Sphalerons and resonance phenomenon in kink-antikink collisions

We show that in some kink-antikink (KAK) collisions sphalerons, i.e., unstable static solutions - rather than the asymptotic free soliton states - can be the source of the internal degrees of freedom (normal modes) which trigger the resonance phenomenon responsible for the fractal structure in the final state formation.


I. INTRODUCTION
Nonlinear field theories supporting kinks in one plus one dimension, and their embeddings in higher dimensions called defects, have a wide range of applications, from protein folding and optical fibers to particle physics and cosmology. Surprisingly, the scattering of kinks reveals a fascinating and rather complex behavior [1-3], despite its conceptual simplicity. The best example is the fractal structure observed in the final state formation in KAK scattering in non-integrable scalar field theories like the φ 4 model [4][5][6][7][8] where, depending on the initial velocity v i of the colliding solitons, two main channels occur: 1) annihilation to the vacuum via the formation of quasi-periodic, slowly decaying states ("bions"), or 2) the reappearance of free kinks after a few bounces. Both the regions where a bion is created (bion chimneys) and the n-bounce regions (bounce windows) show a fractal pattern.
Qualitatively, the existence of the fractal structure is attributed to the so-called resonance phenomenon [4], [6], [8] which couples the kinetic degrees of freedom (DoF) of a soliton with the internal ones. The kinetic (translational) DoF are obviously related to the lightest excitation of the soliton, its zero mode. The internal DoF are typically the massive normal (or quasi-normal [9]) modes of the asymptotic states, i.e., of the free, infinitely separated solitons. During a collision, the energy which is initially stored in the kinetic DoF of the incoming solitons (and possibly also in initially exited massive modes) may be transferred temporarily to the internal DoF, which results in a complicated pattern of final states.
Here it is assumed that the normal modes take the fixed form resulting from the asymptotic kink and antikink during the whole scattering, and no deformation due to the presence of the collision partner is taken into account. We say that the normal modes are "frozen". The reason is arXiv:2109.01834v1 [hep-th] 4 Sep 2021 that for KAK configurations with a finite separation no static solution with a well-defined linear problem (i.e., well-defined normal modes) is available in the φ 4 model. For many years it was believed that the non-integrable mechanical system resulting from a "collective coordinate" (CC) approximation, where only the relative KAK separation s(t) and the two normal-mode amplitudes A 1 (t) and A 2 (t) are promoted to dynamical variables, allows for a reliable quantitative description of KAK scattering and the resonance mechanism. A typographical error [8] in the original paper [4] which propagated through the subsequent literature, however, invalidated those conclusions.
Only very recently it has been shown that the CC approach allows for a reliable description of KAK scattering in the reflection-symmetric case A ≡ A 1 = −A 2 [10]. Here the two crucial steps are i) the removal of a coordinate singularity in the "moduli space" spanned by the CC, and ii) a judicious choice of the initial value of the amplitude A, which must be inferred from the one-kink sector. The same resonance mechanism is assumed to be responsible for the existence of fractal structures also in many other models [11][12][13][14][15][16][17][18][19][20].
In contrast, in some theories there exist static KAK configurations solving the static Euler-Lagrange equations for arbitrary KAK separation, implying the existence of (at least) a oneparameter family of static KAK solutions, parametrized, e.g., by the separation parameter s. In this case, each static KAK solution has its own linear problem, and the resulting normal modes and their frequencies, in general, vary with s. Adiabatic low-energy KAK scattering may be described by an effective motion on the subspace spanned by s, but for more excited initial conditions the normal modes and their variation with s play a crucial role. In particular, whenever a discrete normal mode frequency disappears into the continuum at a particular value s SW of s, this has a huge impact on the scattering process (spectral wall phenomenon [21][22][23]). In an effective CC description, it is mandatory to include the full "dynamical" normal modes, i.e., their variation with s. Any restriction to the frozen normal modes completely misses the spectral walls.
In this letter, we consider an intermediate situation, where in addition to the free infinitely separated solitons there exists a sphaleron [24][25][26], i.e., an unstable static KAK solution at a certain finite separation s b . It turns out that the lowest normal mode of this unstable solution can be the main factor governing the dynamics of KAK collisions in some cases. Specifically, this normal mode may act as the internal DoF capable of storing energy and triggering the fractal structure of final states. This result presents another manifestation of the importance of the temporary form of the field and, therefore, of the temporary properties of the related normal modes, in KAK collisions.
More generally, our results demonstrate the high significance of sphalerons in the dynamics of topological solitons.

II. MODEL WITH UNSTABLE SOLUTION
The family of models which we consider represent a modification of φ 4 theory where is the usual φ 4 model, while is a BPS-impurity modification of the φ 4 model, where the impurity (background field) σ = tanh(x) [22,27], and ∈ [0, 1]. Here BPS (=Bogomolnyi-Prasad-Sommerfield) refers to the fact that the static field equations can be reduced to first-order ("BPS") equations, which possess a oneparameter family of KAK solutions, see eq. (4) below.
We choose the particular model (1) because it has an unstable static solution -a sphaleronfor any ∈ (0, 1). Moreover, it allows to study the competition between the standard mechanism, where the frozen shape mode of the free soliton takes part in the resonant phenomenon, and the new mechanism due to the unstable static solution. The former occurs as → 1, as in the limit = 1 we recover φ 4 theory, while the latter dominates for 2 < ∼ 0.4.
To understand the structure of solutions, let us begin with the = 0 limit, where there are infinitely many static, energetically equivalent solutions [22] φ(x; Their energy is set to 0. They are parametrized by φ 0 ∈ (−∞, 1), which coincides with the value of the field at the origin, and describes a kink and antikink at arbitrary distance [28]. For example, Now, the ground state is φ = −1 with V = 0, while the infinitely separated KAK pair is a local minimum, where V = 8/3. Importantly, there is also a local maximum, occurring at φ b 0 ≈ 0.89167,  Concerning the spectrum of small perturbations, we first observe that the continuum threshold is independent, ω 2 c = 4 [29]. As indicated, there are two configurations, i.e. (asymptotic) static solutions, with a well-defined linear problem, namely the asymptotic KAK pair and the sphaleron.
The asymptotic state is always an infinitely separated φ 4 KAK pair, separated by the vacuum +1, where x 0 → ∞. Therefore, each soliton has one massive normal mode, i.e., the shape mode, η(x) = sinh(x ± x 0 )/ cosh 2 (x ± x 0 ) with frequency ω 2 shape = 3. The vacuum +1 also provides two normal modes located at the origin, as a consequence of the presence of the impurity σ. Their frequencies grow with . Specifically, ω 2 imp (n) = 4 − 1/4( √ 25 − 24 2 − 1 − 2n) 2 , n = 0, 1. They disappear into the continuum spectrum at 2 = 1 and 2 = 2/3 for n = 0, 1 respectively, as it must be (they must be absent in the no-impurity limit → 1). The sphaleron solution, on the other hand, contributes with an obvious unstable mode with ω 2 0 < 0 and several positive frequency normal modes. In Tab. I we list the frequency ω 2 of the most important, first even normal mode for several .

III. FLOW OF THE FRACTAL STRUCTURE
Now we collide the solitons. We choose a largely separated KAK pair as our initial configuration, which are boosted towards each other Here, 2x 0 is the separation distance, v i is the initial velocity of the solitons and γ = (1 − v 2 i ) −1/2 . This is numerically evolved in the EL equation for ∈ [0, 1]. For each model (each ) we vary the initial velocity v i from 0 to 0.3. In Fig. 2 we plot the value of the field at the origin, φ(0, t), after a large time t * = 550 + 40/v i as a function of both the initial velocity v i and . This shows how the fractal structure in the final state formation changes with .
In Fig. 2 the blue regions correspond to the case where the kink and antikink reappear in the final state and therefore represent bounce windows. For example, the big blue region is onebounce scattering. There is also a zero-bounce window -a blue region (for very small v in , below the multicolor region) related to an elastic process resulting from the existence of the unstable static solution and the corresponding local energy maximum which, during the evolution, acts as a kind of barrier. Initially separated solitons must have enough kinetic energy to go through this point. Therefore, there exists a critical velocityṽ crit below which the kinks cannot pass over the sphaleron. The lineṽ crit can be found from the energy conservation law. As → 1, the sphaleron bion, a sort of quasi-oscillating state, decays into the vacuum via the emission of radiation. Typically, one defines another critical initial velocity, v crit , at which the fractal structure accumulates, while above v crit only one-bounce scattering is observed. In Fig. 2 this velocity is defined by the border between the large blue and multicolor regions.
The two-bounce windows, bion chimneys and critical velocities are clearly visible in Fig. 3, where we display the time dependence of the value of the field at the origin, φ(0, t), as a function of the initial velocity v i for 2 = 0.2 and 0.4. They are qualitatively similar to φ 4 theory ( = 1) although the range of v i at which the fractal structure exists strongly decreases as takes smaller values. Specifically, v crit decreases if → 0. One may observe that there are in fact two regimes: (i) a linear decrease of v crit for 0.5 < ∼ 2 ≤ 1; (ii) a much slower decrease of v crit for 0 ≤ 2 < ∼ 0.4.
These two regimes reflect two different origins of the resonance phenomenon.

IV. ORIGIN OF THE RESONANCE PHENOMENON
Let us now concentrate on the second regime where 2 < ∼ 0.4 and the sphaleron is very well approximated by the exact configuration φ(x; φ b 0 ). To understand the bouncing windows in Fig. 3, we take a closer look at the time dependence of the field at the origin φ(x = 0, t). In Fig. 4 we show representative trajectories φ(x = 0, t) for the 2 = 0.2 case. The black curve shows KAK scattering with initial velocity v = 0.0173562, that is, below the critical velocityṽ crit . The result is the elastic (no bounce) reflection at the sphaleron (denoted as the dashed line). As the velocity increases, we enter the region of annihilation via bion formation. Indeed, for v = 0.0173564 (red curve), the trajectory passes the unstable solution.
Then, it is reflected at the repulsive core (repulsive interaction due to the potential (5)), which excites a bound mode. However, the reflected solution has too little energy to climb once again through the sphaleron. As a result, the KAK solution oscillates between the core and the unstable maximum, slowly decaying into the φ = −1 vacuum. For v = 0.0174545, we explore one of the 2-bounce windows. The corresponding trajectory (blue curve) bounces twice at the repulsive core and once at the unstable maximum. Each bouncing at the core affects the amplitude of the bound mode, which finally allows the trajectory to pass the unstable barrier. Thus, in the final state we and has nothing to do with a bound mode of the free (anti)kink.
In the regime where → 1, the well-know situation occurs. Namely, the free solitons provide the internal modes which enter into the resonance phenomenon, as very recently proved in [10].  Our results also indicate that the full understanding of soliton dynamics cannot be achieved only in terms of the asymptotic states, i.e., free solitons and their excitations [30], even though such a frozen approach works well in many cases, see, e.g., [10]. On the contrary, the temporary properties of the field can play the most prominent role and can, in fact, govern the KAK scattering.
Obviously, this leads to an increased complexity of soliton dynamics and should result in the discovery of fractal structures in previously unexpected situations. The importance of transient, temporary states and their normal modes for KAK scattering was already emphasized in [31] for the φ 6 model, although no unstable solution was available in that case.
Moreover, our results explicitly show the high relevance of sphaleron solutions in the nonlinear dynamics of kinks. An important role of sphalerons in the dynamics of nonlinear field theories was conjectured since their discovery. An explicit demonstration, however, is in general difficult because of the complex nature of time-dependent problems in those theories. We would like to remark that unstable kinks (or domain walls) are common in models with two (and more) scalar fields as e.g., the Montonen-Sarker-Trullinger-Bishop model [32][33][34]. They also frequently exist in kink models with background fields where, at some point, the attractive force of a KAK pair can be compensated by external fields [35] or by the presence of boundaries [36], [37]. Finally, they have been recently found in a model with a higher order derivative term [38].