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The decay of Hopf solitons in the Skyrme model
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The decay of Hopf solitons in the Skyrme model
David Foster
†† School of Physics, HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, U.K.
Email: [email protected]
August 16, 2017
Abstract
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 isospinning Hopf solitons decay, they emit a cloud of isospinning radiation.
1
Introduction
The Skyrme model [1] is a nonlinear theory of pions which was identified by Witten as a low energy effective model of QCD [2, 3]. The model has a conserved topological charge that is interpreted as the baryon number B, and the minimal energy static solutions for each 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 the necessary structure to stabilise such configurations. Here we investigate how Hopf soli-tons embedded in the Skyrme model dynamically decay into the vacuum solution. This analysis requires developing an understanding of the geometry, about a point, of configu-ration space. Similar analysis has recently been performed for the Skyrme model, namely understanding 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 Skyrme model. We then numerically show that the first seven Hopf solitons are solutions of Skyrme 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 method
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to identify the prominent direction of breakup. At the end we discuss how the Hopf solitons emit a cloud of isospinning radiation, followed by a short conclusion.
2
The Skyrme model
The 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 the three pion fields, τ = (τ1, τ2, τ3) are the three Pauli matrices and the constraint σ
2
+π·π = 1. Here it is convenient to represent the Skyrme field as a four component unit vector φ(t, x) = (σ(t, x), π1(t, x), π2(t, x), π3(t, x)), where φ(t, x) · φ(t, x) = 1. The model is Lorentz invariant and, in so-called Skyrme units, it can be defined by the Lagrangian density, L = ∂µφ· ∂µφ− 1 2(∂µφ· ∂ µφ)2 + 1 2(∂µφ· ∂νφ)(∂ µφ· ∂νφ). (1) The static energy functional associated with this Lagrangian density is,
E = 1 12π2 Z ∂iφ· ∂iφ+ 1 2(∂iφ· ∂iφ) 2 − 1 2(∂iφ· ∂jφ) 2 d3 x. (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 is chosen to match the boundary condition of the Skyrme-Faddeev model. This boundary condition compactifies R3
∪{∞} to S3
. Hence, a finite energy configuration φ(x) extends to a map φ(x) : S3
→ S3
, and belongs to a homotopy class of π3(S3) = Z and therefore is indexed by an integer B ∈ Z, called the baryon number. B is also the degree of the map φ(x) which can be explicitly calculated as
B≡ Z b(x) d3 x= 1 2π2 Z εabcdφa∂1φb∂2φc∂3φdd 3 x , (3)
where b(x) is the baryon density. Static Skyrmions are solutions of the equations of motion, which are derived from variation of energy, δE(φs) = 0. They are the minimal energy solutions for each value of B. In the figures of Skyrmions we plot level-sets of baryon density b(x). The level sets are coloured as in [8], the centres of the red, green and blue regions are where π3 = 0 and tan−1
π2 π1 = 0,2π 3, 4π
3 respectively. The unseen white region is where π3 = 1 (the boundary) and the black is where π3 = −1 (centre of the Skyrmion). This is the pion colouring scheme used throughout this paper and an example is shown in figure 1 of a B = 1 Hedgehog Skyrmion.
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To relate the B = 1 Skyrmion to the proton or neutron requires including isospin. In the Skyrme model isospin acts on the pion fields, and has had some success in modelling nucleon scattering with spinning Skyrmions [12]. To understand the consequence of isospinning quasi hopfions we implemented a full field simulation with a rigidly isospinning quasi 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, to evolve 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(iωt2 τ3) and such an isospin causes the π1 and π2 components to isospin into each other, where ω = 0.28 is chosen to give the correct spin and isospin value for the B = 1 Skyrmion to model a neutron [12]. Using this as an initial condition gives rise a configuration which is stable to perturbation for a long time, this is shown figure 8. It must be noted that isospinning Skyrminos are only stable if a pion mass term is included (e.g. V (φ) = m2
(1 − σ)), such a term would cause the 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 a spatial 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 is equivalent to spin. This produces two angular momenta, one about each circumference of the quasi hopfion. The conservation of these angular momenta could lead to an enhanced life time of the perturbed hopfion. As the quasi hopfion decays it radiates in the form of
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[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.
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[12] Foster D. and Manton N.S., Scattering of Nucleons in the Classical Skyrme Model, Nucl. Phys. B899(2015) 513–526. 11 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
