Collective coordinate models for 2-vortex shape mode dynamics A. Alonso Izquierdo ,1 N. S. Manton ,2 J. Mateos Guilarte ,3 and A. Wereszczynski 1,4,5 1Departamento de Matematica Aplicada, Universidad de Salamanca, Salamanca, Spain 2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom 3Departamento de Fisica Fundamental, Universidad de Salamanca, Salamanca, Spain 4Institute of Theoretical Physics, Jagiellonian University, Krakow, Poland 5International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, Hiroshima, Japan (Received 26 July 2024; accepted 12 September 2024; published 3 October 2024) Models are developed for the motion of charge-2 Abelian Higgs vortices through the 2-vortex moduli space M, with the vortices excited by their shape mode oscillations. The models simplify to the well- known geodesic flow on M, modified by a potential, when the mode oscillations are fast relative to the moduli space motion and their amplitudes are small. When the lowest-frequency mode is excited with a large amplitude, the geodesic flow is not a correct description. Instead, a chaotic, or even fractal, multibounce structure in vortex-vortex collisions is predicted. DOI: 10.1103/PhysRevD.110.085006 I. INTRODUCTION The vortices in the Abelian Higgs model generally, and in its critical coupling (BPS) version particularly, are prototypical examples of topological solitons in higher dimensions. Since discovered, their static and dynamical properties have been extensively studied from both the analytical as well as the numerical perspective. In particu- lar, low-energy multivortex scatterings reveal various non- trivial features as, e.g., 90° scattering in head-on collisions of two unit charge BPS vortices. This has found an elegant description in terms of the geodesic motion on a corre- sponding moduli space of energy equivalent BPS vortex solutions. The aim of the present work is to understand how the geodesic motion is affected if one departs from the lowest- order regime and allows for excitation of the internal modes hosted by the vortices. It has been known for some time that a unit-charge, critically coupled Abelian Higgs vortex (a BPS vortex) has a unique shape mode—a discrete, normalizable, radial oscillation mode whose frequency is below that of the continuum of radiation modes [1]. More recently, the shape modes of higher-charge, circularly symmetric (i.e., coinci- dent) vortices have been determined. There is a finite number of these modes, the number increasing somewhat irregularly with the charge [2–4]. More recently still, we studied how these shape modes and their frequencies vary over the 2-vortex moduli space [5]. We confirmed that the circularly symmetric 2-vortex has three shape modes, of which the upper two are degenerate. We also found that the degeneracy is broken as the vortices separate, and at a modest separation, the highest mode disappears into the continuum spectrum. The remaining two modes, as the vortices separate further, are the in-phase and 180° out-of- phase combinations of the radial shape modes of the two individual vortices. The in-phase combination has the lower frequency, but the frequencies merge at asymptotically large separation. Over the 2-vortex moduli space, then, there is a non- degenerate lowest-frequency shape mode and two higher- frequency modes, whose frequencies degenerate at vortex coincidence. Here, we first construct a model for the oscillatory dynamics of the lowest mode coupled to 2-vortex motion through the moduli space and then a model for the dynamics of the two higher-frequency modes. We do not attempt to model a simultaneous excitation of all three modes. We will need to use the curved metric on the 2-vortex moduli space. Samols investigated this metric for vortices moving in a plane and discovered a formula for it just involving field data close to the vortex centers. This could be exploited to calculate the metric numerically [6]. The center of mass decouples and has a flat metric, so the nontrivial factor is the moduli space M for the relative motion of the two vortices, with the center of mass fixed. M is a smooth, two-dimensional manifold with Oð2Þ Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 110, 085006 (2024) 2470-0010=2024=110(8)=085006(19) 085006-1 Published by the American Physical Society https://orcid.org/0000-0002-3882-3702 https://orcid.org/0000-0002-2938-156X https://orcid.org/0000-0001-7928-9086 https://orcid.org/0000-0002-0353-4812 https://ror.org/02f40zc51 https://ror.org/013meh722 https://ror.org/02f40zc51 https://ror.org/03bqmcz70 https://ror.org/03t78wx29 https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevD.110.085006&domain=pdf&date_stamp=2024-10-03 https://doi.org/10.1103/PhysRevD.110.085006 https://doi.org/10.1103/PhysRevD.110.085006 https://doi.org/10.1103/PhysRevD.110.085006 https://doi.org/10.1103/PhysRevD.110.085006 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ rotational symmetry (a surface of revolution), and its curvature is such that it can be embedded in R3 as a rounded cone whose apex corresponds to coincident vortices. The mode frequencies vary over M, respecting this rotational symmetry. The model for the lowest shape mode is simpler than the model for the upper modes. Its ingredients are the metric and the lowest mode’s frequency over M. The resulting three-variable dynamics is not integrable, but provided the mode amplitude is small and the motion through M is slow, we can perform an adiabatic analysis because the mode oscillation is fast compared with the motion through M. The reduced, two-variable dynamics is then integrable, using the conserved angular momentum and conserved energy. We find that the geodesic dynamics through moduli space that occurs in the absence of the mode excitation is modified by an induced potential energy function propor- tional to the mode’s frequency. Such a result is standard in the context of adiabatic dynamics. The second model, for the excited upper pair of modes, is more sophisticated because of the conical structure of the frequencies around the point of vortex coincidence (the apex ofM), but it is only applicable when the two vortices are close together, as the highest mode (the third) dis- appears into the continuum spectrum when the vortices reach a modest separation. We can therefore approximate the moduli space as being flat, or of constant curvature, because its relevant inner region is a small neighborhood of the apex. The generic dynamics is still quite complicated and is best studied numerically. This model simplifies if the relative vortex motion is restricted to be head-on, and it naturally joins up to a model for head-on motion in the outer region of moduli space, where the third mode is absent. We can therefore study a head-on collision where the vortices approach from infinity with the second mode excited. In the adiabatic approxi- mation, this is similar to a head-on collision with the lowest mode excited, but the induced potential is repulsive rather than attractive. Both the metric and mode frequencies over the moduli space M are only known numerically, but it is helpful to work with analytical formulas. We start in Sec. II, therefore, by deriving a good analytical approximation to the metric (and a simplified variant of this) and also present simple formulas for the frequencies of the three modes that fit the numerical results. Section III discusses the model for the dynamics with the lowest mode excited and presents numerical results for a head-on collision. The main change, in comparison with the geodesic flow describing the evolution of unexcited BPS vortices, is the appearance of an attractive mode- induced force. This has a great effect on vortex-vortex collisions. When the amplitude of oscillation is fairly large, the dynamics is complicated and exhibits a chaotic struc- ture of multibounce windows reminiscent of what occurs in kink-antikink dynamics in one space dimension [7–9]. In this regime, the geodesic dynamics fails completely. We also perform the small-amplitude adiabatic analysis and derive a reduced dynamics that is integrable. Here, a head- on vortex collision always leads to scattering through 90° (i.e., one bounce), and there are no multibounces. Section IV discusses the dynamics with the higher- frequency modes excited. Here, the adiabatic dynamics for small-amplitude oscillations is more interesting because of the degeneracy of the second and third modes when the vortices coincide. Large-amplitude oscillations lead to complicated dynamics that we do not study in any detail. However, we do present numerical results for head-on collisions where only the second mode is initially excited, and the excitation transfers to the third mode if the vortices pass through the circularly symmetric configuration at the apex of the moduli space and scatter through 90°. In this case, the excitation of the mode gives rise to a force which changes sign at the apex. An initially repulsive interaction as the vortices approach changes into an attractive one as they separate at 90°. As a consequence, the vortices can stop and return—there is then two bounces (i.e., backward) scattering. No further bounces are observed to occur. In an Appendix, we present some details of how our finite-dimensional models for the modes, coupled to the moduli space dynamics, emerge from the underlying field theory. II. APPROXIMATE METRIC AND MODE FREQUENCIES ON THE 2-VORTEX MODULI SPACE The lowest-order dynamics of two unit-charge BPS vortices is captured by geodesic motion on the moduli space M equipped with a curved metric originally found by Samols [6]. Here, the kinetic degrees of freedom of the vortices are excited, but their internal shape modes are not. In the physical 2-plane, we use Cartesian coordinates ðX1; X2Þ. The Higgs field ϕ of a centered 2-vortex has zeros at an unconstrained pair of locations ðX1; X2Þ ¼ �ðx1; x2Þ, so we can denote a point in M (up to a sign) by Cartesian/ polar coordinates ðx1; x2Þ ¼ ðρ cos θ; ρ sin θÞ and combine these into the complex coordinate w ¼ x1 þ ix2 ¼ ρeiθ. We refer to the real and imaginary axes in the w-plane (and also the X1- and X2-axis in the physical plane) as horizontal and vertical, respectively. w is a natural complex coordinate on M, with its magnitude and argument having the ranges ρ ≥ 0 and θ∈ ½− 1 2 π; 1 2 π�. The vortex centers (the Higgs field zeros) are precisely at w and −w in the physical 2-plane. Because vortices are indistinguishable, a shift of θ by π maps a 2-vortex configuration into itself, which explains the limited range of θ. A simple geodesic on M is where the vortices approach head on along the horizontal axis, instantaneously coalesce at the origin, and then separate along the vertical axis. Here, θ jumps by 1 2 π. A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-2 The exact metric on M has the general, circularly symmetric form ds2 ¼ f2ðρÞðdρ2 þ ρ2dθ2Þ: ð1Þ For small and large ρ, the conformal factor is f2ðρÞ ¼ � 2πγρ2 ρ → 0; 2π ρ → ∞: ð2Þ Here, we have absorbed a factor of 2π into Samols’s original metric; γ is approximately 0.433. For large ρ, the metric is asymptotically flat, with an exponentially small correction [10] that we neglect. Because of the range of θ, the moduli space M is not asymptotically a plane but an intrinsically flat cone, whose half-opening angle is 30° (in the embedding in R3). M is globally a rounded cone—a flat cone whose apex is smoothly rounded off. The factor ρ2 ensures that the metric is smooth at ρ ¼ 0. This is verified by changing to a coordinate z proportional to w2. z, whose argument has range 2π, is a better global coordinate than w on M, as it ignores the sign of w, thereby taking into account the identity of the two vortices. If we write z ¼ xþ iy ¼ reiφ, then x ¼ r cosφ and y ¼ r sinφ are useful Cartesian coordinates on M, even though M is curved. Then, ds2 ¼ Ωðx; yÞðdx2 þ dy2Þ ¼ ΩðrÞðdr2 þ r2dφ2Þ; ð3Þ where the conformal factor Ω is a function only of r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 p . A head-on collision, described earlier using the coor- dinate w, becomes a smooth motion along the x axis from þ∞ to −∞. However, because the geodesics on the asymptotically flat cone are simply straight lines in terms of w, we will often work with w as the coordinate onM but at other times with z. For our purposes, it is convenient to have good approx- imations to the metric on M, enabling us to determine geodesics analytically, and hence vortex scattering in the absence of shape mode oscillations. To test these approx- imations, we compare the dependence of the scattering angle on impact parameter with the dependence obtained for the exact metric, presented graphically by Samols [6]. We consider two approximate metrics and will use these later when discussing small-amplitude oscillations of the vortex shape modes and their effects on vortex dynamics and scattering. These approximations to the Samols geom- etry are novel and could be more broadly useful, e.g., for studying vortex quantum states or approximating the moduli space metric for more than two vortices. A striking result of Samols is that the rounded cone has an area deficit of 2π2 relative to the completed flat cone, extrapolated to its pointed apex [6]. We will reproduce this area deficit exactly with our approximate metrics. A. Spherical cap approximation For our first approximation, we attach a spherical cap to a truncated flat cone of opening angle 30°, maintaining a continuous tangent. The join needs to be at ρ ¼ ρ0 ¼ ffiffiffi 6 p . The total metric is ds2ð1Þ ¼ f2ð1ÞðρÞðdρ2 þ ρ2dθ2Þ; ð4Þ where f2ð1ÞðρÞ ¼ 8>< >: 6912πρ2 ð108þρ4Þ2 ρ ≤ ffiffiffi 6 p ; 2π ρ ≥ ffiffiffi 6 p : ð5Þ f2ð1ÞðρÞ is continuous, and both functions in (5) have zero derivative at ρ ¼ ffiffiffi 6 p . To verify that for ρ ≤ ffiffiffi 6 p it is a sphere metric, we introduce1 z ¼ 1ffiffiffiffiffiffiffiffi 108 p w2 ¼ 1ffiffiffiffiffiffiffiffi 108 p ρ2ei2θ: ð6Þ In terms of z we find that, for ρ ≤ ffiffiffi 6 p , ds2ð1Þ ¼ 6912πρ2 ð108þρ4Þ2 ðdρ 2þρ2dθ2Þ¼ 16πdzdz̄ ð1þ zz̄Þ2 : ð7Þ The last expression represents the metric on a sphere with squared radius 4π, with z the stereographic coordinate. Note that the spherical cap is restricted to ρ ≤ ffiffiffi 6 p , which implies that jzj ≤ 1ffiffi 3 p . Hence, on the cap, the maximal polar angle is μ ¼ 60°. (Use jzj ¼ tan 1 2 μ to verify this.) The boundary circles of the cap and the truncated cone have equal lengths ffiffiffiffiffiffiffiffiffiffi 12π3 p . Also, jzj has the correct range for the cap to join the flat cone with a continuous tangent. Finally, the area deficit of the approximate metric (4) is correct. To see this, we compare the area of the missing part of the flat cone, Acone ¼ π Z ffiffi 6 p 0 2πρdρ ¼ 6π2; ð8Þ where the prefactor is the range of θ, with the area of the spherical cap Acap ¼ π Z ffiffi 6 p 0 6912πρ3 ð108þ ρ4Þ2 dρ ¼ 4π2: ð9Þ Acap is one-quarter of the area of a complete sphere of squared radius 4π. The difference of the areas is 1w ¼ ρeiθ is used consistently throughout this paper; z ¼ kw2 for some constant positive multiple k, but k varies between (sub) sections. COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-3 Acone − Acap ¼ 2π2; ð10Þ as required. In this spherical cap approximation, a geodesic on M is formed from a straight line on the flat cone, joined to a segment of a great circle on the spherical cap, joined to another straight line on the cone. It will be convenient to consider the geodesics in the right-hand half-w-plane that are reflection symmetric with respect to the real axis. Any geodesic can be rotated to such a position. Some geodesics on the cone are sufficiently far from the vertex that they do not intersect the spherical cap. These geodesics are com- plete straight lines in the w plane, parallel to the imaginary axis, describing 2-vortex motion without scattering. For the approximate metric (4), with the conformal factor f2ð1ÞðρÞ, a dynamical geodesic trajectory wðtÞ ¼ ρðtÞeiθðtÞ arises as the solution of the equation of motion for a particle with Lagrangian L ¼ 1 2 f2ð1ÞðρÞðρ̇2 þ ρ2θ̇2Þ: ð11Þ There are two constants of motion, the energy E and angular momentum J. E is the same expression as L, and J ¼ f2ð1ÞðρÞρ2θ̇. Asymptotically, f2ð1ÞðρÞ ¼ 2π, as π is the mass of a 1-vortex. The angular momentum of the incoming motion is J ¼ 2πvina, where vin is the speed of each vortex and a is the impact parameter (half the orthogonal separation of the incoming, parallel paths of the two vortices in the physical plane). The initial energy is E ¼ πv2in, so J2=E ¼ 4πa2. Any geodesic has a point of closest approach of the two vortices, where ρ takes its minimum value ρ̃. Here, ρ̇ ¼ 0, so E ¼ 1 2 f2ð1Þðρ̃Þρ̃2θ̇2 and J ¼ f2ð1Þðρ̃Þρ̃2θ̇. θ̇ cancels in J2=E, and conservation of J2=E implies that f2ð1Þðρ̃Þρ̃2 ¼ 2πa2: ð12Þ This relation between the closest approach and the impact parameter will be useful shortly. For a geodesic that does not intersect the spherical cap, ρ̃ is simply a. Geodesics on the spherical cap are segments of great circles on the complete sphere. In terms of the stereo- graphic coordinate z on the cap, these great circles are a family of circles in the z plane, having the algebraic equation zz̄þ βðzþ z̄Þ − 1 ¼ 0; ð13Þ where the parameter β is real and positive. This family is algebraically the linear join of the equatorial great circle zz̄ − 1 ¼ 0 and the line zþ z̄ ¼ 0, which describes a great circle through the pole of the cap (the rounded cone’s apex). They are all great circles because they pass through the antipodal points i and −i on the sphere. As jzj ≤ 1ffiffi 3 p on the spherical cap, the relevant range of β is 1ffiffi 3 p ≤ β ≤ ∞. For β ¼ 1ffiffi 3 p , the geodesic just touches the cap at z ¼ 1ffiffi 3 p , and for β ¼ ∞, the geodesic passes through the pole and represents a head-on collision of vortices. For the conformal factor on the spherical cap (5), the relation giving the closest approach is 6912πρ̃4 ð108þ ρ̃4Þ2 ¼ 2πa2; ð14Þ so ρ̃2 ¼ 12 ffiffiffi 6 p a 1 − ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − a2 8 r ! : ð15Þ From this, we can determine the relation between β and the impact parameter a. The closest approach of the circle (13) to the origin is where the circle crosses the real axis. This is where z2 þ 2βz − 1 ¼ 0, so z ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi β2 þ 1 p − β. As z ¼ w2ffiffiffiffiffiffi 108 p , and w ¼ ρ̃ at closest approach, we deduce that ffiffiffiffiffiffiffiffiffiffiffiffiffi β2 þ 1 q − β ¼ 2 ffiffiffi 2 p a 1 − ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − a2 8 r ! ; ð16Þ which fortunately simplifies to β ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 8 a2 − 1 r : ð17Þ To understand vortex scattering, we focus on the coordinate w. The great circle segments in the z plane become segments of quartic curves in the w plane, with reflection symmetry in the real axis, although we do not need to know these curves in detail. Because the flat-cone parts of a geodesic are straight (in the coordinate w), scattering only occurs on the curved segment, i.e., on the spherical cap. The scattering angle depends on the change of direction of this segment, between its start and end points. The direction of an infinitesimal part of a segment is the argument of dw, and by reflection symmetry, the change of direction along this segment is directly related to the difference between the arguments of dw and dw̄, that is, to the argument of dw dw̄, evaluated at the end point of the segment. To calculate the argument of dw dw̄, we note that w ¼ ð108Þ1=4 ffiffiffi z p , so dw ¼ ð108Þ1=4dz 2 ffiffiffi z p and dw̄ ¼ ð108Þ1=4dz̄ 2 ffiffiffī z p : ð18Þ Therefore, A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-4 dw dw̄ ¼ ffiffiffī z z r dz dz̄ : ð19Þ Next, taking the differential of Eq. (13) for a great circle segment, we find that dz dz̄ ¼ − β þ z β þ z̄ ; ð20Þ so everywhere along the segment, dw dw̄ ¼ − ffiffiffī z z r β þ z β þ z̄ : ð21Þ The end points in the z plane are where the circle (13) intersects the boundary circle of the spherical cap, jzj ¼ 1ffiffi 3 p . The end point with positive imaginary part is z ¼ 1 3β ð1þ i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3β2 − 1 q Þ: ð22Þ After some manipulation, we find from (21) the end point value arg dw dw̄ ¼ πþ arctan ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3β2−1 q −2arctan � 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3β2−1 q � ; ð23Þ and simple geometry shows that the scattering angle Θ along this geodesic is Θ ¼ π − arg dw dw̄ ¼ 2 arctan � 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3β2 − 1 q � − arctan ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3β2 − 1 q : ð24Þ Expressed in terms of the impact parameter a, using (17), this becomes Θ ¼ 2 arctan ffiffiffiffiffiffiffiffiffiffiffiffiffi 6 a2 − 1 r − arctan 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 6 a2 − 1 r ! : ð25Þ Finally, using the subtraction and double-angle formulas for the tangent function, we conclude that tanΘ ¼ ð6 − a2Þ3=2 að9 − a2Þ : ð26Þ This is valid for a ≤ ffiffiffi 6 p , and for larger impact parameters, there is no scattering. As a → 0, the scattering angle approaches 90°, the expected result for two vortices in a head-on collision [6]. Figure 1 shows the scattering angle as a function of impact parameter, using this spherical cap approximation to the 2-vortex moduli space. B. Flat cap approximation For our second, cruder approximation to the metric on M, we attach a flat cap—a disk—to the top of a truncated flat cone, with the join at ρ ¼ ρ0 ¼ 2. As before, w ¼ ρeiθ with ρ ≥ 0, and θ∈ ½− 1 2 π; 1 2 π�. The total metric is ds2ð2Þ ¼ f2ð2ÞðρÞðdρ2 þ ρ2dθ2Þ; ð27Þ where f2ð2ÞðρÞ ¼ 8>< >: 1 2 πρ2 ρ ≤ 2; 2π ρ ≥ 2: ð28Þ This matches the asymptotic, flat-cone metric for ρ ≥ 2. For ρ ≤ 2, the change of coordinate z ¼ 1 4 w2 ¼ 1 4 ρ2ei2θ con- verts the metric to ds2 ¼ 2πdzdz̄, with jzj ≤ 1, a flat-disk metric. The missing part of the cone has area 4π2, whereas the disk has area 2π2. The area deficit is again 2π2, as required. FIG. 1. The scattering angle Θ and tanΘ as functions of the impact parameter a: the spherical cap approximation (blue), the flat cap approximation (orange), and the asymptotic approximation (green) [10]. COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-5 In this approximation, a geodesic has straight segments on the cone joined to a straight segment on the flat cap. Intrinsically, the tangent to the geodesic is continuous, even though the complete surface has a delta-function curvature at the join. In the z plane, the geodesic segment on the flat cap is zþ z̄ ¼ 2X; ð29Þ where the parameter X is real and non-negative, and in the range 0 ≤ X ≤ 1. This segment, parallel to the imaginary axis, has the reflection symmetry that we imposed earlier. In the w plane, this geodesic segment becomes w2 þ w̄2 ¼ 8X; ð30Þ which is a rectangular hyperbola. Its closest approach to the origin is ρ̃ ¼ 2 ffiffiffiffi X p . To find the relation between X and the impact parameter a, we again use the conservation of J2=E, leading to f2ð2Þðρ̃Þρ̃2 ¼ 2πa2, the analog of (12), which implies that 1 2 πρ̃4 ¼ 2πa2: ð31Þ Therefore, X ¼ 1 2 a: ð32Þ The scattering angle of the vortices depends only on the change of direction in the w plane of the flat-cap geodesic segment between its end points, again given by dw dw̄. The differential of Eq. (30) for this segment implies that dw dw̄ ¼ − w̄ w : ð33Þ The end point (with positive imaginary part) in the z plane is where jzj ¼ 1, so z ¼ X þ i ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − X2 p , which converts to w ¼ ffiffiffi 2 p ð ffiffiffiffiffiffiffiffiffiffiffiffi 1þ X p þ i ffiffiffiffiffiffiffiffiffiffiffiffi 1 − X p Þ. Therefore, arg � − w̄ w � ¼ π − 2 arctan � ffiffiffiffiffiffiffiffiffiffiffiffi 1 − X pffiffiffiffiffiffiffiffiffiffiffiffi 1þ X p � ; ð34Þ so the scattering angle is Θ ¼ 2 arctan � ffiffiffiffiffiffiffiffiffiffiffiffi 1 − X pffiffiffiffiffiffiffiffiffiffiffiffi 1þ X p � ; ð35Þ and from the double-angle formula, tanΘ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − X2 p X : ð36Þ As X ¼ 1 2 a, the scattering angle as a function of impact parameter a in the flat cap approximation is Θ¼ arctan � ffiffiffiffiffiffiffiffiffiffi 2− a pffiffiffiffiffiffiffiffiffiffiffi 2þ a p � ; so tanΘ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 4− a2 p a ; ð37Þ both functions being shown in Fig. 1. This is not such a good approximation for the scattering angle as that obtained using the spherical cap approximation. In par- ticular, the scattering angle (37) has an unwanted square root singularity as a → 2. C. Approximations for the mode frequencies The squared frequency ω2 1 of the lowest mode varies with the vortex separation. It monotonically increases from approximately ω2 1ð0Þ ¼ 0.5378 to ω2 1ð∞Þ ¼ 0.7747 (the squared frequency of the 1-vortex radial shape mode) as ρ increases from 0 to ∞.2 ω2 1ðρÞ can be approximated by the rather simple function ω2 1ðρÞ¼ 8>< >: ω2 1ð0Þþ 1 R3ðRþ2Þðω2 1ð∞Þ−ω2 1ð0ÞÞρ4 ρ≤R ω2 1ð∞Þ− 2 Rþ2 ðω2 1ð∞Þ−ω2 1ð0ÞÞe2ðR−ρÞ ρ≥R; ð38Þ which is continuous and has continuous first derivative. It can be expressed in terms of the coordinate r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 p using the relation ρ2 ¼ ffiffiffiffiffiffiffiffi 108 p r. The form of (38) is motivated by the facts that near the origin the squared frequency grows quadratically with r, i.e., quartically with ρ, and that it approaches ω2 1ð∞Þ exponentially with the vortex separation 2ρ. Here, R is a scale parameter, and a good fit is achieved with R ¼ 2. Because ω2 1 increases with the vortex separation, it will generate an attractive interaction. The squared frequency of the second mode can be approximated as ω2 2ðρÞ ¼ ω2 2ð0Þ − � ω2 2ð0Þ − ω2 2ð∞Þ�ð1 − e−0.2ρ 2Þ; ð39Þ where ω2 2ð0Þ ¼ 0.9747 is the degenerate frequency of the second and third modes at coincidence, i.e., at the apex of the moduli space, and ω2 2ð∞Þ ¼ ω2 1ð∞Þ ¼ 0.7747. ω2 depends linearly on ρ2 near the apex and continues smoothly to negative ρ2 to give the frequency ω3 of the third mode. (In this context, r is proportional to jρ2j.) Close to ρ2 ¼ −1, the third mode hits the continuum threshold ω ¼ 1, and disappears. 2More precise frequencies, together with numerical error estimates, are given in Ref. [5]. The continuum spectrum has frequencies ω ≥ 1. A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-6 III. MODEL FOR THE EXCITED, LOWEST-FREQUENCY MODE A. Collective coordinate model Here, we consider a collective coordinate model for 2-vortex dynamics with the lowest mode excited. For vortices approaching from a large separation, this mode represents an in-phase superposition of the radial shape modes on each vortex. This model is quite simple. Over the 2-vortex moduli space M with metric ds2 ¼ Ωðx; yÞðdx2 þ dy2Þ, we assume there is defined a harmonic oscillator with normal coordinate η and position-dependent frequency ω1ðx; yÞ. The following Lagrangian couples the excited oscillator to motion through M: L ¼ 1 2 Ωðx; yÞðẋ2 þ ẏ2Þ þ 1 2 η̇2 − 1 2 ω2 1ðx; yÞη2: ð40Þ There are no cross-terms in the kinetic energy because the moduli space directions are zero modes of the 2-vortex fields, whereas the oscillator direction is a positive- frequency shape mode, and these modes are orthogonal. The equations of motion derived from L are d dt ðΩẋÞ − 1 2 ∂xΩðẋ2 þ ẏ2Þ þ ω1∂xω1η 2 ¼ 0; ð41Þ d dt ðΩẏÞ − 1 2 ∂yΩðẋ2 þ ẏ2Þ þ ω1∂yω1η 2 ¼ 0; ð42Þ η̈þ ω2 1η ¼ 0: ð43Þ Equations (41) and (42) can be expanded out, giving Ωẍþ1 2 ∂xΩẋ2þ∂yΩẋ ẏ− 1 2 ∂xΩẏ2þω1∂xω1η 2¼ 0; ð44Þ Ωÿ− 1 2 ∂yΩẋ2þ∂xΩẋ ẏþ 1 2 ∂yΩẏ2þω1∂yω1η 2¼ 0: ð45Þ Here, the coefficients of the quadratic terms in velocity (divided by Ω) encode the Levi-Civita connection on M. B. Numerical results For the numerical analysis of this model, we use the spherical cap approximation to Ω, i.e., to the geometry of FIG. 2. Number of bounces N as a function of initial velocity vin. Here, N ¼ 10 denotes 10 or more bounces. Upper left: multibounces immersed in one bounce windows, vin ∈ ½0; 0.1�. Upper right: multibounces immersed in two-bounce windows, vin ∈ ½0.0862; 0.088�. Bottom left: multibounces immersed in three-bounce window, vin ∈ ½0.0870892; 0.0871576�. Bottom right: multibounces immersed in four-bounce window, vin ∈ ½0.087122032; 0.0871248364�. COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-7 M, and the approximation (38) for ω2 1. Both have rotational symmetry. For simplicity, we restrict ourselves to head-on collisions. Thus, it is consistent to put y≡ 0 identically in (44) and identify positive xwith motion of the vortex pair along the horizontal axis while negative x corresponds to motion along the vertical axis. Each passage through x ¼ 0 corresponds to 90° scattering, and we call this a bounce. For this restricted motion, Ωẍþ 1 2 ∂xΩẋ2 þ ω1∂xω1η 2 ¼ 0; ð46Þ η̈þ ω2 1η ¼ 0: ð47Þ We assume that at t ¼ 0 the vortices are well separated along the horizontal axis and approaching each other. In our numerics, we assumed xð0Þ ¼ 3, corresponding to an initial vortex separation 2ρ ≈ 11, and vin ¼ −ẋð0Þ > 0. Because the mode-mediated force between the vortices is always attractive, there is at least one collision, i.e., xðtÞ ¼ 0 at least once. The lowest mode is excited with initial ampli- tude ηð0Þ ¼ 2 and η̇ð0Þ ¼ 0. vin is then varied. One should remember that vin is not the initial velocity of vortices in the physical plane, although the difference is rather small. The relation between the variables ρ and x gives vphysin ¼ � 27 4 � 1=4 vinffiffiffiffiffiffiffiffiffi xð0Þp ¼ � 3 4 � 1=4 vin: ð48Þ Our main result is that we find a chaotic structure in the scattering as vin increases. The usual 90° scattering (one bounce) arising from the geodesic approximation is, in a rather chaotic way, replaced by multibounce windows, where colliding vortices form a quasibound state perform- ing N bounces—each being a 90° scattering. For suffi- ciently large amplitude, the interchanging sequence of one bounce and multibounce windows starts from arbitrary small initial velocity and ends when vin exceeds a critical velocity vcr. For ηð0Þ ¼ 2, we find that vcr ¼ 0.0988, and for vin > vcr, we observe only single bounces; see Fig. 2, upper left. The figure shows the number of bounces (from 1 to N ≥ 10) for initial velocities in the range vin ∈ ½0; 0.1�. This chaotic structure has an approximately self- similar pattern. In Fig. 2, upper right, we show the N ≥ 3 bounces immersed among two-bounce scatterings, for vin ∈ ½0.0862; 0.088�. This structure repeats. The lower panels of Fig. 2 show N ≥ 4 bounces immersed among three-bounce scatterings, for vin ∈ ½0.0870892; 0.0871576� and N ≥ 5 bounces immersed among four-bounce scatter- ings, for vin ∈ ½0.087122032; 0.0871248364�. In Fig. 3, left, we show examples of 1-, 2-, 3- and four-bounce solutions xðtÞ for vin ¼ 0.086, 0.087, 0.08757 and 0.087571687055, respectively. Clearly, a tiny change in the initial conditions can lead to a dramatic change in the scattering. In Fig. 3, right, we plot the time evolution ηðtÞ of the mode amplitude for this four-bounce solution. The maximum amplitude is almost constant but briefly grows during the collisions. In Fig. 4, we show how the structure of bounces changes if we vary the initial mode amplitude ηð0Þ but fix the initial velocity vin. For example, for vin ¼ 0.05, the first multi- bounce occurs when ηð0Þ ¼ 0.504, whereas for vin ¼ 0.15, it occurs when ηð0Þ ¼ 0.826. For sufficiently small values of the amplitude, there is always a regime with one bounce, i.e., a single 90° scattering. In this quasigeodesic regime, the geodesic dynamics is only softly modified by the mode excitation, making the vortex-vortex collision faster; see Fig. 5, where we plot the time T at which the trajectory reaches x ¼ −3, starting from x ¼ 3. T decreases as the initial amplitude of the mode grows, a consequence of the attractive force triggered by the nonzero mode amplitude. However, above a critical value of ηð0Þ, the chaotic multibounce behavior starts. At this point, T jumps, and the geodesic approximation breaks down completely. Not surprisingly, the quasigeodesic regime is larger if vin is larger. In the next subsection, we will analyze this regime from an adiabatic point of view. FIG. 3. Left: examples of trajectories xðtÞ of the 2-vortex modulus: one-bounce, vin ¼ 0.086; one-bounce, vin ¼ 0.087; three-bounce, vin ¼ 0.08757; and four-bounce, vin ¼ 0.087571687055. Right: evolution of the mode amplitude ηðtÞ for the four-bounce solution. A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-8 We remark that, although the positions of the one-bounce and multibounce collisions are quite sensitive to details of our collective model, that is, to changes of ω1 and Ω, the chaotic multibounce structure is robust. Therefore, we expect that vortex scattering in the full field theory with the lowest mode excited will exhibit the same features. In fact, results from numerical simulations of the field theory confirm the validity of our collective model [11]. All these results resemble what is observed in kink- antikink scattering inϕ4 theory in (1þ 1) dimensions. There also, there is a chaotic sequence of multibounce windows and annihilation regions called bion chimneys [7,8], with a strong dependence on the initial relativevelocity of the kinks and on the initial shape mode amplitude. In particular, the accumulation of 2-vortex multibounces as vin tends to 0 (for fixed initial mode amplitude) possesses its counterpart in scatterings between a wobbling kink and antikink [12]. Such a chaotic pattern is explained in terms of a resonant energy transfer mechanism between the kink kinetic energy and shape mode energy [7,8], and it has recently been established that a collective model with two degrees of freedom explains the observed dynamics well [9]. The main difference from vortices is that the chaotic behavior in kink- antikink collisions occurs even without an initial excitation of the shape mode because there is automatically an attractive force between kink and antikink. For the BPS 2-vortices, an excitation of the lowest shape mode is needed to generate an attraction. C. Adiabatic approximation We now treat the oscillator as a fast variable and the motion through M as slow. It is also important that the mode amplitude is relatively small. More precisely, we assume that ω1 and Ω, together with their x and y derivatives, are Oð1Þ and that ẋ and ẏ are OðεÞ, with ε small. We wish ẍ and ÿ to be Oðε2Þ and see from Eqs. (44) and (45) that the oscillator amplitude η needs to be OðεÞ. With these assumptions, the solution of Eq. (43) for the oscillator, in the adiabatic approximation, takes the form ηðtÞ ¼ AðxðtÞ; yðtÞÞ cos �Z t 0 ω1ðxðt0Þ; yðt0ÞÞdt0 � ; ð49Þ FIG. 4. Number of bounces as a function of initial amplitude of the mode in the range ηð0Þ∈ ½0; 2�. Left: vin ¼ 0.005; right: vin ¼ 0.015. FIG. 5. Collision time T as a function of the initial amplitude of the mode ηð0Þ. T is the time for the trajectory to reach x ¼ −3, starting from x ¼ 3. Left: vin ¼ 0.010; right: vin ¼ 0.015. COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-9 where we have chosen the time origin to coincide with an instantaneous maximal amplitude of η. The integral is along a path through M that is still to be determined using the equations for x and y. The amplitude A is also yet to be determined but only depends on the position inM at time t and not on the path through M. Clearly, A needs to be OðεÞ. Differentiating (49) twice with respect to time, we find that η̈þ ω2 1η ¼ Ä cos �Z t 0 ω1 � xðt0Þ; yðt0Þ�dt0� − ð2ω1Ȧþ Aω̇1Þ sin �Z t 0 ω1 � xðt0Þ; yðt0Þ�dt0�; ð50Þ where ω̇1 is the total time derivative of ω1 along the path. The first term on the right-hand side is Oðε3Þ and can be neglected relative to the remaining terms, which are Oðε2Þ. Equation (43) for the oscillator is therefore satisfied, provided that 2ω1Ȧþ Aω̇1 ¼ 0: ð51Þ The solution is A ¼ Cffiffiffiffiffiffi ω1 p ð52Þ with C a constant, so A is indeed like ω1, being just a function of the instantaneous position ðxðtÞ; yðtÞÞ. C is the adiabatic invariant of the oscillator, remaining constant despite the oscillator having a varying frequency ω1 along the path in M. For consistency, C is OðεÞ. We now derive the reduced, adiabatic equations of motion for x and y, using the approximate solution for the oscillator ηðtÞ¼ Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω1ðxðtÞ;yðtÞÞ p cos �Z t 0 ω1 � xðt0Þ;yðt0Þ�dt0�: ð53Þ We need to take the time average of η2 over one period of the oscillator, to derive the average force that acts. This is hη2i ¼ 1 2 A2 ¼ 1 2 C2 ω1 . The reduced equations are therefore Eqs. (44) and (45) with ω1η 2 replaced by 1 2 C2. It is more convenient to give the reduced Lagrangian, from which these equations follow, namely, Lred ¼ 1 2 Ωðx; yÞðẋ2 þ ẏ2Þ − 1 2 C2ω1ðx; yÞ: ð54Þ The constant C2 is determined by the oscillator’s initial conditions, and implicitly, ẋ2; ẏ2, and C2 are all Oðε2Þ. In summary, the oscillator dynamics generates a poten- tial energy 1 2 C2ω1ðx; yÞ that, together with the conformal factor Ω, governs adiabatic motion through M. This motion has the conserved energy Ered ¼ 1 2 Ωðx; yÞðẋ2 þ ẏ2Þ þ 1 2 C2ω1ðx; yÞ; ð55Þ which matches the conserved energy for the original Lagrangian (40) if we use the time-averaged oscillator energy. The latter is hEosci ¼ 1 2 A2ω2 1 ¼ 1 2 C2ω1; ð56Þ where we have ignored the contribution of Ȧ relative to that of A. Note that the adiabatic invariant C2 is 2 ω1 hEosci. The discussion so far has needed to assume no symmetry property for ω1 and Ω. However, in the context of the 2-vortex moduli space M (the rounded cone), both the frequency ω1 of the lowest shape mode and the metric conformal factor Ω have Oð2Þ rotational sym- metry. It is therefore convenient to use the polar coor- dinates x ¼ r cosφ; y ¼ r sinφ on M and the radial functions ω1ðrÞ and ΩðrÞ. r ¼ 0 is the apex of M, where the vortices are coincident, and r extends to þ∞ as the vortices separate. ωðrÞ and ΩðrÞ are positive and have zero derivative at r ¼ 0. The angle φ has the range φ∈ ½−π; π�. The Lagrangian (40) for the shape mode dynamics coupled to the moduli space motion simplifies as a result of the rotational symmetry, and in particular, the reduced Lagrangian (54) simplifies to Lred ¼ 1 2 ΩðrÞðṙ2 þ r2φ̇2Þ − 1 2 C2ω1ðrÞ: ð57Þ There are two constants of motion for Lred, the energy and angular momentum, E ¼ 1 2 ΩðrÞðṙ2 þ r2φ̇2Þ þ 1 2 C2ω1ðrÞ; ð58Þ J ¼ ΩðrÞr2φ̇: ð59Þ After eliminating φ̇ from the energy in favor of the angular momentum, the radial motion can be found by quadrature, provided we know both ω1ðrÞ and ΩðrÞ, and all the initial data, including that for the oscillator. As ω1ðrÞ is an increasing function of r, the potential is attractive, but the effective potential in the reduced dynamics, Vred ¼ 1 2 C2ω1ðrÞ þ J2 2ΩðrÞr2 ; ð60Þ A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-10 includes a centrifugal term, so it can be can be attractive or repulsive. There can therefore be both bounded and scattering solutions for the adiabatic 2-vortex dynamics when the lowest shape mode is excited. This contrasts with the dynamics in the absence of shape mode oscil- lations, where the motion on M follows geodesics, and consists purely of scattering trajectories [6]. The existence of bound orbits depends on the ratio between the adiabatic invariant and the angular momentum, C=J. The simplest motion, using this adiabatic approximation, is a head-on collision, with the vortices approaching from a large separation at some finite velocity. The timing for this is shown in Fig. 5. IV. MODEL FOR HIGHER-FREQUENCY SHAPE MODES A. Collective coordinate model Recall that the higher-frequency shape modes (the second and third modes) become degenerate at the apex of the moduli space M and that the third mode enters the continuum close to the apex. A model for the higher- frequency modes needs to allow for their interaction, as their frequencies are close together, but it needs to be constructed only in the inner region ofM, around the apex. Here, we can approximate the geometry ofM by a flat cap (the spherical cap approximation would be a refinement), and the mode frequencies can be approximated as having a linear dependence on jzj, where z ¼ xþ iy ¼ reiφ is the suitably normalized complex coordinate centered at the apex of M. The coordinate z is a parametrization for the 2-vortex gauge and Higgs fields over M; similarly, the vector space of higher-frequency shape modes and their ampli- tudes, involving deformations of the gauge and Higgs fields and a background gauge condition, can be para- metrized by an abstract pair of amplitudes ζ and χ. These are initially complex, but we will impose a reality condition below. There is Oð2Þ rotational symmetry about z ¼ 0, and the modes transform at z ¼ 0 as a doublet of Oð2Þ. The potential energy for the modes is constructed using the 2 × 2 Hermitian matrix M¼ � λ αðx− iyÞ αðxþ iyÞ λ � ¼ � λ αz� αz λ � ; ð61Þ where λ and α are real and positive constants; λ is the degenerate eigenvalue of M when z ¼ 0. The eigenvalues of M split for z ≠ 0, becoming λ� αjzj. The complete Lagrangian of the model, with standard kinetic terms and a quadratic potential obtained using M, is L¼1 2 � ż�żþ ζ̇�ζ̇þ χ̇�χ̇−ðζ�χ�Þ � λ αz� αz λ �� ζ χ �� ¼1 2 � ṙ2þr2φ̇2þ ζ̇�ζ̇þ χ̇�χ̇−ðζ�χ�Þ � λ αre−iφ αreiφ λ �� ζ χ �� : ð62Þ ζ and χ are the (undiagonalized) amplitudes of the modes whose frequencies are controlled by the z-dependent matrix M. Before looking at the equations of motion, it helps to say more about the eigenvectors of M. A generic 2 × 2 Hermitian matrix is of the form M ¼ λþ a · σ ¼ λþ a1σ1 þ a2σ2 þ a3σ3, where σ1, σ2, and σ3 are the Pauli matrices. This has eigenvalues λ� jaj, so the eigenvalues are degenerate only when a ¼ 0. Generally, in a family of 2 × 2 Hermitian matrices, the eigenvalues degen- erate on a submanifold of real codimension 3 in the parameter space. But for our problem, we have only two real moduli, and degeneracy still occurs. The reason is that M is a family ofHermitianmatriceswith a “real” structure. If a2were zero, thenMwould bemanifestly real, andwe could seek real eigenvectors. Degeneracy would then occur when a1 ¼ a3 ¼ 0, a codimension-2 condition. It is more con- venient in our model to set a3 ¼ 0, as this simplifies the eigenvectors, but there is still a real structure, and eigenvalue degeneracy occurs at z ¼ 0, a single point in the two- dimensionalmoduli space. The real structure occurs because for the vortices in the Abelian Higgs model the shape mode eigenfunctions are derived from eigenfunctions of a scalar Schrödinger operator with a real potential [5]. More concretely, the model Lagrangian (62) is invariant under ζ� ↔ χ, and we can impose the “reality” condition ζ� ¼ χ. This gives a consistent truncation of the equations of motion. Moreover, it is consistent to impose this condition on the Lagrangian. We therefore restart from the Lagrangian (62), with ζ� replaced by χ (and ζ replaced by χ�), L ¼ 1 2 ż�żþ χ̇�χ̇ − λχ�χ − 1 2 αz�χ2 − 1 2 αzχ�2: ð63Þ The equations of motion now simplify to ̈zþ αχ2 ¼ 0; ð64Þ χ̈ þ λχ þ αzχ� ¼ 0; ð65Þ and there is a conserved energy E ¼ 1 2 ż�żþ χ̇�χ̇ þ λχ�χ þ 1 2 αz�χ2 þ 1 2 αzχ�2: ð66Þ Equation (64) implies that the mode oscillations affect the moduli space motion, and Eq. (65) implies that the mode oscillation frequencies at each instant depend on the COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-11 location z in moduli space. These coupled equations can probably not be solved analytically but only numerically. B. Adiabatic approximation However, we can treat the dynamics adiabatically, assuming that z varies on a timescale much longer than the inverse of the oscillation frequencies ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ� αjzjp . This is just a little more sophisticated than the adiabatic treatment of the lowest-frequency oscillation mode, in Sec. III. The scaling that makes an adiabatic analysis possible is to suppose that z, λ, and α are Oð1Þ, with λ� αjzj remaining bounded away from zero, and that ż ¼ OðεÞ and ̈z ¼ Oðε2Þ, with ε small. This requires the mode amplitudes jζj and jχj to be OðεÞ, i.e., small, but their oscillation frequencies are Oð1Þ. To proceed, we need a basis of eigenvectors of the matrix M appearing in Eq. (62). The eigenvectors/eigenvalues are given by � λ αre−iφ αreiφ λ �� ζ χ � ¼ ν � ζ χ � ; ð67Þ so the eigenvalues are νþ ¼ λþ αr and ν− ¼ λ − αr, with respective eigenvectors � ζ χ � ¼ � 1 eiφ � and � ζ χ � ¼ � 1 −eiφ � ð68Þ of squared norm 2. However, these eigenvectors do not satisfy the reality condition ζ� ¼ χ until we multiply by a suitable phase factor (which does not affect their ortho- normality). The real eigenvectors are VþðφÞ¼� � e− 1 2 iφ e 1 2 iφ � and V−ðφÞ¼� � ie− 1 2 iφ −ie1 2 iφ � ; ð69Þ where the phases are fixed, but there remains an ambiguity in sign. Note that when φ increases by 2π, the sign of each of these eigenvectors reverses. For a given r ≠ 0, the eigenspaces are therefore Möbius bundles over the circle parametrized by φ. In fact, these bundles extend to the entire punctured plane r ≠ 0. It is significant that these Möbius bundles join up smoothly at the origin, where the eigenvalues degenerate. There is continuity of the eigenvectors and eigenvalues along any line in the z plane that passes through the origin. Consider, for example, the line y ¼ 0with x running from a positive to a negative value. There is a constant eigenvector ð1; 1ÞT along this line with eigenvalue λþ αx and another constant eigenvector ði;−iÞT with eigenvalue λ − αx. To check this, one has to identify a point with negative x as having r ¼ jxj and φ ¼ π. Along this line, the lower eigenvalue runs smoothly into the upper eigenvalue and vice versa. Such eigenvalue crossing naturally occurs because the eigenvalue spectrum exhibits a conical struc- ture over a neighborhood of the origin. We next express the dynamical mode amplitudes in terms of the eigenvectors as � ζ χ � ¼ aþðtÞVþ � φðtÞ�þ a−ðtÞV− � φðtÞ�; ð70Þ where aþ and a− are real. (An arbitrary initial sign choice for the eigenvectors is made.) The time derivative is d dt � ζ χ � ¼ ȧþVþþaþ∂φVþφ̇þ ȧ−V−þa−∂φV−φ̇: ð71Þ The eigenvectors have the simple φ derivatives ∂φVþ ¼ − 1 2 V− and ∂φV− ¼ 1 2 Vþ; ð72Þ so d dt � ζ χ � ¼ � ȧþþ1 2 a−φ̇ � Vþþ � ȧ− − 1 2 aþφ̇ � V−: ð73Þ The Lagrangian (62) therefore takes the form, in terms of the real amplitudes aþ and a−, L ¼ 1 2 ðṙ2 þ r2φ̇2Þ þ � ȧþ þ 1 2 a−φ̇ � 2 þ � ȧ− − 1 2 aþφ̇ � 2 − ðλþ αrÞa2þ − ðλ − αrÞa2−; ð74Þ and the corresponding equations of motion are ̈r − rφ̇2 þ αða2þ − a2−Þ ¼ 0; ð75Þ r2φ̇þ ȧþa− − ȧ−aþ þ 1 2 ða2þ þ a2−Þφ̇ ¼ l; ð76Þ d dt � ȧþþ1 2 a−φ̇ � þ1 2 � ȧ− − 1 2 aþφ̇ � φ̇þðλþαrÞaþ ¼ 0; ð77Þ d dt � ȧ− − 1 2 aþφ̇ � − 1 2 � ȧþþ1 2 a−φ̇ � φ̇þðλ−αrÞa−¼ 0: ð78Þ Equation (76), for φ, has been integrated once, and l is the conserved angular momentum. The equations so far are all exact for the model Lagrangian (62), but now we make the adiabatic approxi- mation.We assume that r;φ areOð1Þ, ṙ; φ̇ areOðεÞ, and ̈r; φ̈ areOðε2Þ, where ε is small. For Eq. (75) to be consistent, the oscillator amplitudes aþ and a− need to beOðεÞ or smaller. A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-12 The following discussion is rather schematic, as the detailed formulas are not very illuminating. The basic solution of Eqs. (75) and (76), ignoring the oscillator contribution, is a straight-line motion. Let us orient this line so that x ¼ b and y ¼ vt in Cartesians, where b is a positive Oð1Þ constant and the velocity v is OðεÞ. Then, l ¼ bv, so rðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ v2t2 p and φ̇ ¼ bv b2 þ v2t2 : ð79Þ The straight line needs to miss the origin for φ̇ to remain bounded. (We shall consider a head-on collision below, using the Cartesian coordinate x.) Using the slowly time- dependent rðtÞ in the oscillator equations (77) and (78) and ignoring all the subleading terms that depend on φ̇, we deduce that the adiabatic solutions for the oscillator amplitudes are aþðtÞ¼ Cþ ðλþαrðtÞÞ14 cos �Z t 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λþαrðt0Þ p dt0 þ γþ � ; a−ðtÞ¼ C− ðλ−αrðtÞÞ14 cos �Z t 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ−αrðt0Þ p dt0 þ γ− � : ð80Þ The constants Cþ and C−, together with the phases γþ and γ−, are adiabatic invariants that depend on the initial data. We can now deduce the modification, due to the mode oscillations, of the straight-line motion. In Eq. (75), we substitute the time-averaged values ha2þi ¼ C2 þ 2 ffiffiffiffiffiffiffiffiffiffiffiffi λþαrðtÞ p and ha2−i ¼ C2 − 2 ffiffiffiffiffiffiffiffiffiffiffi λ−αrðtÞ p . This results in a modified radial accel- eration that can be attributed to a radial potential, Vrad ¼ C2þ ffiffiffiffiffiffiffiffiffiffiffiffiffi λþ αr p þ C2 − ffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αr p þ const: ð81Þ This potential is repulsive for the second (a−) oscillator, as its frequency increases approaching r ¼ 0, and attractive for the third oscillator. In Eq. (75), the term ȧþa− − ȧ−aþ is oscillatory and has zero average, so we ignore it. However, the time-averaged value of a2þ þ a2− is C2 þ 2 ffiffiffiffiffiffiffiffiffiffiffiffi λþαrðtÞ p þ C2 − 2 ffiffiffiffiffiffiffiffiffiffiffi λ−αrðtÞ p , and from Eq. (76), we can find the Oðε2Þ modification to φ̇ due to the mode oscillations. The analysis has not yet led to any mixing of the oscillation modes. Mixing occurs if we retain the leading mode-coupling terms in (77) and (78), giving äþ þ ðλþ αrÞaþ ¼ −ȧ−φ̇; ð82Þ ä− þ ðλ − αrÞa− ¼ ȧþφ̇: ð83Þ Using the OðεÞ solutions for the oscillator amplitudes and for φ̇ on the right-hand side, the particular integrals give Oðε2Þ corrections to the previously determined OðεÞ homogeneous solutions. In this calculation, it is sufficient to regard the oscillator frequencies as unvarying. There is no resonance because the frequency of each oscillator differs from the frequency of its forcing term by �2r, and we are assuming r remains Oð1Þ. A special case is if the second mode (a−) is initially excited but the third mode is not. This can occur if the vortices approach from a large separation, where the third mode has disappeared into the continuum. The analysis above goes through, and the third mode essentially does not contribute. There is a small excitation of the third mode due to the forcing by the second mode if the collision is not head on, but the amplitude generated is Oðε2Þ. However, the third mode is much more strongly excited in a head-on collision, and this is the most interesting case. Let us suppose the initial motion is along the x axis (the horizontal axis) in the moduli space, approaching from þ∞, and that the second mode is excited. By symmetry, the motion remains on the x axis and can pass through x ¼ 0, leading to 90° scattering of the vortices, although the repulsive potential generated adiabatically by the mode oscillation may prevent this. Now, recall that because of the conical structure of the oscillator spectrum it is purely the third mode that becomes excited if x becomes negative. The frequency of the excited oscillator, for either sign of x, is the smoothly varying function λ − αxðtÞ, provided jxj is not large, rather than λ − αrðtÞ (with r ¼ jxj). The relevant equations of motion along the x axis are ẍ − αu2 ¼ 0; ð84Þ üþ ðλ − αxÞu ¼ 0; ð85Þ obtained from Eqs. (64) and (65) by setting z ¼ x and χ ¼ −iu, where x and u are real. For this reduced system, there is a conserved energy, E ¼ 1 2 ẋ2 þ u̇2 þ ðλ − αxÞu2: ð86Þ Again, this is a consistent truncation, and an adiabatic treatment is feasible if we assume the previous scaling with ε:u oscillates with the slowly varying frequencyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αxðtÞp . The adiabatic solution of (85) is then uðtÞ ¼ C ðλ − αxðtÞÞ14 cos �Z t 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αxðt0Þ p dt0 � ; ð87Þ where the constant C is OðεÞ. After time averaging the oscillatory driving force u2, Eq. (84) becomes ẍ − α 2 C2ffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αx p ¼ 0; ð88Þ with first integral COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-13 1 2 ẋ2 þ C2 ffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αx p ¼ E: ð89Þ The constant E can be identified with the total conserved energy. This is Oðε2Þ and has comparable contributions from the motion in moduli space and from the oscillating mode. The oscillating mode contributes an effective poten- tial proportional to ffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αx p to the moduli space dynamics, modifying what would otherwise be geodesic dynamics with ẋ constant. Recall that λ is the degenerate value of the oscillator frequency at vortex coincidence. Equation (89) can be integrated once more to give t ¼ Z dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2E − 2C2 ffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αx pp : ð90Þ The integral here is elementary. In terms of the spatial variable q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − C2 E ffiffiffiffiffiffiffiffiffiffiffiffiffi λ − αx pr ; ð91Þ we find the implicit solution q − 1 3 q3 ¼ αC4 ð2EÞ32 t: ð92Þ The solution runs between the stationary points of the cubic q − 1 3 q3, i.e., between q ¼ �1, which is where αx ¼ λ. Over a finite time interval, αx decreases from λ, climbs the potential and stops at t ¼ 0, then increases back to λ. The stopping point is where q ¼ 0, i.e., where αx ¼ λ − E2 C4. The motion may or may not pass through x ¼ 0, depending on the energy. In practice, this model is only valid for x near zero because it ignores the third mode entering the continuum. Also, the squared frequency is only approximately linear in x. The calculated dependence of both squared frequencies on x is shown in Fig. 1 of Ref. [5]. This shows the crossover of the mode frequencies at x ¼ 0. In summary, to model a 2-vortex collision with the second shape mode excited, in the adiabatic approximation, one should ignore the third shape mode entirely until the separation reduces to r ¼ λ=α, then for smaller r use the model above, where the two upper modes are coupled. If the collision is considerably away from head on, then the third mode will be only slightly excited; the vortices will scatter; and when r becomes larger than λ=α, the third mode can again be ignored. Excitation of the second mode generates a repulsive potential, which increases the scatter- ing angle relative to that for purely geodesic motion. On the other hand, in a head-on collision, the second mode converts entirely into the third mode if the vortices reach coincidence at r ¼ 0. If they do, and scatter through 90°, then the oscillations of the third mode can hit the spectral wall where that mode enters the continuum, and it is not clear what happens next; the simplified models we have proposed will break down, and a full field-theory simu- lation of the 2-vortex collision is probably necessary. Finally, in a collision that is near to, but not exactly, head on, it is necessary to solve the equations for the coupled second and third modes numerically, as the coupling becomes strong when the frequencies become close to degenerate. We have not investigated this. C. Numerical results As we observed in the previous subsection, during a head-on collision of two vortices, the second and third modes interchange, but only the second mode (the out-of- phase superposition of the radial mode on each vortex) can be excited if the vortices are initially well separated. Our model for vortex dynamics with the second and third modes excited can be extended to the regime of well-separated vortices. There, it has a form very similar to the lowest- mode case, L ¼ 1 2 Ωðx; yÞðẋ2 þ ẏ2Þ þ 1 2 η̇2 − 1 2 ω2 2ðx; yÞη2; ð93Þ where η is now the amplitude of the second mode and the squared frequency ω2 2 is a monotonically decreasing function of the vortex separation parameter ρ, approxi- mately given by the expression (39). For head-on collisions, we can consistently put y ¼ 0. Then, the Lagrangian (93) reduces to L ¼ 1 2 ΩðxÞẋ2 þ 1 2 η̇2 − 1 2 ω2 2ðxÞη2; ð94Þ with ω2 2ðxÞ¼ω2 2ð0Þ− � ω2 2ð0Þ−ω2 2ð∞Þ� 1−e−0.2 ffiffiffiffiffiffi 108 p x : ð95Þ Because ω2 2 increases as x decreases from a positive value and becomes negative, the intervortex force is always toward positive x. Therefore, one can say that the mode generated force changes its sign depending if the vortices are before or after the first collision, i.e., the point where they are on top of each other. Initially repulsive force becomes attractive. After the second collision, the force once again changes into a repulsive force. This has the following dynamical consequences for vortices approach- ing from x ¼ 3. If the initial amplitude of the second mode is sufficiently large, the vortices scatter back before coalescing; i.e., they do not reach x ¼ 0. With a smaller amplitude, they may pass through x ¼ 0 and reach a negative x before scattering back; i.e., the vortices scatter from the horizontal to the vertical axis, stop, and return to the horizontal axis. This is a two-bounce solution. These possibilities are plotted in Fig. 6, in which vin ¼ 0.015. If A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-14 the amplitude is smaller still, the vortices separate suffi- ciently along the vertical axis that the third mode enters the continuum spectrum, and the so-called spectral wall phe- nomenon can be expected to occur [13]. This possibility is not taken into account in our collective coordinate model. To conclude, from this adiabatic modeling, we expect to observe at most two-bounce scatterings of vortices in the full field theory dynamics when the second mode is initially excited, and we do not expect any chaotic multibounce pattern. V. CONCLUSION For the last 40 years, it has been believed that scattering of BPS vortices is completely governed by geodesic flow on the pertinent moduli space. As an effect, the famous π=2 scattering in head-on collision has been found. In our paper, we found arguments that this is a highly simplified picture, valid only if vibrational modes hosted on the vortices are not excited too much. Specifically, to get some semianalytical understanding,we proposed collective coordinate models for the dynamics of BPS 2-vortex solutions excited by their shape modes. The models generalize the standard geodesic flow on the 2-vortex moduli spaceM by including the shape mode amplitudes as additional collective coordinates. Importantly, in contrast to the force-free geodesic dynamics of unexcited BPS vortices, excitation of the shape modes introduces intervortex forces whose sign depends on how the relevant mode’s frequency varies overM. This can significantly change the dynamics, leading to a complete breakdown of the original geo- desic flow. The most striking result is the appearance of a chaotic, probably self-similar, fractal-like pattern of multibounce windows in vortex-vortex collisions if the lowest shape mode is excited. This is explained in terms of the well- known resonant energy transfer mechanism. During colli- sions, the kinetic energy of vortex motionmay be temporary transferred into shape mode energy. The vortices may then be unable to overcome the attractive interaction triggered by the mode excitation and instead collide once again. This process repeats until the kinetic energy is again sufficiently large for the vortices to separate. A similar mechanism is very well understood in kink-antikink collisions in (1þ 1) dimensions, but here, for the first time, we have shown that it can also influence the dynamics of higher-dimensional solitons. In particular, we have found that an even number of collisions (bounces) changes the famous 90° scattering of vortices into 180° backward scattering. Backward scatter- ing can also occur if the higher-frequency (second) mode is excited because in this case the excitation triggers a repulsive force between the vortices. Additionally, due to level crossing, this force becomes attractive after the vortices pass through the circularly symmetric configura- tion in a head-on collision (one bounce), which makes a second bounce more likely, resulting in backward scatter- ing. No further bounces are possible in this channel. Importantly, all our predictions, based on the collective coordinate approach, completely agree with numerical simulations of excited 2-vortex scattering in the full field theory [11]. In fact, our construction qualitatively explains all phenomena observed in such scatterings. These are the existence of chaotic multibounce scatter- ings, if the lowest mode is excited, and appearance of maximally two-bounce scatterings in the case in which the higher mode is excited [11]. Observed variety of distinct types of the behavior of the excited vortices in the scattering processes has its origin in the flow of the spectral structure on the original moduli space of two BPS vortices. Of course, to get quantitative agreement, one has to further refine the collective coordinate models. This is because the models make several simplifications which for a chaotic system may not be negligible. For example, we used simplified, analytical expressions for both the metric function and the dependence of the frequencies on the vortex separation. Also, we did not take into account a possible modification of the moduli space metric due to the amplitudes of the modes, even though it is known that FIG. 6. Left: trajectories of the vortex position xðtÞ for the model with the second mode excited, for various values of the initial mode amplitude, and vin ¼ 0.015. Right: evolution of the mode amplitude ηðtÞ for the motion with the initial amplitude ηð0Þ ¼ 0.14. COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-15 vibrational (massive) degrees of freedom can affect the metric of kinetic, zero modes, see, e.g., the description of a single, vibrating kink [9]. Such a metric modification is obviously a subleading effect in comparison with the appearance of an intervortex force, but nonetheless, it can affect the locations of bounce windows. Also, inclusion of nonquadratic terms in the effective potential may slightly change the dynamics. Looking from a wider perspective, it is fascinating that effects previously associated only with one-dimensional kinks find their counterparts in the dynamics of higher- dimensional solitons. ACKNOWLEDGMENTS We acknowledgeMorgan Rees for informing us of results concerning the scattering of excited vortices prior to pub- lication. This research was supported by the Spanish MCIN with funding from European Union NextGenerationEU (Grant No. PRTRC17.I1) and Consejeria de Educacion from JCyL through theQCAYLEproject, aswell asMCINProject No. PID2020–113406GB-I0. This research has made use of the high-performance computing resources of the Castilla y León Supercomputing Center (SCAYLE), financed by the European Regional Development Fund (ERDF). N. S.M. is partially supported byUKSTFC consolidatedGrant No. ST/ T000694/1. A.W. was supported by the Polish National Science Centre, Grant No. NCN 2019/35/B/ST2/00059. APPENDIX: FIELD THEORY DERIVATION OF THE EFFECTIVE MODELS In this Appendix, we sketch the derivation of the two effective models, identified in the main sections of this paper, governing the dynamics of BPS 2-vortices when shapes modes are excited. The Abelian Higgs field theory action is [14] S¼ Z � − 1 4 FμνFμνþ1 2 DμϕDμϕ− 1 8 ð1− ϕ̄ϕÞ2 � dx1dx2dt ¼ Z fT−Vgdt; ðA1Þ such that the kinetic and potential energy of the system in the temporal gauge A0 ¼ 0 are, respectively, T ¼ 1 2 Z f∂tAi∂tAiþ∂tϕ̄∂tϕgdx1dx2; V¼ 1 2 Z � FijFijþDiϕDiϕþ1 8 ð1− ϕ̄ϕÞ2 � dx1dx2: ðA2Þ It is well known that BPS 2-vortex solutions satisfy the first-order equations F12¼ 1 2 ð1− ϕ̄ϕÞ; D1ϕþ iD2ϕ¼ 0 ðA3Þ and describe configurations where two unit magnetic flux vortices have an arbitrary separation. In the center-of-mass frame, the 2-vortex solution is denoted as [5] ÃjðX1;X2;x1;x2Þ; ϕ̃ðX1;X2;x1;x2Þ; ðA4Þ where ðX1; X2Þ are the spatial coordinates of a general point in the spacetime. �ðx1; x2Þ specify the locations of the constituent vortices (zeros of the scalar field ϕ̃) and play the role of real coordinates on the centered 2-vortex moduli space M. They can be combined into the complex coordinate w ¼ x1 þ ix2 ¼ ρeiθ introduced in Sec. II, although as we have seen it is more convenient to work with a complex coordinate z ¼ xþ iy ¼ reiφ proportional to w2, that is, z ¼ kw2. Then, r ¼ kρ2 and φ ¼ 2θ. Note that the angle θ varies only in the range ½− 1 2 π; 1 2 π� because adding π moves the unit vortex at ðx1; x2Þ to ð−x1;−x2Þ and vice versa, but these two configurations are identical. On the other hand, φ ¼ 2θ has the normal angular range. In terms of the real moduli space coordinates ðx; yÞ, the 2-vortex configurations (A4) can be assembled as the column vector Φ̃ðX1; X2; x; yÞ ¼ � Ã1ðX1; X2; x; yÞ; Ã2ðX1; X2; x; yÞ; ϕ̃1ðX1; X2; x; yÞ; ϕ̃2ðX1; X2; x; yÞ � T; ðA5Þ where ϕ̃1 and ϕ̃2 are the real and imaginary parts of ϕ̃. 1. Collective coordinate model for 2-vortex dynamics with the lowest-frequency shape mode excited The configuration for moving vortices with the lowest- frequency shape mode excited is written as Ψ̃ � X1; X2; xðtÞ; yðtÞ � ¼ Φ̃ � X1; X2; xðtÞ; yðtÞ � þ ηðtÞξ1 � X1; X2; xðtÞ; yðtÞ � ; ðA6Þ where ξ1 denotes the lowest (normalized) shape mode of the second-order fluctuation operator H, Hξ1ðX1; X2; x; yÞ ¼ ω2 1ðrÞξ1ðX1; X2; x; yÞ; ðA7Þ and ηðtÞ is the real shape mode amplitude. Note that the frequency ω1 depends on the point in the moduli space where the spectral problem is studied, but because of rotational symmetry, it only depends on r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 p . The flow over the moduli space of both the eigenfunction ξ1 and the eigenvalue ω2 1 has been numerically described in Ref. [5]. x, y, and η are the collective coordinates, whose effective dynamics is implemented by assuming that the A. ALONSO IZQUIERDO et al. PHYS. REV. D 110, 085006 (2024) 085006-16 temporal dependence of the fields in R2 is only through ẋ, ẏ, and η̇. Under this hypothesis, the effective dynamical system is constructed as follows. Neglecting contributions of the form η R ˙̃ΦT ξ̇1dX1dX2, η R ξ̇T1 ˙̃ΦdX1dX2, and η2 R ξ̇T1 ξ̇1dX1dX2, the effective kinetic energy reads Teff ¼ 1 2 gxxðx; yÞẋ2 þ 1 2 gyyðx; yÞẏ2 þ 1 2 η̇2; ðA8Þ where the metric factors are gxxðx; yÞ ¼ Z � ∂Φ̃ ∂x � T � ∂Φ̃ ∂x � dX1dX2 ¼ ΩðrÞ gyyðx; yÞ ¼ Z � ∂Φ̃ ∂y � T � ∂Φ̃ ∂y � dX1dX2 ¼ ΩðrÞ: ðA9Þ These integrals define the conformal factor ΩðrÞ of the Samols metric [6]. We finally obtain Teff ¼ 1 2 ΩðrÞðẋ2 þ ẏ2Þ þ 1 2 η̇2: ðA10Þ The contribution to the Lagrangian which does not depend on time derivatives is evaluated up to second order in η and is 1 2 η2 Z ξT1 ðX1; X2; x; yÞHξ1ðX1; X2; x; yÞdX1dX2: ðA11Þ Using (A7) and taking into account the rotational sym- metry, we find the effective potential over the moduli space, Veffðx; y; ηÞ ¼ 1 2 ω2 1ðrÞη2: ðA12Þ Because ω2 1ðrÞ has its minimum value ω2 1ð0Þ at the origin and increases to a finite value ω2 1ð∞Þ > ω2 1ð0Þ at infinity, attractive forces arise between the vortices when the lowest shape mode is excited. Moreover, this potential is propor- tional to the squared amplitude of the mode, opening the door to the transfer of energy from the mode into the kinetic energy of vortices moving through the moduli space. Thus, the effective Lagrangian Leff ¼Teff −Veff ¼ 1 2 ΩðrÞðẋ2þ ẏ2Þþ1 2 η̇2− 1 2 ω2 1ðrÞη2; ðA13Þ the starting point of Sec. III, is obtained from the field theory as an effective collective coordinate model. We stress finally that incorporating the effect of the shape mode up to second order generalizes to the quantized theory as a one-loop/semiclassical correction to moduli space dynamics. 2. Collective coordinate model for 2-vortex dynamics with the two higher-frequency shape modes excited Consider the configuration Σ � X1;X2;xðtÞ;yðtÞ �¼ Φ̃ � X1;X2;xðtÞ;yðtÞ � þη2ðtÞξ2 � X1;X2;xðtÞ;yðtÞ � þη3ðtÞξ3 � X1;X2;xðtÞ;yðtÞ � ; ðA14Þ which describes a dynamical 2-vortex excited by the shape modes ξ2 and ξ3 having the higher frequencies ω2 and ω3 ≥ ω2, respectively, and amplitudes η2 and η3. These modes are mutually orthogonal eigenfunctions of the second-order fluctuation operator H, see Ref. [5], HξiðX1;X2;x;yÞ¼ω2 i ðrÞξiðX1;X2;x;yÞ; i¼ 2;3: ðA15Þ The spectral flow of ω2 2 and ω2 3 over M is summarized as follows: (i) The frequencies are rotationally symmetric over M and only depend on r. (ii) At the apex of M, the two frequencies degener- ate: ω2 2ð0Þ ¼ ω2 3ð0Þ. (iii) ω2 2ðrÞ decreases linearly with r near r ¼ 0 and approaches the limit ω2 2ð∞Þ ¼ ω2 1ð∞Þ > 0. (iv) ω2 3ðrÞ increases linearly with r near r ¼ 0, reaching 1 at the boundary of a disk inM where the mode ξ3 disappears into the continuum. Arguing exactly as in the previous derivation of the effective model for the lower-frequency shape mode, we envisage the following effective Lagrangian for the upper- frequency modes: Leff ¼ 1 2 ΩðrÞðẋ2 þ ẏ2Þ þ 1 2 η̇22 þ 1 2 η̇23 − 1 2 ω2 2ðrÞη22 − 1 2 ω2 3ðrÞη23: ðA16Þ Investigation of the effective dynamics including the higher-frequency modes is particularly worthwhile inside the disk where both modes are present. Here, one can approximate any BPS 2-vortex solution by adding a zero mode of suitable amplitude to the circularly-symmetric, coincident 2-vortex solution. This produces a splitting of the double zero of the Higgs field into two single zeros with a small separation 2ρ. The collective coordinate procedure prescribes that the motion through M is only due to the time dependence of ρðtÞ. If the shape modes of frequencies ω2 and ω3 are also excited, we are led to study the effective dynamics of the configurations Σ � X1;X2;xðtÞ;yðtÞ �¼ Φ̃ðX1;X2;0;0ÞþϵðtÞξ0ðX1;X2;0;0ÞÞ þη2ðtÞξ2ðX1;X2;0;0Þ þη3ðtÞξ3ðX1;X2;0;0Þ: ðA17Þ COLLECTIVE COORDINATE MODELS FOR 2-VORTEX SHAPE … PHYS. REV. D 110, 085006 (2024) 085006-17 ϵ, η2, and η3 are, respectively, the amplitudes of the zero mode and these two shape modes, the new collective coordinates. Plugging the expression (A17) into the sec- ond-order action, we obtain Sð2Þ ¼ 1 2 Z fkξ0k2ϵ̇2ðtÞ þ kξ2k2η̇22ðtÞ þ kξ3k2η̇23ðtÞ − ω2 2ðrÞkξ2k2η22ðtÞ − ω2 3ðrÞkξ3k2η23ðtÞgdt; ðA18Þ where the orthogonality of the eigenfunctions ξi of H has been employed. We can assume that the shape modes ξ2 and ξ3 are normalized, but we shall use the non-normalized zero mode ξ0ðX1;X2;0;0Þ¼R � hðRÞsinΘ;hðRÞcosΘ;− h0ðRÞ f2ðRÞ ;0 � T ðA19Þ described in Refs. [4,5], where ðR;ΘÞ here denote spatial polar coordinates, i.e.,X1 ¼ R cosΘ andX2 ¼ R sinΘ. This choice of the zero mode involves the splitting of the vortex zeros in the x2 direction (vertical direction) away from the apex of the moduli space, such that x2ðtÞ determines the intervortex distance. The expression (A19) allows us to find a relation between the amplitude ϵ of the zeromode ξ0 and x2 as 3 ϵðtÞ ¼ ðδ2Þ2 2jc2j ðx2ðtÞÞ2 ¼ C1ðx2ðtÞÞ2 ¼ C1 k rðtÞ; ðA20Þ where δ2 ≈ 0.236146 and c2 ≈ −0.277308. These values come from the local behavior of the coincident 2-vortex Higgs field profile f2ðRÞ ≈ δ2R2 and of the zeromode (A19) with hðRÞ ≈ 1þ c2R2. Additionally, ω2 2ðrÞ¼ λ− C2 k r; ω2 3ðrÞ¼ λþC2 k r; ðA21Þ where λ ¼ ω2 2ð0Þ ¼ ω2 3ð0Þ ≈ 0.97303 and C2 ≈ 0.025873. Working with the expressions (A20) and (A21), the action becomes Sð2Þ ¼ 1 2 Z � C2 1 k2 kξ0k2ṙ2ðtÞ þ η̇22ðtÞ þ η̇23ðtÞ − � λ − C2 k rðtÞ � η22ðtÞ − � λþ C2 k rðtÞ � η23ðtÞ � dt; ðA22Þ with kξ0k2 ≈ 62.4936. We now combine the two real shape mode amplitudes into a single complex amplitude χ, η3 ¼ 1ffiffiffi 2 p � e i 2 φχ� þ e− i 2 φχ � ¼ η�3; η2 ¼ 1ffiffiffi 2 p i � e i 2 φχ� − e− i 2 φχ � ¼ η�2; ðA23Þ which involves the argumentφ of the coordinate z ¼ reiφ on the moduli space. This induces a rotation which generalizes the dynamics. Now, the motion can be in any direction through the origin of the moduli space, and the Lagrangian in (A22) becomes Leff ¼ 1 2 C2 1 k2 kξ0k2ż�żþ χ̇�χ̇−λχ�χ− 1 2 C2 k z�χ2− 1 2 C2 k zχ�2: ðA24Þ Finally, if the value of k is set as k ¼ C1kξ0k and we define α ¼ C2=ðC1kξ0kÞ, then the effective Lagrangian becomes Leff ¼ 1 2 ż�żþ χ̇�χ̇ − λχ�χ − 1 2 αz�χ2 − 1 2 αzχ�2; ðA25Þ which reproduces the expression (63) introduced in the main text. [1] H. 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