assume that 2 ξ a ϕ ) follow immediately. the because the discrete symmetry explained in the previous Lem , = 0 2 k ϕ D The structure of the PDE system ( Regarding vortex zero modes the main result is as follows: ) into x n, k ~x, → ~x ( ( 1 ⊥ 0 0 thogonality between dual cylindrically symmetric ξ Proof: fluctuation operator which is orthogonal and linearly independent of Proposition: Lemma: symmetric vortices. First, we statesymmetry a in helpful BPS Lemma vortex which un fluctuation space. cally symmetric vortices, with the contour conditions ϕ interested in the construction of the Proof: order ODE because it replaces ( ξ In this section weeigenfunctions shall which belong prove to a the proposition kernel of characterizing the operator 4.1 Analytical investigation of the algebraic kernel of the first-or where JHEP05(2016)074 ) ) r ( 4.6 ) = 0 nk r (4.12) (4.15) (4.11) (4.14) (4.13) g ( )–( nk g 4.3  ) r ( . 2 n near the origin, j pansion around , ) are separated, f ) is regular, and r ) = 0 (4.16) ℓ ) 2 + r 2 ) r ( i r +1 integrability of the kθ j , ℓ n n,k r nk 2 j ( 2 ) x n, k ~x, h c r )) + ∞ ( L ( r 0 ( + n,k ) sin( = 0 is a regular singular +2+2 ξ =0 O ∞ j ( n r j n ) = 0 (4.10) ) = 0 X c r , ( 2 atz into the set ( r r < + ) DE’s remain: of e ( ( ) n β s nk i r i 2 t j ) = nk nk ( ) =0 + r r g g r ∞ ℓ − ), and taking into account that 6= 0, i.e., ( ) ( nk ) ) n n ). In this situation there exists a 2 ) g ). Now we shall investigate the ) r ) = , and the P n nk ) 2 nk ( r r − n,k − t + Z j ( ( ( n n 4.14 n,k − c +2 0 ( n n r, θ k k the origin 4.14 c n ( ) ∈ k β − r f 2 2 s ) functions. Continuity in the angular ) + + r f r k r r k ϕ j ( ( + + ) + )(1 + , h 2 2 n nk nk r ) into ( n ) t ( r g (1 + ) and r 2 h − − ( r e nk ) + (1 + )(1 + ( t k k − 4.15 r s n nk ] ( ∼ – 12 – ) f r dr h k ) + + r ) ) and r s nk ( j r r ( at the origin: t (1 + − r Z n ), we obtain ) we end with a single second-order ODE for ( (  n ) ′ ) nk ) n π f r = r g − nk r β ( 1 + ( ( g )+ j = 0 although the second linearly independent solution k 4.10 n 4.11 n − r r − = 2 ( nk ); ( r ) − g to be an integer, r f 2 and 2 n β ′ nk j k kθ ) = g ℓ n,k k ) j ( ) in ( r 2 n e ( c r ) + r )] ( = 0. Plugging ( r 2 r ) + [ ℓ 1 + ( ( nk r =0 r ∞ t ) cos( nk n − ( − j X +2 t r ( nk x n, k ~x, n s ( h ( dr nk d n β r 0 − dg dr nk ξ dt =2 ∞ t h r k j =0 X +2 ∞ ℓ r ) = n =0 + ∞ r 2 j X P ( ) = − n nk r ) has a singularity. The analytic solution admits a series ex r, θ [1 g ( r ∼ 1 − ) ϕ 4.14 ) r ( r is chosen as the minimum value that selects = 0 of the form: ( n point of the second-order differential equation ( Regularity of the function f single analytic solution at r of ( s does not vanish, at we obtain the identity ′′ nk g • 2 r − Solving for the function in the search for the BPS vortex zero modes. Plugging this ans behaviour of this function: of four ODE’s, only two independent but coupled first-order O fluctuations demands that which govern the behaviour of the in terms of the self-dual vortex profiles and plugging this expression into ( part of the solution requires where the radial and angular dependencies of the components JHEP05(2016)074 ) ), r ( .  4.16 nk (4.20) (4.18) (4.19) (4.17) 2 ): ) is h )) ) 1 in ( r r − ( ( r coefficient. 4.16 k compatible 2 4.18 = 0. More- n , given that ) ′ nk − f ) s ) n h +1 r ( k n,k n,k n 2 ( n,k , in ( 1 ( 2 ( c . c c . ) ) = ) + r r ) 1 ( ( )+1 + 1 grand of ( − k i n,k , . . . , n en coefficients is read . nk 2 nk shes: 2 ( k − 2 a pole at the origin, we g h n . , . he series expansion coef- +2). Thus, we shall stick  = 0 for the characteristic + ) n c 2( 2 1 1 k in the equation above are ) r = 0 not being null. Thus, in a n ( − − ( i ) = 1 n,k 1) ) ... ) 1 − 2 ( ) n,k k i = O − 1 ( +2 i − n = 0 up to the n,k + − n,k , c 2 ( s k n,k 0 ( n c i n i ) ↔ − + 2 ( c 2 c ≤ k c 2( ) for the odd terms proportional 1 +2 r (2 ··· k ) 1) k 2 k − s , 2 ) e = − k r k and dr r + 4.16 i ) ) − ( ) ) adopts the form: n n +2 ∞ = 2 2 r n,k k +1 0 n,k 2( ⇒ ( ) 6 ( i e 2 ( n,k − r Z 2 ( c c +1 ), let us recall that c nk 2 k i = n,k π r ) 2 ( h = + also vanish, at least up to ( ) ) c 0 ) ) ) nk i n,k n,k +1 ≥ 0 ( 2 ( h n,k n,k – 13 – i ) = 2 n,k )(1 + 0 ( c 4 ( c ) is null if (1+2 2 ( r 1 + 1) ( c c s c ( 1 or B: i ], − 2 ≈ − = nk 2 1 4.16 ) − + 1) ) ) ) = n − k r r i k − ( ( +1. Annihilation of the term independent of k n,k + in ( − 2 ( − n − rdr ρ 2 nk nk [ c n k r n k h n h ( ∞ 1 2( i 0 = − Z + 1)(2 ) (1 + ,..., s k i 2 π − , ) imply (2 n 2( = 2 A: r = 1 2 ) of the vortex zero modes is 4.19 6= 0, the two characteristic exponents, the two values of i k up to order 2 ) ) + 1 the left member vanishes despite ) k . In terms of the function 4.12 n,k ), are: 0 ( n,k A c j ( = , with c x n, k ~x, i ( 4.17 = 0 and the constants multiplying +1 0 i ) ξ 2 k r n,k 1 ( neighborhood of the origin the function The term proportional to non-null. The two-termfrom recurrence the annihilation relations of between the the terms proportional ev to When show that all the odd coefficients c demand that In order to avoid singularities in the integrand coming from to exponent A. Thus, the second coefficient in the expansion vani ficients unveils the indicial equation from which we extract recurrence relations that determine t Both possibilities are equivalent: simply redefine to choice Near the origin, the behaviour of the first summand of the inte the norm ( The relations ( Because with ( over, the recurrence relations extracted from ( JHEP05(2016)074 (4.21) ) in the functions. . Together 4 4.21 ing around R . ) zero modes. ) r r ⊕ − ) +3 n, k √ being an integer, e 2 k ; 2 2 R ~x r Z ( C ( ( 2 Bessel functions with ) = 0 ⊥ 1 into the differential 0 y symmetric self-dual ∈ O + L r in are prevented if the ξ ( k r ∈ r + → e nk ) √ ) g trictions. We conclude the 1 ] +1 r symmetric vortices induced we can get the exponentially 2 ( k C ) 2 ly decaying tail ( r ); we end with the differential n ns in the linear approximation r 1, as stated in the proposition. 0 +2 β ∞ k + n,k 2 x n, k ~x, 2 ( − + ( ≥ 2 ≃ c 4.14 0 ) n → k ξ r n + 2) ≤ ) 1 and and − k r the following step is to investigate the k ( ) 1, the univaluedness of the BPS vortex k  (2 n : ), equivalent to → ⇒ ) ≤ 2 . − − n,k r ) ) far from the origin. Replacing the asymp- 0 ( ) ) k ) ( r ) c r n r r ( ( ( ( +2 1+ nk n 0 k n ′ nk n,k ≤ f g nk 2 ( K π, n, k f h c g 2 k ≥ 1 – 14 – , ( ) + [(1 + C ) behaves as: − ) of the vortex zero modes according to the above r ) and therefore + 2 k ∞ ( ) into equations ( ≈ − to being 0 r + 1 + 4.7 ′ n ( nk ) + 2 4.18 k k n, k r r, θ r )) ) ( ; 2 ( nk rg → − 0 r r n ~x h ( ( ξ ( 1 − r − 2 n 0 ) zero modes, mutatis mutandis the k − ) ′ nk ξ f ) = k r h ), perpendicular to them in field space, they form a system r ( 1+ − ( arbitrary constants. Therefore, near the origin the second ) = ( I that are normalizable. The orthogonality between the zero n, k n 1 1 ′′ nk ) of ; r ) nk g n, k − C ~x n t +2 ) 2 ; ( k k n,k r 0 ~x 2 ( ∞ 4.18 ) = − ( ξ c − + r n ) arises directly from the angular dependence of these eigen ⊥ ≃ 0 r, θ, n, k ( 2( ξ ) the modified Bessel equation → ( r r 0 nk 4.7 ξ ) g and r 4.14 )( ( ) nk ) and the general form ( n,k g 0 ( n, k c ; 4.9 ~x ( 0 ξ orthogonal zero modes of fluctuation around any rotationall magnetic flux quanta inequality Singularities in the integrand coming from a pole at the orig summand of the integrand in ( holds. Together with the inequality with zero modes totic vortex profiles when appears. The solution isvery a well known linear asymptotic combination properties: of the modified equation ( asymptotic behaviour of the function restricts the possible values of Asymptotic behaviour of the function n • decaying particular solution. We thus keep the exponential By adequately tuning the integration constants Now let us insert equation ( Proposition. Notice that vortex zero mode eigenfunctionsfiniteness that of does the norm not ( impose new res with their partners 4.2 Weakly deformed cylindrically symmetric BPS vortices:We have mov just found newobtained solutions by of the means self-duality of equatio by perturbations adding to of them the the BPS cylindrically modes of 2 vortex of magnetic flux JHEP05(2016)074 k )-  k ) − . Of +2 , − n ) and k k ] n,k ] 1 π − 2 ( n θ θ n c i − ) (4.23) , 1) n, k n 1) e r ; k d ( 1 − ~x − + 2) es ( − − nk k 0 k k dr ξ ) ) (4.22) − dh − (2 ) = 1 kθ kθ n ) ǫ n − . +2 -vortex obtained as k , . . . , n , k − θ n,k n and that other ) − 2 ( 2 1 cos( sin( θ . n n c , ) ) ) ) k r iV k − ) ) tesimal regular ( ) sin[( ) cos[( k r r · r r − r r ( ( elf-dual vortices migrate − ( ( π ] + +1 ( ( = 0 n n ( n n θ k ′ nk ′ nk 1 i f-dual ( ) f f 2( nk nk i pture the essential features e h h V e unveiled by studying the k e e associated magnetic field. h h k = sign( 1 1 rbed along the 1 1 ) − ion induced by the zero mode − − = − − θ r − − ) ) n ikθ k k n ( k k k self-dual rotationally symmetric r e − − V ns: − − n − n n nk ) , j n n ) n d r h ( r π i (  1 ǫ r ǫ r cos[( ( ǫ r ǫ r k n, k, j e and ǫ − n ( ′ nk − +1) ikθ − − k f j θ 2 n h e ) zero modes gives rise to a phase change ) ) − (2 )+ ) + ) + k 1 n i ) with multiplicity iψ r r − e ( ) = k n, k + x n ~x, x n ~x, k n = x n ~x, x n ~x, − ; ( ( i ǫ r 1 − 1 ( ( n 2 1 dr ~x n 1 2 dβ ψ ( + ψ ψ x n, k ~x, ikθ V V  ǫ r quanta of magnetic flux at the origin but moves ( ⊥ r – 15 – 0 n | e n, k, j = iθ ξ e ) k ( and ψ k e − r +2 ) = ) = ⊥ ψ ) ) = ) = k )= ) is a decreasing function near the origin. Therefore, ) n,k θ r 2 ( r +2 inθ ( c ( : k | n,k e n n, k n, k 2 n 2 ( n, k n, k π ) n, k ; ; β nk c ; ; d r ; 2 n ~x ~x h ( ~x ~x 2 n ~x ( ( n ( ( n r + 2) − ( d 2 1 1 2 | i f k + 2) ) 12 ǫ ϕ ǫ ϕ +2 ǫ a ǫ a k (2 ǫf k n,k ǫ 2 ( ) = ) = (2 -vortex leaves c  | ) + ) + )+ ) + ) + n ǫ 2 n n = d n, k n, k ; − ; ; ) x n ~x, x n ~x, ~x x n ~x, x n ~x, + 2) ( ( ~x ~x ( ( ( ( ( )-roots of the unit multiplied by an infinitesimal factor tim 2 1 1 2 inθ k 12 e k e e ψ ψ V V ψ n,k,j V ( (2 F n − r ǫ iθ e n n ) ) = ) = ) = ) = d ) = = k − n, k n, k n, k n, k n, k ) zero modes are: ) = n ) on a self-dual ; ; ; ; ; r n, k, j ~x ~x ~x ~x ~x ( ( ( ( ( ( zeroes located at the origin in the rotationally symmetric s 0 n, k 2 1 1 2 Higgs field zeroes away to be placed on the vertices of an infini n, k r ; e e e e 12 Although the locations of the zeroes of the Higgs field of a sel k V V ; ψ ψ k ~x e x n, k ~x, F ~x ( ( ( − − -vortex are still self-dual solutions of the vortex equatio e ⊥ 0 0 ψ under this perturbation to the positions n course, the perturbation produced by the which tells us that the origin is a zero of The main propertiesbehaviour of of the the Higgs perturbed field self-dual near the solutions origin: ar where we have used the fact that n Therefore, the following infinitesimal deformations of the infinitesimally apart from theξ origin. In sum, the perturbat zeroes appear characterized by the conditions in the new positions shifted by or, in complex variables in field space, polygon, the ( n ξ of these stringyThe objects, magnetic fields it of is a rotationally also symmetric interesting vortex pertu to look at th a zero mode fluctuation of a rotationally symmetric vortex ca JHEP05(2016)074 ) ) 1 of c r ( e dual V n, k nk ; (4.24) (4.25) ) that ) = h ~x 0 infinity. ( , r 0 4.7 ) ( ξ r ) for every ( nk h nk 1 radial form n, k dr ) with the aim ; − dh ~x instead of taking 1 ( 0 ese functions were , i.e., − ǫ 4.22 ξ 0 k r . Vector plots of the − n k ) describes the neutral , . . . , n th columns of figure 1, r = ] 1 ate region by interpolat- ) or ( θ r s in such a way that the and vector potential , ) he self-dual vortex profiles ep, we obtain a numerical 4.24 k precise as permitted by the e ortex zero modes . We warn the reader that ψ 3.2 c self-dual vortices are still s. The first row in figure 1 ) are displayed respectively as it should be the case. We lue of ly generated solutions -vortex not merely as the free = 0 ~x − se a finite )–( ( n hics of the value of its moduli. ons. It is clear from ( . s us how the scalar, vector and zero modes n a k c ) range. Starting from the same ) field r y per unit surface area reads 4.22 ( ∞ ) to vary. Using a shooting numer- , 0 cos[( 0 nk r ǫ r ( dr ) and derived analytically. dh ~x + defined in ( ′ nk 1 ( r h
− ϕ ) e V k 4 are collected in figure 1, which arranges r ) is a homogeneous linear differential equa- ] range. The numerical results confirm the ( − , 0 2 n 3 n 4.9 , r ) in the [ f , r and 2 · r ) to obtain the exponentially decaying solution, – 16 – ( , ] 0 − e 1 ψ θ r 1 nk , ) ( h k ′ nk 1 2 h − = 0 ≃ n k  2 | sin[( ) ǫ ) also in the [0 r zero modes of fluctuations of a rotationally symmetric self- ( − n, k ; ) n nk ~x r h ( ( e n ψ ) as a normalizable function, otherwise the solution goes to dr dβ r − | ( r n 1 nk  ). Taking into account that ( h 2 1 r = 0 and very far from the origin. ) = ). In the previous subsection, analytical expressions of th ( r n r ( β ) = n, k nk ; h ~x ( n, k ; ⊥ 12 ) and It remains to describe the vortex zero modes in the intermedi We shall now offer graphic representations of the self-dual v ~x e r F ( ( -vortex is limited to the numerical computation of the V n ing between these two regimes. Thus, we search for numerical factors found near using the same proceduref as that previously employed to find t and allow the value of its derivative at this point value of the angular momentum n in order to reach an appreciable visualization we shall choo of gaining some insight into the meaning of the new solutions tion, we fix the value of the function at some determined point the construction of the 2 equilibrium fluctuations of any BPS cylindrically symmetri Because the zero modesolutions of fluctuations the of first-order rotationally equations, their symmetri energy densit some infinitesimal value. Graphics describing the 5-vortex which is exactly the magneticemphasize field that of the the information deformed contained solution, in formulas ( limited analytical knowledge of the BPS multivortex soluti motion of the Higgs zeroes,magnetic the fields vortex are centers, deformed but in also these tell processes in a manner as ical technique, we tune the value of theoretical behavior for small and large values of profile of the function integration constants and the same scheme with a negative st which defines This procedure allows us to construct scalar and vector field zero mode fluctuations the relevant plots in a table format in order of decreasing va and the corresponding perturbed fields however, we insert respectively vector plots for the scalar in the first and third columns of figure 1. In the second and four the perturbed vortex solutions togetherHere with the a darker density the grap single color vortex is centers the emerge less as the shadowed value regions of in the the modulus plot i JHEP05(2016)074 he ) and ~x ( + λ is another ξ t PS cylin- 2 λ ) = 4 and the 1 ω t difference is and scatter- radians. Next, ϕ k − π 3 , 2 ) = 2 ~x ( = 1 fluctuation fields , ϕ -axes, i.e., one of the r + λ 1 k a ξ x + − second row, figure 1, the rturbed vector potentials n of the original 5-vortex , H θ ys at the original location. origin while the other three a e shall take profit from the 3), and its influence on the regates from the other ones, ero mode fluctuation one of ndrically symmetric self-dual , ng the diagonal lines of every tagon. ed scalar field that two of the inal configuration are expelled r, separate following opposite e angles are 5 d BPS vortices in the Abelian n modes in the pure point spec- he discrete symmetry arising in stead of copying with first-order ~x, ) = ( ( ~x 0 ( ) is shown in the last figure plotted ξ ~x ⊥ ( + λ e ξ V = 0 zero mode scalar field fluctuation is = 3, 4) of angular momentum k come in pairs, ) moves along the , k 5 ~x ( + ~x, ψ . Then, H ( 1) of angular momentum – 17 – 2 0 λ , be an eigenfunction of the second-order vortex 2) scalar and vector fields are respectively shown ξ ω 5 0. This symmetry prompts the following Lemma: , t 5 ) ~x, > 2 = 2 is illustrated in the third row, figure 1. Graphical ( ~x, 2 0 λ ( k , ϕ ξ 0 ω 1 ξ , ϕ θ , a r a with eigenvalue ), which are degenerate and orthogonal to each other. The PDE + ~x direction from the remaining ones which stay at the origin. T ( H = 5-vortex scalar field ) = ( ⊥ 2 ~x x + n λ ( ξ + λ 2 λ -operator. With respect to the vortex zero modes an importan ξ ω + H let ) = ~x ( ing modes ⊥ + λ The corresponding perturbed vector potential drically symmetric vortices ξ + fluctuation operator Lemma: system equivalent to the spectralthe analysis condition of exhibits zero also modes t even if H Higgs model. 5.1 Particles on BPS topological vortices: translational,We internal first note that the eigenfunctions of the need of solvingdifferential second-order equations differential equations, systems. in supersymmetric To structure overcome of this the difficulty variational w problem aroun We shall investigate now the existencetrum of of the excited fluctuatio are plotted. Four quanta ofquadrant magnetic in flux the are plane, ejectedFinally, in followi while the the last row remaining andvortex second single solution column quanta perturbed in sta figure by 1, the the action cyli of the fourth row, figure 1, the zero mode illustrates the profile of the zero mode plotted. We observe that allin the five the single directions quanta determined in the by orig the vertices of a5 regular pen Boson-vortex bound states. Internal fluctuation modes on B fluctuation for angular momentum in this row.are The not differences so between prominent thezero as perturbed mode for and the fluctuation non-pe companion of scalar angular fields. momentum In the representations of the zero mode in the first andconfiguration third only columns. two constituent Under quantamagnetic this remain flux fixed zero quanta at mode are the fluctuatio ejected in directions whose relativ BPS 5-vortex solution is displayed.vortex quanta, It is initially seen amalgamatedsenses in in the along the perturb the 5-vortex cente behavior of the correspondingthe perturbed zeroes fields. of the Under this z five single quanta forming thewhich BPS remain 5-vortex located configuration at seg the origin of the plane. JHEP05(2016)074 ) ~x ( ǫ a ) + l 5-vortex per- ~x ( V ) 4. ~x 0. ( , a 1 , 2 , 3 , = 4 k ) 3. ~x – 18 – ( ǫ ϕ ) + ~x ( ) for the values ψ , k ; 5 ~x ( 0 ξ ) zero modes of fluctuation around a 5-vortex and the self-dua ) 2. ~x , k ( ; 5 ϕ ~x ( 0 1. ξ

.

0 0 0 0 0 5 ~x, ( ξ 5. ξ 4. 0) , 1) , 5 ~x, ( 2) , 5 ~x, ( ξ 3. 3) , 5 ~x, ( ξ 2. ~x, ( ξ 1. 4) , 5 Figure 1 turbed by the zero mode JHEP05(2016)074 . ). r  a ~x ) is . ( (5.3) (5.1) (5.2) (5.4) ) are hold. + λ 5.1 ξ 2 2 λ ) is also ψ 1 2 1 2 ω 1 ~x −→ − governing ( a a ϕ ϕ x n, k ~x, ,..., 2 2 2 2 ⊥ λ λ λ λ ( θ + 2 −∇ + a n ω ω ω ω ,.... , + λ ) = ξ H 2 1 . χ ~x 2 , , λ = ( = = = = ω 1 itive spectrum of + λ 2 2 2 2 and ξ ψ = 1 = 0 ϕ ϕ ϕ ϕ 2 + θ k ) = 0 1 1 ) =  ) and a ∇ ∂ r H , we shall impose the ψ ψ ~x ( k 1) ( , k , k 1 2 + V if ⊥ nk → ∇ ∇ 2 − H             v and + n 2 λ k ) )  ξ r x n, k ~x, − 1 ω ) ) V 2 ( 2 a + 2 + 2 + 1 ψ kθ kθ k r k + λ auge fluctuations. 1 1 2 kθ kθ , ϕ H V ξ 1 s, i.e., the PDE system ( ϕ ϕ ∇ +  sin( cos( ϕ 2 2 sin( 2 λ θ cos( + 2 θ 1) ) is determined in both cases as a = ψ ψ ) ) ) ) θ ω θ r 1 2 2 2 ( | − kθ kθ kθ kθ ∇ ∇ − ψ ψ k → − ) sin nk | 2 2 2 ) sin ) -vortex also includes excited (positive) r V ) cos v r ) cos ) is tantamount to the PDE system r ( 2 r ( ∇ k (3 n -term of a cosine Fourier series, while r − − ( ~x ( ( ) sin( V ϕ k 1 2 2 ) sin( ) cos( ) cos( ( n nk 1 2 nk , which is orthogonal to, henceforth linearly λ f v nk v nk a a ) of the operator , 2 nθ λ ξ nθ nθ with eigenvalue v nθ + ) )  v 2 r k ~x r k + 2 2 λ 2 2 ω r ( k 2 r k | | ϕ + 2 + ω − + + λ | ψ ψ − | | ξ + ) ) – 19 – H ) ψ ) sin( ) cos( ) sin( r ) cos( r ) r | −∇ → ) ( r ( r r r r ( + + r ) =  ( ( ( ( ( ( (3 ∂r nk ∂r 1 nk ~x 2 2 dr ) are the nk ∂r nk ( ∂r nk 1 2 nk nk nk + nk ϕ ∂v ∂v v λ v v v 1 ∂v ) dv ∂v ) ξ ) ) ) ) + ) −∇ −∇ ) ϕ r r r r r 1 + ( ( ), provided that kθ ( 2 ( ( kθ k ( kθ n kθ n n n zero modes, the spectrum of the operator ∂ H x n, k ~x, − f f f f ( k 5.1 cos( sin( ) −∇ n V + λ sin( cos( r θ θ  ξ ( θ θ 2 ) are clearly mutually orthogonal. All this proves the Lemma + + 2 ~x nk cos cos ( dr 2 2 sin sin v 2 a a − − + λ d ξ 2 1             ). ψ ψ with the same eigenvalue − 2 2 ~x ( + ∇ ∇ ) = ) = ) is an eigenfunction of + λ ) and 2 besides the 2 ξ H ~x ) arises in a sine Fourier series. Moreover, ~x ( there is a class of fluctuations belonging to the strictly pos ( − + 2 ⊥ ⊥ 1 1 + λ x n, k ~x, x n, k ~x, a a + λ ( ( ξ ξ 2 1 of the form: x n, k ~x, the spectral condition ψ ψ A+ λ A+ λ ( 1 1 ξ + χ + λ Fluctuations of type ∇ ∇ χ solution of the 1D Sturm-Liouville problem linearly independent. The radial form factor H 2 2 Searching for positive eigenfunctions − 1. Class A: Proof: the fluctuations of a rotationally symmetric BPS background gauge condition in order to eliminate spuriousProposition: g orthogonal eigen-modes of two generic classes: Moreover, independent of, invariant under the transformation These relations are satisfied by the self-dual (BPS) vortice equivalent to the PDE system ( Therefore, eigenfunction of by definition. One easily checks that the spectral condition JHEP05(2016)074 . ) − 5.2 H emerges } , and 0 . 0 the SUSY + −{ these excited (0) = 0 and H + > 4 R nk , 2 λ dr is a positive real ) = 0 ∈ †        R dv ω p ] r        ] D p ( } θ ] . ] ⊕ θ ) satisfies scattering , θ θ ) nk confined to the open 1) r ) 1) ) built from the solu- ~x ( ection we shall exploit 2 D + 1 h unctions at the origin 1) 1) 2 j ( ) − 2 )] ) R ) ) − nk ω − λ ( p 5.3 − r rs − k v ξ kθ 2 ( { k kθ kθ kθ † k 2 k ) satisfies L − )–( r f − D ( − − sin( n 2 λ 1. sin( n + 1, where cos( cos( − ) 5.2 n n nk ) ω ) r ) ) 2 ) r 2 ) λ ) ( r > r h r ( r p r ( 1. r ( ( ( ω ( ( n 2 λ ′ nk [ n f n n ′ nk ) sin[( ′ nk ′ nk h ) = r − f ω f f are isospectral. If h ) cos[( h r h ~x ) sin[( ) cos[( ) = ( r − n ( r 2 r − − ( − ( ( − λ p ) + nk ≤ ( H ξ r nk h nk † nk 2 ( λ h k h − h ω = 0 or D come in pairs and are related through the ′ nk ≤ and 2 λ h +               H ω )] 1 1 + r − − ( k k H – 20 – n − − ⇒ H n n nβ r r ) ) that belong to either the continuous or the discrete -spectrum if the form factor ~x ( + + 2 ) behaves as an scattering function at infinity. 5.4 − ) = ) = λ H k r ξ , are admissible. The corresponding eigenfluctuations be- ( 2 2 λ } 0 ω nk − 1 and the radial form factor all the eigenvalues h 1) such that the associated eigenfunctions of the form ( 1 x n, k ~x, x n, k ~x, , ( ( − { − ) = − ⊥ there exists a discrete set of eigenvalues (0 + ~x ) B+ λ ( ) = 0. thoroughly described in the previous section. R ξ r − λ ∈ ) + [ B+ ( λ ξ . ∈ r ξ 2 j − + ( ( , . . . , n nk p ω ) are normalizable. In order to belong to v 1 H H ′′ nk , 5.3 rh →∞ there are also eigenfunctions of the form = 0 r k long to the continuous boundary conditions. Thus, the continuous spectrum number, from the threshold value 1. interval and ( Scattering states: lim eigenfunctions must comply with the boundary conditions Bound states: Zero modes: Scattering states: in order to prove the Proposition stated in the previous subs • • • • tions of the radial equationspectrum ( of There are excited modes of fluctuations of the form ( where Note that in thisrestricts class the of angular momentum fluctuation to regularity be of 0 the wave f There are two types, arising for either 2. Class B: supercharges, i.e., structure implies that the eigenfunctions of Apart from the zero modes, the operators the SUSY quantum mechanical structure hidden in the operato Proof: JHEP05(2016)074 − − + H H H with (5.6) (5.7) of 2 λ ω . to find the . ) solves the t ), which are ~x  ) both if the − λ ( t ) r ξ 1 ( 5.6  = 0 ~x a ( ) 4 nk 1 if ~x v a )] ( ) is null. Second, 1 ~x + ~x ( ) (5.9) a ( − H ) 2 kθ ~x A+ λ H ψ ( ξ = 0 (5.8) 1 ) )[ 1 operator to ψ ckground gauge condition. ~x , as shown in the previous ~x a ( ) sin( ] † ( − tions demand that the wave + 1 , the spectral problem of r 2 1 λ ) = 0, which is only solved by ) of the form ( ( ) D a ~x ω ψ H ) (5.5) ) ~x ilities separately: ~x ( ( nk ~x ( ~x 1 t − + ( ( 1 v a 1 1  ) a 3 ) A+ ach class associated with one of the λ a ) r 2 ψ ξ )] | ( ~x 2 λ ) = ) 2 ~x ( n ψ ω ), i.e., ( | 2 ~x ~x f ( ( ψ 1 1 + A+ λ 5.5 ) 0 0 0 a + [ ) a 2 ) = ξ 1 ~x ~x 1 ~x )[ ( ( 2 ∂ ( a 1 1 and apply the ~x 1 2 −∇ − ( a a ∂θ a ) for the radial form factor 2 ∂ 2 − )  ) – 21 – ∂ ψ 2 ~x 2 1 ) or H | 5.4 r ( 1 blocks, prompting a complete decoupling of the − 1 − ψ kθ | ) = ) a -eigenfunction, which shares the eigenvalue × − 2 2 ~x ~x 2 block arising because of the decoupling of the scalar ) + )] 1 ( ) satisfies the background gauge: + ( ∂ ~x 1 ~x ~x − × H 2 ( ) are admissible eigenfunctions of ∂r  ( ( a ∂a ) requires ( A ) cos( λ ⊥ 1 λ r ) = 0. These modes of fluctuation are not allowed by the ξ ~x r 1 1 ∂ 5.6 ( A+ λ ( A+ ω λ −∇ ~x A+ λ ξ ( ( ξ − ⊥ [ − ξ nk -eigenfunctions of the form 2 ⊥ ) satisfies the PDE ( v  1 ∂ − 2 A+ ~x λ ) = a λ ( ξ A+ λ ) and our strategy in the search for positive eigenfunctions 1 2 ~x H 1 − ) and belong to the spectrum of ω ξ ∂r ( ~x ∂ ) = a ~x 1 ( − ) into the ODE ( ~x ( )] − ( λ if A λ − . The reasons are twofold: first, the kernel of 1 ξ ~x ξ ) = † 5.8 A+ ( λ † a + + ~x ξ ( D D H H A+ ⊥ λ λ λ 1 ξ 1 ω [ ω ) solves the PDE: 1 A+ λ there exist ) applied to ~x ξ ∂ ( 1 ) = ) = − 5.7 a ~x ~x ). The fluctuations ( ( + λ 5.5 ξ ), is: A+ λ ~x ξ ) = 0 and therefore ( ~x − ( Class A: 1 A λ ξ The corresponding SUSY partner Lemma. These eigenfunctions,Equation however, ( do not satisfy the ba such that is divided into three classesdiagonal of positive sub-blocks. eigenfunctions, We e shall now investigate these• possib is block-diagonal: it consistsvector of field two fluctuations, 1 and one 2 eigenfunctions of will be to solve the spectral problem of field fluctuations from the vector field perturbations. In sum background gauge. We are left with the fluctuations converts the PDE ( are orthogonal to PDE ( eigenfunctions of It is immediate to check that angular dependence is in cosine or in sine. Univalued fluctua when written in polar coordinates. The separation ansatz a Thus, Thus, fluctuations of the form ( JHEP05(2016)074 ) ) j k = + r r ) ) ( ± ... . i 5.9 +2 1) k ∞ 2 nr k ) , n,k = , , (5.10) v ( j dr r ~x 1 ) for s v (0 [1 ( ) , r ) = 0-term 2 ( ∈ ∈ n,k ) between ) given in =0 A+ ( λ 2 with a po- − j ∞ j λ i λ = 0 ξ n,k 2 v nk ( 2 k ω ω 5.3 k P thus requires v + 5.11 r v 2 λ k s r r ω if ) if ∞ ) = 0 (5.11) , A+ λ 0 ). This procedure 0. The recurrence ξ = Z n,k ). The norm of the lity of +2 ( 0 ) = ) ≥ ) and ( ~x n 5.4 π v ( 2 r k k ( r ) = 0 i n,k 5.2 C ≈ ( , ( 2 r A+ λ nr r SUSY nature: ( v )  O v χ ) r r nk ( k ) = + 1 v ~x , nk  i + ( ) means that these coefficients j v − 2 2 2 1  i r ) in the vicinity of the regular r ) and r k ( 2 λ ) i , ~x 2 λ 2 ω x a in the sine case and 4 ( a sufficiently smooth interpolation 5.11 + 5.4 2 − ω n,k 2 2 ( j − q d 2 λ r k v A+ λ − 2 ω ... 2 λ ξ − R near the origin, one solves for 1 2 ω Z −  , the asymptotic behavior of the function 6= 0 into the ODE ( ) + k q 1 r − ) : = ( Y   ) h 2 k n = 1 2 2 r n,k f k ( ( + 0 = 0. Normalizability of K k C =2 = 0. The characteristic exponents are ∞ v ) 6= 0, is bounded below and tends to a positive ) j 2 X nk imply that ~x k 2 r + v ( k – 22 – ( C ) = k +  − dr j nk r + + A λ ··· r ξ , n, k 1 ) produces the expressions ( ) 2  dv ; k 2 s at the origin: r r , − r 1 n,k = 2 if ( 2 5.6 − λ ) = 1. Substitution of the factorized expressions ( ( j = 2 + 1 coefficient. The recurrence relations ( are grouped by powers. Annihilation of the λ r ω v nk − eff 2 ( k ω ) i v n = 1 because our solutions are such that V

) 2 − C eff r q ) k n,k k 1 ( V ( j ~x 2 i, i ( − v + = nk q − dr 2 v  ) for the two term recurrence relation  s ) →∞ A 2 λ k k = 2 r j ξ d I J † 5.4 j = 0, i.e., one plugs the series expansion 1 1 + − D 6= 0 and r C C s ). We shall now investigate these regimes: λ ( 1 r ) is no more than a radial differential Schr¨odinger equation ω k    ( −

nk 5.4 h v = →∞ −→ r 2 . Therefore, near the origin we find the solution =0 ∞ = 0 starting from the condition j X k ) ) ) near infinity is governed by the differential equation = 1 if r r must be a positive integer number r ~x ( which is regular and presents no problems in the normalizabi ,... ( ( k k 2 nk C , nk v ) eigenfunctions, for example, is simplified by means of thei A+ λ but we shall choose relations between the odd coefficients arising from ( induces the indicial equation where the coefficients v vanish at least up toeven the coefficients 2 singular point trades the ODE ( by applying the Frobenius method to the ODE ( which is a Bessel or modified Bessel equation. Thus, Asymptotic behavior of the function near 1 ... Regularity of the funci´on ξ ~x The ODE ( ( k • • A+ λ which includes a centrifugal barrier when ξ constant at infinity: lim into the fluctuations ofthe the Proposition for form the ( excited modes of fluctuation tential well in the cosine case. number where regularity at the origin, exponential decay atin infinity, and between, to JHEP05(2016)074 ) ) 1) ∞ it is , , 5.12 , i.e., ) = 0. ), has (5.13) (5.12) ± [1 (0 − ~x ~x ( ( ∈ H ∈ t H 2 2 λ λ  t a a ω ω ) )  2 n ) and ( 2 ~x ) | ω ( if ~x if 2 ψ Hilbert space. ( 5.6 | a 2 ) 4 1. A continuous a + ) = ~x ) R = 0 in the asymp- 2 ( ~x > ~x 1 1 ( ( the modified Bessel ⊕  2 ψ 2 2 λ C ) 2 π a −∇ ω ψ 2 ) ) ) 2 1 ~x 2 ) R | ( ( ~x ype of bound state fluctu- + 2 2 tail ( ( ψ coupled blocks in k | a acts only on the scalar field 2 ( L r near the origin is regular as ) me type of eigenfluctuations wn in the last Lemma. This a − ~x + ) 1) enabling their existence. In − ) and consequently its SUSY ( r ), however, , t ~x e form 2 ~x 2 H 1 ( ~x  ( (0 ψ ( 1 ) are both solutions of the same t − ) ~x ⊥ ψ  2 λ − ~x −∇ ∈ −⊥ ( ( ω 1) range, boson-vortex bound states ) 2 ) A λ 2 2 j , A+ λ ). But by comparing ( a ~x ζ p ~x ω ϕ ζ ( ( (0 2 i 2 − ) 0 0 ) 5.7 r a a  ~x ∈ ~x 1 2 λ 2 ( ( exhibiting a non-normalizable oscillatory ∂ 2 ω λ 2 ∂ 1 ) and sin ) becomes the PDE: ( -eigenfunction adopt the form a ) and k ω − ϕ ) − ~x + r 2 ~x Y ( ~x 0 1 ) ( C √ ) belonging to the 5.7 ( 1 H we expect to find SUSY pairs of positive modes ~x −  2 p a – 23 – ( + 0 0 h a A λ 2 5.6  ± 1 ξ  and a 2 π ∂ H 2 ) = ) range, i.e., for energies above the scattering thresh- ) exp k ∂ 2 1  ~x r 2 J ) is accordingly ( ∞ ) = ) = 0 and the background gauge discards this type of C √  + λ r , − 1 ~x ~x k ( ω λ ( ( + A [1 λ 1 2 ω − ζ nk i belong to the Bessel function catalogue. The asymptotic +( a r ∈ B v λ r k ξ 2 λ ) = 1 λ Y ω ~x ) = − ω 2 block-diagonal matrix inside ( 2 λ ~x − − ( . To find the eigenvalues in the point spectrum of both ω × 1 k A λ and p ζ p −⊥ K  † h k A λ J D ζ cos † exp λ , 1 and k r r 1 1 ω -eigenfunctions of the form D C C √ √ k λ − K 1 I      ω , H the lower 2 ) = k ). = 1. Below this threshold, in the I ~x →∞ −→ ( r ) must be a solution of the spectral ODE ( 2 λ ) = ) 5.5 ~x r ~x ω ( ( A+ ( λ 2 ζ ⊥ nk as those described before because a v The genuine Bessel function sum, in the discrete spectrum of totic solutions given before). Theations difficulty is in the identification finding of this the t discrete set of where behaviour of form factor asymptotic behaviour are solutions of the above ODE when of fluctuation of the generic form ( may exist becausefunctions of the exponential asymptotic behavior of spectrum arises in the necessary to identify a particular solution whoseindicated behaviou previously and exhibit a decreasing exponential old + A+ λ ζ H Class B: equation ( such that the background gauge condition ( partners. In particular, the SUSY partner fluctuations. There exist another class of fluctuations of th already been discussed becausesymmetry of connects the discrete the symmetry eigenfunctions sho we conclude that the resultof found encompasses exactly the sa • fluctuation mode. The perpendicular eigenfunction Note that the second choice in our strategy of studying the de The unique regular solution is automatically satisfies the background condition ( where the search for JHEP05(2016)074 . - ) + − 5.7 H H ! ) ) erges. (5.16) (5.15) (5.14) (5.17) re are ~x ~x 2 block . ( ( 1 2 n. ×       ϕ ϕ , ! ) )

k ~x       ( ~x ete eigenfunc- 2 V λ of the spectral ( ) ) 2 k 2 ω ~x ~x 2 λ ϕ V ( ( ) ϕ ω 2 2 ) ~x = ( ~x ϕ ϕ ( ) ) 2 ! 1 ) ) ~x ~x V ), the corresponding ( ( V ) ) k ~x ~x 1 2 ( ( ) − ∂ ~x ~x + 1) + ) 2 2 V V ( ( , paired with its SUSY ~x ) k 1 2 ( ~x 5.14 ) + 2 ϕ ϕ ~x − ( | V 2 ) are admissible as candi- ( ~x ) ) ϕ ϕ 2 ( ) + ) + 1 ~x ϕ 2 ψ ~x ~x H ! | 1 ϕ ( es with the zero modes: ( ( ~x ~x ) ϕ xactly equal to the scattering ( − ) ( ( ) 2 1 ) ϕ ⊥ ~x ) 1 1 ) 1 2 ( ~x ~x ~x ψ ψ ! ( ( ~x 1 ( ~x ϕ ϕ 2 ( ( B+ ) ) 1 1 k λ − ψ + 2 2 ξ ~x ~x V ψ V V ) ) + ( ( V 2 V k ) verify the background gauge ( ~x ~x + 1 (5.18) 2 1 − + ( ( V ) + − V V + ) ) + 1 1 E ~x 2 hold. But these first-order PDE are ) † ( −∇ ~x ~x ϕ ϕ 1 − ∂ ( ( ~x 1 5.16 E ) ) 2 ∂ ) and ( ) ) + 2 1 k it is easy to check that the 2 ϕ − 2 ~x ~x ~x ψ ~x ~x ϕ ϕ k V ( ( ) and consequently in the spectrum of ( ( ( = 1 ϕ 1 ) ) : ) 1 2 k ∂ 2 2 ~x 2 + 1) + ∂ ) ~x ~x ( ~x ψ ψ V + k ∂ − ϕ ϕ 2 ( ( ( 2 B+ λ 2 2 1 1 | V 1 2 − − ξ −∇ H ) means that the eigenvalues ,ϕ ψ ) and ( H ∂ ∂ ) + 1 ψ V V 2 ψ ) + − | ϕ ~x – 24 – − ( = ( ( − ~x − + | ( ) + ) 1 5.18 k 2 1 5.15 and 1 2 1 1 − ~x ~x + 1) + ϕ ∂ are obtained by solving ( ( ( ∂ ∂ ϕ 2 ψ 2 + 1 1 k H 1 − 2 ∂ | ∂ − ϕ ϕ

V 2 ψ − H 1 2 ∇ | 2 H ∂ ∂ ( =       . The scalar fluctuations must be solutions of the 1 2 −∇       E λ + 1 and k ω λ + 1 H V ω 1 2 k ψ V ) = 1 ~x ) = −∇ ( ∇ ~x ( ) in the spectrum of

−⊥ − = -bound states of this class and only a continuous spectrum em k B B λ λ = ) are greater than or equal to the threshold value 1. Thus, the + 1) + − ξ ξ ∂ 2 = 1 is unclear at this point and demands a closer investigatio 5.13 ) † † -eigenfunctions are: 2 k ψ 2 H | 2 1 D D + 2 V acting on the scalar fluctuations ,ϕ ψ ω 5.14 -eigenfunctions. Now we shall demonstrate the lack of discr λ λ | 1 2 1 1 H ∇ ( ω ω − + ϕ ( 2 1 | H H − + ) = ) = H 2 ~x ~x ( ( ⊥ and its adjoint. The factorization ( inside problem ( no positive Nevertheless, the nature of eigenfunctions ofthreshold eigenvalue e in terms of the first order partial differential operator Lack of discrete spectrum of can be factorized as B+ λ −∇ ξ B+ λ • ξ

tions of the form ( obeyed by the self-dual vortices, and hence dates to be provided that SUSY partner We shall also analyze the connection of this type of eigenmod Assuming that eigenfunctions of which are a priori candidates to belong to the spectrum of spectral PDE system: partners in the spectrum of It is easy to check that the fluctuations ( JHEP05(2016)074 . ) } n )- 1 ) r , − r { ( r ( } ( } R 1 +1 n 1 { nk k ) = 0. { nk . r are the u r ) = 0 u nβ ( † r } ( − . A closer 1 L } { nk k + →∞ 1 u r ) = 0 (5.20) ) = 1 { nk H r r ] (5.19) u )] ≃ ( ( θ and r  ) ( B 2 eff nk r L ( V u . nk )) ) = 1+ r } + 1)  r k 1 W . ( 2 λ { nk dr k 2 ) within this class B of ω →∞ u lim r nk + r )] ( r l barrier. Thus, ectrum of ± − n k = 1. The operator he continuous spectrum (1 + W d β dr 2 , where 2 2 H )-eigenfunction belong to λ r − R L ) cos[( ) tends to a constant at in- r − ω )) † ( r r ) n , k ( ( } (1 + r L − 1 1 s living at the threshold of the operator are easily integrated: ( 2 e { nk nk
− n − = ) = [ u ) } + u ) +1 1 r = 1 correspond to the k } R (1 + r { n,n k ( nβ ) with H 2 1 ( ( } r { u r 2 − 1 n ) = = 1 { nk ; R ω ) = ~x u 5.20 (1 + r ( ) nβ 2 λ [ (  2 L r +1, 1) + ω ) n 2 ( − = 0 or infinitely far apart of the vortex 1 k r : r r 2 ( − r nk L nβ r = ) ( 2 nk r W r n ( )] + . These modes are thus akin to the half-bound 2 W n + r n e n , ϕ f ( r 1 )) + – 25 – r ( ] 2 n . If r dr 2 θ 2 1 + f ) one modes close to infinity ( 1 + r 2 r n ) as unknown. This is a radial differen- Schr¨odinger d ( Z H + f dr r } + ) into the spectral ODE: (  1 + 1) [1 +
{ nk d = dr nk k n 1 2 u and † 2 (1 + u r 1 = 1. Indeed the 5.14 exp r , denoting the absence of lower eigenfunctions. This means L 2 1 − n 2 λ − + ∝  respectively near ) = d ω H r ) 2 ) sin[( R 2 + ( B r r eff d ( ( dr B V eff 1 + } nk 1 , nk V { nk dr − ) u 2 1 du u r  ( 0 r 1 − r ≃ → nk − r ) = are: ) ) = r ) W ( r r ~x ( } ( + ( ∞ 2 1 1 B { nk nk eff d ϕ dr = dr u u V 2 } r − 1 d { = − R a relation immediately derived from the ODE ( the kernel of the second-order differential operator which reinforces the previous claimon about the the threshold emergence of value t The asymptotics of core admits a (Infeld-Hull) factorization in the form tial equation with an effective potential well: It is thus clear that the eigenfunctions with The zero at the origin is needed to compensate the centrifuga The separation in the polar coordinates ansatz eigenfunctions belonging to the kernel of first-order differential operators L with the radial form factor converts the PDE system ( The radial form factors of these “one” modes of the lacks physical nodes in fluctuations that only includes thelook zero at modes the in behaviour the of point the sp that there are no bound states in the discrete spectrum of reveals interesting features of these radialcontinuous wave spectrum function of finity and go to zero at the origin as JHEP05(2016)074 , x + + = H x +2 )=0 k r . The (5.21) ( 2 )        ) r r ] r 2 ( ) nk ) ] ( r , ) g n +2 nk f , k   g n,k k θ m θ 2 ( 2 λ r k θ m θ c 1 or 2 at ) around the ω 1 2 ± + r , see e.g. Refer- sin( ) = ) − cos( ) sin[ 3 ) = 0 r ) − ) 1 5.21 ) cos[ ( r r r ) 2 λ ) = 0 √ r ) ( ) ( ( n,k r r ( r r 0 ( r ω ( ( nk ( ( n ( ′ nk c nk nk f ass B eigenfunctions n 2 u n ′ nk h q nk f 0 h nk h f n the spectrum of h D Levinson theorem. 2 h g laborated in section 4, tion ( ) thus read − ) satisfies the ODE )] → − r − r r )] arctan r ust mentioned in a form o modes close to the ori- ( ( 2  λ        ≈ ± 2 n n ω 5.16 nk )) + ) f m − r = h r now we shall unveil the con- k ) is determined by the Bessel r 1D kinks, respectively tanh ( − ( − r n x 2 decays to zero at infinity like 4 ( ), which confirms the existence 1+ 2 λ nk (1 φ r nβ Y − 2 ) = ω h ( nk 2 [ r 2 ) and ( g ~x r ) in the form: ( nk , n − C + g ⊥ n 2 + 5.15 ) + ) ··· + 5.20 B+ λ r  , n k ( r 2 , − 1 ′ nk 1 h k , − )(1 + )] , ξ 2 1. λ n r ( = 0 ω , and these states could be interpreted as a new – 26 – −        n ≥ 1 k ) k ) ] ] q = 0 or two nodes at 2 λ − nβ r ω x − ) except for the term proportional to the frequency k θ k θ ) + [(1 + m θ m θ (1 + r  + 2  ) appearing in ( ( n r sin( − cos( k 4.14 − ( ′ nk ) 2 ) ) k ) r ) sin[ ) ), where we also saw that ) cos[ r r r ( nk r ( ( r r rg r ( ( − 1+ ( = 0 reveals similar behaviour near the origin of the positive ( ( u n n ′ nk ′ r nk ′ nk J 1 f f − r g h h nk nk nk 1 ) − )] h h h 6= 0. The eigenfunctions ( − , at most as r − C r 1 ( r ) ( 1. In this case the radial form factor − n        k ′′ nk ) + [ n,k − g →∞ − 0 ( r m n β −→ 1. Note that the half-bound states tend respectively to r 2 c n ( r k r r ) − +2 ′′ nk r − − x ( n ) becomes ) = 2 2 n r h ) = nk ~x − r ( g ) with ]. This type of modes with ( = 5.20 B+ +3 nk λ 35 m k and have either one node at ξ g )+[1 2 r ( r . Therefore, one sees that the positive eigenmodes of clas B i ( 2 λ ′′ nk eigenfunctions in this class Bgin: to the behaviour of vortex zer O negative powers of type of “fractionary”bound states by some refinement of the 2 BPS vortex zero modesnection versus between class the B zero eigenfunctions: modesdiscussed in studied in this section section.as 4 In close and order the as to Cl possible performwe to express the the the analysis description function j of vortex zero modes e with ±∞ regular singular point ω g states arising in the fluctuations of sine-Gordon and belong to the same class asThe the asymptotic BPS behavior vortex of zero the modes. radial form factor ence [ revealing the oscillatory asymptotic behavior of ODE ( and 3 tanh which is exactly the ODE ( ODE of only continuous spectrum for such that: Application of the Frobenius method to the differential equa • JHEP05(2016)074 ond, (5.22) uctua- reduces j ; vortices ) ) requires + i ( nk ~x H v ( ). The exis- j with positive ⊥ ; 5.4 2 + nk B+ λ . Orthogonality ω ξ H + -term. Regularity 2 λ = H ω j ; ) i ) and ( nk ~x v omentum is low enough. (  er finite-difference scheme s as well as to investigate so the discrete eigenvalues ving in a central potential. 2 = 0 B+ ) λ odes of ry of the radial Schr¨odinger ξ E x crete spectrum of pectrum of or vortex zero modes because tood analysing the properties ) in the ODE ( e rotationally symmetric BPS 2 ) r e equations can also be under- k nd to understand the structure gular dependence is established ~x d to boson-vortex bound states ( aranteed by the conservation of ing different eigenvalue are or- (∆ modes obey to certain combina- ( 2 − nk i 2 2 ′ v B λ r k )+ , ξ ) x ) + ~x ∆ ( r i ( − ( 2 V n A λ f ξ ) by the recurrence relations  D are clearly orthogonal. Orthogonality between ) = + 5.4 = – 27 – − 1) j E ; H − ) i 2 ( nk ~x ) x n, k ~x, ( v x ( + ′ − eff B λ (∆ V i j , ξ ; 2 ) +1) i ( ~x nk ( v + − A λ ξ 1) j ; D 1, as in the zero modes of fluctuation case. This proves the sec − i ( nk − v n + ≤ 2 j ; ) ) k i x ( nk v 1) interval. ≤ , 2 (∆  − tion of bound state type j ; +1) which differs from the vortex zero mode equation only in the at the origin, however, of the positive eigenfluctuations class B, proposal in the Proposition. that 0 i ( nk v in the (0 2 j − tence of boson-vortex boundof states the can effective be potential physically wells unders Accordingly, we expect toOur find bound strategy states to when achievewhich the this simulates angular the finding m differential is equation to ( employ a second-ord which arise in thestood appropriate equation. as Schr¨odinger governing Thes the quantum planar motion of a particle mo 5.2.1 Numerical procedure for finding BPS vortex excited modes of fl and our goal istheir to properties. identify This some taskwe is of need more these to difficult possible than findω the bound numerically search not state f only the eigenfunctions but al eigenfunctions lower than 1.tions Physically, these of the fluctuation scalar andvortex. vector field Thus, fluctuations fluctuations trapped with by these th properties correspon to the numerical computation of the radial form factor The search for and the analysis of some fluctuations in the dis eigenfunctions belonging to the same class with different an In this last section weof shall excited try fluctuations to of elucidate class the A existence a belonging to the discrete s by the Fourier theorydifferential and, equations in guarantees general,thogonal. that Sturm-Liouville eigenfunctions theo hav 5.2 Portrait of bosonic bound states on BPS cylindrically symmetric the scalar product in the SUSY partnership: between eigenfunctions belonging to different classes is gu because class A and B eigenfunctions of Finally it remains to prove the orthogonality of the eigen-m JHEP05(2016)074 . ) 3 = = , tes 2 . In 2 5.4 2 , = 0. ) ) f the n 1 53859 → ∞ 1 , . , k = 0 = 8000 tructed r A 51 A j 41 = 1 ) ; 0 ω ω N = 0 the Lapla- 94252. n ( N nk . 2 points with v enough. We ) 1 , = 0 N A 20 2 max tinguished: (a) If ) ω r 1 labels the discrete , j (arrayed in different A 52 = 2 with eigenvalues = 1 respectively. For 00004. The reason is . ω = 0 with eigenvalues n n 0 938456 and ( k and (2) . at tends to 1 at f vortex zero modes. In k ∼ l eigenvalue problem ( we have chosen generically ) ). We show the eigenvalues = 0 2 j ; x 2 s with the magnetic flux ] for a large ) (1) nk y mentioned meshes have been 5.22 nvalues are near the scattering 2 = 5. In this table we display the = 0 and , v values of (∆ ayed in table 1. Different figures . The index j using the mesh with 2 A n 50 ; k max O n bit quite slow decaying asymptotic ω 6 y -vortex bound state for 2 nk , r − n ω is obtained through diagonalization of 10 = j ; , and choose a mesh of × j = 8000 grid points. The results achieved . ; 2 nk 4 0 . (1) nk /N ω 3 v N max r  ∼ 2 max ) – 28 – 263679 and ( r = 2 ) in table 1. In particular, we have implemented . 2 x 2 = 2 arises with eigenvalue ( = 50 such that k 98835 while the remaining eigenvalue ( 83025. We find three bound states for the case x 77747, which arises for angular momentum . k . N = k . (∆ ) (blue lines), the boson-vortex bound states energies = 1. This last situation is also suitable for the case = 0 (∆ r max x ( r 2 k = 0 = 0 = 0 O 2 and ) n ) + 1 f 2 , x 2 2 n ) ) ) A 50 + 1 2 1 = 1, we find only one boson vector-vortex bound state. The , , , (∆ 2 ω 2 2 A r 10 k n A n A 40 31 1, there is no question about the existence of scattering sta f ω ω ω = 4000 and  ), with ∆ > 0 x ) = + 2 λ r , but we also find one boson- N ∆ ( and the quantized magnetic flux j ω i ; + ) is of order ( = eff = 2, we find two boson vector-vortex bound states ( k j (1) H nk ; 2 ) in a Mathematica environment with a choice of two grids cons V v ) n . The eigenfunction and the eigenvalue depend on the values o N 5.22 nk x 97303 whose Fourier wave numbers are v − = 3 the situation is completely similar to the case . 5.22 j ,N = 15 in the numerical scheme in such a way that the precision of ; (∆ n = 319288 and ( 402708 and ( . . (2) = 0 nk (arrayed in different columns). Two types of behavior are dis ··· v j matrix in the left member of the linear system ( 2 ; max , k ) ) 4 3 r i 2 1 ( nk = 0 = 0 , N , v for the lowest values of − 1 A 21 2 2 × , ) ) ω + 1 1 Table 2 summarizes all the spectral information of the radia There is a logical structure almost so rich as the structure o , , N (1) H = 4, two of them corresponding to the angular momentum A A = 5 with the eigenvalues ( 30 40 = 0 = 0, we have a central potential with a minimum at the origin th 601272 although a fourth state with 701767 corresponds to the value . . ω ω k Above this threshold, in the spectrum of eigenfunctions. The contour conditions are: angular momentum 0 eigenvalue of this state is ( and ( For magnetic flux n general, we observe that the number of bound states increase effective potentials the particular, for magnetic flux A good estimation of the discrete eigenvalues rows) and manifest a satisfactory reliability shownin in the the displayed data results displ correspondingemphasized by to enclosing the them two inthe previousl parentheses. value We remark that i points. Special treatment isthreshold required value 1 for where states we whose choose eige (red lines) and the radial eigenfunction profile for several analyzed in this section for vortex solutions with vorticit respectively with of denote ( where we have confined the problem to the interval [0 0 magnetic flux n that eigenfunctions close to the scatteringbehavior, threshold which exhi demands a greater value of ( cian operator in ( the algorithm ( JHEP05(2016)074 ) k of 5.4 + H ------= 4 and = 3 k = 4000 and n 0 N are plotted for 0 the centrifugal ± rved that when =0 and to 1 when . These results have H + r k > . For the H n 94252(29) 94252(36) b) If . . ------= 2 ing figures arising from either + s within the class A eigenfunc- k = 0 = 0 t there is still a minimum of H 2 2 ) ) mulating the ODE system ( ) radial wave functions plugged ng with functions of r ( A A 52;1 52;1 ω ω nk ( ( v ) and searching for the eigenvectors. In table ) that tends to infinity at 97303(22) 97303(58) 83025(60) 83025(73) 701767(2) 701767(6) 601272(6) 601272(9) ...... k - - – 29 – = 1 5.22 for finding bound states. k = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 2 2 2 2 2 2 2 2 n ) ) ) ) ) ) ) ) is small enough. All together we observe a shallower ) with two different meshes of respectively -vortex small fluctuation operator k A A A A A A A A 51;1 51;1 21;1 21;1 31;1 31;1 41;1 41;1 n ω ω ω ω ω ω ω ω ( ( ( ( ( ( ( ( 5.22 Eigenvalues of the discrete spectrum of ) unveils the details of the bound state eigenfunctions of 5.3 98835(24) 98835(36) 263679(8) 263679(9) 93845(63) 93845(87) 777476(2) 53859(69) 53859(71) 402708(7) 402708(8) 319288(3) 319288(4) 777476(0) ...... = 0 k = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 ) or ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) of the second-order 5.2 A A A A A A A A A A A A A A 40;2 40;2 50;1 50;1 50;2 50;2 10;1 10;1 20;1 20;1 30;1 30;1 40;1 40;1 ω ω ω ω ω ω ω ω ω ω ω ω ω ω ( ( ( ( ( ( ( ( ( ( ( ( ( ( x n, k ~x, ( 0 0 0 0 0 0 0 . Numerical estimation of the discrete spectrum eigenvalue . In the second, third, and fourth columns of table 1 it is obse 0 0 0 0 0 0 0 N N N N N N N A+ N λ N N N N N N N 2 2 2 2 2 2 2 ξ = 8000 points. Parentheses are used to point out the disagree 0 = 5 BPS vortices, a second boson-vortex bound state appear. ( → ∞ 5 4 2 3 1 n N into formulae ( r well (that becomes a wall for a big enough each eigenvalue. The information obtained from these barrier in the effectivethe potential potential gives away rise from the to core a if hard core bu n grows we need a higher magnetic flux with an eigenvalue below the scattering threshold decreasi the thicker or the thinner meshes. 2 The boson-vortex bound state eigenfunctions are found by si 5.2.2 BPS vortex bound state wave functions by the finite-difference equations system2 ( the radial form factors characterizing the positive eigen Table 1 been obtained using the algorithm ( tions JHEP05(2016)074 d is 0) , k ; 5 ) ~x ( + A 1 x n, k ~x, ξ ( = 3 eff k V ) in the differen- = 0 fluctuation at = 5 quanta of mag- k l use this case as a x n, k ~x, ( n eff V ing fluctuations corresponding isplayed in table 1 because ector bosons trapped at the + ates because they come from examine the details: (1) Both A /walls a entical angular momentum . We superimpose the lower positive = 2 k k and n and vector displayed on the potential wells + A + ϕ – 30 – H 263679 are plotted, as well as the perturbed fields . = 1 containing only vector particles. k = 0 − 2 ) -spectrum on these figures. H + A 50;1 . ) for the lower values of H 0) have zeroes at the origin, grow in the middle region and ten ω n , corresponding to the bound state positive fluctuation 5.4 . The BPS 5-vortex does not vary under this ; 5 + ~x A ∞ ( + ǫ a = A + a r = 0 V k and 0) and , + . Graphical representation of the effective potential wells A ; 5

Eigenvalues of the discrete spectrum of ~x

( n 5 = n 3 = n 4 = 2 = n 1 = n In the first row of figure 2 the scalar Properties of the boson vector-vortex bound states support ǫ ϕ + n + A to the lowest eigenvalue ( of a BPS 5-vortex centeredϕ at the origin. It is interesting to ψ tial equation Schr¨odinger ( to zero again at netic flux areparadigmatic shown example in of figure thethe 2 discrete qualitative in bound behavior a state of spectrum table the d format. eigenfunctions carrying Indeed id we shal similar for every vorticity Table 2 eigenvalues in the pure point class A. Although thesevortex wave core, functions we describe shall scalar callthe and them positive v eigenfunctions boson of vector-vortex bound st JHEP05(2016)074 . = 0, 1) is -axis , x 2 94252. x . ; 5 ~x d density ( = 0 + 93845. The t row) while . 2 A ) ϕ 1 , -axis, inducing 1), see figure 2. 2 = 0 , A 52 x 2 5 ω third row of figure ) y along the 601272. The radial 2 but this frozen picture . ure, with frequency , ~x, merges for the vector ( 0 A 50 t + ω -spectrum is the eigen- = 0 A = + a 2 ) t profile function involves a and vector bosons in this ts other bound state in the ociated with them can be H 1 r of the fluctuation eigen- , ymmetry and gets thinner 1, far away from the kink center ar fluctuation ain near infinity, see figure 2 A 51 t of the previously described ≃ y the periodic time-dependent tionary”process, a grow of the ω in fact an oscillatory process of φ d. Several positive normalizable d bound state of a meson trapped by well as their corresponding eigen- -vortex zero modes in the kernel of 2) is the squeezing of the 2-vortex n mpg/General/Mathematicatools ~x, ( ∼ ψ -spectrum takes the value ( ) in the last row of table 2. = 3. Clearly, the kink remains unchanged at of fluctuation has been constructed and their 1. All this happens at + r -axis but outwards along the 2 3 ( 1 + H – 31 – 0 = 0 whose eigenvalue is ( ω x , , ∗ 5 H k ≪ − x ρ x 0, and diminishes until 2 x sinh cosh -axis when it differs appreciably from zero for interme- 3. = 1 whose eigenvalue is ( x < 1 if 1 √ k x ) = = ≃ − x 2) is described in the last row of figure 2. Focusing our 3 ( http://campus.usal.es/ , ω 3 φ 5) and the corresponding fluctuation ϕ ; 5 ~x, ~x ( ( and V 2) we observe that the perturbation pressures the scalar fiel A+ 1 x , ξ ; 5 ) corresponding to this eigenfunction is shown in table 2 (las ~x r ( ( 7 . 1 , t 0, but diminishes when A+ ) = tanh 1 1) vector-vortex bound state is graphically depicted in the , 51 -axis. Of course, there is an oscillation of this static pict x ϕ , -axis both from the left and from the right, but less intensel v ( 1 601272 with a certain dipolar character. A similar process e 50 1 . x > x K ; 5 x ω 0 φ ~x ( √ + = A 1 Animations illustrating the previously mentioned behavio The next excited vector-boson bound state in the discrete In addition to the first mentioned fluctuation mode there exis The highest eigenvalue in the discrete spectrum with angular momentum ξ 1 It is enlightening to compare this bound state with the excite , -kink: 7 4 + = 0-eigenmode, see figure 2 (second row), although the radial A 51 stronger in a strip enclosing the in both senses, seebound figure state 2 the (thirdalong BPS row). the rotationally Capturing symmetric the 5-vortex scalar loses s diate distances from the origin. The net effect on 2 (third row). We observe a new class of fluctuations. The scal along the mode with Fourier wave number modes together with thedownloaded energy at density the web and page magnetic field ass ω eigenfunctions, bound states, of this Hessian operator, as mathematical and physical properties thoroughly describe perturbed fields behave followingk a similar pattern than tha 6 Brief summary and outlookAn ortho-normal basis of BPS cylindrically symmetric This peculiar state new node, see the probability function the matrix second-order PED operator λφ form factor increases at concentrated in a disk inwards along the H a quadrupolar density distribution oscillating in time. the (first row). (2) We seevortex this density instantaneous at picture middle as distances an frominflation/deflation “infla the due origin. to the But temporal it dependenceterm: is induced cos b its center, but grows at the core’s middle, and remain fixed ag self-dual 5-vortex field on the right, but increases until oscillates in time, with frequency attention in JHEP05(2016)074 ]. 2) , ) 34 ~x ; 5 ( ~x ( + + A A 1 ξ ǫ a )+ ~x ( 1) and V , ; 5 ~x ( + s-Higgs model, see [ A t the concepts developed 1 ξ ) to be estimated by the r g some boson-vortex bound ( l structure is available. The ) 4. ns vortex zero mode fluctu- 0), , ~x nk ( ecision. This novel type of BPS h ing will work in more sophisti- ; 5 + ~x by this fluctuation. A ( nd state fluctuations exhibits very a + . A 2 ξ 0), , ; 5 ~x ( ) 3. ~x + ( A 1 – 32 – ξ + A ǫ ϕ )+ ~x ( ψ ) 2. ~x ( + A ϕ 1.

. Vector boson-vortex bound states

1 1 2 1 The techniques and procedures used in this paper suggest tha 2) , 5 ; ~x ( ξ 12. ( ξ 11. 10. 1) , 5 ; ~x 0) , 5 ; ~x ( ξ 0) , 5 ; ~x ( ξ 9.

+ + + + A A A A same method. More dubious, however, is the prospect of findin One expects a similaration, structure demanding of only the an self-dual appropriate Chern-Simo radial form factor Figure 2 cated superconducting media providedfirst that a system BPS of or this self-dua type that comes to mind is the Chern-Simon values, have been identified within ancylindrically unexpected symmetric level positive of vector pr boson-vortexintriguing bou properties and deserves further investigation and the results attained in this basic superconducting sett on a BPS 5-vortex and the self-dual 5-vortex fields perturbed JHEP05(2016)074 ]. , 37 , 36 Sov. ommons Yad. Fiz. , (1973) 45 (1988) 669 ]. (1993) 5961 group B 61 B 296 ]. SPIRE D 48 IN (1976) 449 [ , ][ 24 SPIRE = 2 supersymmetry on IN ]. Nucl. Phys. sigma models, see [ N , Nucl. Phys. , dex theorem in this context. redited. , Phys. Rev. , s about the zero modes of the ro modes and bound states in ed , SPIRE enario to play around with these , Cambridge University Press, IN [ hep-th/9507025 (1957) 1442] [ [ 32 Sov. J. Nucl. Phys. , Boston Birkh¨auser, U.S.A. (1980). , Cambridge University Press, Cambridge ]. ]. , (1975) 760 ] seems to be plausible, even though the high (1996) 650 35 – 33 – SPIRE SPIRE 38 ]. ]. IN IN [ [ 37 An exact classical solution for the ’t Hooft monopole A classical vortex solution of the Abelian Higgs model Zero modes of rotationally symmetric generalized vortices SPIRE SPIRE IN IN [ [ ), which permits any use, distribution and reproduction in Vortex line models for dual strings Zh. Eksp. Teor. Fiz. Topological solitons ]. (1978) 213 (1978) 235 Vortices and monopoles Phys. Rev. Lett. ]. 62 62 Some remarks on the paper of Callias , J. Math. Phys. SPIRE , Asymptotics of Nielsen-Olesen vortices IN CC-BY 4.0 SPIRE (1976) 1100 (1979) 3008 Stability of classical solutions ][ On the magnetic properties of superconductors of the second IN Multivortex solutions of the Ginzburg-Landau equations Classical solutions in quantum field theory (1957) 1174 [ This article is distributed under the terms of the Creative C Vortex string motion in the Abelian Higgs model 5 D 14 D 19 Axial anomalies and index theorems on open spaces ]. ]. (1976) 861] [ SPIRE SPIRE IN IN hep-ph/9310264 U.K. (2004). and vortex scattering 24 Phys. JETP [ Commun. Math. Phys. Phys. Rev. Phys. Rev. Commun. Math. Phys. and the Julia-Zee dyon [ [ Cambridge U.K. (2012). [8] E.J. Weinberg, [1] A.A. Abrikosov, [3] L. Perivolaropoulos, [9] C. Callias, [5] M.K. Prasad and C.M. Sommerfield, [6] A. Jaffe and C.[7] Taubes, N. Manton and P. Sutcliffe, [2] H.B. Nielsen and P. Olesen, [4] E.B. Bogomolny, [14] J. Burzlaff and D.H. Tchrakian, [10] R. Bott and R. Seeley, [11] H.J. de Vega and F.A. Schaposnik, [12] E.J. Weinberg, [13] P.J. Ruback, References derivative terms look harmful. Open Access. any medium, provided the original author(s) and source are c Severely constrained by the possibility of building extend BPS defects in the system discussed in [ these systems, the mainthis difficulty framework in appears investigating to vortex beEven ze the more precise tantalizing, development the of development an of in a similar analysi Attribution License ( ideas is the broad domain of U(1)-gauged massive non-linear state in this necessarily planar model. Another possible sc JHEP05(2016)074 , ] , , Acta , , ]. (2007) 266402 PoS(ISFTG)013 , , SPIRE (2014) 321 98 One loop ggs model IN (2004) 061702 , , cond-mat/9909426 ][ efects 1606 , D 70 , He: effect of bound states in 3 ]. .M. Guilarte, ]. ]. yado and J. Mateos Guilarte, yado and J. Mateos Guilarte, (1999) 601] [ eon, M. de la Torre Mayado, ]. ]. ]. Phys. Rev. Lett. logical solitons , sen vortices 70 SPIRE , M. de la Torre Mayado and M.J. Phys. Rev. SPIRE SPIRE IN arXiv:1307.0740 , [ IN IN AIP Conf. Proc. SPIRE ][ SPIRE SPIRE , ][ IN ][ IN IN Lectures on the mass of topological solitons ][ ][ ][ Tunnel determinants from spectral zeta Kink fluctuation asymptotics and zero modes . Bound fermion states on a vortex line in a type II (2014) 125 – 34 – . Quantum-induced interactions in the moduli space of ]. ]. On the solitons of the Chern-Simons-Higgs model 01 Bound states and instabilities of vortices Mutual friction in superfluid (1991) 9667 SPIRE arXiv:1207.0942 SPIRE hep-th/0504143 [ [ IN arXiv:0707.4592 IN [ [ hep-th/9812102 JHEP hep-ph/9503457 hep-th/9506021 [ [ [ ]. (1964) 307 , ]. ]. 9 ]. B 44 Pisma Zh. Eksp. Teor. Fiz. [ ]. Radiation from excited vortex in the Abelian Higgs model Zero modes of the vortex-fermion system SPIRE Chiral gauge theory for graphene SPIRE SPIRE IN SPIRE IN IN (2012) 2170 SPIRE ][ (1981) 681 (2008) 431 (1991) 511 [ IN (1999) 167 ][ ][ IN (1995) 4621 (1996) 4004 (2005) 125010 [ ][ Phys. Lett. Phys. Rev. (1999) 609 , , Fermion zero modes on vortices in chiral superconductors C 9 C 72 B 22 B 190 B 797 70 D 52 D 54 D 71 Quantum fluctuations around low-dimensional topological d Bound states of the vector field with a vortex in the Abelian Hi ]. SPIRE cond-mat/0701760 IN arXiv:1402.0703 arXiv:0909.2107 hep-th/0406129 Phys. Polon. superconductor the vortex core Phys. Rev. Phys. Rev. [ JETP Lett. [ functions. Instanton effects in[ quantum mechanics Nucl. Phys. corrections to the mass ofNucl. self-dual Phys. semi-local planar topo Senosiain, [ Phys. Rev. J. Mateos Guilarte andhep-th/0611180 J.M. Castaneda, Mu˜noz Quantum corrections to the mass[ of self-dual vortices Quantum oscillations of self-dual Abrikosov-Nielsen-Ole Eur. Phys. J. Eur. Phys. J. degenerate BPS domain walls [27] C. Caroli, P. de Gennes and J. Matricon, [30] H. Arodz, [31] H. Arodz and L. Hadasz, [28] N.B. Kopnin and M.M. Salomaa, [29] M. Goodband and M. Hindmarsh, [25] R. Jackiw and S.Y. Pi, [26] G.E. Volovik, [23] A. Alonso-Izquierdo and J. Mateos Guilarte, [24] R. Jackiw and P. Rossi, [22] J.M. Guilarte, A. Alonso-Izquierdo, W. Garcia Fuertes [20] A. Alonso Izquierdo, W. Garcia Fuertes, M.A. Gonzalez L [21] A.A. Izquierdo, W.G. Fuertes, M. de la Torre Mayado and J [18] A. Alonso Izquierdo, W. Garcia Fuertes, M. de la Torre Ma [19] A. Alonso Izquierdo, W. Garcia Fuertes, M. de la Torre Ma [16] A. Alonso-Izquierdo and J. Mateos-Guilarte, [17] A. Alonso-Izquierdo and J.M. Guilarte, [15] W. Garcia Fuertes and J.M. Guilarte, JHEP05(2016)074 One ] upled to (2004) 163 ]. ]. B 681 ]. SPIRE IN ][ , SPIRE SPIRE IN A note on BPS vortex arXiv:1504.08123 IN [ ][ ][ , Nucl. Phys. , ]. ilarte, s eon and J. Mateos Guilarte, Two species of vortices in massive SPIRE arXiv:1409.8419 IN [ (2015) 125025 ][ hep-ph/0702014 [ arXiv:1509.06632 D 91 [ Peristaltic modes of single vortex in the Abelian ]. – 35 – (2015) 139 02 SPIRE IN (2016) 29 [ Phys. Rev. (2007) 105015 Selfdual Chern-Simons vortices arXiv:1108.5742 , [ JHEP ]. , B 753 D 75 Decomposing instantons in two dimensions Classifying BPS states in supersymmetric gauge theories co SPIRE (1990) 2234 -models IN σ ][ 64 (2012) 175401 Phys. Lett. Phys. Rev. , , A 45 ]. SPIRE IN hep-th/0304125 gauged non-linear [ loop correction to classical masses[ of quantum kink familie J. Phys. Higgs model Phys. Rev. Lett. bound states higher derivative chiral models [37] A. Alonso-Izquierdo, W.G. Fuertes and J.M. Guilarte, [38] M. Nitta and S. Sasaki, [35] A. Alonso Izquierdo, W. Garcia Fuertes, M.A. Gonzalez L [36] M. Nitta and W. Vinci, [33] A. Alonso-Izquierdo, W. Garcia Fuertes and J. Mateos[34] Gu R. Jackiw and E.J. Weinberg, [32] T. Kojo, H. Suganuma and K. Tsumura,