Exploring the generalized loosely bound Skyrme model
EExploring the generalized loosely bound Skyrme model
Sven Bjarke Gudnason
Department of Physics, and Research and Education Center for Natural Sciences, Keio University,Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
E-mail: gudnason(at)keio.jp
Abstract:
The Skyrme model is extended with a sextic derivative term – called the BPS-Skyrme term – and a repulsive potential term – called the loosely bound potential. A largepart of the model’s parameter space is studied for the 4-Skyrmion which corresponds to theHelium-4 nucleus and emphasis is put on preserving as much of the platonic symmetriesas possible whilst reducing the binding energies. We reach classical binding energies forHelium-4 as low as 0 . .
6% – still with the cubic symmetryintact.
Keywords:
Skyrmions, solitons, nuclei, low binding energies a r X i v : . [ h e p - ph ] N ov ontents The Skyrme model is an interesting approach to nuclear physics which is related to funda-mental physics, i.e. it is a field theory [1, 2]; indeed the soliton in the model is identifiedwith the baryon in the large- N limit of QCD [3, 4]. The soliton is called the Skyrmion. Themost interesting aspect of the Skyrmions is that although a single Skyrmion is identifiedwith a single nucleon, multi-Skyrmion solutions exist which can be identified with nuclei ofhigher baryon numbers. This fact distinguishes the Skyrme model from basically all otherapproaches to nuclear physics; the nuclei are no longer bound states of interacting pointparticles. What is more interesting is that since even the single Skyrmion is a spatiallyextended object (as opposed to a point particle), the multi-Skyrmions become extendedobjects with certain platonic symmetries [5]. A particular useful Ansatz for light nucleiwas found using a rational map [6]. For larger nuclei, however, there is some consensusthat when a pion mass term is included in the model, they are made of B/ c ) is very small and hence there are certain points/lines in the Skyrmionsolutions where the solution can afford to have very large field derivatives – of the orderof 1 / √ c . In ref. [18] the order of magnitude of c for the classical binding energies to bein the ballpark of the experimental values was estimated to be around c ∼ .
01; whereastheir numerics was trustable only down to about c ∼ . σ = Tr [ U ], where U is the chiral field, then tosecond order, the loosely bound potential is the potential that can reduce the bindingenergy the most while preserving platonic symmetries of the Skyrmions.In ref. [27], we expanded the model by making a hybrid model out of the BPS-Skyrme-type models and the loosely bound model. Let us define the generalized Skyrme model asthe standard Skyrme model with the addition of the sextic BPS-Skyrme term. Thus inref. [27] we studied the generalized Skyrme model with the pion mass term and the looselybound potential. More precisely, we studied the model using the rational map Ansatz forthe 4-Skyrmion in the regime where the coefficients for the BPS-Skyrme term, c , and theloosely bound potential, m , were both taken to be small (i.e. smaller or equal to one).Physical effects on the observables of the model could readily be extracted without theeffort of full numerical PDE (partial differential equation) calculations. Of course, thatcompromise had the consequence that we could not detect the change of symmetry in theSkyrmion solutions and hence we investigated only a very restricted part of the parameterspace. Point Particle Model Limit B P S - S k y r m e M o d e l L i m it c m this paper [27] 0 0.5 1 1.5 2 0 2 4 6 8 10 Figure 1 . Parameter space of the model explored in this paper (green) as well as what was studiedin ref. [27] (region inside the blue box). The BPS-Skyrme Model Limit (BSML) is defined as c ∝ m → ∞ and the Point Particle Model Limit (PPML) is c = 0, m → ∞ . In this paper, we perform the full PDE calculations for the 4-Skyrmion in the gener-alized Skyrme model with the loosely bound potential and a standard pion mass term, forsmall values of the coefficient of the BPS-Skyrme term and up to large values of the massparameter of the loosely bound potential, see fig. 1. In particular, in this paper we studythe following region of parameter space: c ∈ [0 ,
1] and m ∈ [0 , m thatenters the Lagrangian and hence 6 = 36 is much larger than the other coefficients in theLagrangian (which are all of order one). In fact, we are close to the limit of how far we canpush m with the current numerical codes. We have not considered the direction of large c in this work, as it will not reduce the binding energy unless we also turn on a large coef-ficient of the potential. That situation, however, yields two possibilities: either we do notgo beyond the limit where the numerics becomes difficult as discussed above, or one has totake the near-BPS limit carefully, which will require overcoming further technical obstacles– 3 –han dealt with here. Nevertheless, in the part of parameter space we have studied in thispaper, we are able to obtain a classical binding energy of the 4-Skyrmion as low as 0 .
2% –about a factor of four smaller than the experimental value for Helium-4. After taking intoaccount the quantum correction to the mass of the nucleon due to the spin contribution(as it is a spin- state in the ground state), the binding energy increases to about 3 .
6% –about a factor of 4 . λφ model in 1+1 dimensions L kink = −
12 ( ∂ µ φ ) − λ (cid:126) (cid:18) φ − (cid:126) m λ (cid:19) . (1.1)We can estimate the kink mass with the following back-of-an-envelope estimate: we rescalethe length scale x µ → (cid:126) x µ /m and rescale the field φ → (cid:113) (cid:126) λ mφ ; the Lagrangian density isnow dimensionless with an overall dimensionful prefactor of m (cid:126) λ . Assuming the kink exists,its mass must be an order one number times m λ where the (cid:126) /m came from integratingover x . Considering now the quantum correction due to massive modes, one obtains anorder-one number times (cid:126) ω , where ω is the curvature of the effective potential createdby the kink solution [31]. That is, the eigenvalue of the fluctuation around the kink is ω ∼ m / (cid:126) and it follows in the harmonic approximation that the quantum energy is (cid:126) ω ∼ m . To realize this, it suffices to note that the second variation of the potential withrespect to the field is − m (cid:126) + 3 λ (cid:126) φ and that the soliton solution is proportional to (cid:126) λ − m ; the resultant effective potential for the fluctuations is thus independent of λ . Inthis example the mass dimensions of m and λ are 1 and 2, respectively. If λ (cid:28) m then theperturbation series makes sense and furthermore, the quantum correction is much smallerthan the classical contribution M classical + δM ∝ m λ (cid:18) O (cid:18) λm (cid:19)(cid:19) . (1.2)The situation is more complicated in the case of the 3-dimensional Skyrmions than in thecase of simple 1-dimensional kinks (which are integrable). First of all, it is not known tothe best of the author’s knowledge, how weakly coupled the fluctuation spectrum really is.Comparing to the kinks, the question would be how small λ eff is for the Skyrmions.The scope in this paper, however, will be to focus on reducing the classical bindingenergy of the Skyrmions under the constraint of preserving as much symmetry of theoriginal Skyrmions as possible. This is thus in the spirit of assuming that the fluctuationspectrum of the Skyrmions is weakly coupled and thus the classical mass is the largest– 4 –ontribution to the mass by far; the zero-mode quantization gives the most importantquantum corrections and the remaining modes give corrections to the mass of the orderof magnitude of the zero-mode contributions or less. The reason for preserving as muchsymmetry as possible is first of all to be able to keep some of the phenomenological successesof the Skyrme model already obtained; such as the Hoyle state and its correspondingrotational band having a slope a factor of 2.5 lower than that of the ground state [9].Taking into account the vibrational spectrum of the 4-Skyrmions vibrating between a flatsquare configuration and a tetrahedral arrangement was crucial in obtaining the rightspectrum of Oxygen-16, having a large energy splitting between states of the same spinsand opposite parity [10]. Both of these results would fall apart if the 4-Skyrmion loses itscubic symmetry. Finally, and perhaps even more importantly, the larger the symmetry, thebetter the chances are that the symmetry can eliminate unwanted degeneracies, e.g. theparity doubling found in the B = 5 cluster system in the model of ref. [30].The paper is organized as follows. In the next section we will introduce the model,define the observables and finally propose an order parameter for a quantitative measureof the symmetry change. In sec. 3 we will present the numerical results. Finally, sec. 4concludes the paper with a discussion of the results and what to do next. The model we study in this paper is the generalized Skyrme model – consisting of a kineticterm, the Skyrme term and the BPS-Skyrme term – with a pion mass term and the so-calledloosely bound potential. In physical units we have L = ˜ f π L + 1 e L + 4 c c c e ˜ f π L − ˜ m π ˜ f π m V, (2.1)where the kinetic term, the Skyrme term [1, 2] and the BPS-Skyrme term [15, 16] are givenby L = 14 Tr ( L µ L µ ) , (2.2) L = 132 Tr ([ L µ , L ν ][ L µ , L ν ]) , (2.3) L = 1144 η µµ (cid:48) ( (cid:15) µνρσ Tr [ L ν L ρ L σ ])( (cid:15) µ (cid:48) ν (cid:48) ρ (cid:48) σ (cid:48) Tr [ L ν (cid:48) L ρ (cid:48) L σ (cid:48) ]) , (2.4)with the left-invariant current defined as L µ ≡ U † ∂ µ U, (2.5)in terms of the chiral Lagrangian or Skyrme field U which is related to the pions as U = σ + i τ · π , (2.6)with the nonlinear sigma model constraint det U = 1 or σ + π · π = 1 and τ is a 3-vectorof the Pauli matrices. The Greek indices µ, ν, ρ, σ = 0 , , , E = ˜ λE and ˜ x i = ˜ µx i , respectively, where we have the units [27]˜ λ = ˜ f π e √ c c , ˜ µ = (cid:114) c c e ˜ f π , (2.7)and finally the pion mass in physical units [27]˜ m π = √ c c e ˜ f π m . (2.8)Hence in dimensionless units, the Lagrangian (2.1) reads L = c L + c L + c L − V. (2.9)For a positive definite energy density, we require c > c > c ≥ The potential we will consider in this paper is due to the results of ref. [19], whichshowed that the pion mass term squared lowers the binding energy further than the pionmass term to the fourth power while keeping the symmetries of the 4-Skyrmion. We willalso include the standard pion mass term and thus the total potential is V = V + V , (2.10)where we have defined V n ≡ n m n (1 − σ ) n , (2.11)and σ = Tr [ U ]. Only V gives a contribution to the pion mass. Both V and V breakexplicitly the chiral symmetry SU(2) L × SU(2) R to the diagonal SU(2) L+R . The target spaceis thus SU(2) L × SU(2) R / SU(2)
L+R (cid:39)
SU(2) (cid:39) S . Since the Skyrmion, which is identifiedwith the baryon, is a texture [35], it is characterized by the topological degree, B , π ( S ) = Z (cid:51) B, (2.12)where B is called the baryon number. The baryon number or topological degree of aSkyrmion configuration can be calculated by B = 12 π (cid:90) d x B , B µ = − (cid:15) µνρσ Tr ( L ν L ρ L σ ) . (2.13)We will throughout the paper denote Skyrmions of degree B as B -Skyrmions.Finally, for the numerical calculations we have to settle on a choice of normalizationof the units and we follow that of refs. [19, 26, 27], c = 14 , c = 1 , (2.14) As long as the Skyrmion is not emanating from a black hole horizon [32, 33], we could also take c > c ≥
0; in this paper, however, we will fix c > – 6 –nd hence the energies and lengths are given in units of ˜ f π /e and 1 / ( e ˜ f π ), respectively,while the physical pion mass is given by˜ m π = √ c c e ˜ f π (cid:114) − ∂V∂σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ =1 = 2 e ˜ f π m , (2.15)where we have used the coefficients (2.14). In this paper we will use m = 1 / In this section we will list the observables to be measured in the numerical calculations.Since they are greatly overlapping with our previous studies, we will only review thembriefly here and refer to ref. [27] for details. As in refs. [26, 27] we will only consider the4-Skyrmion in this paper, as it plays a unique role in the alpha-particle interpretation of theSkyrme model and it is the building block of the lattice structure appearing for large nuclei[7–11]. More importantly, it is where to look for the change in symmetry that inevitablykicks in for strongly repulsive potentials, see e.g. the point particle model [18, 24] and alsoref. [19].As usual we are interested in the classical and spin/isospin quantum-corrected bind-ing energies. The energy of the 1-Skyrmion is obtained by minimizing the static energycorresponding to (minus) the Lagrangian (2.9) for the hedgehog U = cos f ( r ) + i ˆ x · τ sin f ( r ) , (2.16)where ˆ x ≡ x /r is the unit 3-vector at the origin and r = √ x · x is the radial coordinate.We will call the energy of the B -Skyrmion E B . As the initial condition for the 4-Skyrmion,we will use the rational map Ansatz [6] U = cos f ( r ) + i n R · τ sin f ( r ) , (2.17) n R = (cid:18) R + ¯ R | R | , i ( ¯ R − R )1 + | R | , − | R | | R | (cid:19) , (2.18) R ( z ) = z + 2 √ iz + 1 z − √ iz + 1 , (2.19)for the c = 0, m = 0 solution and z = e iφ tan( θ ) is the Riemann sphere coordinate. Oncea numerical solution has been obtained, we can calculate the classical relative bindingenergy (CRBE) of the 4-Skyrmion as δ = 1 − E E , (2.20)and the quantum-corrected relative binding energy (QRBE) as δ tot4 = 1 − E E + (cid:15) ) , (2.21)where (cid:15) is the quantum correction due to the isospin quantization of the 1-Skyrmion. TheCRBE (2.20) is independent of the physical units and thus independent of the calibration– 7 –f the model. The QRBE (2.21), on the other hand, after factoring out the energy units,still depends on the Skyrme coupling e .Calibrating the Skyrme-like models can be done in many ways and often ways are in-vented to minimize the problem of over-binding by compensating with a better calibration.In this paper, we will not turn to the calibration for compensating the over-binding, buttry to reduce the binding energy by varying the parameters of the Lagrangian (2.9). Thus,we will stick with a simple calibration where we set the mass and size of the 4-Skyrmion tothose of Helium-4. In order to calculate the electric charge radius of the 4-Skyrmion, wenote that the ground state of Helium-4 is an isospin 0 state and thus the charge radius inthe Skyrme model is given entirely by the baryon charge radius [27] r = r ,E = r ,B = 18 π (cid:90) d x r B . (2.22)The calibration now reads [27]˜ f π = 2 √ c (cid:115) r ˜ M He ˜ r He E = (cid:115) r ˜ M He ˜ r He E , e = 1 √ c (cid:115) r E ˜ r He ˜ M He = (cid:115) r E ˜ r He ˜ M He , (2.23)where in the latter expressions, we plugged in the normalization (2.14) and the experimentaldata used here are ˜ M He = 3727 MeV and ˜ r He = 8 . × − MeV − .With the calibration in place, we can now determine the quantum correction to themass of the 1-Skyrmion due to (spin/)isospin quantization (with the normalization (2.14))˜ m N ≡ ˜ M = ˜ f π e ( E + (cid:15) ) = ˜ E + ˜ (cid:15) , (cid:15) = e J ( J + 1) , (2.24)where Λ is the diagonal component of the isospin inertia tensor for the 1-Skyrmion: U ij =Λ δ ij where U ij is given in the next section in eq. (2.31). For the hedgehog Ansatz (2.16)the expression for Λ readsΛ = 8 π (cid:90) dr r sin f (cid:18) c + c f r + c r sin f + 2 c sin ( f ) f r r (cid:19) . (2.25)Finally, for the ground state of the proton, J = and thus (cid:15) = e . We will also considerthe ∆ resonance as a spin- excitation of the 1-Skyrmion [36] yielding˜ m ∆ = ˜ f π e ( E + 5 (cid:15) ) . (2.26)The ∆ resonance is nevertheless problematic in the Skyrme model, see the discussion.As the ground state of Helium-4 is a spin-0, isospin-0 state, there is no quantumcorrection to the mass due to zero-mode quantization (although there are corrections dueto massive modes, see the discussion).Finally, we will consider the electric charge radius of the proton and the axial coupling.The details and tensor expressions are given in ref. [27] and we will just state the final results– 8 –ere r ,E = 12 r ,B + (cid:82) dr r (cid:0) c r sin f + c sin ( f ) (cid:0) sin f + r f r (cid:1) + 2 c sin ( f ) f r (cid:1) (cid:82) dr (cid:0) c r sin f + c sin ( f ) (cid:0) sin f + r f r (cid:1) + 2 c sin ( f ) f r (cid:1) , (2.27) g A = − π (cid:90) dr r (cid:20) c (sin 2 f + rf r ) + c (cid:18) sin f sin 2 fr + 2 sin ( f ) f r r + sin(2 f ) f r (cid:19) + 2 c sin fr (cid:18) sin ( f ) f r r + sin(2 f ) f r (cid:19) (cid:21) , (2.28)where the baryon charge radius is r ,B = − π (cid:90) dr r sin ( f ) f r , (2.29)and the axial coupling in physical units is given by˜ g A = g A c e = g A e , (2.30)which is obtained by multiplying the dimensionless expression by ˜ λ ˜ µ and in the last ex-pression we have used the normalization (2.14).Some geometric observables that we will calculate, are the tensors of inertia corre-sponding to the spin and the isospin of the 4-Skyrmion. For the generalized model (2.1)they can be written as U ij = − (cid:90) d x (cid:16) c Tr ( T i T j ) + c L k , T i ][ L k , T j ]) − c T i [ L k , L l ])Tr ( T j [ L k , L l ]) (cid:17) , (2.31) V ij = − (cid:90) d x (cid:15) imn (cid:15) jpq x m x p (cid:16) c Tr ( L n L q ) + c L k , L n ][ L k , L q ]) − c L n [ L k , L l ])Tr ( L q [ L k , L l ]) (cid:17) , (2.32) W ij = 12 (cid:90) d x (cid:15) jmn x m (cid:16) c Tr ( T i L n ) + c L k , T i ][ L k , L n ]) − c T i [ L k , L l ])Tr ( L n [ L k , L l ]) (cid:17) , (2.33)and they enter the kinetic energy of the Lagrangian (2.9) as T = 12 a i U ij a j − a i W ij b j + 12 b i V ij b j , (2.34)where the isospin and spin angular momenta, respectively, are defined as a i ≡ − i Tr [ τ i A − ˙ A ] , b i ≡ i Tr [ τ i ˙ BB − ] , (2.35)and they act on the static Skyrme field, U ( x i ), as U = AU ( R ij x j ) A − , R ij = Tr [ τ i Bτ j B − ] , (2.36)where B ( t ) ( A ( t )) is an SU(2) matrix that transforms the static Skyrmion to the (iso)-spinning Skyrmion. Although in the form displayed here the contribution due to the (sextic) BPS-Skyrme term appearswith a relative minus sign, all terms contribute to the tensors with the same sign (positive for U and V while W vanishes for the 4-Skyrmion). – 9 – .2 Symmetry One could contemplate how to extract the symmetries from a numerical Skyrmion con-figuration. One guess could be to use the tensors of inertia which encode geometricalinformation about the soliton, in particular related to its spinning and isospinning. An-other more brute-force attempt could be to take moments of the energy; schematically (cid:82) ( x ) n ( x ) n ( x ) n E . This, in principle, could extract further geometrical informationfrom the numerical Skyrmion.However, we are interested in a particular symmetry, namely octahedral symmetryand would like to know when it is broken to its tetrahedral subgroup. Therefore we can usethe transformations of the octahedral symmetry which are not symmetry transformationsof the tetrahedral subgroup. In the following, we place a cube such that the Cartesianaxes are perpendicular to 3 of its faces and the origin is at the center of the cube. Thetetrahedral symmetry group contains the following transformations: the identity, three C transformations and eight C transformations. The C transformations rotate the cubeby π around one of the Cartesian axes and the C transformations rotate the cube by ± π/ , , − , − , , − , −
1) or ( − , , −
1) direction. Theoctahedral symmetry group includes another six C transformations as well as three C transformations. The C transformations rotate the cube by π in the (1 , , − , , , , , , − , − ,
1) or (0 , − , −
1) direction and the C transformations rotate thecube by π/ C symmetry transformations, say about the x -axis. The meaningof the Skyrmion possessing such a discrete symmetry is of course that after rotating theSkyrmion by π/ x -axis, we must subsequently perform an appropriate rotationin isospin space to get back to the original Skyrmion. The appropriate isospin rotation tofollow the (spatial) C ,x rotation is a rotation by π in isospin space about the π -axis. The4-Skyrmion possessing octahedral symmetry will be invariant under these two subsequenttransformations, while the tetrahedrally symmetric 4-Skyrmion will not. We can thusconstruct the order parameter for octahedral symmetry as follows. We perform the C transformation as well as the C transformation in isospin space on the Skyrmion and thensubtract off the original Skyrmion, take a 2-norm of the resulting field and finally integrateover space. If the symmetry is preserved, this integral vanishes. We thus define σ O h ≡ V (cid:90) d x Tr (cid:20)(cid:16)(cid:16) e iπ ˆ I e iπ ˆ J − (cid:17) U (cid:17) † (cid:16) e iπ ˆ I e iπ ˆ J − (cid:17) U (cid:21) , (2.37)where we have divided by the volume of the Skyrmion, V ≡ πr , in order to get adimensionless result.With all observables at hand we are now ready to turn to the numerical calculations. In this paper, we have used the term “cubic” symmetry referring loosely to the symmetry of the cube,which is octahedral symmetry. This is because the dual of a cube is a octahedron and the two latter objectsshare the symmetry: octahedral symmetry. Throughout the paper we will use both terms interchangeably. – 10 –
Numerical results
The numerical calculations in this paper are all carried out on cubic lattices of size 121 with a spatial lattice constant of about h x = 0 .
08 and the derivatives are approximated by afinite difference method using a fourth-order stencil. In previous works we were able to usethe relaxation method with a forward-time algorithm, but that turns out to be too slow forthe generalized Skyrme model when including the BPS-Skyrme term (with nonvanishing c ). In this paper, therefore, we used the method of nonlinear conjugate gradients tofind the numerical solutions. Although standard implementations of the algorithm worksmoothly for small values of the potential parameter m (cid:46)
1, some nontrivial tweaking anda sophisticated line search algorithm was needed for ensuring convergence in the large- m part of the parameter space. In particular, we found a viable solution based on switchingbetween the Newton-Raphson algorithm and a line search using a quadratic fit along thesearch direction of the conjugate gradients method.As a handle on the precision we checked that the numerically integrated baryon chargeis captured by the solution to within 0 . . × − was obtained.The solutions obtained and presented here are made on a square grid in parameterspace with c = 0 , . , . , . . . , m = 0 , . , . , . . . ,
6, yielding a total of 671 numericalsolutions. The baryon charge density isosurfaces are shown in figs. 11–13. As mentioned inthe previous section, the initial condition for the 4-Skyrmion at the point ( m , c ) = (0 , m being theabscissa and c being the ordinate. On all figures, we will overlay three lines with the orderparameter, σ O h , which measures if the solutions possess octahedral symmetry ( σ O h = 0)or remain only tetrahedrally symmetric ( σ O h > σ O h = 0 . , . , E
0 1 2 3 4 5 6 m c E (a) E
0 1 2 3 4 5 6 m c E (b) Figure 2 . Energies of the (a) 1-Skyrmion and (b) 4-Skyrmion in Skyrme units. The dashed linesshow contours of σ O h = 0 . , . , – 11 – ∼π [ MeV ]
0 1 2 3 4 5 6 m c f ∼π [ MeV ] (a) e
0 1 2 3 4 5 6 m c e (b) Figure 3 . Calibration constants (a) ˜ f π [MeV] and (b) the Skyrme coupling constant e . The dashedlines show contours of σ O h = 0 . , . , First we plot the classical energies (masses) of the 1-Skyrmion and the 4-Skyrmion inSkyrme units in fig. 2 just to get a feel for how the energy changes in parameter spacebefore calibrating the model. Both figures show isocurves according to our expectation,i.e. the energy increases roughly in quadrature from the contribution due to the looselybound potential with coefficient m and from the BPS-Skyrme term with coefficient c .The increase in energy, nevertheless, is quite drastic. If we compare the ( m , c ) = (0 , m , c ) = (6 ,
1) point, the energy increases with a factor of 5 .
42 for the1-Skyrmion and 5 .
95 for the 4-Skyrmion.In fig. 3 the calibration constants, i.e. the pion decay constant, ˜ f π and the Skyrmecoupling constant e , are shown in the parameter space. It is interesting that for small m <
1, the pion decay constant (see fig. 3(a)) with our calibration convention is almostindependent of c . What happens in this regime is that the sextic term increases both thesize and the energy of the 4-Skyrmion such that the ratio is almost constant r ( c ) /E ( c ) ∝ const. For large m , however, the mass of the 4-Skyrmion increases faster than the radiusand hence the above-mentioned linear relation no longer holds; as a result the pion decayconstant decreases for increasing c . If we now hold c fixed, an increase in m increases themass and reduces the size of the Skyrmions and hence always leads to a decrease in the piondecay constant. The combined behavior is displayed in fig. 3(a). The pion decay constantis underestimated everywhere, since in the pion vacuum its experimentally measured valueis about 184 MeV.The Skyrme coupling constant e is shown in fig. 3(b) and depends on the product ofthe size and the energy of the 4-Skyrmion and hence displays a different behavior. Forsmall c (cid:28)
1, the increase in m has a mild behavior since the loosely bound potentialboth increases the mass and reduces the size of the Skyrmions. For c = 0 and m (cid:46) . m , whereas for m > . c is continued but the turning point (0 .
7– 12 – ∼ N [ MeV ]
0 1 2 3 4 5 6 m c m ∼ N [ MeV ] (a) m ∼∆ [ MeV ]
0 1 2 3 4 5 6 m c m ∼∆ [ MeV ] m ∼∆ [ MeV ] (b) Figure 4 . (a) Nucleon mass, ˜ m N [MeV] and (b) ∆ mass, ˜ m ∆ [MeV]. The thick red dashed line in(b) is the experimentally measured mass of the ∆ resonance ( ˜ m exp∆ (cid:39) σ O h = 0 . , . , m ∼π [ MeV ]
0 1 2 3 4 5 6 m c m ∼π [ MeV ] m ∼π [ MeV ] (a) g ∼ A
0 1 2 3 4 5 6 m c g ∼ A g ∼ A (b) Figure 5 . (a) pion mass, ˜ m π [MeV] and (b) axial coupling ˜ g A . The thick red dashed lines in(a) are the experimentally measured pion masses ( ˜ m exp π (cid:39) . , . g exp A (cid:39) .
27) [38]. The dashed lines showcontours of σ O h = 0 . , . , above) moves slightly downwards as c is increased. For finite c the increase in m nowleads to a larger increase in the coupling e .We will now show the spectrum of the model, starting in fig. 4 with the nucleon massand the ∆ mass. The nucleon mass (fig. 4(a)) in our calibration scheme is tightly relatedto the total binding energy (QRBE), because we fit the mass and size of the 4-Skyrmionto those of Helium-4. Therefore, once the binding energy is right, then so is the nucleonmass. We will thus discuss this in more detail shortly, but let us mention that in the top-right part of the parameter space, i.e. for large m and large c , the nucleon mass is only– 13 –verestimated by about 28 MeV, which is about 3% above the experimentally measuredvalue.The ∆ mass is shown in fig. 4(b). First of all, we should warn the reader aboutidentifying this spin excitation of the 1-Skyrmion with the ∆ resonance, as it may beinherently inconsistent (see also the discussion). Nevertheless, we will show the results forcompleteness. As usual in a Skyrme-like model with this interpretation of the ∆, its massis underestimated. What is worse is that where the binding energy and nucleon mass tendto their experimentally measured values, the ∆ mass tends to be the smallest and hencefarthest from its true value.The pion mass is shown in fig. 5(a). In our chosen calibration scheme, the dependenceon c is mild and it mostly depends on the value of m ; that is, the pion mass decreases forincreasing m . For m (cid:39) . m = 1 /
4. In order to get a more physical value of the pion massin the top-right corner of the parameter space, we could increase the value of m to sayabout 0.5 in order to compensate the decrease in the physical value caused by the looselybound potential. We have not done this, as this is not a pressing issue for the model at themoment; the value of the pion decay constant is very far from its experimental value andthe QRBE is not quite at the physical values either. One attitude about the low-energyconstants (LECs) is that they should be “renormalized” to the in-medium conditions thatthe inside of the baryons possess. If this really justifies the pion decay constant to differby more than factors of two (four) from its experiment value, then the same may apply tothe pion mass.The axial coupling of the nucleon is shown in fig. 5(b). In the region of parameterspace where the sextic term dominates ( c ∼ m = 0), the axial coupling is generallytoo large (however, this may change for larger values of c than studied here), whereasfor large m , c = 0 the axial coupling is generally too small. In the top-right corner ofparameter space where the binding energy turns out to be smallest, the axial couplingtakes on intermediate values. Unfortunately, the experimentally measured value (thick reddashed line) is reached just after the cubic symmetry of the 4-Skyrmion is lost.The quantum mass correction to the nucleon as a 1-Skyrmion is shown in fig. 6(a) andthe diagonal value of the isospin inertia tensor, Λ is shown in fig. 6(b). The two quantitiesare related by eq. (2.24). As already mentioned, there are two limits where the classicalbinding energy can become infinitesimally small: the point particle model limit (PPML)which corresponds to c fixed, m → ∞ ; and the BPS-Skyrme model limit (BSML) whichcorresponds to c /m fixed, m → ∞ . It is interesting to see – within the calibrationscheme adopted here – that the direction of the BPS-Skyrme model limit reaches quantummass corrections to the 1-Skyrmion which are almost half the values obtained in the pointparticle model limit. If this strategy for obtaining a physically sensible Skyrme modelis correct, it is important to see where the quantum correction to the 1-Skyrmion canbe suppressed enough to reach physically measured values of the binding energies. Thisviewpoint is typical for a purist particle physicist who prefers as little fine tuning as possible.Other possibilities are of course that the physics at the atomic scale is highly fine tunedand there are big corrections to both the 1-Skyrmions and the B -Skyrmions canceling each– 14 – ∼1 [ MeV ]
31 323232.5 32.532.532.5 333333 343434 353535 404040 4545 5050 5560
0 1 2 3 4 5 6 m c ε ∼1 [ MeV ] (a) Λ
75 70 65 60606060 55555555 50505050 454545 404040 353535 303030 252525 2020 15
0 1 2 3 4 5 6 m c Λ (b) Figure 6 . (a) Quantum isospin correction to the nucleon mass, ˜ (cid:15) [MeV] and (b) the diagonal ofthe isospin inertia tensor of the 1-Skyrmion, Λ, in Skyrme units. The dashed lines show contoursof σ O h = 0 . , . , δ
0 1 2 3 4 5 6 m c δ δ (a) δ tot
0 1 2 3 4 5 6 m c δ tot (b) Figure 7 . (a) (CRBE) Classical and (b) (QRBE) quantum relative binding energy of the 4-Skyrmion. The thick red dashed line in (a) is the experimental binding energy of Helium-4 ( δ exp4 (cid:39) . σ O h = 0 . , . , other out almost precisely, leaving behind binding energies at the 1-percent level (see alsothe discussion).The quantity ˜ (cid:15) contains both the energy unit, the value of the Skyrme coupling and theisospin inertia tensor’s diagonal value. In order to disentangle the various effects, we showthe value of the diagonal of the isospin inertia tensor, Λ, in fig. 6(b). As ˜ (cid:15) ∝ Λ − , we cansee that, overall, the above mentioned behavior indeed stems from Λ and not peculiaritiesof the calibration. That is, the smallness of the quantum correction in the BPS-Skyrmemodel limit comes from the fact that Λ is bigger than in the point particle model limit.– 15 –his fact, in turn, can be traced directly to the fact that the Skyrmions become largerwhen a sizable sextic term (BPS-Skyrme term) is included and they become smaller whena strong loosely bound potential is turned on.We are now ready to present one of the main results of the paper, namely the relativebinding energies in fig. 7. In this part of parameter space, the CRBE (fig. 7(a)) is almostindependent of c . That is, only cranking up the sextic term does not reduce the CRBEbut in fact – in this calibration scheme – it leads to a slight increase in the binding energy.The effect is quite mild though. The explanation is that the sextic term just makes theSkyrmions larger and heavier; however, after the calibration this effect is almost swallowedup. The loosely bound potential, on the other hand, does its job very well. The CRBEis reduced to the 1-percent level around m ∼ . c , and itreaches values slightly below 0 .
2% for m ∼
6. Now the sextic term is crucial for whathappens. If we solely include the loosely bound potential, the model loses the platonicsymmetries of the Skyrmions and the Skyrmions become well separated point particles.However, if we turn on the BPS-Skyrme term, the 4-Skyrmion can retain its cubic symmetryand possess a CRBE below the 1-percent level. The upper black dashed line shows thesymmetry order parameter for σ O h = 0 . particle in the ground state. This means that if itis a consistent treatment not to include any other quantum corrections (which is probablynot the case), then we must find a point in the parameter space where the quantum masscorrection to the 1-Skyrmion is at the 1-percent level. In fig. 7(b) we can see that thediscussion of the quantum correction ˜ (cid:15) carries directly over to the QRBE. In particular,in the direction of the point particle model limit ( c = 0, m → ∞ ) the QRBE we reachin the parameter space is only as low as slightly below 6%, whereas in the direction ofthe BPS-Skyrme model limit ( c ∝ m → ∞ ) the QRBE reaches values as low as 3 . .
8% (experimental value), i.e. about2 .
8% over-binding. This is, however, the level of over-binding also sometimes present innuclear models like the ab initio no-core shell model, see e.g. Ref. [37].We will now turn to the electric charge radius of the proton, which is shown in fig. 8(a).In the Skyrme model it has half its contribution from the baryon charge density andthe other half from the vector charge corresponding to the isospin, see eq. (2.27). It isinteresting to see that we can obtain the best QRBE for large m and large c and hencealso the best nucleon mass, but the physically measured value of the electric charge radiusof the proton is reached spot-on in the direction of the point particle model limit; that isfor c = 0 and m ∼ B -Skyrmions tend to be too smalland the 4-Skyrmion is no exception. Due to the calibration where we fit the size of the 4-Skyrmion to that of Helium-4, the 1-Skyrmion is thus generally too large. It is interesting,nevertheless, to see that the point particle model limit gets the proton size right.– 16 – ∼1 ,E [ fm ]
0 1 2 3 4 5 6 m c r ∼1 ,E [ fm ] r ∼1 ,E [ fm ] (a) r ∼1 ,B [ fm ]
0 1 2 3 4 5 6 m c r ∼1 ,B [ fm ] (b) Figure 8 . (a) (˜ r ,E ) Electric and (b) (˜ r ,B ) baryon charge radii of the proton. The thick red dashedline in (a) is the experimentally measured charge radius of the proton (CODATA) (˜ r exp1 ,E (cid:39) . σ O h = 0 . , . , σ O h
333 2.52.52.5 222 1.51.51.5 111 0.50.50.5 0.10.10.1
0 1 2 3 4 5 6 m c σ O h (a) v
0 1 2 3 4 5 6 m c v (b) Figure 9 . (a) Order parameter for octahedral symmetry, σ O h , and (b) diagonal component v of thespin inertia tensor of the 4-Skyrmion, V ij = vδ ij , in Skyrme units. The dashed lines show contoursof σ O h = 0 . , . , The baryon charge radius is not physically measurable, but it is a component of theelectric charge radius of the proton. For large m , we can see that their behaviors arecomparable, see fig. 8(b).The last but important observable we will study here is our proposal for an order pa-rameter for the symmetry breaking of the cubic (octahedral) symmetry of the 4-Skyrmion,see eq. (2.37). By comparing the symmetries observed in the figures 11–13 with the valuesseen in fig. 9(a), we see that for σ O h < . σ O h is very close to zero ev-– 17 –rywhere, except close to the dashed line and the deviation here is merely numerical error.For σ O h (cid:38) . V ij = vδ ij is diagonal and W ij = 0 vanishes, whereas the two nonzero values of the isospintensor of inertia are U = U and U . We can see from fig. 10 that U is in generaldifferent from U except in the region of parameter space where c is small and m islarge. U
220 200 180 160160160160160 140 140 140140140140 120120120 100100100 8080 6060
0 1 2 3 4 5 6 m c U (a) U
0 1 2 3 4 5 6 m c U (b) Figure 10 . (a) U = U and (b) U of the isospin inertia tensor of the 4-Skyrmion in Skyrmeunits. The dashed lines show contours of σ O h = 0 . , . , In this paper, we have made an extensive parameter scan of the generalized Skyrme modelwith the loosely bound potential by performing full PDE calculations. The full numericalcalculations are necessary for detecting the symmetries of the Skyrmions, in particular,whether the 4-Skyrmion possesses the sought-for cubic symmetry or it loses it becoming atetrahedrally symmetric object of “point” particles. In ref. [27] we made an initial study ofthe parameter space limited to m ≤ m , the (square-root of the) coefficientof the loosely bound potential.The philosophy that we have used as a guiding principle is to keep as much symmetry aspossible whilst reducing the binding energies as much as possible; we try to cling on to the– 18 –latonic symmetries possessed by the Skyrmions of small B ( B <
8) and in particular thecubic symmetry of the 4-Skyrmion. Next we note that the classical relative binding energy(CRBE) is almost independent of the sextic term with coefficient c , but depends stronglyon the loosely bound potential with coefficient m . Now as a direct test for whether theabove stated philosophy is justified experimentally, we can compare two limits: the pointparticle model limit (PPML) ( c = 0 and m → ∞ ) and the BPS-Skyrme model limit(BSML) ( c ∝ m → ∞ ). Of course we have not taken any strict limit and just considered m large ( m (cid:39)
6) and the BSML case here will refer to c = 1, m = 6. First of all,we find that although the CRBE is almost the same in the two cases, the QRBE receivesa larger quantum contribution in the PPML case than in the BSML case. This fact isintimately related to the value of the isospin inertia tensor being larger when the sexticterm is turned on, which in turn is due to the sextic term enlarging the Skyrmions. Thenucleon mass correlates with the binding energy in our calibration scheme and hence iscloser to the measured value in the BSML case (but still overestimated) than in the PPMLcase. On the other hand, the pion decay constant is larger (but still underestimated) inthe PPML case than in the BSML case and the electric charge radius is almost spot-onthe experimental value in the PPML corner of our parameter space. Finally there is a tiebetween the two cases for the axial coupling of the nucleon, for which it is overestimatedin the BSML case and underestimated in the PPML case. We present in table 1 fourbenchmark points compared to experimental data, including the PPML and BSML cases. Table 1 . Benchmark points compared to experimental data. The model points are: A [SSM]Standard Skyrme Model; B [GSM] Generalized Skyrme Model; C [PPML] Point Particle ModelLimit; D [BSML] BPS-Skyrme Model Limit.
Point A [SSM] Point B [GSM] Point C [PPML] Point D [BSML]( m , c ) = (0 ,
0) ( m , c ) = (0 ,
1) ( m , c ) = (6 ,
0) ( m , c ) = (6 , f π − − − − m N +12.0% +13.9% +6.2% +2.9%˜ m ∆ − − − m ± π +14.7% +39.9% − − m π +18.6% +44.7% − − g A +17.4% +30.3% − δ +970.3% +1174.5% − − δ tot4 +1331.4% +1514.6% +725.7% +348.9%˜ r ,E +27.8% +25.3% − (cid:15) small then so is 5˜ (cid:15) ,see ref. [39]; more concretely, if we want the spin/isospin contribution to the mass of thenucleon, ˜ (cid:15) , to be less than the binding energy of roughly 16 MeV, which is approximatelythe binding energy of nuclear matter, then it is impossible for 5˜ (cid:15) to be as large as 366MeV [39] . As further pointed out in ref. [39], the ∆ resonance probably needs a fullyrelativistic treatment and should be considered as a resonance with a complex mass pole.Considering larger coefficients of the sextic BPS-Skyrme term is a natural continuationof this work and it may show qualitatively interesting new behavior of the model. If theBPS-Skyrme term and the potential term become too large, however, one enters the near-BPS regime of the BPS-Skyrme model which is known to be technically difficult. In thelight of the discussions in this paper, the more important question is whether it is necessaryto obtain solutions with very low classical binding energies – like in this paper – or thetrue solution to the quantum physics of nuclei lies in sizable corrections that perhaps viabeautiful symmetries somehow all balance in such a way as to give small binding energiesat the 1-percent level for all nuclei. This will be left as work for the future.We should remind the reader that we did not get the physical pion mass right in theregion of parameter space with small binding energies. This can easily be fixed, but thechange for the rest of the physics is expected to be insignificant and as long as the piondecay constant is so far from its measured value, it remains a question whether the pionmass should be close to its experimental value or not. Lattice-QCD simulations often getgood results even though their pion mass is typically much too large compared with theexperimental value.There are lots of directions to consider for improving the Skyrme model in order for itto become a full-fledged high-precision model of nuclear physics. If the program succeeds, In nuclear matter, the rigid body quantization can be neglected, hence the binding energy of nuclearmatter gives an upper bound on the mass contribution of the spin to the nucleon, ˜ (cid:15) (cid:46)
16 MeV. – 20 –t will become a few-parameter model which basically can cover all nuclei. The list ofproblems is however not so short. The problem of the binding energy that we have workedon in this paper is not solved yet and we are probably getting closer to a point where wecan determine whether the Skyrme model is natural and hence the quantum correctionssomehow are small as expected in systems with semi-classical quantization, or there arerelatively large quantum corrections that just happen to balance out almost perfectly overa large variety of nuclei – many possessing different symmetries.The small isospin breaking present in Nature still remains largely unincorporated inthe Skyrme model. The recent suggestion by Speight [40] is based on including the ω mesonand an explicit symmetry breaking term. This direction of improving the Skyrme model isalso considered for solving the binding energy problem. That is, including vector mesonsin the model, see e.g. [14] where ρ mesons are considered.A further improvement to be considered, which will become more important for thestudies of large nuclei, is to include the effects of the Coulomb energy. Although it is knownhow to calculate the Coulomb force for multi-Skyrmions, it should ideally be back-reactedonto the Skyrmions. This would require some partial gauging and further complicate themodel.It would also be interesting to consider other Skyrmions than the 4-Skyrmion, in orderto check our claims about the preservation of symmetries in the model. A preliminarystudy suggests that for the 8-Skyrmion, the two cubes retain their separate octahedralsymmetries but become more weakly bound to each other with the result that, in the low-binding energy regime, the chain and twisted chain become almost degenerate in energies.There are plenty of other Skyrmions that would be interesting to study.The question, however, remains: how to resolve the quantum part of the binding-energy problem for Skyrmions as nuclei? Hopefully, this question may be answered in thefuture. Acknowledgments
S. B. G. thanks Chris Halcrow for discussions. S. B. G. is supported by the Ministry of Ed-ucation, Culture, Sports, Science (MEXT)-Supported Program for the Strategic ResearchFoundation at Private Universities “Topological Science” (Grant No. S1511006) and by aGrant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science”(KAKENHI Grant No. 15H05855) from MEXT, Japan. The calculations in this work werecarried out using the TSC computing cluster of the “Topological Science” project at KeioUniversity.
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T h e v e r t i c a l a x i s d e n o t e s t h e v a l u e s o f c w h il e t h e h o r i z o n t a l a x i s i s m . T h e d a s h e d li n e ss h o w c o n t o u r s o f σ O h = . , . , f r o m t o p t o b o tt o m . T h ece n t e r o f m a ss o f t h e S ky r m i o n c o rr e s p o nd s t o i t s p o s i t i o n i np a r a m e t e r s p a ce .. – 25 – i g u r e . B a r y o n c h a r g e d e n s i t y i s o s u r f a ce s o f t h e - S ky r m i o n s o l u t i o n s o v e r p a r t o f t h e s c a nn e dp a r a m e t e r s p a ce . T h e v e r t i c a l a x i s d e n o t e s t h e v a l u e s o f c w h il e t h e h o r i z o n t a l a x i s i s m . T h e d a s h e d li n e ss h o w c o n t o u r s o f σ O h = . , . , f r o m t o p t o b o tt o m . T h ece n t e r o f m a ss o f t h e S ky r m i o n c o rr e s p o nd s t o i t s p o s i t i o n i np a r a m e t e r s p a ce ..