TTopological inner structures of baryons in hypersphere soliton model
Soon-Tae Hong ∗ Center for Quantum Spacetime and Department of Physics, Sogang University, Seoul 04107, Korea (Dated: January 23, 2020)Exploiting a soliton on a hypersphere we construct baryon charge density profiles possessingspherical symmetry. In this topological soliton model, we study cohomology group structure asso-ciated with a nilpotent BRST charge, and knot structure related with gluon effect and confinementproblem in the quark model. We also predict baryon properties such as masses, charge radii andmagnetic moments by using hypersphere Skyrmion Lagrangian with pion mass correction term.
PACS numbers: 12.39.Dc, 13.40.Em, 14.20.-cKeywords: hypersphere Skyrmion, baryon, charge density, BRST
I. INTRODUCTION
In 1962, Dirac proposed [1] that the electron should be considered as a charged conducting surface and its shapeand size should pulsate. Here the surface tension of the electron was supposed to be needed to prevent the electronfrom flying apart under the repulsive forces of the charge. Moreover he proposed that such an electron shape shouldhave spherical symmetry. Motivated by this, we will consider the Skyrmion model on a hypersphere [2, 3], to studytopological profile structures inside baryons.On the other hand, the Dirac Hamiltonian scheme has been developed [4–7], to convert the second class constraintsinto the first class ones. Moreover, exploiting the Hamiltonian quantization method, there have been attempts toquantize the constrained systems to investigate BRST [8] symmetries involved in the systems.In particular, the first order tetrad gravity have been investigated in the Dirac Hamiltonian scheme to see that thesecond class constraints reduce the theory to second order tetrad gravity and the first class constraints are differentfrom those in the second order formalism, but satisfy the same gauge algebra if one uses Dirac brackets [5]. Therehave been also attempts to study the quantum BRST charge for quadratically nonlinear Lie algebras, and to derivea condition which is necessary and sufficient for the existence of a nilpotent BRST charge [6].The standard Skyrmion model [7, 9, 10] is defined on ordinary three dimensional Euclidean space R so that themapping U : R → S cannot fulfill homotopy group [11]. Here S is the group manifold for the SU(2) group structure.In the standard Skyrmion model, the three dimensional Euclidean physical space R is assumed to be topologicallycompactified to S and the spatial infinity is located on the northpole of S . However in order to define strictly thehomotopy group Π ( S ) = Z related with the rigorous mapping U : S → S , we need the physical space S , namelythe hypersphere manifold, instead of R which is assumed to be topologically compactified to S . We notice thatthe Skyrmion on the hypersphere has been considered [2, 3] to obtain a topological lower bound on the energy. Itis also interesting to note that, in a (1+1)-dimensional supersymmetric soliton, topological boundary conditions andthe BPS bound have been investigated to fix ambiguities in the quantum mass of the soliton [12].In this paper, exploiting the hypersphere Skyrmion model, we will probe theoretically the inner structures of thebaryons to investigate their charge density distributions and knot structure of a M¨obius strip associated with thetopological soliton. After constructing the first class Hamiltonian of the hypersphere Skyrmion, we also study thecohomology group related with the BRST charge of the geometrically constrained system.In Sec. 2, we will study the pion mass effects on the physical properties of the baryons. In Sec. 3, we will analyzethe profile and knot structures inside the nucleons. In Sec. 4, we will construct the BRST invariant Hamiltonian ofthe hypersphere Skyrmion to study its cohomology group structure. Sec. 5 includes conclusions. ∗ Electronic address: [email protected] In the standard Skyrmion model the homotopy group is not as bad as the text implies, because the homotopy group is employed asa convenient way to realize the fact that with all known soliton solution in flat space one can integrate the baryon density to yield adiscrete interger [3]. a r X i v : . [ phy s i c s . g e n - ph ] J a n II. BARYON PREDICTIONS IN HYPERSPHERE SKYRMION MODEL WITH PION MASS
We start with the Skyrmion Lagrangian of the form L S = (cid:90) d x (cid:20) f π ∂ µ U † ∂ µ U ) + 132 e tr[ U † ∂ µ U, U † ∂ ν U ] + 12 m π f π (tr U − (cid:21) , (2.1)where U is an SU(2) chiral field, f π is the pion decay constant, e is a dimensionless Skyrme parameter, and m π =139 M eV is the pion mass. We note that the quartic term is necessary to stabilize the soliton in the baryon sector.The last term is the pion mass term, and this term contribution is small so that we will treat it perturbatively. For theminimum soliton energy of the Skyrmion we take a hedgehog ansatz U ( (cid:126)x ) = e iτ i ˆ x i f ( r ) where τ i are Pauli matrices,ˆ x = (cid:126)x/r and for unit winding number lim r →∞ f ( r ) = 0 and f (0) = π .Now we proceed to investigate baryon phenomenology by using the hyperspherical metric derived in Appendix A ds ( S ) = λ dµ + λ sin µ ( dθ + sin θ dφ ) . (2.2)In the hypersphere Skyrmion, we obtain the soliton energy of the form E = f π e (cid:34) πL (cid:90) π dµ sin µ (cid:32)(cid:18) dfdµ + 1 L sin f sin µ (cid:19) + 2 (cid:18) L dfdµ + 1 (cid:19) sin f sin µ (cid:33) + 4 πm π L ( ef π ) (cid:90) π dµ sin µ (1 − cos f ) + 6 π (cid:21) , (2.3)where L = ef π λ is a radius expressed in dimensionless units. The soliton energy E has the lower bound which isthe topological lower bound plus the pion mass correction. The chiral angle f in the lower bound energy satisfiesequations of motion, dfdµ + 1 L sin f sin µ = 0 , L dfdµ + 1 = 0 . (2.4)One of the simplest solutions of Eq. (2.4) is the identity map f ( µ ) = π − µ with the condition L = 1.Using the above identity map we obtain the soliton energy given by E = f π e (cid:104) π (cid:0) L + L (cid:1) + π m π L e f π (cid:105) . We notethat L = ef π λ is fixed to satisfy the condition L = 1 so that the above soliton energy can have a minimum value E = f π e (cid:18) π + 2 π m π e f π (cid:19) . (2.5)From now on we will use the condition L = 1 to predict the physical properties of baryons in the hypersphere Skymionmodel.In the hypersphere Skyrmion, spin and isospin states can be treated by collective coordinates a µ = ( a , (cid:126)a ) , ( µ = 0 , , , , (2.6)corresponding to the spin and isospin collective rotation A ( t ) ∈ SU(2) given by A ( t ) = a + i(cid:126)a · (cid:126)τ . Using the chiralfield U ( (cid:126)x, t ) = A ( t ) U ( (cid:126)x ) A † ( t ) with U ( (cid:126)x ) being the hedgehog ansatz, we obtain the Skyrmion Lagrangian on thehypersphere L S = − E + 2 I ˙ a µ ˙ a µ , (2.7)where E is the soliton energy given by Eq. (2.5) and I is the moment of inertia of the form I = 3 π e f π . (2.8)Here we note that the moment of inertia is not affected by the pion mass correction since the pion mass Lagrangianin Eq. (2.1) does not contain derivative terms.Performing Legendre transformation we obtain the canonical Hamiltonian H = E + 18 I π µ π µ , (2.9) TABLE I: The physical properties of baryons in the hypersphere Skyrmion model possessing the pion mass term, comparedwith experimental data. The input parameters are indicated by ∗ .Quantity Prediction Experiment Quantity Prediction Experiment (cid:104) r (cid:105) / M,I =0 M N
939 MeV ∗
939 MeV (cid:104) r (cid:105) / M,I =1 M ∆ (cid:104) r (cid:105) / M,p ∗ µ p (cid:104) r (cid:105) / M,n µ n − . − . (cid:104) r (cid:105) / E,I =0 µ ∆ ++ . − . (cid:104) r (cid:105) / E,I =1 µ N ∆ (cid:104) r (cid:105) p (0.780 fm) (0.805 fm) f π
55 MeV 93 MeV (cid:104) r (cid:105) n − (0 .
179 fm) − (0 .
361 fm) e 4.10 − where π µ are canonical momenta conjugate to the collective coordinates a µ . After the canonical quantization, wearrive at the Hamiltonian H = E + 12 I I ( I + 1) , (2.10)where I (= 1 / , / , ... ) are isospin quantum numbers. Exploiting Eq. (2.10) we find the nucleon mass M N for I = 1 / M ∆ for I = 3 /
2, respectively M N = ef π (cid:18) π e + e π (cid:19) + 2 π f π e (cid:18) m π ef π (cid:19) ,M ∆ = ef π (cid:18) π e + 5 e π (cid:19) + 2 π f π e (cid:18) m π ef π (cid:19) . (2.11)Since the pion mass does not affect on the charge radii, as in the massless pion case of hypersphere Skyrmion, wehave the charge radii [3] (cid:104) r (cid:105) / M,I =0 = (cid:104) r (cid:105) / M,I =1 = (cid:104) r (cid:105) / M,p = (cid:104) r (cid:105) / M,n = (cid:104) r (cid:105) / E,I =1 = (cid:114)
56 1 ef π , (2.12) (cid:104) r (cid:105) / E,I =0 = √
32 1 ef π , (cid:104) r (cid:105) p = 1924 1( ef π ) , (cid:104) r (cid:105) n = −
124 1( ef π ) . (2.13)Next we obtain the magnetic moments µ p = 2 M N ef π (cid:18) e π + π e (cid:19) , µ n = 2 M N ef π (cid:18) e π − π e (cid:19) ,µ ∆ ++ = 2 M N ef π (cid:18) e π + 9 π e (cid:19) , µ N ∆ = 2 M N ef π · √ π e , (2.14)where the nucleon mass M N is changed due to the pion mass contribution as in Eq. (2.11).From the charge radii in Eq. (2.12) we choose (cid:104) r (cid:105) / M,p = 0 .
80 fm as an input parameter. One can then have ef π = 225 .
23 MeV = (0 .
876 fm) − and, with this fixed value of ef π , one can proceed to calculate the other chargeradii as shown in Table I. Exploiting values of M N = 939 MeV as another input parameter, we evaluate the deltahyperon mass to yield M ∆ = 1132 MeV. The predictions for M ∆ and the other quantities are listed in Table I. Wenote that the predictions with the massive pion are comparable to the corresponding experimental data as shown inTable I. In particular, one can realize that the predictions for M N , M ∆ and µ p in the massive pion case are muchmore improved compared to those in the massless pion one [3].Now we investigate the physical meaning of the radius λ of the hypersphere. Inserting the above value ef π =(0 .
876 fm) − into the condition L = ef π λ = 1, we obtain the value of λλ = 0 .
876 fm . (2.15) FIG. 1: The density distributions ρ N ( µ ) for (a) proton and (b) neutron are plotted versus µ . Here we emphasize that the above value is the size of hypersphere radius. To make its meaning clear, we considerone dimensional case. The radius of circle S is finite but a stereographic projection maps the circle to the straightline including infinity as shown in Figure 2 below. Moreover one can readily see that the circle with any radius sizecontains the property of infinity since one can trace the circle infinitely many times. Next we consider two dimensionalspherical shell S of radius R . The shell can be made of circles S with radius R sin θ where θ is the angle definingthe position of S . Here we note that all the radii of the circles are less than the radius of the spherical shell ofradius R . Similarly, in the three dimensional hypersphere of radius λ , all the spherical shell radii λ sin µ are alsoless than the radius λ of the hypersphere as shown in Figure 7 in Appendix A. The hypersphere is made of twodimensional spherical shell S and one dimensional circle S of radius λ . Moreover the maximum value of the radiusof the spherical shell is λ . The hypersphere Skyrmion is thus defined inside the radius λ whose value is given in Eq.(2.15). This radius value λ = 0 .
876 fm is comparable to the predictions for the charge radii in Eq. (2.12) given inTable I. Moreover the relations in Eq. (2.12) strongly support that the baryon phenomenology is well described interms of the Skyrmion defined on S embedded in Euclidean physical space R . III. TOPOLOGICAL INTERNAL STRUCTURES OF NUCLEONS
Now we investigate charge density profiles, knot structure, gluon effect and confinement problem. Using thehypersphere metric in Eq. (2.2), we construct the proton and neutron charge densities of the form ρ p ( µ ) = 1 π sin µ (cid:18) µ (cid:19) , ρ n ( µ ) = 1 π sin µ (cid:18) −
43 sin µ (cid:19) . (3.1)We then find the proton and neutron charges as follows Q p = (cid:90) π dµ ρ p ( µ ) , Q n = (cid:90) π dµ ρ n ( µ ) , (3.2)which yield Q p = 1 , Q n = 0 , (3.3)as expected.The proton and neutron charge densities in Eq. (3.1) are depicted in Figure 1 (a) and Figure 1 (b), respectively.From Eq. (3.1) we find two root values of ρ n ( µ )=0: µ = π and µ = π . We now have three regions I, II and III inthe neutron charge density as shown in Figure 1 (b) and for each region, we obtain the charge fractions as follows Q I n = √ π , Q II n = − √ π , Q III n = √ π , (3.4) FIG. 2: The stereographic projection from S to R is defined in terms of µ , r and λ . which are in good agreement with the result Q n = 0. Here we note that the charge density distribution of neutron hasno dependence on the coordinates θ and φ and it has dependence only on the coordinate µ , radial distance analoguedefined on S . The charge density distributions of the neutron and proton in the hypersphere Skyrmion model are thus(hyper)spherically symmetric, similar to the mass density distribution inside the Earth. This feature is contradictoryto the quark model, where the charge density distributions are localized on three points without spherical symmetryinside the nucleons. Moreover, in the neutron the charge density distribution curve is not flat with zero value. Thismeans that the neutron has nontrivial charge density distribution along the radial direction. Here we emphasize thatthe spherical symmetry of the charge density distributions of the nucleons is similar to that of the electron proposedby Dirac [1].Next we investigate the stereographic projection which relates S = S × S geometry to R = S × R one inthe hypersphere Skyrmion. To do this, we define the stereographic projection in terms of the third angle µ and thehypersphere radius λ cos µ λ ( λ + r ) / , sin µ r ( λ + r ) / . (3.5)Here one notes that the northpole located at the third angle µ = π corresponds to the spatial infinity which is notcontained in R . The stereographic projection from S to R is depicted in Figure 2.Using Eq. (3.5) we obtain the chiral angle in the identity map f ( µ ) = π − µ in terms of r and λ , f ( r ) = π − cos − λ − r λ + r . (3.6)Here one notes that f ( r ) in Eq. (3.6) is different from the chiral angle in the standard Skyrmion model since f ( r ) isobtained through the stereographic projection (3.5) using the chiral angle f ( µ ) defined on S hypersphere. The chiralangle f ( r ) in Eq. (3.6) is depicted in Figure 3.Now exploiting the stereographic projection we obtain the proton and neutron charge densities in terms of r and λ , ρ p ( r ) = 1 π λλ + r (cid:18) λrλ + r (cid:19) (cid:34) (cid:18) λrλ + r (cid:19) (cid:35) ,ρ n ( r ) = 1 π λλ + r (cid:18) λrλ + r (cid:19) (cid:34) − (cid:18) λrλ + r (cid:19) (cid:35) , (3.7)which are depicted in Figure 4 (a) and Figure 4 (b), respectively. Similar to the above results mentioned about ρ n ( µ ) = 0, we obtain two root values of ρ n ( r )=0: r = λ √ and r = √ λ . Exploiting the nucleon charge densities inEq. (3.7), we obtain Q p = (cid:82) ∞ dr ρ p ( r ) and Q n = (cid:82) ∞ dr ρ n ( r ) which reproduce Q p = 1 and Q n = 0 as in Eq. (3.3). FIG. 3: The chiral angle f ( r ) is plotted versus radial distance r in unit of fm.FIG. 4: The charge density distribution ρ N ( r ) for (a) proton and (b) neutron are plotted versus radial distance r in unit of fm.FIG. 5: The (a) ordinary and (b) M¨obius strips correspond to the boson and fermion, respectively. FIG. 6: The M¨obius strip with knot structure explains the gluon effect and confinement problem.
Moreover, we have again three regions I, II and III in the neutron charge density as shown in Figure 4 (b). For eachregion, we obtain consistently the charge fractions in Eq. (3.4), as expected.Now we investigate the topological structure of Skyrmion on the hypersphere S . To to this, we exploit the simplifiedcompact manifold S instead of S . In soliton physics in one dimension, it is well known that an ordinary strip shownin Figure 5 (a) represents a boson. However, to describe a fermion, we need a M¨obius strip possessing writhingnumber one as shown in Figure 5 (b). To relate the hypersphere Skyrmion model to the quark model where, in thefermion such as proton and neutron, there exist three quarks, we separate the M¨obius strip into three pieces as shownin Figure 5 (b) to yield two M¨obius strips which are linked to each other, as shown in Figure 6. More specifically, wehave the M¨obius strip with the circumference length equal to that of the original M¨obius strip in Figure 5 (b) andwith writhing number one. This small M¨obius strip originates from the middle piece of the M¨obius strip in Figure 5(b). Next we have the other M¨obius strip with the circumference length of two times that of the original M¨obius stripin Figure 5 (b) and with writhing number two. This large M¨obius consists of the first and third pieces of the M¨obiusstrip in Figure 5 (b). Here the writhing number two originates from addition of those of the first and third pieces ofthe original M¨obius strip. Moreover the large strip corresponds to uu in uud and dd in udd for proton and neutrondefined in quark based model, respectively.We emphasize that two M¨obius strips are linked to produce a knot structure. Due to this knot structure, thesoliton is unbroken to yield the strong interaction feature. In other words, the gluon effects in the quark model can beexplained by the knot structure of the M¨obius strips in the hypersphere Skyrmion model. The confinement problemassociated with the gluon effects among quarks has been described in terms of asymptotic freedom in the theory ofthe strong interaction [13]. We emphasize again that the confinement problem can be explained by the knot structureof the M¨obius strips associated with the topological soliton which delineates the inner structures of the baryons.Next we can classify the boson and fermion in terms of the strip structures. The boson performs 2 π rotation toreturn its starting situation. This feature can be explained by exploiting 2 π rotation on the ordinary strip shownin Figure 5 (a). Next the fermion performs 4 π rotation to return its initial situation according to the relativisticquantum mechanics. This feature can be explained by using 4 π rotation on the M¨obius strip shown in Figure 5(b). The baryonic inner structure is thus successfully explained on the S hypersphere manifold embedded in the R Euclidean space. This statement is one of main themes of our result. In contrast, in the case of bosonic innerstructure, we do not need to consider such a hypersphere geometry.
IV. COHOMOLOGY
In this section, we investigate the cohomology involved in the hypersphere Skyrmion model. Before we get startedon the cohomology in the hypersphere Skyrmion model, we briefly introduce the de Rham cohomology in algebraictopology [7, 14–16]. In this formalism we consider a d operator, namely exterior differentiation d : ω p → ω p +1 , (4.1)where ω p is a p -form. The d operator then satisfies, for all forms ω p , d ω p = 0 . (4.2)In R , d acts like the gradient on 0-forms, the curl on 1-forms and the divergence on 2-forms. We thus have theidentity d = 0 in Eq. (4.2) as follows ∇ × ( ∇ f ) = 0 , ∇ · ( ∇ × (cid:126)v ) = 0 , (4.3)for any scalar function f and vector (cid:126)v [15]. A p -form ω p is defined to be closed if dω p = 0, while it is defined to beexact if ω p = dω p − for some form ω p − . Exploiting the property (4.2), one readily checks that, for a given exactform ω p (= dω p − ), dω p = d ω p − = 0, which means that every exact form is closed.By using the d operator in Eq. (4.1), we define p -th de Rham cohomology group C p ( M, R ) of the manifold M andthe field of real number R with the following quotient group [7, 14–16] C p ( M, R ) = Z p ( M, R ) B p ( M, R ) , (4.4)where Z p ( M, R ) are the collection of all d -closed p -forms ω p for which dω p = 0, and B p ( M, R ) are the collection ofall d -exact p -forms ω p for which ω p = dω p − . Moreover the d -closed p -form ω p is deformed into the other d -closed p -form ω (cid:48) p = ω p + dω p − . Namely ω p is homologous to ω (cid:48) p under the d operator: ω p ∼ ω (cid:48) p , since dω p − = ω (cid:48) p − ω p .Next we consider the first class Dirac formalism. In the hypersphere Skyrmion, we have the second class constraintsΩ = a µ a µ − ≈ = a µ π µ ≈
0, so that we have the Poisson algebra ∆ kk (cid:48) = { Ω k , Ω k (cid:48) } = 2 (cid:15) kk (cid:48) a µ a µ .The hypersphere Skyrmion model thus becomes a second class constrained Hamiltonian system. Following the DiracHamitonian quantization procedure, we find the first class constraints ˜Ω i ( i = 1 , = a µ a µ − θ = 0 , ˜Ω = a µ π µ − a µ a µ π θ = 0 , (4.5)where ( θ, π θ ) are the St¨uckelberg fields. The first class constraints now satisfy { ˜Ω , ˜Ω } = 0. The first class physicalfields ˜ a µ and ˜ π µ are given in terms of the original physical fields a µ and π µ , and the St¨uckelberg fields θ and π θ asfollows [7] ˜ a µ = a µ (cid:34) − ∞ (cid:88) n =1 ( − n (2 n − n ! θ n a ν a ν (cid:35) = a µ (cid:18) a ν a ν + 2 θa ν a ν (cid:19) / , ˜ π µ = ( π µ − a µ π θ ) (cid:34) ∞ (cid:88) n =1 ( − n (2 n − n ! θ n a ν a ν (cid:35) = ( π µ − a µ π θ ) (cid:18) a ν a ν a ν a ν + 2 θ (cid:19) / , (4.6)which fulfill { ˜Ω i , ˜ a µ } = 0 and { ˜Ω i , ˜ π µ } = 0. We next obtain the first class Hamiltonian of the form˜ H = E + 18 I ( π µ − a µ π θ )( π µ − a µ π θ ) a ν a ν a ν a ν + 2 θ . (4.7)Here we note that the first class Hamiltonian is strongly involutive with the first class constraints { ˜Ω i , ˜ H } = 0.Now, in order to obtain the BRST invariant gauge fixed Lagrangian, we introduce two canonical sets of ghost andanti-ghost fields ( C i , ¯ P i ) and ( P i , ¯ C i ) ( i = 1 ,
2) together with St¨uckelberg fields ( N i , B i ) ( i = 1 , C i and P i areghost number (+1)-forms while ¯ C i and ¯ P i are ghost number ( − {C i , ¯ P j } = {P i , ¯ C j } = {N i , B j } = δ ij .Next we define the BRST charge Q = C i ˜Ω i + P i B i , (4.8)with which we define δ operator as follows δ · = { Q, ·} . (4.9)Explicitly Q is the generator of the following infinitesimal BRST transformations δa µ = −C a µ , δπ µ = 2 C a µ + C ( π µ − a µ π θ ) ,δθ = C a µ a µ , δπ θ = 2 C ,δ C i = 0 , δ ¯ P i = ˜Ω i ,δ P i = 0 , δ ¯ C i = B i ,δ N i = −P i , δ B i = 0 . (4.10)Moreover we find for all the above fields δ · = { Q, { Q, ·}} = 0 , (4.11)which is related with the nilpotent BRST charge Q .Now we define the fermionic gauge fixing function ΨΨ = ¯ C i χ i + ¯ P i N i . (4.12)Choosing the unitary gauge: χ = Ω and χ = Ω , we have { Q, { Q, Ψ }} = 0, which implies δ Ψ = δ { Q, Ψ } = 0 . (4.13)The gauge fixed BRST invariant Hamiltonian is then given by H eff = ˜ H (cid:48) − { Q, Ψ } , (4.14)where ˜ H (cid:48) = ˜ H + 14 I π θ ˜Ω − I C ¯ P . (4.15)Here ˜ H is given by Eq. (4.7) and the last term is Faddeev-Popov ghost term associated with the ghost and anti-ghostfields, C and ¯ P , respectively.Next we exploit the BRST operator δ in Eq. (4.9) to define the cohomology associted with Qδ : α p → α p +1 , (4.16)where α p is a ghost number p -form. We then can construct the p -th de Rham type cohomology group C p ( M, R ) ofthe manifold M and the field of real number R with the quotient group C p ( M, R ) = Z p ( M,R ) B p ( M,R ) . We note that thiscohomology group is similar to the de Rham one in Eq. (4.4), and Z p ( M, R ) are now the collection of all Q -closedghost number p -forms α p for which δα p = 0, and B p ( M, R ) are the collection of all Q -exact ghost number p -forms α p for which α p = δα p − . Next, in terms of the cohomology, the Hamiltonians ˜ H (cid:48) and H eff are readily shown to be theghost number 0-forms, and they are Q -closed δ ˜ H (cid:48) = { Q, ˜ H (cid:48) } = 0 , δH eff = { Q, H eff } = 0 , (4.17)where we have used the identities in Eq. (4.10).Moreover we notice that Ψ is the ghost number ( − { Q, Ψ } is Q -exact ghost number 0-form since { Q, Ψ } can be rewritten as δ Ψ, which satisfies the identity in Eq. (4.13). Here one notes that in order to guarantee theBRST invariance of H eff we have included in H eff the Q -exact term { Q, Ψ } , and in ˜ H (cid:48) the term possessing π θ andthe Faddeev-Popov ghost term. Moreover the term { Q, Ψ } fixes the particular unitary gauge corresponding to thefixed point ( θ = 0 , π θ = 0) in the gauge degrees of freedom associated with two dimensional internal phase spacecoordinates ( θ, π θ ), which are two canonically conjugate St¨uckelberg fields.Now the Hamiltonians ˜ H (cid:48) and H eff thus can be used to define Z ( M, R ). Here M is the non-compact Hilbertspace of the hypersphere Skyrmion model and R is the real number field. The { Q, Ψ } can be used to define the B ( M, R ). With these Z ( M, R ) and B ( M, R ), we construct the 0-th de Rham type cohomology group C ( M, R )for the hypersphere Skyrmion model C ( M, R ) = Z ( M, R ) B ( M, R ) . (4.18)It is interesting to note that the ghost number 0-form H eff is deformed into the other ghost number 0-form ˜ H (cid:48) .Namely H eff is homologous to ˜ H (cid:48) under the BRST transformation δ associated with Q , H eff ∼ ˜ H (cid:48) , (4.19)since { Q, Ψ } = ˜ H (cid:48) − H eff .0 V. CONCLUSIONS
In summary, we have evaluated the physical properties, such as masses, charge radii and magnetic moments, ofthe baryons, and also have obtained their charge density profiles possessing spherical symmetry. To do this we haveused the hypersphere Skyrmion with pion mass correction term. In this model, we have constructed the first classHamiltonian to study the cohomology group structure associated with the nilpotent BRST charge.In particular, we have noticed that the pion mass effects on the physical properties of the baryons improve theirpredictions so that these predicted values are quite in good agreement with their corresponding experimental data.The interesting feature of the hypersphere Skyrmion is that the baryons are delineated in terms of the knot structureof the M¨obius strips. The knot can thus explain the corresponding gluon effect and confinement problem in the quarkrelated model such as quantum chromodynamics.It seems appropriate to comment on the quark-gluon plasma state in the quark model. This state means that theknot structure is broken up in the hypersphere Skyrmion model. The Alice detector of the Large Hadron Collideris scheduled to detect the quark-gluon plasma state, which is assumed to exist in an extremely hot soup of massivequarks and massless gluons. The quark-gluon plasma state is supposed to occur immediately after the Big Bang of thetiny early universe manufactured in the Large Hadron Collider. In the standard cosmology, the quark-gluon plasmastate can exist shortly and disappear eventually to enter the radiation dominated phase. It is expected that the Alicewill be able to detect the procedure of particle states along with the evolution of the tiny universe planned to occurat the Large Hadron Collider. We recall that as far as radiation and matter are concerned, the mixture of these twoexists together in the current universe.
Acknowledgments
The author was supported by Basic Science Research Program through the National Research Foundation of Koreafunded by the Ministry of Education, NRF-2019R1I1A1A01058449. He would like to thank Peter van Nieuwenhuizenfor helpful correspondence and kind encouragement.
Appendix A: Hypersphere geometry
Before we investigate the S hypersphere geometry, we briefly recapitulate the spherical coordinates by introducingthe coordinates ( x, y, z ) in R x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. (A.1)Here r is the radius of S which is described in terms of the spherical shell coordinates ( θ, φ ). The coordinates ( x, y, z )thus satisfy S spherical shell condition x + y + z = r , (A.2)which shows that S spherical shell with radius r is embedded in three dimensional Euclidean space R . The rangesof the spherical coordinates are given by 0 ≤ r < ∞ , 0 ≤ θ ≤ π and 0 ≤ φ ≤ π .In R , the three metric is given by ds ( R ) = dx + dy + dz = dr + r ( dθ + sin θ dφ ) . (A.3)The line element is given by d(cid:126)x = ˆ e r dr + ˆ e θ r dθ + ˆ e φ r sin θ dφ, (A.4)where (ˆ e r , ˆ e θ , ˆ e φ ) are the unit vectors along the three directions. Similarly, the area elements are given by d(cid:126)a r = ˆ e r r sin θ dθ dφ, d(cid:126)a θ = ˆ e θ r sin θ dr dφ, d(cid:126)a φ = ˆ e φ r dr dθ, (A.5)and the volume element is defined as d x = r sin θ dr dθ dφ. (A.6)1 FIG. 7: The S geometry is depicted in terms of S × S . Here S is simplified by S , for convenience. In the spherical coordinates system, the gradient operator is given by ∇ = ˆ e r ∂∂r + ˆ e θ r ∂∂θ + ˆ e φ r sin θ ∂∂φ . (A.7)Next we consider an S hypersphere embedded in R R ⊂ S (= R + {∞} ) ⊂ R . (A.8)Here one notes that even though S is embedded in a four dimensional manifold R , S itself is a three dimensionalmanifold. This statement can be readily understood if we consider two dimensional situation. Namely a sphericalshell S is a two dimensional manifold even though it is embedded in three dimensional manifold R . Moreover thereason why S needs three dimensional embedding manifold R is that S is defined as follows R ⊂ S (= R + {∞} ) ⊂ R . (A.9)In order to investigate the Skyrmion on the hypersphere of radius λ , we introduce the coordinates ( x, y, z, w ) in R to describe a point on S x = λ sin µ sin θ cos φ, y = λ sin µ sin θ sin φ, z = λ sin µ cos θ, w = λ cos µ, (A.10)The ranges of the hyperspherical coordinates are given by 0 ≤ µ ≤ π , 0 ≤ θ ≤ π and 0 ≤ φ ≤ π , and λ is the radiusof S which is described in terms of the hypersphere coordinates ( µ, θ, φ ).Now we note that the coordinates ( x, y, z, w ) satisfy the S hypersphere condition x + y + z + w = λ , (A.11)which shows that the S hypersphere of radius λ is embedded in the four dimensional Euclidean space R . The threemetric on S in R is given by ds ( S ) = dx + dy + dz + dw = λ dµ + λ sin µ ( dθ + sin θ dφ ) . (A.12)The radial direction from the origin of S defines the forth direction of R . The coordinates of the R thus consistof θ , φ , µ and λ . On the S hypersphere, the three metric is given by the line element d(cid:126)x = ˆ e µ λ dµ + ˆ e θ λ sin µ dθ + ˆ e φ λ sin µ sin θ dφ, (A.13)where (ˆ e µ , ˆ e θ , ˆ e φ ) are the unit vectors along the three directions. The relation between R = S × R geometry and S = S × S one is depicted in Figure 7. Here S of radius λ sin µ is simplified by S of radius λ sin µ , for convenience.Moreover the radial direction, associated with ˆ e r , from center of the simplified S with radius λ sin µ represents theradial direction of R in R = S × R . In this picture, S spherical shell of the radius λ sin µ meets S of radius λ .Now we have ˆ e θ , ˆ e φ and ˆ e µ which are perpendicular to one another.2Next we find the following relation ˆ e r = ˆ e µ cos µ + ˆ e λ sin µ. (A.14)Here we note that ˆ e r is defined only in R ⊂ R , but ˆ e µ and ˆ e λ are defined in the part of R on which S of radius λ sin µ does not reside.The area elements are given by d(cid:126)a µ = ˆ e µ λ sin µ sin θ dθ dφ, d(cid:126)a θ = ˆ e θ λ sin µ sin θ dµ dφ, d(cid:126)a φ = ˆ e φ λ sin µ dµ dθ, (A.15)and the volume element is defined as d x = λ sin µ sin θ dµ dθ dφ. (A.16)In the hyperspherical coordinates system, the gradient operator is given by ∇ = ˆ e µ λ ∂∂µ + ˆ e θ λ sin µ ∂∂θ + ˆ e φ λ sin µ sin θ ∂∂φ . (A.17) [1] P.A.M. Dirac, Proc. Roy. Soc. Lond. A , 57 (1962).[2] N.S. Manton, P.J. Ruback, Phys. Lett. B , 137 (1986).[3] S.T. Hong, Phys. Lett. B , 211 (1998).[4] P.A.M. Dirac, Lectures in Quantum Mechanics (Yeshiva University Press, New York, 1964).[5] L. Castellani, P. van Nieuwenhuizen, M. Pilati, Phys. Rev. D , 352 (1982).[6] K. Schoutens, A. Sevrin, P. van Nieuwenhuizen, Commun. Math. Phys. , 87 (1989).[7] S.T. Hong, BRST Symmetry and de Rham Cohomology (Springer, Heidelberg, 2015), and references therein.[8] C. Becchi, A. Rouet, R. Stora, Phys. Lett. B , 344 (1974); C. Becchi, A. Rouet and R. Stora, Ann. Phys. , 287 (1976);I.V. Tyutin, Levedev preprint, LEBEDEV-75-39 (1975).[9] T.H.R. Skyrme, Proc. Roy. Soc. A , 127 (1961).[10] G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B , 552 (1983).[11] H. Toda, Composition Methods in Homotopy Groups of Spheres (Princeton University Press, Princeton, 1962).[12] H. Nastase, M. Stephanov, P. van Nieuwenhuizen and A. Rebhan, Nucl. Phys. B , 471 (1999), hep- th/9802074.[13] D. Gross, F. Wilczek, Phys. Rev. Lett. , 1343 (1973); H.D. Politzer, Phys. Rev. Lett. , 1346 (1973).[14] J.J. Rotman, An Introduction to Algebraic Topology (Springer-Verlag, Heidelberg, 1988); C.A. Weibel,
An Introduction toHomological Algebra (Cambridge University Press, Cambridge, 1994); T. Frankel,
The Geometry of Physics (CambridgeUniversity Press, Cambridge, 1997).[15] J. Baez, J.P. Muniain,
Gauge Fields, Knots and Gravity (World Scientific, Singapore, 1994).[16] S.T. Hong, Mod. Phys. Lett. A25