It is understood that the Skyrme model has a topologically interesting
baryonic excitation which can model nuclei. So far no stable knotted solutions,
of the Skyrme model, have been found. Here we investigate the dynamics of Hopf
solitons decaying to the vacuum solution in the Skyrme model. In doing this we
develop a matrix-free numerical method to identify the minimum eigenvalue of
the Hessian of the corresponding energy functional. We also show that as the
Hopf solitons decay, they emit a cloud of isospinning radiation.Comment: 10 pages, 12 image

Foster, David

English

arXiv

David Foster

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Journal of Physics A Mathematical and Theoretical

The decay of Hopf solitons in the Skyrme model

It is understood that the Skyrme model has a topologically interesting baryonic excitation which can model nuclei. So far no stable knotted solutions, of the Skyrme model, have been found. Here we investigate the dynamics of Hopf solitons decaying to the vacuum solution in the Skyrme model. In doing this we develop a matrix-free numerical method to identify the minimum eigenvalue of the Hessian of the corresponding energy functional. We also show that as the Hopf solitons decay, they emit a cloud of isospinning radiation

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Foster, D. (2017). The decay of Hopf solitons in the Skyrme model.Journal of Physics A: Mathematical and Theoretical, 50(40), Article405401. https://doi.org/10.1088/1751-8121/aa86c2Peer reviewed versionLicense (if available):CC BY-NC-NDLink to published version (if available):10.1088/1751-8121/aa86c2Link to publication record on the Bristol Research PortalPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia IOP Publishing at http://iopscience.iop.org/article/10.1088/1751-8121/aa86c2. Please refer to terms of use ofthe publisher.University of Bristol – Bristol Research PortalGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/brp-terms/Home Search Collections Journals About Contact us My IOPscienceThe decay of Hopf solitons in the Skyrme modelThis content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 137.222.149.11This content was downloaded on 22/08/2017 at 14:32Manuscript version: Accepted ManuscriptFosterTo cite this article before publication: Foster, 2017, J. Phys. A: Math. Theor, at press:https://doi.org/10.1088/1751-8121/aa86c2This Accepted Manuscript is: © 2017 IOP Publishing LtdDuring the embargo period (the 12 month period from the publication of the Version of Record of thisarticle), the Accepted Manuscript is fully protected by copyright and cannot be reused or repostedelsewhere.As the Version of Record of this article is going to be / has been published on a subscription basis,this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 monthembargo period.After the embargo period, everyone is permitted to copy and redistribute this article for non-commercialpurposes only, provided that they adhere to all the terms of the licencehttps://creativecommons.org/licences/by-nc-nd/3.0Although reasonable endeavours have been taken to obtain all necessary permissions from third parties toinclude their copyrighted content within this article, their full citation and copyright line may not bepresent in this Accepted Manuscript version. Before using any content from this article, please refer tothe Version of Record on IOPscience once published for full citation and copyright details, aspermission will likely be required. All third party content is fully copyright protected, unlessspecifically stated otherwise in the figure caption in the Version of Record.When available, you can view the Version of Record for this article at:http://iopscience.iop.org/article/10.1088/1751-8121/aa86c2The decay of Hopf solitons in the Skyrme modelDavid Foster†† School of Physics, HH Wills Physics Laboratory,University of Bristol, Tyndall Avenue, Bristol BS8 1TL, U.K.Email: dave.foster@bristol.ac.ukAugust 16, 2017AbstractIt is understood that the Skyrme model has a topologically interesting baryonicexcitation which can model nuclei. So far no stable knotted solutions, of the Skyrmemodel, have been found. Here we investigate the dynamics of Hopf solitons decayingto the vacuum solution in the Skyrme model. In doing this we develop a matrix-free numerical method to identify the minimum eigenvalue of the Hessian of thecorresponding energy functional. We also show that as isospinning Hopf solitonsdecay, they emit a cloud of isospinning radiation.1 IntroductionThe Skyrme model [1] is a nonlinear theory of pions which was identified by Witten as alow energy effective model of QCD [2, 3]. The model has a conserved topological chargethat is interpreted as the baryon number B, and the minimal energy static solutions foreach integer B are topological soliton solutions called Skyrmions.No known stable knotted solutions have been found in the classical Skyrme model.This is because, unlike the Skyrme-Faddeev model [4, 5], the model does not possess thenecessary structure to stabilise such configurations. Here we investigate how Hopf soli-tons embedded in the Skyrme model dynamically decay into the vacuum solution. Thisanalysis requires developing an understanding of the geometry, about a point, of configu-ration space. Similar analysis has recently been performed for the Skyrme model, namelyunderstanding the vibrational modes of the Skyrmions which correspond to Lithium-7[6] and Oxygen-16 [7]. The format of this paper is as follows: We introduce the Skyrmemodel. We then numerically show that the first seven Hopf solitons are solutions ofSkyrme model, but not minimum energy solutions. Next we discuss how the Hopf soli-tons decay into the vacuum solution, where we introduce a matrix-free numerical method1Page 1 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Acceped Manuscriptto identify the prominent direction of breakup. At the end we discuss how the Hopfsolitons emit a cloud of isospinning radiation, followed by a short conclusion.2 The Skyrme modelThe Skyrme field, U(x), is an SU(2)-valued scalar and can be expressed as U(t,x) =σ(t,x) I2+ iπ(t,x) ·τ , where σ(t,x) is a constraint field, π1(t,x), π2(t,x), π3(t,x) are thethree pion fields, τ = (τ1, τ2, τ3) are the three Pauli matrices and the constraint σ2+π·π =1. Here it is convenient to represent the Skyrme field as a four component unit vectorφ(t,x) = (σ(t,x), π1(t,x), π2(t,x), π3(t,x)), where φ(t,x) · φ(t,x) = 1. The model isLorentz invariant and, in so-called Skyrme units, it can be defined by the Lagrangiandensity,L = ∂µφ · ∂µφ−12(∂µφ · ∂µφ)2 +12(∂µφ · ∂νφ)(∂µφ · ∂νφ). (1)The static energy functional associated with this Lagrangian density is,E =112π2∫(∂iφ · ∂iφ+12(∂iφ · ∂iφ)2 −12(∂iφ · ∂jφ)2)d3x. (2)At fixed time, φ(x) is a map φ : R3 → S3, where φ → (0, 0, 0, 1) at spatial infinity;this is different to the boundary condition usually chosen for the Skyrme model and ischosen to match the boundary condition of the Skyrme-Faddeev model. This boundarycondition compactifies R3∪{∞} to S3. Hence, a finite energy configuration φ(x) extendsto a map φ(x) : S3 → S3, and belongs to a homotopy class of π3(S3) = Z and thereforeis indexed by an integer B ∈ Z, called the baryon number. B is also the degree of themap φ(x) which can be explicitly calculated asB ≡∫b(x) d3x =12π2∫εabcdφa∂1φb∂2φc∂3φd d3x , (3)where b(x) is the baryon density. Static Skyrmions are solutions of the equations ofmotion, which are derived from variation of energy, δE(φs) = 0. They are the minimalenergy solutions for each value of B. In the figures of Skyrmions we plot level-sets ofbaryon density b(x). The level sets are coloured as in [8], the centres of the red, greenand blue regions are where π3 = 0 and tan−1(π2π1)= 0, 2π3, 4π3respectively. The unseenwhite region is where π3 = 1 (the boundary) and the black is where π3 = −1 (centreof the Skyrmion). This is the pion colouring scheme used throughout this paper and anexample is shown in figure 1 of a B = 1 Hedgehog Skyrmion.2Page 2 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 3 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 4 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 5 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 6 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 7 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 8 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted Manuscript6 Isospinning quasi hopfionsTo relate the B = 1 Skyrmion to the proton or neutron requires including isospin. Inthe Skyrme model isospin acts on the pion fields, and has had some success in modellingnucleon scattering with spinning Skyrmions [12]. To understand the consequence ofisospinning quasi hopfions we implemented a full field simulation with a rigidly isospinningquasi hopfion as initial condition,σ(t,x) = σ(x), π1(t,x) = π1(x) cos(ωt) + π2(x) sin(ωt),π3(t,x) = π3(x), π2(t,x) = π2(x) cos(ωt)− π1(x) sin(ωt). (5)The full field simulation used a leap frog algorithm, discretised on a numerical lattice, toevolve the field equations. The initial conditions were derived from the above configura-tion for t = 0 and t = δt. This spinning, (5), is analogous to the more common SU(2)formulation U(t,x) = A(t)U0(x)A†(t) where A(t) = exp( iωt2τ3) and such an isospin causesthe π1 and π2 components to isospin into each other, where ω = 0.28 is chosen to givethe correct spin and isospin value for the B = 1 Skyrmion to model a neutron [12]. Usingthis as an initial condition gives rise a configuration which is stable to perturbation for along time, this is shown figure 8. It must be noted that isospinning Skyrminos are onlystable if a pion mass term is included (e.g. V (φ) = m2(1− σ)), such a term would causethe quasi hopfion to shrink and is therefore neglected here.Figure 8: Isospinning Q′ = 1 quasi hopfions.The isospin, similar to the spherically symmetric B = 1 Skyrmion, also causes aspatial rotation. This is apparent because the colours twist around the quasi hopfion,so as the colour rotate into each other they cause the ring to spin. Hence, isospin isequivalent to spin. This produces two angular momenta, one about each circumference ofthe quasi hopfion. The conservation of these angular momenta could lead to an enhancedlife time of the perturbed hopfion. As the quasi hopfion decays it radiates in the form of9Page 9 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 10 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted Manuscript[2] Witten, E., Global Aspects of Current Algebra, Nucl. Phys. B223 (1983) 422–432.[3] Witten, E., Current Algebra, Baryons, and Quark Confinement, Nucl. Phys. B223(1983) 433–444.[4] Faddeev, L. D., Quantization of solitons, Princeton preprint IAS-75-QS70, (1975).[5] Sutcliffe, P., Knots in the Skyrme-Faddeev model, Proc. Roy. Soc. Lond. A463 (2007)3001–3020.[6] Halcrow, C. J., Vibrational quantisation of the B = 7 Skyrmion, Nucl. Phys. B904(2016) 106–123.[7] Halcrow, C. J., King, C. and Manton, N. S., A dynamical α-cluster model of 16O,arXiv:1608.05048.[8] Manton, N.S., Classical Skyrmions – Static Solutions and Dynamics, Math. MethodsAppl. Sci. 35 (2012) 1188–1204.[9] Foster, D., Massive Hopfions, Phys. Rev. D83 (2011) 085026.[10] Cho, Y., Park, B. and Zhang, P., New interpretation of skyrme theory, Int. J. Mod.Phys. A23 (2008) 267–298.[11] Sutcliffe P. M. and Krusch S., Sphalerons in the Skyrme model, J. Phys. A. 37 (2004)9037.[12] Foster D. and Manton N.S., Scattering of Nucleons in the Classical Skyrme Model,Nucl. Phys. B899 (2015) 513–526.11Page 11 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted Mauscript

                          Foster, D. (2017). The decay of Hopf solitons in the Skyrme model. Journalof Physics A: Mathematical and Theoretical, 50(40), [405401].https://doi.org/10.1088/1751-8121/aa86c2Peer reviewed versionLicense (if available):CC BY-NC-NDLink to published version (if available):10.1088/1751-8121/aa86c2Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia IOP Publishing at http://iopscience.iop.org/article/10.1088/1751-8121/aa86c2. Please refer to terms of use ofthe publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-termsHome Search Collections Journals About Contact us My IOPscienceThe decay of Hopf solitons in the Skyrme modelThis content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 137.222.149.11This content was downloaded on 22/08/2017 at 14:32Manuscript version: Accepted ManuscriptFosterTo cite this article before publication: Foster, 2017, J. Phys. A: Math. Theor, at press:https://doi.org/10.1088/1751-8121/aa86c2This Accepted Manuscript is: © 2017 IOP Publishing LtdDuring the embargo period (the 12 month period from the publication of the Version of Record of thisarticle), the Accepted Manuscript is fully protected by copyright and cannot be reused or repostedelsewhere.As the Version of Record of this article is going to be / has been published on a subscription basis,this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 monthembargo period.After the embargo period, everyone is permitted to copy and redistribute this article for non-commercialpurposes only, provided that they adhere to all the terms of the licencehttps://creativecommons.org/licences/by-nc-nd/3.0Although reasonable endeavours have been taken to obtain all necessary permissions from third parties toinclude their copyrighted content within this article, their full citation and copyright line may not bepresent in this Accepted Manuscript version. Before using any content from this article, please refer tothe Version of Record on IOPscience once published for full citation and copyright details, aspermission will likely be required. All third party content is fully copyright protected, unlessspecifically stated otherwise in the figure caption in the Version of Record.When available, you can view the Version of Record for this article at:http://iopscience.iop.org/article/10.1088/1751-8121/aa86c2The decay of Hopf solitons in the Skyrme modelDavid Foster†† School of Physics, HH Wills Physics Laboratory,University of Bristol, Tyndall Avenue, Bristol BS8 1TL, U.K.Email: dave.foster@bristol.ac.ukAugust 16, 2017AbstractIt is understood that the Skyrme model has a topologically interesting baryonicexcitation which can model nuclei. So far no stable knotted solutions, of the Skyrmemodel, have been found. Here we investigate the dynamics of Hopf solitons decayingto the vacuum solution in the Skyrme model. In doing this we develop a matrix-free numerical method to identify the minimum eigenvalue of the Hessian of thecorresponding energy functional. We also show that as isospinning Hopf solitonsdecay, they emit a cloud of isospinning radiation.1 IntroductionThe Skyrme model [1] is a nonlinear theory of pions which was identified by Witten as alow energy effective model of QCD [2, 3]. The model has a conserved topological chargethat is interpreted as the baryon number B, and the minimal energy static solutions foreach integer B are topological soliton solutions called Skyrmions.No known stable knotted solutions have been found in the classical Skyrme model.This is because, unlike the Skyrme-Faddeev model [4, 5], the model does not possess thenecessary structure to stabilise such configurations. Here we investigate how Hopf soli-tons embedded in the Skyrme model dynamically decay into the vacuum solution. Thisanalysis requires developing an understanding of the geometry, about a point, of configu-ration space. Similar analysis has recently been performed for the Skyrme model, namelyunderstanding the vibrational modes of the Skyrmions which correspond to Lithium-7[6] and Oxygen-16 [7]. The format of this paper is as follows: We introduce the Skyrmemodel. We then numerically show that the first seven Hopf solitons are solutions ofSkyrme model, but not minimum energy solutions. Next we discuss how the Hopf soli-tons decay into the vacuum solution, where we introduce a matrix-free numerical method1Page 1 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accep ed Manuscriptto identify the prominent direction of breakup. At the end we discuss how the Hopfsolitons emit a cloud of isospinning radiation, followed by a short conclusion.2 The Skyrme modelThe Skyrme field, U(x), is an SU(2)-valued scalar and can be expressed as U(t,x) =σ(t,x) I2+ ipi(t,x) ·τ , where σ(t,x) is a constraint field, pi1(t,x), pi2(t,x), pi3(t,x) are thethree pion fields, τ = (τ1, τ2, τ3) are the three Pauli matrices and the constraint σ2+pi·pi =1. Here it is convenient to represent the Skyrme field as a four component unit vectorφ(t,x) = (σ(t,x), pi1(t,x), pi2(t,x), pi3(t,x)), where φ(t,x) · φ(t,x) = 1. The model isLorentz invariant and, in so-called Skyrme units, it can be defined by the Lagrangiandensity,L = ∂µφ · ∂µφ−12(∂µφ · ∂µφ)2 +12(∂µφ · ∂νφ)(∂µφ · ∂νφ). (1)The static energy functional associated with this Lagrangian density is,E =112pi2∫ (∂iφ · ∂iφ+12(∂iφ · ∂iφ)2 −12(∂iφ · ∂jφ)2)d3x. (2)At fixed time, φ(x) is a map φ : R3 → S3, where φ → (0, 0, 0, 1) at spatial infinity;this is different to the boundary condition usually chosen for the Skyrme model and ischosen to match the boundary condition of the Skyrme-Faddeev model. This boundarycondition compactifies R3∪{∞} to S3. Hence, a finite energy configuration φ(x) extendsto a map φ(x) : S3 → S3, and belongs to a homotopy class of pi3(S3) = Z and thereforeis indexed by an integer B ∈ Z, called the baryon number. B is also the degree of themap φ(x) which can be explicitly calculated asB ≡∫b(x) d3x =12pi2∫εabcdφa∂1φb∂2φc∂3φd d3x , (3)where b(x) is the baryon density. Static Skyrmions are solutions of the equations ofmotion, which are derived from variation of energy, δE(φs) = 0. They are the minimalenergy solutions for each value of B. In the figures of Skyrmions we plot level-sets ofbaryon density b(x). The level sets are coloured as in [8], the centres of the red, greenand blue regions are where pi3 = 0 and tan−1(pi2pi1)= 0, 2pi3, 4pi3respectively. The unseenwhite region is where pi3 = 1 (the boundary) and the black is where pi3 = −1 (centreof the Skyrmion). This is the pion colouring scheme used throughout this paper and anexample is shown in figure 1 of a B = 1 Hedgehog Skyrmion.2Page 2 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 3 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 4 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 5 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 6 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 7 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 8 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted Manuscript6 Isospinning quasi hopfionsTo relate the B = 1 Skyrmion to the proton or neutron requires including isospin. Inthe Skyrme model isospin acts on the pion fields, and has had some success in modellingnucleon scattering with spinning Skyrmions [12]. To understand the consequence ofisospinning quasi hopfions we implemented a full field simulation with a rigidly isospinningquasi hopfion as initial condition,σ(t,x) = σ(x), pi1(t,x) = pi1(x) cos(ωt) + pi2(x) sin(ωt),pi3(t,x) = pi3(x), pi2(t,x) = pi2(x) cos(ωt)− pi1(x) sin(ωt). (5)The full field simulation used a leap frog algorithm, discretised on a numerical lattice, toevolve the field equations. The initial conditions were derived from the above configura-tion for t = 0 and t = δt. This spinning, (5), is analogous to the more common SU(2)formulation U(t,x) = A(t)U0(x)A†(t) where A(t) = exp( iωt2τ3) and such an isospin causesthe pi1 and pi2 components to isospin into each other, where ω = 0.28 is chosen to givethe correct spin and isospin value for the B = 1 Skyrmion to model a neutron [12]. Usingthis as an initial condition gives rise a configuration which is stable to perturbation for along time, this is shown figure 8. It must be noted that isospinning Skyrminos are onlystable if a pion mass term is included (e.g. V (φ) = m2(1− σ)), such a term would causethe quasi hopfion to shrink and is therefore neglected here.Figure 8: Isospinning Q′ = 1 quasi hopfions.The isospin, similar to the spherically symmetric B = 1 Skyrmion, also causes aspatial rotation. This is apparent because the colours twist around the quasi hopfion,so as the colour rotate into each other they cause the ring to spin. Hence, isospin isequivalent to spin. This produces two angular momenta, one about each circumference ofthe quasi hopfion. The conservation of these angular momenta could lead to an enhancedlife time of the perturbed hopfion. As the quasi hopfion decays it radiates in the form of9Page 9 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted ManuscriptPage 10 of 11AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted Manuscript[2] Witten, E., Global Aspects of Current Algebra, Nucl. Phys. B223 (1983) 422–432.[3] Witten, E., Current Algebra, Baryons, and Quark Confinement, Nucl. Phys. B223(1983) 433–444.[4] Faddeev, L. D., Quantization of solitons, Princeton preprint IAS-75-QS70, (1975).[5] Sutcliffe, P., Knots in the Skyrme-Faddeev model, Proc. Roy. Soc. Lond. A463 (2007)3001–3020.[6] Halcrow, C. J., Vibrational quantisation of the B = 7 Skyrmion, Nucl. Phys. B904(2016) 106–123.[7] Halcrow, C. J., King, C. and Manton, N. S., A dynamical α-cluster model of 16O,arXiv:1608.05048.[8] Manton, N.S., Classical Skyrmions – Static Solutions and Dynamics, Math. MethodsAppl. Sci. 35 (2012) 1188–1204.[9] Foster, D., Massive Hopfions, Phys. Rev. D83 (2011) 085026.[10] Cho, Y., Park, B. and Zhang, P., New interpretation of skyrme theory, Int. J. Mod.Phys. A23 (2008) 267–298.[11] Sutcliffe P. M. and Krusch S., Sphalerons in the Skyrme model, J. Phys. A. 37 (2004)9037.[12] Foster D. and Manton N.S., Scattering of Nucleons in the Classical Skyrme Model,Nucl. Phys. B899 (2015) 513–526.11Page 11 of 11 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-107976.R2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960 Accepted Mauscript

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The decay of Hopf solitons in the Skyrme model

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