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=ρ0=6. The total metric is ds(1)2=f(1)2(ρ)(dρ2+ρ2dθ2),(4)where f(1)2(ρ)={6912πρ2(108+ρ4)2ρ≤6,2πρ≥6.(5)f(1)2(ρ) is continuous, and both functions in (5) have zero derivative at ρ=6. To verify that for ρ≤6 it is a sphere metric, we introduce¹

w=ρeiθ is used consistently throughout this paper; z=kw2 for some constant positive multiple k, but k varies between (sub)sections.

Note that the spherical cap is restricted to ρ≤6, which implies that |z|≤13. Hence, on the cap, the maximal polar angle is μ=60°. (Use |z|=tan12μ to verify this.) The boundary circles of the cap and the truncated cone have equal lengths 12π3. Also, |z| 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=π∫062πρdρ=6π2,(8)where the prefactor is the range of θ, with the area of the spherical cap Acap=π∫066912πρ3(108+ρ4)2dρ=4π2.(9)Acap is one-quarter of the area of a complete sphere of squared radius 4π. The difference of the areas is Acone-Acap=2π2,(10)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 complete 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 f(1)2(ρ), a dynamical geodesic trajectory w(t)=ρ(t)eiθ(t) arises as the solution of the equation of motion for a particle with Lagrangian L=12f(1)2(ρ)(ρ˙2+ρ2θ˙2).(11)There are two constants of motion, the energy E and angular momentum J. E is the same expression as L, and J=f(1)2(ρ)ρ2θ˙.

Asymptotically, f(1)2(ρ)=2π, as π is the mass of a 1-vortex. The angular momentum of the incoming motion is J=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=πvin2, so J2/E=4πa2. Any geodesic has a point of closest approach of the two vortices, where ρ takes its minimum value ρ˜. Here, ρ˙=0, so E=12f(1)2(ρ˜)ρ˜2θ˙2 and J=f(1)2(ρ˜)ρ˜2θ˙. θ˙ cancels in J2/E, and conservation of J2/E implies that f(1)2(ρ˜)ρ˜2=2πa2.(12)This relation between the closest approach and the impact parameter will be useful shortly. For a geodesic that does not intersect the spherical cap, ρ˜ is simply a.

Geodesics on the spherical cap are segments of great circles on the complete sphere. In terms of the stereographic coordinate z on the cap, these great circles are a family of circles in the z plane, having the algebraic equation zz¯+β(z+z¯)-1=0,(13)where the parameter β is real and positive. This family is algebraically the linear join of the equatorial great circle zz¯-1=0 and the line z+z¯=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 |z|≤13 on the spherical cap, the relevant range of β is 13≤β≤∞. For β=13, the geodesic just touches the cap at z=13, and for β=∞, 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 6912πρ˜4(108+ρ˜4)2=2πa2,(14)so ρ˜2=126a(1-1-a28).(15)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+2βz-1=0, so z=β2+1-β. As z=w2108, and w=ρ˜ at closest approach, we deduce that β2+1-β=22a(1-1-a28),(16)which fortunately simplifies to β=8a2-1.(17)

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 dw¯, that is, to the argument of dwdw¯, evaluated at the end point of the segment.

To calculate the argument of dwdw¯, we note that w=(108)1/4z, so dw=(108)1/4dz2zanddw¯=(108)1/4dz¯2z¯.(18)Therefore, dwdw¯=z¯zdzdz¯.(19)Next, taking the differential of Eq. (13) for a great circle segment, we find that dzdz¯=-β+zβ+z¯,(20)so everywhere along the segment, dwdw¯=-z¯zβ+zβ+z¯.(21)The end points in the z plane are where the circle (13) intersects the boundary circle of the spherical cap, |z|=13. The end point with positive imaginary part is z=13β(1+i3β2-1).(22)After some manipulation, we find from (21) the end point value argdwdw¯=π+arctan3β2-1-2arctan(123β2-1),(23)and simple geometry shows that the scattering angle Θ along this geodesic is Θ=π-argdwdw¯=2arctan(123β2-1)-arctan3β2-1.(24)Expressed in terms of the impact parameter a, using (17), this becomes Θ=2arctan6a2-1-arctan(26a2-1).(25)Finally, using the subtraction and double-angle formulas for the tangent function, we conclude that tanΘ=(6-a2)3/2a(9-a2).(26)This is valid for a≤6, 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.

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=ρ0=2. As before, w=ρeiθ with ρ≥0, and θ∈[-12π,12π]. The total metric is ds(2)2=f(2)2(ρ)(dρ2+ρ2dθ2),(27)where f(2)2(ρ)={12πρ2ρ≤2,2πρ≥2.(28)This matches the asymptotic, flat-cone metric for ρ≥2. For ρ≤2, the change of coordinate z=14w2=14ρ2ei2θ converts the metric to ds2=2πdzdz¯, with |z|≤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.

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 z+z¯=2X,(29)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+w¯2=8X,(30)which is a rectangular hyperbola. Its closest approach to the origin is ρ˜=2X. To find the relation between X and the impact parameter a, we again use the conservation of J2/E, leading to f(2)2(ρ˜)ρ˜2=2πa2, the analog of (12), which implies that 12πρ˜4=2πa2.(31)Therefore, X=12a.(32)

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 dwdw¯. The differential of Eq. (30) for this segment implies that dwdw¯=-w¯w.(33)The end point (with positive imaginary part) in the z plane is where |z|=1, so z=X+i1-X2, which converts to w=2(1+X+i1-X). Therefore, arg(-w¯w)=π-2arctan(1-X1+X),(34)so the scattering angle is Θ=2arctan(1-X1+X),(35)and from the double-angle formula, tanΘ=1-X2X.(36)As X=12a, the scattering angle as a function of impact parameter a in the flat cap approximation is Θ=arctan(2-a2+a),sotanΘ=4-a2a,(37)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 particular, the scattering angle (37) has an unwanted square root singularity as a→2.

The squared frequency ω12 of the lowest mode varies with the vortex separation. It monotonically increases from approximately ω12(0)=0.5378 to ω12(∞)=0.7747 (the squared frequency of the 1-vortex radial shape mode) as ρ increases from 0 to ∞.²

More precise frequencies, together with numerical error estimates, are given in Ref. [5]. The continuum spectrum has frequencies ω≥1.

The squared frequency of the second mode can be approximated as ω22(ρ)=ω22(0)-(ω22(0)-ω22(∞))(1-e-0.2ρ2),(39)where ω22(0)=0.9747 is the degenerate frequency of the second and third modes at coincidence, i.e., at the apex of the moduli space, and ω22(∞)=ω12(∞)=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 |ρ2|.) Close to ρ2=-1, the third mode hits the continuum threshold ω=1, and disappears.

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=Ω(x,y)(dx2+dy2), we assume there is defined a harmonic oscillator with normal coordinate η and position-dependent frequency ω1(x,y). The following Lagrangian couples the excited oscillator to motion through M: L=12Ω(x,y)(x˙2+y˙2)+12η˙2-12ω12(x,y)η2.(40)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 ddt(Ωx˙)-12∂xΩ(x˙2+y˙2)+ω1∂xω1η2=0,(41)ddt(Ωy˙)-12∂yΩ(x˙2+y˙2)+ω1∂yω1η2=0,(42)η¨+ω12η=0.(43)Equations (41) and (42) can be expanded out, giving Ωx¨+12∂xΩx˙2+∂yΩx˙y˙-12∂xΩy˙2+ω1∂xω1η2=0,(44)Ωy¨-12∂yΩx˙2+∂xΩx˙y˙+12∂yΩy˙2+ω1∂yω1η2=0.(45)Here, the coefficients of the quadratic terms in velocity (divided by Ω) encode the Levi-Civita connection on M.

For the numerical analysis of this model, we use the spherical cap approximation to Ω, i.e., to the geometry of M, and the approximation (38) for ω12. 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 x with motion of the vortex pair along the horizontal axis while negative x corresponds to motion along the vertical axis. Each passage through x=0 corresponds to 90° scattering, and we call this a bounce. For this restricted motion, Ωx¨+12∂xΩx˙2+ω1∂xω1η2=0,(46)η¨+ω12η=0.(47)

We assume that at t=0 the vortices are well separated along the horizontal axis and approaching each other. In our numerics, we assumed x(0)=3, corresponding to an initial vortex separation 2ρ≈11, and vin=-x˙(0)>0. Because the mode-mediated force between the vortices is always attractive, there is at least one collision, i.e., x(t)=0 at least once. The lowest mode is excited with initial amplitude η(0)=2 and η˙(0)=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 vinphys=(274)1/4vinx(0)=(34)1/4vin.(48)

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 performing N bounces—each being a 90° scattering. For sufficiently 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 η(0)=2, we find that vcr=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∈[0,0.1].

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∈[0.0862,0.088]. This structure repeats. The lower panels of Fig. 2 show N≥4 bounces immersed among three-bounce scatterings, for vin∈[0.0870892,0.0871576] and N≥5 bounces immersed among four-bounce scatterings, for vin∈[0.087122032,0.0871248364].

In Fig. 3, left, we show examples of 1-, 2-, 3- and four-bounce solutions x(t) for vin=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 η(t) 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 η(0) but fix the initial velocity vin. For example, for vin=0.05, the first multibounce occurs when η(0)=0.504, whereas for vin=0.15, it occurs when η(0)=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=-3, starting from x=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 η(0), 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.

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 in ϕ4 theory in (1+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 relative velocity 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.

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 O(1) and that x˙ and y˙ are O(ϵ), with ϵ small. We wish x¨ and y¨ to be O(ϵ2) and see from Eqs. (44) and (45) that the oscillator amplitude η needs to be O(ϵ). With these assumptions, the solution of Eq. (43) for the oscillator, in the adiabatic approximation, takes the form η(t)=A(x(t),y(t))cos(∫0tω1(x(t′),y(t′))dt′),(49)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 in M at time t and not on the path through M. Clearly, A needs to be O(ϵ).

Differentiating (49) twice with respect to time, we find that η¨+ω12η=A¨cos(∫0tω1(x(t′),y(t′))dt′)-(2ω1A˙+Aω˙1)sin(∫0tω1(x(t′),y(t′))dt′),(50)where ω˙1 is the total time derivative of ω1 along the path. The first term on the right-hand side is O(ϵ3) and can be neglected relative to the remaining terms, which are O(ϵ2). Equation (43) for the oscillator is therefore satisfied, provided that 2ω1A˙+Aω˙1=0.(51)The solution is A=Cω1(52)with C a constant, so A is indeed like ω1, being just a function of the instantaneous position (x(t),y(t)). 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 O(ϵ).

We now derive the reduced, adiabatic equations of motion for x and y, using the approximate solution for the oscillator η(t)=Cω1(x(t),y(t))cos(∫0tω1(x(t′),y(t′))dt′).(53)We need to take the time average of η2 over one period of the oscillator, to derive the average force that acts. This is ⟨η2⟩=12A2=12C2ω1. The reduced equations are therefore Eqs. (44) and (45) with ω1η2 replaced by 12C2. It is more convenient to give the reduced Lagrangian, from which these equations follow, namely, Lred=12Ω(x,y)(x˙2+y˙2)-12C2ω1(x,y).(54)The constant C2 is determined by the oscillator’s initial conditions, and implicitly, x˙2,y˙2, and C2 are all O(ϵ2).

In summary, the oscillator dynamics generates a potential energy 12C2ω1(x,y) that, together with the conformal factor Ω, governs adiabatic motion through M. This motion has the conserved energy Ered=12Ω(x,y)(x˙2+y˙2)+12C2ω1(x,y),(55)which matches the conserved energy for the original Lagrangian (40) if we use the time-averaged oscillator energy. The latter is ⟨Eosc⟩=12A2ω12=12C2ω1,(56)where we have ignored the contribution of A˙ relative to that of A. Note that the adiabatic invariant C2 is 2ω1⟨Eosc⟩.

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 O(2) rotational symmetry. It is therefore convenient to use the polar coordinates x=rcosφ,y=rsinφ on M and the radial functions ω1(r) and Ω(r). r=0 is the apex of M, where the vortices are coincident, and r extends to +∞ as the vortices separate. ω(r) and Ω(r) are positive and have zero derivative at r=0. The angle φ has the range φ∈[-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=12Ω(r)(r˙2+r2φ˙2)-12C2ω1(r).(57)There are two constants of motion for Lred, the energy and angular momentum, E=12Ω(r)(r˙2+r2φ˙2)+12C2ω1(r),(58)J=Ω(r)r2φ˙.(59)After eliminating φ˙ from the energy in favor of the angular momentum, the radial motion can be found by quadrature, provided we know both ω1(r) and Ω(r), and all the initial data, including that for the oscillator.

As ω1(r) is an increasing function of r, the potential is attractive, but the effective potential in the reduced dynamics, Vred=12C2ω1(r)+J22Ω(r)r2,(60)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 oscillations, 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.

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 of M, around the apex. Here, we can approximate the geometry of M 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 |z|, where z=x+iy=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 amplitudes, involving deformations of the gauge and Higgs fields and a background gauge condition, can be parametrized by an abstract pair of amplitudes ζ and χ. These are initially complex, but we will impose a reality condition below. There is O(2) rotational symmetry about z=0, and the modes transform at z=0 as a doublet of O(2).

The potential energy for the modes is constructed using the 2×2 Hermitian matrix M=(λα(x-iy)α(x+iy)λ)=(λαz*αzλ),(61)where λ and α are real and positive constants; λ is the degenerate eigenvalue of M when z=0. The eigenvalues of M split for z≠0, becoming λ±α|z|. The complete Lagrangian of the model, with standard kinetic terms and a quadratic potential obtained using M, is L=12{z˙*z˙+ζ˙*ζ˙+χ˙*χ˙-(ζ*χ*)(λαz*αzλ)(ζχ)}=12{r˙2+r2φ˙2+ζ˙*ζ˙+χ˙*χ˙-(ζ*χ*)(λαre-iφαreiφλ)(ζχ)}.(62)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=λ+a·σ=λ+a1σ1+a2σ2+a3σ3, where σ1, σ2, and σ3 are the Pauli matrices. This has eigenvalues λ±|a|, so the eigenvalues are degenerate only when a=0. Generally, in a family of 2×2 Hermitian matrices, the eigenvalues degenerate 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 of Hermitian matrices with a “real” structure. If a2 were zero, then M would be manifestly real, and we could seek real eigenvectors. Degeneracy would then occur when a1=a3=0, a codimension-2 condition. It is more convenient in our model to set a3=0, as this simplifies the eigenvectors, but there is still a real structure, and eigenvalue degeneracy occurs at z=0, a single point in the two-dimensional moduli 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*=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=12z˙*z˙+χ˙*χ˙-λχ*χ-12αz*χ2-12αzχ*2.(63)The equations of motion now simplify to z¨+αχ2=0,(64)χ¨+λχ+αzχ*=0,(65)and there is a conserved energy E=12z˙*z˙+χ˙*χ˙+λχ*χ+12αz*χ2+12αzχ*2.(66)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 location z in moduli space. These coupled equations can probably not be solved analytically but only numerically.

However, we can treat the dynamics adiabatically, assuming that z varies on a timescale much longer than the inverse of the oscillation frequencies λ±α|z|. 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 O(1), with λ±α|z| remaining bounded away from zero, and that z˙=O(ϵ) and z¨=O(ϵ2), with ϵ small. This requires the mode amplitudes |ζ| and |χ| to be O(ϵ), i.e., small, but their oscillation frequencies are O(1).

To proceed, we need a basis of eigenvectors of the matrix M appearing in Eq. (62). The eigenvectors/eigenvalues are given by (λαre-iφαreiφλ)(ζχ)=ν(ζχ),(67)so the eigenvalues are ν+=λ+αr and ν-=λ-αr, with respective eigenvectors (ζχ)=(1eiφ)and(ζχ)=(1-eiφ)(68)of squared norm 2. However, these eigenvectors do not satisfy the reality condition ζ*=χ until we multiply by a suitable phase factor (which does not affect their orthonormality). The real eigenvectors are V+(φ)=±(e-12iφe12iφ)andV-(φ)=±(ie-12iφ-ie12iφ),(69)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=0 with x running from a positive to a negative value. There is a constant eigenvector (1,1)T along this line with eigenvalue λ+αx and another constant eigenvector (i,-i)T with eigenvalue λ-αx. To check this, one has to identify a point with negative x as having r=|x| and φ=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 structure over a neighborhood of the origin.

We next express the dynamical mode amplitudes in terms of the eigenvectors as (ζχ)=a+(t)V+(φ(t))+a-(t)V-(φ(t)),(70)where a+ and a- are real. (An arbitrary initial sign choice for the eigenvectors is made.) The time derivative is ddt(ζχ)=a˙+V++a+∂φV+φ˙+a˙-V-+a-∂φV-φ˙.(71)The eigenvectors have the simple φ derivatives ∂φV+=-12V-and∂φV-=12V+,(72)so ddt(ζχ)=(a˙++12a-φ˙)V++(a˙--12a+φ˙)V-.(73)The Lagrangian (62) therefore takes the form, in terms of the real amplitudes a+ and a-, L=12(r˙2+r2φ˙2)+(a˙++12a-φ˙)2+(a˙--12a+φ˙)2-(λ+αr)a+2-(λ-αr)a-2,(74)and the corresponding equations of motion are r¨-rφ˙2+α(a+2-a-2)=0,(75)r2φ˙+a˙+a--a˙-a++12(a+2+a-2)φ˙=ℓ,(76)ddt(a˙++12a-φ˙)+12(a˙--12a+φ˙)φ˙+(λ+αr)a+=0,(77)ddt(a˙--12a+φ˙)-12(a˙++12a-φ˙)φ˙+(λ-αr)a-=0.(78)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 approximation. We assume that r,φ are O(1), r˙,φ˙ are O(ϵ), and r¨,φ¨ are O(ϵ2), where ϵ is small. For Eq. (75) to be consistent, the oscillator amplitudes a+ and a- need to be O(ϵ) or smaller. 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=b and y=vt in Cartesians, where b is a positive O(1) constant and the velocity v is O(ϵ). Then, ℓ=bv, so r(t)=b2+v2t2andφ˙=bvb2+v2t2.(79)The straight line needs to miss the origin for φ˙ to remain bounded. (We shall consider a head-on collision below, using the Cartesian coordinate x.) Using the slowly time-dependent r(t) in the oscillator equations (77) and (78) and ignoring all the subleading terms that depend on φ˙, we deduce that the adiabatic solutions for the oscillator amplitudes are a+(t)=C+(λ+αr(t))14cos(∫0tλ+αr(t′)dt′+γ+),a-(t)=C-(λ-αr(t))14cos(∫0tλ-αr(t′)dt′+γ-).(80)The constants C+ and C-, together with the phases γ+ 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 ⟨a+2⟩=C+22λ+αr(t) and ⟨a-2⟩=C-22λ-αr(t). This results in a modified radial acceleration that can be attributed to a radial potential, Vrad=C+2λ+αr+C-2λ-αr+const.(81)This potential is repulsive for the second (a-) oscillator, as its frequency increases approaching r=0, and attractive for the third oscillator. In Eq. (75), the term a˙+a--a˙-a+ is oscillatory and has zero average, so we ignore it. However, the time-averaged value of a+2+a-2 is C+22λ+αr(t)+C-22λ-αr(t), and from Eq. (76), we can find the O(ϵ2) modification to φ˙ 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 a¨++(λ+αr)a+=-a˙-φ˙,(82)a¨-+(λ-αr)a-=a˙+φ˙.(83)Using the O(ϵ) solutions for the oscillator amplitudes and for φ˙ on the right-hand side, the particular integrals give O(ϵ2) corrections to the previously determined O(ϵ) 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 O(1).

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 O(ϵ2).

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 +∞, and that the second mode is excited. By symmetry, the motion remains on the x axis and can pass through x=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-αx(t), provided |x| is not large, rather than λ-αr(t) (with r=|x|).

The relevant equations of motion along the x axis are x¨-αu2=0,(84)u¨+(λ-αx)u=0,(85)obtained from Eqs. (64) and (65) by setting z=x and χ=-iu, where x and u are real. For this reduced system, there is a conserved energy, E=12x˙2+u˙2+(λ-αx)u2.(86)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 frequency λ-αx(t). The adiabatic solution of (85) is then u(t)=C(λ-αx(t))14cos(∫0tλ-αx(t′)dt′),(87)where the constant C is O(ϵ). After time averaging the oscillatory driving force u2, Eq. (84) becomes x¨-α2C2λ-αx=0,(88)with first integral 12x˙2+C2λ-αx=E.(89)The constant E can be identified with the total conserved energy. This is O(ϵ2) and has comparable contributions from the motion in moduli space and from the oscillating mode. The oscillating mode contributes an effective potential proportional to λ-αx to the moduli space dynamics, modifying what would otherwise be geodesic dynamics with x˙ 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=∫dx2E-2C2λ-αx.(90)The integral here is elementary. In terms of the spatial variable q=1-C2Eλ-αx,(91)we find the implicit solution q-13q3=αC4(2E)32t.(92)The solution runs between the stationary points of the cubic q-13q3, i.e., between q=±1, which is where αx=λ. Over a finite time interval, αx decreases from λ, climbs the potential and stops at t=0, then increases back to λ. The stopping point is where q=0, i.e., where αx=λ-E2C4. The motion may or may not pass through x=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=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=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 scattering 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=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 simulation 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.

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=12Ω(x,y)(x˙2+y˙2)+12η˙2-12ω22(x,y)η2,(93)where η is now the amplitude of the second mode and the squared frequency ω22 is a monotonically decreasing function of the vortex separation parameter ρ, approximately given by the expression (39).

For head-on collisions, we can consistently put y=0. Then, the Lagrangian (93) reduces to L=12Ω(x)x˙2+12η˙2-12ω22(x)η2,(94)with ω22(x)=ω22(0)-(ω22(0)-ω22(∞))(1-e-0.2108x).(95)Because ω22 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 approaching from x=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=0. With a smaller amplitude, they may pass through x=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=0.015. If the amplitude is smaller still, the vortices separate sufficiently along the vertical axis that the third mode enters the continuum spectrum, and the so-called spectral wall phenomenon 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.