# Skyrmion-Skyrmion Scattering and Nuclear Physics

Lead Research Organisation:
University of Kent

Department Name: Sch of Maths Statistics & Actuarial Scie

### Abstract

To understand a complex system such as the world we live in, it is useful to understand its parts. An atom, for example, consists of electrons and the atomic nuclei, which in turn consist of protons and neutrons. As particle accelerators have revealed, protons and neutrons are also made up of smaller particles, each one contains three quarks. And so, now that we know the protagonists, we need to understand how they interact. Apart from gravity, there are three interactions: the electromagnetic interaction which acts only on charged particles (such as electrons and protons); the weak interaction which only affects certain particles and is very weak indeed; and the strong interaction which holds atomic nuclei together. The strong interaction is much stronger (and much more complicated) than the other two, but it falls off very rapidly with distance. In the theory of the strong interaction known as quantum chromodynamics (QCD), quarks interact via yet another type of particle called gluons. At the very high energies found in particle accelerators such as LHC in Geneva, QCD explains experimental findings to an amazing accuracy. However, at lower energy levels, quarks like to stick together, either in threes to form protons and neutrons or in twos to form particles known as pions. As a consequence, a single quark can never be observed on its own. This means that QCD becomes very complicated at low energies and it is extremely difficult to calculate the properties of a proton from QCD.An alternative approach to this problem was devised in 1961 by the British physicist Tony Skyrme (1922-1987), before QCD was discovered. Skyrme proposed a theory which involves only the particles that are directly observable, the protons, neutrons and pions. The theory is very elegant, since protons and neutrons are described as topological solitons, which can be thought of as knots in a field of pions. In Skyrme's honour, these solitons are known as Skyrmions. In 1979, the American theoretical physicist Edward Witten showed that the Skyrme model captures the essential features of QCD for low energies. This means that we can use the Skyrme model to describe the phenomena in nuclear physics.In standard nuclear physics, atomic nuclei are bound states of protons and neutrons under rules which have to be determined for each case. This approach gives accurate results but does not provide a unified picture coming from first principles such as QCD. The Skyrme model lies between standard nuclear physics and QCD, and thus provides a different perspective on nuclear physics experiments.Although the Skyrme model is much simpler than QCD, many difficulties remain. Since protons and neutrons obey the laws of quantum mechanics, the Skyrme model with its Skyrmions also needs to be quantized. There are some technical issues (namely the calculation of what are called Finkelstein-Rubinstein constraints) which I have recently resolved. Using these techniques masses and excitation energies of atomic nuclei have been calculated and achieved remarkable qualitative agreement with nuclear physics. Despite its motivation from physics, this project is firmly rooted in mathematical physics using sophisticated mathematical tools devised by pure and applied mathematicians.A vast amount of nuclear physics experiments is concerned with the scattering of atomic nuclei - for example hitting a helium atom with a proton and measuring how the atomic nuclei break up. Such experiments can be compared to Skyrmion-Skyrmion scattering, an area of research which is currently in need of development. The aim of my project is to use both analytical approximations and numerical calculations to understand Skyrmion-Skyrmion scattering. There are early indications that the Skyrme model can help us to make experimental predictions in situations that are difficult to address with the standard techniques of nuclear physics.

### Planned Impact

My project is about using Skyrmion-Skyrmion scattering to describe the scattering of atomic nuclei. The UK is internationally leading in the field of topological solitons, and my work will have a big impact in the soliton community. The project is highly interdisciplinary, addressing problems in pure mathematics and numerical analysis. So, there will also be a lot of interest from the mathematical community. The main aim of the project is to use the Skyrme model to calculate the likelihood of various scattering events and compare the results to nuclear physics experiments. If these calculations were successful, this could have a major impact on nuclear physics. Obviously, having a better understanding of nuclear physics is of great practical importance for example for the safety of nuclear power. The project will give a PDRA the opportunity to further develop his numerical and analytical skills and become an expert on topological solitons and their application. The project will also give rise to interesting PhD projects which will further build knowledge in this area and contribute to the UK continuing to be a world leader in this field. Finally, the area of topological solitons is of a particular beauty both in terms of visualizing the various field configurations and in terms of the geometric concepts involved. This lends itself to promoting science in schools and via general talks to non-specialists.

## People |
## ORCID iD |

Steffen Krusch (Principal Investigator) |

### Publications

Ashcroft J
(2015)

*Baby Skyrme models without a potential term*in Physical Review D
Battye R
(2014)

*Classically isospinning Skyrmion solutions*in Physical Review D
Battye R
(2013)

*Isospinning baby Skyrmion solutions*in Physical Review D
Foster D
(2015)

*Scattering of Skyrmions*in Nuclear Physics B
Foster D
(2013)

*Negative baryon density and the folding structure of the B = 3 skyrmion*in Journal of Physics A: Mathematical and Theoretical
Haberichter, M.

*Isospinning Skyrmions*in Proceedings of Quarks-2014
Krusch S
(2015)

*Moduli spaces of lumps on real projective space*in Journal of Mathematical PhysicsTitle | Photos/Images for the EPSRC Science Photo Competition 2014 |

Description | We are producing photos and images to be submitted for this year's EPSRC Science photo competition. We show images that illustrate new discoveries and understanding arising from the research conducted within this grant. |

Type Of Art | Image |

Year Produced | 2014 |

Impact | We want to share our research outcomes through pictures with the general audience. |

URL | http://www.epsrc.ac.uk/newsevents/events/photocomp2014/ |

Description | The Skyrme model is a model for nuclear physics which describes atomic nuclei. This project made progress in calculating ground and excited states of nuclei. We obtained a better understanding of scattering in the Skyrme model which is likely to help explain scattering experiments in nuclear physics. |

Exploitation Route | The project develops sophisticated mathematics and computer simulations to solve problems in nuclear physics. This has impact on Education, namely the next generation of UK researchers (the project suported two PhD students and two PostDocs). The research gives rise to a better understanding of nuclear physics which in turn can lead to better energy secrurity. (A conference with nuclear physicists is being organised in China by Prof Manton.) |

Sectors | Education,Energy |

URL | http://www.kent.ac.uk/smsas/personal/skyrmions/index.html |

Description | IoP Conference Grant |

Amount | £700 (GBP) |

Organisation | Institute of Physics (IOP) |

Sector | Learned Society |

Country | United Kingdom |

Start | 08/2014 |

End | 08/2014 |

Description | LMS Research in Pairs: Dave Foster and Derek Harland (Leeds) |

Amount | £600 (GBP) |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 05/2014 |

End | 06/2014 |

Description | School of Mathematics (Research Visit of Dr Wereszczynski) |

Amount | £500 (GBP) |

Organisation | University of Kent |

Sector | Academic/University |

Country | United Kingdom |

Start | 03/2014 |

End | 09/2014 |

Description | New Research links with Jagiellonian University, Krakow, Poland |

Organisation | Jagiellonian University |

Country | Poland |

Sector | Academic/University |

PI Contribution | The grant led to a development of a research collaboration Mareike Haberichter, Dave Foster and Steffen Krusch (supported by the grant) and Andrzej Wereszczynski at the Jagiellonian University, Krakow, Poland. David Foster and Steffen Krusch visited Krakow in November 2013. Andrzej Wereszczynski visited Kent in April 2014. This collaboration has led to new research ideas. Further research visits took place in May and July. We are planning to apply for further research funding. |

Collaborator Contribution | Andrzej Wereszczynski organised workshops in Krakow which were attended by David Foster, Mareike Haberichter and Steffen Krusch. He also gave talks at Kent. He is applying for funding for a research visit by Mareike Haberichter for three months in summer 2015. |

Impact | Workshops and Conferences, publication in preparation |

Start Year | 2013 |

Description | New Research links with University of Cambridge, Cambridge, UK |

Organisation | University of Cambridge |

Department | Department of Applied Mathematics and Theoretical Physics (DAMTP) |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | The grant led to a development of a research collaboration between Steffen Krusch, Dave Foster, Mareike Haberichter and Prof. Nick Manton at the University of Cambridge, UK. This collaboration has led to new research ideas and several ongoing joint projects on Skyrmion solutions (higher charge solutions, multilayer rational map ansätze, different calibration choices). |

Collaborator Contribution | Joint research projects on Skyrmions; new inghts into Skyrmion solutions, Publications in preparation. |

Impact | Mareike Haberichter works as a Research Associate at DAMTP. Publications in preparation. |

Start Year | 2013 |

Description | New Research links with University of Leeds, Leeds, UK |

Organisation | University of Leeds |

Department | School of Mathematics Leeds |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | The grant led to a development of a research collaboration between Dave Foster and Derek Harland at the University of Leeds, UK. This collaboration has led to new research ideas and an ongoing collaboration on gauged Skyrmion solutions. This collaboration got additional funding by a LMS Research in Pairs grant. |

Collaborator Contribution | Joint research project on gauged Skyrmions; new inghts into gauged Skyrmion solutions, Publication in preparation. |

Impact | This collaboration will results in new insights on gauged Skyrmion solutions |

Start Year | 2014 |

Description | LMS 150th Anniversary Mathematics Festival: What's your angle? |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Public/other audiences |

Results and Impact | The London Mathematical Society collaborated with the Science Museum, the interactive theatre company non zero one, and a number of UK universities including the topological soliton group at Kent to create an immersive mathematics festival, titled 'What's Your Angle? Uncovering Maths'. The festival took place between the 25 and 29 November in the Science Museum's Wellcome Wing Basement Gallery. The four-day event comprised an adults-only (Calcu)Lates event, a schools day, and two public days. Over 1400 people visited. |

Year(s) Of Engagement Activity | 2015 |

URL | http://www.lms.ac.uk/2015/footfall_festival |

Description | Research Placements for Sixth Form Students |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | Yes |

Geographic Reach | National |

Primary Audience | Schools |

Results and Impact | The students found out more about what STEM subjects can offer. The students made genuine contributions to our projects and felt a real sense of ownership about their projects. The students decided to study STEM. |

Year(s) Of Engagement Activity | 2014 |

URL | http://www.nuffieldfoundation.org/nuffield-research-placements |

Description | Talk at Kent Sciences Research Festival 2014 |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | Yes |

Geographic Reach | Local |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | Presentation sparked questions and interest. Possible Collaborations and links with other Kent research groups were investigated. After talk, possible joint research projects with the condensed matter group in Kent were explored. |

Year(s) Of Engagement Activity | 2014 |

URL | http://www.kent.ac.uk/stms/researchfestival/ |