arXiv subjects

N. S. Manton

Publications and source records attributed to N. S. Manton.

At least 37 records · Page 2Linked to original sources

arXiv2020

Kink Moduli Spaces -- Collective Coordinates Reconsidered

Moduli spaces - finite-dimensional, collective coordinate manifolds - for kinks and antikinks in $ϕ^4$ theory and sine-Gordon theory are reconsidered. The field theory Lagrangian restricted to moduli space defines a reduced Lagrangian, combining a potential with a kinetic term that can be interpreted as a Riemannian metric on moduli space. Moduli spaces should be metrically complete, or have an infinite potential on their boundary. Examples are constructed for both kink-antikink and kink-antikink-kink configurations. The naive position coordinates of the kinks and antikinks sometimes need to be extended from real to imaginary values, although the field remains real. The previously discussed null-vector problem for the shape modes of $ϕ^4$ kinks is resolved by a better coordinate choice. In sine-Gordon theory, moduli spaces can be constructed using exact solutions at the critical energy separating scattering and breather (or wobble) solutions; here, energy conservation relates the metric and potential. The reduced dynamics on these moduli spaces accurately reproduces properties of the exact solutions over a range of energies.

hep-th↗

arXiv2020

Kink-Antikink Interaction Forces and Bound States in a Biharmonic $ϕ^4$ Model

We consider the interaction of solitons in a biharmonic, beam model analogue of the well-studied $ϕ^4$ Klein-Gordon theory. Specifically, we calculate the force between a well separated kink and antikink. Knowing their accelerations as a function of separation, we can determine their motion using a simple ODE. There is good agreement between this asymptotic analysis and numerical computation. Importantly, we find the force has an exponentially-decaying oscillatory behaviour (unlike the monotonically attractive interaction in the Klein-Gordon case). Corresponding to the zeros of the force, we predict the existence of an infinite set of field theory equilibria, i.e., kink-antikink bound states. We confirm the first few of these at the PDE level, and verify their anticipated stability or instability. We also explore the implications of this interaction force in the collision between a kink and an oppositely moving antikink.

nlin.PS↗

arXiv2020

Evidence for Tetrahedral Structure of Calcium-40

More than 100 excited states of the Calcium-40 nucleus, with isospin 0, are classified into rotational-vibrational bands of an intrinsic tetrahedral structure. Almost all observed states below 8 MeV can be accommodated, as well as many high-spin states above 8 MeV. The bands have some similarity to those classifying states of Oxygen-16, but the A-mode vibrational frequency is lower relative to the E-mode and F-mode frequencies than in Oxygen-16. Previously identified rotational bands up to spin 16 and energy above 20 MeV are unified here into a smaller number of tetrahedral bands.

nucl-th↗

arXiv2019

Iterated $ϕ^4$ Kinks

A first order equation for a static $ϕ^4$ kink in the presence of an impurity is extended into an iterative scheme. At the first iteration, the solution is the standard kink, but at the second iteration the kink impurity generates a kink-antikink solution or a bump solution, depending on a constant of integration. The third iterate can be a kink-antikink-kink solution or a single kink modified by a variant of the kink's shape mode. All equations are first order ODEs, so the nth iterate has n moduli, and it is proposed that the moduli space could be used to model the dynamics of n kinks and antikinks. Curiously, fixed points of the iteration are $ϕ^6$ kinks.

hep-th↗

arXiv2019

Oxygen-16 Spectrum from Tetrahedral Vibrations and their Rotational Excitations

A reinterpretation of the complete energy spectrum of the Oxygen-16 nucleus up to 20 MeV, and partly beyond, is proposed. The underlying intrinsic shape of the nucleus is tetrahedral, as in the naïve alpha-particle model and other cluster models, and A, E and F vibrational phonons are included. The A- and F-phonons are treated in the harmonic approximation, but the E-vibrations are extended into a two-dimensional E-manifold of D2-symmetric, four-alpha-particle configurations, following earlier work. This allows for the underlying tetrahedral configuration to deform through a square configuration into the dual tetrahedron, so there is tunnelling between the tetrahedron and its dual, with the associated breaking of parity doubling. However, E-manifold states can still be interpreted in terms of E-phonons. Rotational excitations of the vibrational states are analysed as in the classic work of Dennison, Robson and others, with centrifugal corrections to the rotational energy spectrum included. States with F-phonons require Coriolis corrections too. The first-excited $0^+$ state at 6.05 MeV is modelled as a state with two E-phonons; this allows a good fit of the lowest $2^+$ and $2^-$ states as excitations with one E-phonon. The Coriolis parameter $ζ$ is chosen positive to ensure the right splitting of the $3^+$ and $3^-$ states near 11 MeV. Altogether, about 80 states with isospin zero are predicted below 20 MeV, and these match rather well the more than 60 experimentally tabulated states. Several high-spin states are predicted, up to spin 9 and energy 30 MeV, and these match some of the high-spin, natural parity states in this energy range that have been observed. The model proposed here is mainly phenomenological but it receives some input from analysis of Skyrmions with baryon number 16.

nucl-th↗

arXiv2019

The Inevitability of Sphalerons in Field Theory

The topological structure of field theory often makes inevitable the existence of stable and unstable localised solutions of the field equations. These are minima and saddle points of the energy. Saddle point solutions occurring this way are known as sphalerons, and the most interesting one is in the electroweak theory of coupled W, Z and Higgs bosons. The topological ideas underpinning sphalerons are reviewed here.

hep-th↗

arXiv2018

Forces between Kinks and Antikinks with Long-range Tails

In a scalar field theory with a symmetric octic potential having a quartic minimum and two quadratic minima, kink solutions have long-range tails. We calculate the force between two kinks and between a kink and an antikink when their long-range tails overlap. This is a nonlinear problem, solved using an adiabatic ansatz for the accelerating kinks that leads to a modified, first-order Bogomolny equation. We find that the kink-kink force is repulsive and decays with the fourth power of the kink separation. The kink-antikink force is attractive and decays similarly. Remarkably, the kink-kink repulsion has four times the strength of the kink-antikink attraction.

hep-th↗

arXiv2018

Force between Kinks with Long-range Tails

In a scalar field theory that has a symmetric octic potential with a quartic minimum and two quadratic minima, kink and mirror kink solutions have long-range tails. We calculate the force between these kinks when their long-range tails overlap. This is a nonlinear problem, solved using an adiabatic ansatz for the accelerating kinks that leads to a modified, first-order Bogomolny equation. We find the force is repulsive and decays with the fourth power of the kink separation.

hep-th↗

arXiv2017

Quantized Skyrmions from SU(4) Weight Diagrams

Starting from solutions of the lightly-bound Skyrme model, we construct many new Skyrmion solutions of the standard Skyrme model with tetrahedral or octahedral symmetry. These solutions are closely related to weight diagrams of the group SU(4), which enables us to systematically derive some geometric and energetic properties of the Skyrmions, up to baryon number 85. We discuss the rigid body quantization of these Skyrmions, and compare the results with properties of a selection of observed nuclei.

hep-th↗

arXiv2016

Complex Geometry of Nuclei and Atoms

We propose a new geometrical model of matter, in which neutral atoms are modelled by compact, complex algebraic surfaces. Proton and neutron numbers are determined by a surface's Chern numbers. Equivalently, they are determined by combinations of the Hodge numbers, or the Betti numbers. Geometrical constraints on algebraic surfaces allow just a finite range of neutron numbers for a given proton number. This range encompasses the known isotopes.

hep-th↗

arXiv2016

A dynamical $α$-cluster model of $^{16}$O

We calculate the low-lying spectrum of the $^{16}$O nucleus using an $α$-cluster model which includes the important tetrahedral and square configurations. Our approach is motivated by the dynamics of $α$-particle scattering in the Skyrme model. We are able to replicate the large energy splitting that is observed between states of identical spin but opposite parities, as well as introduce states that were previously not found in other cluster models, such as a $0^-$ state. We also provide a novel interpretation of the first excited state of $^{16}$O and make predictions for the energies of $6^-$ states that have yet to be observed experimentally.

nucl-th↗

arXiv2016

Electromagnetic Transition Strengths for Light Nuclei in the Skyrme model

We calculate reduced $B(E2)$ electromagnetic transition strengths for light nuclei of mass numbers $B=8,12,16,20,24$ and $32$ within the Skyrme model. We find that the predicted transition strengths are of the correct order of magnitude and the computed intrinsic quadrupole moments match the experimentally observed effective nuclear shapes. For the Hoyle state we predict a large $B(E2)\!\uparrow$ value of $0.0521\, \rm{e}^2\rm{b}^2$. For Oxygen-16, we can obtain a quantitative understanding of the ground state rotational band and the rotational excitations of the second spin-0 state, $0_2^+$.

nucl-th↗

arXiv2015

Electron Scattering Intensities and Patterson Functions of Skyrmions

The scattering of electrons off nuclei is one of the best methods of probing nuclear structure. In this paper we focus on electron scattering off nuclei with spin and isospin zero within the Skyrme model. We consider two distinct methods and simplify our calculations by use of the Born approximation. The first method is to calculate the form factor of the spherically averaged Skyrmion charge density; the second uses the Patterson function to calculate the scattering intensity off randomly oriented Skyrmions, and spherically averages at the end. We compare our findings with experimental scattering data. We also find approximate analytical formulae for the first zero and first stationary point of a form factor.

nucl-th↗

arXiv2015

Analytic vortex solutions on compact hyperbolic surfaces

We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations.

hep-th↗

arXiv2014

States of Carbon-12 in the Skyrme Model

The Skyrme model has two Skyrmion solutions of baryon number $12$, with $D_{3h}$ and $D_{4h}$ symmetries. The first has an equilateral triangular shape and the second an extended linear shape, analogous to the triangle and linear chain structures of three alpha particles. We recalculate the moments of inertia of these Skyrmions, and deduce the energies and spins of their quantized rotational excitations. There is a good match with the ground-state band of Carbon-12, and with the recently established rotational band of the Hoyle state. The ratio of the root mean square matter radii also matches the experimental value.

nucl-th↗

arXiv2014

A Skyrme model approach to the spin-orbit force

The spin-orbit force is a vital tool in describing finite nuclei and nucleon interactions; however its microscopic origin is not fully understood. In this paper we study a model inspired by Skyrmions which provides a classical explanation of the force. To simplify the calculations the Skyrmions are approximated as two-dimensional rigid discs which behave like quantum cogwheels.

hep-th↗

arXiv2014

Rational Relativistic Collisions

If two point particles collide relativistically in one dimension, and the masses, velocities and gamma factors of the incoming particles are rational numbers, then the velocities and gamma factors of the outgoing particles are rational. Numerous examples can be found using Pythagorean triples. At all velocities, the collision results in a Lorentzian reflection of the 2-momenta of the particles.

physics.pop-ph↗

arXiv2014

Quantization of T_d- and O_h-symmetric Skyrmions

The geometrical construction of rational maps using a cubic grid has led to many new Skyrmion solutions, with baryon numbers up to 108. Energy spectra of some of the new Skyrmions are calculated here by semi-classical quantization. Quantization of the B=20 T_d-symmetric Skyrmion, which is one of the newly found Skyrmions, is considered, and this leads to the development of a new approach to solving Finkelstein-Rubinstein (F-R) constraints. Matrix equations are simplified by introducing a Cartesian version of angular momentum basis states, and the computations are easier. The quantum states of all T_d-symmetric Skyrmions, constructed from the cubic grid, are classified into three classes, depending on the contribution of vertex points of the cubic grid to the rational maps. The analysis is extended to the larger symmetry group O_h. Quantum states of O_h-symmetric Skyrmions, constructed from the cubic grid, form a subset of the T_d-symmetric quantum states.

hep-th↗