Is topological Skyrme Model consistent with the Satandard Model

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光学模型介绍英文作文

光学模型介绍英文作文英文：Optical model is a mathematical model that describesthe behavior of light in various media, such as air, water, and glass. It is widely used in the field of optics to predict and analyze the propagation, reflection, refraction, and absorption of light.The optical model is based on the principles of geometrical optics and wave optics. Geometrical optics assumes that light travels in straight lines and obeys the laws of reflection and refraction at the interface between two media. Wave optics, on the other hand, considers light as a wave that can diffract, interfere, and undergo polarization.One of the most important parameters in the optical model is the refractive index, which is a measure of how much a material slows down the speed of light. Therefractive index is different for different materials and can be used to calculate the angle of refraction when light passes through a medium.Another important parameter is the absorption coefficient, which measures how much light is absorbed by a material per unit distance. This is important for designing optical devices such as filters and lenses.In addition to these parameters, the optical model also considers factors such as the thickness of the medium, the angle of incidence, and the polarization state of light. By combining all these factors, the optical model can accurately predict the behavior of light in various scenarios.中文：光学模型是一种描述光在各种介质中行为的数学模型，例如空气、水和玻璃。

Momentum-Space Topology of Standard Model

Байду номын сангаас
where F is any matrix function of any operator which commutes with the Green’s function G . For example, in the Standard Model above the electroweak transition there is a U (1)Y symmetry generated by the hypercharge Y . Since the hypercharge matrix Y commutes with the Green’s function, the matrix F can be any power of the hypercharge, F = Y n . One can easily verify that the perturbations of the Green’s function, which conserve the U (1)Y symmetry, do not change the integral N (Y n ). In the planar phase of 3 He the corresponding U (1) symmetry is combined rotations in spin and orbital space with generator J3 = S3 + L3 . Let us introduce the generating function for all the topological invariants containing powers of the hypercharge N (θY ) = tr eiθY Y N . (4)

TCDMATH 05-08, hep-th0509098 A zero-mode quantization of the Skyrmion

3
1 1 − Tr (Ri Ri ) − ([Ri , Rj ][Ri , Rj ]) − 2 16
2mπ Fπ e
2
Tr (U − 1) .
(2)
We wish to quantize the rotational and isorotational degrees of freedom. Rather than acting on a speciﬁc Skyrme conﬁguration, we want to consider the zero-mode space of ﬁelds generated from a general static conﬁguration Us (x) by isorotation A and rotation B : U (x) = AUs xB A† , where A is in the 2 × 2 representation, xB i = Bij xj , (4) and Bij is a three-dimensional matrix representation of B . Since rotation and isorotation are symmetries of the original Lagrangian, these conﬁgurations are all energy-degenerate. The eﬀective Lagrangian on this restricted space of conﬁgurations can be calculated by allowing A and B to depend on time, giving L = −M + Lrot , where Lrot is the kinetic Lagrangian 1 1 (5) Lrot = Ωi Uij Ωj + ωi Vij ωj − ωi Wij Ωj , 2 2 with the rotational and isorotational angular velocities ω and Ω given by ˙ , Ωi = −iTr σi A† A ˙ , ωi = −iTr σi B† B (6) and the moment of inertia tensors Uij , Vij and Wij given by the following integrals Uij = − d3 x Tr (Ti Tj ) + 1 Tr ([Rk , Ti ][Rk , Tj ]) , 4 2 (3)

A Note in the Skyrme Model with Higher Derivative Terms

a r X i v :h e p -p h /9402257v 1 9 F eb 1994A NOTE IN THE SKYRME MODEL WITHHIGHER DERIVATIVE TERMSJorge Ananias NetoCentro Brasileiro de Pesquisas F´ısicas,R.Dr.Xavier Sigaud 15022290-180Rio de Janeiro,BrazilFebruary,1994AbstractAnother stabilizer term is used in the classical Hamiltonian of the Skyrme Model that permits in a much simple way the generalization of the higher-order terms in the pion derivative ﬁeld.Improved numerical results are obtained.PACS number:12.40.-y1It is well established that the Skyrme Model[1]reproduces with relative suc-cess most of the static properties of Nucleon(within approximately∼30%). The idea consists of treating baryons as soliton solutions in the Non-Linear Chiral SU(2)Sigma Model whose original Lagrangian isF2πL1=32e2 d3r T r U+∂µU,U+∂νU 2,(2) where e is a dimensionless parameter.The physical properties are then calculated making use of the semi-classical expansion of the Quantum Hamiltonian where we perform a rota-cional collective coordinate expansion[3]U(r,t)=A(t)U0(r)A+(t),where A is a SU(2)matrix and U0a static soliton solution.Adopting the hedgehog ansatz U=exp iτ.ˆr F(r)whereτis the Pauli matrix and F(r)is called the chiral angle,the Hamiltonian with the rotational mode can be written asl(l+2)H=M+where n=1,2,....It must be noted that this form is also invariant under chiral transformation (U transforming under U →AUB −1,A and B are SU(2)matrices),and does not destabilize the soliton solution,since we can to deﬁne a positive static Hamiltonian.To simplify the calculations we will adopt the Sugawara form,L µ=U +∂µU,whose the static component can be written as L i =iτa L a i .Using the hedgehog ansatz,the kinetic static term (1)becomes then,L 1=−c 1d 3r T r ∂i U∂i U +=c 1d 3r tr [L i L i ]=−2c 1d 3r L a i L ai ,(6)where L a i L ai =2sin 2Fc 1(2L a i L ai )+c 3c 1(2L a i L a i )n −1.(7)There are many particular values of the relationscnc 1=1(2e 2F 2π)n −1,(8)the series of the classical energy(7)converges in an exponential formE =d 3r2c 1L a i L aiexpL a i L a i16d 3r 22sin 2Fr 2+F ′2eπx 2+F′2exp2sin 2FK !and a dimensionless variablex deﬁned by3x=eFπr.(10) It is interesting to point out that with the choice of standard form(5)and with the deﬁnition(10),all the coupling constants are absorbed in the new dimensionless variable x.From(9)the variational equation is[2x2+2x2S+8x2F′2+4x2SF′2]F′′+8sin2F F′2+4S sin2F F′2+4xF′−16sin2F F′x−2sin2F−2Ssin2F=0,(11) where S≡ 2sin2Fx2,(12) where B is a constant,to obtain using numerical integration the soliton solution.Inﬁgure1,we show the numerical behaviour of parameter B,whichis direclty proportional to the axial vector constant coupling,g A=2πe2.The inertia moment is calculated similarly to the series of the classi-cal energy.Performing the rotacional collective coordinate expansion of the classical Lagrangian and picking up only terms linear in T r(∂0A+∂0A),we obtain the expression of the series of the inertia moment given byI= d3r8I=2πFπe3 ∞0dxx2sin2F1+F′2+2sin2Fx2 ,(14)where we have used the series representation exp y(1+y)= ∞k=0y k(k+1)TABLE1-Physical parameters in the Skyrme ModelFπ(Mev)129146144186e 5.458.69 6.82-<r2>1Our results indicate an improvement in the physical values of the pion decay constant,Fπ,and the axial coupling constant,g A.The others values,i.e., the magnetic moments and the isoescalar mean square radius remain basi-caly the same obtained by Adkins[5]et al.and Marleau[4].This procedure, without doubt,improves the physical results.In order to obtain better re-sults for quantities like magnetic moments and the isoescalar mean square radius,we should treat the quantization of the classical Hamiltonian(collec-tive coordinate quantization[3])in more formal way,using the information about constraint that is present in the system[6].This particular study will be object of a forthcoming paper[7].I would like to thank M.G.do Amaral for critical reading.The work is supported by CNPq,Brasilian Research Council.References[1]T.H.Skyrme,Proc.Roy.Soc.A260,127(1961).[2]G.Derrick,J.Math.Phys.5,1252(1964).[3]G.S.Adkins,C.R.Nappi and E.Witten,Nucl.Phys.B228,553(1983).6[4]S.Dub´e and L.Marleau,Phys.Rev.D41,5,1606(1990).[5]G.S.Adkins,Chiral Solitons,Proceedings of the Lewes Workshop(WorldScientiﬁc,Singapore)pag.47.[6]K.Fujii,K.I.Sato,N.Toyota and A.P.Kobushkin,Phys.Rev.Lett.58,7(1987),Jorge Ananias Neto,Preprint IF/UFRJ/94,hep-th9401001. [7]Jorge Ananias Neto,work in progress.7This figure "fig1-1.png" is available in "png" format from: /ps/hep-ph/9402257v1。

Skyrmion

SkyrmionIn particle theory,the skyrmion (/ˈskɜrmi.ɒn/)is a hy-pothetical particle related originally [1]to baryons .It was described by Tony Skyrme and consists of a quantum su-perposition of baryons and resonance states.[2]Skyrmions as topological objects are also important in solid state physics ,especially in the emerging technology of spintronics .A two-dimensional magnetic skyrmion ,as a topological object,is formed,e.g.,from a 3D eﬀective-spin “hedgehog”(in the ﬁeld of micromagnetics :out of a so-called "Bloch point "singularity of homotopy degree +1)by a stereographic projection ,whereby the positive north-pole spin is mapped onto a far-oﬀedge circle of a 2D-disk,while the negative south-pole spin is mapped onto the center of the disk.1Mathematical deﬁnitionIn ﬁeld theory,skyrmions are homotopically non-trivial classical solutions of a nonlinear sigma model with a non-trivial target manifold topology –hence,they are topological solitons .An example occurs in chiral models [3]of mesons,where the target manifold is a homogeneous space of the structure group (SU (N )L ×SU (N )RSU (N )diag)where SU (N )L and SU (N )R are the left and right parts of the SU (N )matrix,and SU (N ) ₐ is the diagonal subgroup .If spacetime has the topology S 3×R ,then classical con-ﬁgurations can be classiﬁed by an integral winding num-ber [4]because the third homotopy groupπ3(SU (N )L ×SU (N )R SU (N )diag ∼=SU (N ))is equivalent to the ring of integers,with the congruence sign referring to homeomorphism .A topological term can be added to the chiral Lagrangian,whose integral depends only upon the homotopy class ;this results in superselection sectors in the quantised model .A skyrmion can be approximated by a soliton of the Sine-Gordon equation ;after quantisation by the Bethe ansatz or otherwise,it turns into a fermion inter-acting according to the massive Thirring model .Skyrmions have been reported,but not conclu-sively proven,to be in Bose-Einstein condensates ,[5]superconductors ,[6]thin magnetic ﬁlms [7]and also chiral nematic liquid crystals .[8]2Skyrmions in an emerging tech-nologyOne particular form of the skyrmions is found in mag-netic materials that break the inversion symmetry and where the Dzyaloshinskii-Moriya interaction plays an im-portant role.They form “domains”as small as a 1nm (e.g.in Fe on Ir(111)[9]).The small size of mag-netic skyrmions makes them a good candidate for fu-ture data storage solutions.Physicists at the University of Hamburg have managed to read and write skyrmions using scanning tunneling microscopy.[10]The topological charge,representing the existence and non-existence of skyrmions,can represent the bit states “1”and “0”.3References[1]At later stages the model was also related to mesons .[2]Wong,Stephen (2002).“What exactly is a Skyrmion?".arXiv :hep-ph/0202250[hep/ph ].[3]Chiral models stress the diﬀerence between “left-handedness”and “right-handedness”.[4]The same classiﬁcation applies to the mentioned eﬀective-spin “hedgehog”singularity":spin upwards at the north-pole,but downward at the southpole.See also Döring,W.(1968).“Point Singularities in Mi-cromagnetism”.Journal of Applied Physics 39(2):1006.Bibcode :1968JAP....39.1006D .doi :10.1063/1.1656144.[5]Al Khawaja,Usama;Stoof,Henk (2001).“Skyrmionsin a ferromagnetic Bose–Einstein condensate”.Nature 411(6840):918–20.Bibcode :2001Natur.411..918A .doi :10.1038/35082010.PMID 11418849.[6]Baskaran,G.(2011).“Possibility of Skyrmion Superconductivity in Doped Antiferromagnet K$_2$Fe$_4$Se$_5$".arXiv :1108.3562[cond-mat.supr-con ].[7]Kiselev,N.S.;Bogdanov,A.N.;Schäfer,R.;Rößler,U.K.(2011).“Chiral skyrmions in thin magnetic ﬁlms:New objects for magnetic storage technologies?".Journal of Physics D:Applied Physics 44(39):392001.arXiv :1102.2726.Bibcode :2011JPhD...44M2001K .doi :10.1088/0022-3727/44/39/392001.[8]Fukuda,J.-I.;Žumer,S.(2011).“Quasi-two-dimensional Skyrmion lattices in a chiralnematic liquid crystal”.Nature Communica-tions 2:246.Bibcode :2011NatCo...2E.246F .doi :10.1038/ncomms1250.PMID 21427717.123REFERENCES [9]Heinze,Stefan;Von Bergmann,Kirsten;Menzel,Matthias;Brede,Jens;Kubetzka,André;Wiesen-danger,Roland;Bihlmayer,Gustav;Blügel,Ste-fan(2011).“Spontaneous atomic-scale magneticskyrmion lattice in two dimensions”.Nature Physics7(9):713–718.Bibcode:2011NatPh...7..713H.doi:10.1038/y summary(Jul31,2011).[10]Romming,N.;Hanneken, C.;Menzel,M.;Bickel,J. E.;Wolter, B.;Von Bergmann,K.;Kubet-zka, A.;Wiesendanger,R.(2013).“Writing andDeleting Single Magnetic Skyrmions”.Science341(6146):636–9.Bibcode:2013Sci...341..636R.doi:10.1126/ysummary–(Aug8,2013).3 4Text and image sources,contributors,and licenses4.1Text•Skyrmion Source:/wiki/Skyrmion?oldid=637550141Contributors:Michael Hardy,Charles Matthews,Phys, Icairns,Lumidek,Pjacobi,Jag123,Fwb22,Rjwilmsi,Conscious,Wikid77,Headbomb,Lincoln F.Stern,Tarotcards,KylieTastic,Pix-elBot,Doprendek,Addbot,Luckas-bot,Yobot,Citation bot,Obersachsebot,Omnipaedista,Citation bot1,Merongb10,Meier99,Korepin, EmausBot,JSquish,ZéroBot,StringTheory11,AManWithNoPlan,Isocliﬀ,Parcly Taxel,Bibcode Bot,BattyBot,ChrisGualtieri,Andy-howlett,1andreasse,Nicohoho,NorskMaelstrom,Noah Van Horne and Anonymous:74.2Images•File:Portal-puzzle.svg Source:/wikipedia/en/f/fd/Portal-puzzle.svg License:Public domain Contributors:?Original artist:?4.3Content license•Creative Commons Attribution-Share Alike3.0。

Reasoning about space the modal way

2.2
Topological completeness
The ﬁrst semantic completeness proof for S4 did not use the by now dominant relational modal models, which go back to Kanger, Hintikka, and Kripke in the 1950s. It was actually an earlier spatial completeness argument of [16], in terms of the following notions. Recall that a topological space is a pair W, τ , where W is a non-empty set and τ a collection of subsets of W satisfying the following three conditions: • ∅, W ∈ τ , • if U, V ∈ τ , then U ∩ V ∈ τ , • if {Ui }i∈I ∈ τ , then
2
2.1
Modal language and topological semantics
Language and axioms
Let us ﬁrst set the scene where we will operate. The basic language L of propositional modal logic is composed of • a countable set of proposition letters, • boolean connectives ¬, ∨, ∧, →, • modal operators 2, 3. The standard axiomatization of our central logic S4 is 2(ϕ → ψ ) → (2ϕ → 2ψ ) 2ϕ → 22ϕ 2ϕ → ϕ (K) (4) (T)

The mass and dynamical state of Abell 2218

a r X i v :a s t r o -p h /9806230v 2 18 J u n 1998Mon.Not.R.Astron.Soc.000,1–16(1998)Printed 1February 2008(MN La T E X style ﬁle v1.4)The mass and dynamical state of Abell 2218D.B.Cannon,T.J.Ponman &I.S.HobbsSchool of Physics and Space Research,University of Birmingham,Birmingham B152TT,UKAccepted 1998??.Received 1998??;in original form 1997??ABSTRACTAbell 2218is one of a handful of clusters in which X-ray and lensing analyses of thecluster mass are in strong disagreement.It is also a system for which X-ray data and radio measurements of the Sunyaev-Zel’dovich decrement have been combined in an attempt to constrain the Hubble constant.However,in the absence of reliable information on the temperature structure of the intracluster gas,most analyses have been carried out under the assumption of isothermality.We combine X-ray data from the ROSAT PSPC and the ASCA GIS instruments,enabling us to ﬁt non-isothermal models,and investigate the impact that this has on the X-ray derived mass and the predicted Sunyaev-Zel’dovich eﬀect.We ﬁnd that a strongly non-isothermal model for the intracluster gas,which im-plies a central cusp in the cluster mass distribution,is consistent with the available X-ray data and compatible with the lensing results.At r <1′,there is strong evi-dence to suggest that the cluster departs from a simple relaxed model.We analyse the dynamics of the galaxies and ﬁnd that the central galaxy velocity dispersion is too high to allow a physical solution for the galaxy orbits.The quality of the radio and X-ray data do not at present allow very restrictive constraints to be placed on H 0.It is apparent that earlier analyses have under-estimated the uncertainties involved.However,values greater than 50km s −1Mpc −1are preferred when lensing constraints are taken into account.Key words:galaxies:clusters:individual:A2218–X-rays:galaxies –dark matter –cosmic microwave background –gravitational lensing –distance scale1INTRODUCTIONThe masses of galaxy clusters can be determined in three main ways:using the velocity dispersion of the galaxies,the pressure gradient in the hot intracluster gas derived from X-ray imaging and spectroscopy,and by analysing the lens-ing of background galaxies by the cluster potential.Each of these approaches involve assumptions and are vulnerable to various systematic errors.It is therefore useful to compare the results of the diﬀerent techniques.The X-ray and lens-ing approaches are generally considered the most reliable.Results from them have now been compared for a number of clusters (Fort &Mellier 1994).The agreement is often reasonable,but there are a few spectacular exceptions,of which Abell 2218(hereafter A2218)is the most well studied example.A2218is an optically compact (core radius ≈1′;Dressler 1978)cluster of galaxies,located at a redshift of 0.171(Kristian,Sandage &Westphal 1978),and classiﬁed as richness class 4(Abell,Corwin Jr &Olowin 1989).The clus-ter appears well relaxed,with the majority of the galaxies centred around the sole cD galaxy.However,detailed photo-metric studies (Pello-Descayre et al.1988;Pello et al.1992)suggest the existence of a second,smaller galaxy concen-tration,displaced from the cD by 67′′.Spectroscopic study (Le Borgne,Pello &Sanahuja 1992),performed on the cen-tral region (<4′)of A2218,has provided redshift informa-tion for 66of the objects within the core and shown that the average velocity dispersion is 1370km s −1.A succession of X-ray telescopes have allowed the prop-erties of the hot gas within A2218to be established.With the Einstein IPC &HRI (Perrenod &Henry 1981)and ROSAT PSPC (Siddiqui 1995),the emission was found to be smooth (on scales ∼1′),azimuthally symmetric and cen-tred on the cD galaxy.Fitting a polar proﬁle of the surfacebrightness with a King model gave a core radius of 58′′+16−16and a β-value of 0.63(Boynton et al.1982).Integrated spec-tral analyses with Ginga gave a gas temperature of 6.72+0.5−0.4keV and metallicity of 0.2+0.2−0.2Z ⊙(McHardy et al.1990).By virtue of Ginga’s bandwidth,this determination is com-monly accepted as the most accurate estimate of the mean gas temperature.Most recently,deep observations with the ROSAT HRI (Markevitch 1997)have shown the presence of signiﬁcant X-ray substructure within the cluster core,sug-c1998RAS2 D.B.Cannon,T.J.Ponman&I.S.Hobbsgesting that the cluster may have undergone a recent merger event.This may account for the absence of any signs of a coolingﬂow in the cluster(Arnaud1991;White1996).Several previous comparisons(Miralda-Escude&Babul 1995;Kneib et al.1995;Kneib et al.1996;Natarajan& Kneib1996)between strong lensing and X-ray analyses have found a factor of2discrepancy in the gravitating masses predicted by the two methods.Suggested explanations have centred upon the assumption of hydrostatic equilibrium for the cluster gas,the possibility that magneticﬁelds may pro-vide signiﬁcant pressure support to the gas and the presence of substructure within the cluster.The Sunyaev-Zel’dovich decrement associated with A2218has been extensively studied(Jones et al.1993; Birkinshaw&Hughes1994;Saunders1996).These results have been used,in conjunction with X-ray data,to con-strain the Hubble constant(Silk&White1978;McHardy et al.1990;Birkinshaw&Hughes1994).These analyses have been made in the absence of reliable information about tem-perature variations in the intracluster gas and have there-fore been forced to make simplifying assumptions,such as that of isothermality(McHardy et al.1990;Birkinshaw& Hughes1994;Kneib et al.1995).This assumption is without a strong theoretical foundation and conﬂicts with the results of most cosmological simulations(Navarro,Frenk&White 1995;Navarro,Frenk&White1996;Tormen,Bouchet& White1996),which show temperature declining with radius, and mass distributions which have a central cusp.The ques-tion then arises as to whether simplifying assumptions have signiﬁcantly biased the conclusions of previous X-ray anal-yses.For example,is the apparent discrepancy between the X-ray and lensing masses unavoidable or does it arise sim-ply from the use of inappropriate assumptions in the X-ray analysis?Motivated by the desire to avoid such restrictive as-sumptions,we have carried out an X-ray analysis which combines the capabilities of ROSAT and ASCA.The lim-ited spectral bandwidth and resolution of the ROSAT PSPC is compensated for by the superior spectral properties of ASCA.Conversely,the poor spatial performance of ASCA is complemented by the higher spatial resolution of ROSAT. This approach has never before been applied to A2218.The central aim of this paper is to improve our under-standing of A2218by comparing the results of our X-ray analysis with lensing,SZ and galaxy velocity studies.It also serves as a case study on the possible dangers of assum-ing an isothermal gas,when one has no information to the contrary.Throughout the paper we assume an Einstein-de Sitter cosmology withΩ=1,q0=0.5and H0=50km s−1 Mpc−1,except where otherwise stated.2X-RAY ANALYSISThe objective of the analysis is to use spatially and spec-trally resolved X-ray data to constrain models of the distri-bution of gas properties in the cluster.For an in-depth dis-cussion of the procedures covered in this section,see Cannon (1997).2.1Spectral-image modellingWe work with X-ray spectral images,which constitute blurred records of the spectral properties of the cluster pro-jected along the line of sight.Since information about the disposition of material perpendicular to the plane of the sky is not available,it is necessary to make some assumption about the geometry of the source.We assume that the clus-ter is spherically symmetric.In practice,A2218is slightly elliptical,with an axis ratio of0.8(Siddiqui1995).How-ever this modest ellipticity should not introduce any serious errors into our derived masses(Fabricant,Rybicki&Goren-stein1984).It is important to allow for the spatial and spectral blurring introduced by the telescope,as described by the instrument point spread function(psf)and energy response matrix.We adopt a forwardﬁtting approach(Eyles et al. 1991;Watt et al.1992),in which the properties of the gas are parameterised as analytical functions of cluster radius. The emission from each spherical shell in the cluster is com-puted using a Raymond&Smith(1977),hereafter RS,hot plasma code.After correcting for cluster redshift,the spec-tral emissivity proﬁles are folded through the instrument spectral response,projected along the line-of-sight,rebinned into an xy grid and blurred with the psf.This produces a predicted spectral image which can be directly compared to the observed data,using a maximum-likelihood statistic.It-eratively adjusting the model parameters results in a best-ﬁt to the data.Using analytical forms for the radial distribution of gas properties has the advantage of regularising the solution(i.e. suppressing instabilities in the deprojection and deblurring processes),however one runs the risk that the solution may be dictated by the mathematical function imposed.This can lead to overconﬁdence in derived results,as acceptable al-ternatives which mightﬁt the data are ruled out by the lim-itations of the available models.The commonly employed restriction of isothermality is an extreme example of this. We attempt to avoid this problem by using a range of radial functions.This is particularly important for the tempera-ture and we use not only a number of parametric forms for T gas(r),but also an alternative approach in which T gas(r)is determined indirectly,byﬁtting a model for the mass distri-bution,as discussed below.The gas density proﬁle is much more readily determined by the X-ray data,so we haveﬁtted only two radial forms.Assuming that the intracluster gas is in hydrostatic equilibrium in the potential well of the cluster,the total gravitating mass within radius r,from the centre of the clus-ter,is related to the gas temperature and density by:M grav(r)=−kT gas(r)d ln r+d ln T gas(r)The mass and dynamical state of Abell22183generally wellﬁtted by core-index type models(King1962; King1972):ρgas(r)=ρgas,0 1+(r/r c)2 −αρ(2) whereρgas,0is the central gas density normalisation(amu cm−3),r c the core radius(arcmin)andαρthe density in-dex(unitless).The main deviations from this form occur at small radii,where coolingﬂows give rise to surface bright-ness cusps in many clusters,though not in A2218.Recent N-body studies(Navarro,Frenk&White1995; Navarro,Frenk&White1996;Tormen,Bouchet&White 1996)have achieved goodﬁts to dark matter(DM)and gas proﬁles in simulated clusters with an alternative descrip-tion.The proﬁles are found to steepen progressively,from ρgas(r)∝r−1in the core,to r−3near the virial radius, following the formρgas(r)=ρgas,0 x(1+x)2 −1(3) where x=r/r s,r s being the scale radius(arcmin).Both of the above analytical forms have beenﬁtted to the X-ray data for A2218.2.1.2Gas temperatureThe gas temperature distribution is less well determined, since this requires a combination of spatial and spectral res-olution which has not generally been available in the past. We consider a variety of simple models:a linear temperature ramp(LTF),T gas(r)=T gas,0−βr(4) where T gas,0is the gas temperature(keV)at the cluster centre,βthe temperature gradient(keV arcmin−1)and r the radius(arcmin);a King-type temperature description (KTF),T gas(r)=T gas,0 1+(r/r T)2 −β(5) where r T is the temperature core radius(arcmin)andβthe temperature index(unitless);and a polytropic temperature description(TTF),T gas(r)=T gas,0 1+(r/r c)2 αρ(1−γ)(6) where r c is the gas density core radius(arcmin)andγ,the polytropic index(unitless),isﬁtted as a free parameter vary-ing between isothermality(γ=1)and adiabaticity(γ=5/3).2.1.3Gravitating massAn alternative toﬁttingρgas(r)and T gas(r)is toﬁtρgas(r) and M grav(r).The corresponding temperature proﬁle can then be inferred,via Equation1.We use several alternative forms,motivated by the distribution of galaxies in clusters (Rood et al.1972),and by the results of N-body simulations. These include:a core-index description(DMF),ρDM(r)=ρDM,0 1+(r/r c)2 −αDM(7) whereρDM,0is the central dark matter density normalisation (amu cm−3),r c the core radius(arcmin)andαDM the den-sity index(unitless);a model based upon the simulations of Navarro,Frenk&White(1995)(DNF),ρDM(r)=ρDM,0 x(1+x)2 −1(8) where x=r/r s and r s is the scale radius(arcmin);and a Hernquist proﬁle(DHF),ρDM(r)=ρDM,0/ 2πb(1+b)3 (9) where b=r/r s and r s is the scale radius(arcmin).2.1.4Fitting the modelsDetermination of the best-ﬁt parameters for a cluster model proceeds in the way commonly employed for spectralﬁtting. Theﬁt statistic employed is maximum likelihood,rather than chi-squared,since the data are generally strongly Pois-sonian.Theﬁt and its local slope are determined at some initial position in the parameter space.This information is used to predict an improved set of model parameters and theﬁt statistic re-determined.The process is iteratively re-peated until the statistic slope falls below a pre-determined value.One limitation of this method is,however,that the ﬁtting tends to follow the local gradient in the statistic,until it encounters a minimum.Thus theﬁt can become trapped in a“valley”,which it regards as the best-ﬁt result,even though a more suitable combination of parameters may oc-cur elsewhere.To avoid this,we randomly perturb models during analysis and force them to re-ﬁt(to check if the same minimum is produced).Conﬁdence regions can be derived for each best-ﬁt pa-rameter,by oﬀsetting the parameter of interest from its best-ﬁt value(both above and below the best-ﬁt),and reoptimis-ing the other parameters.The resulting increase in theﬁt statistic,from its optimum value,is used to determine what oﬀset would need to be applied in order to create a user-deﬁned change in the statistic.This deﬁnes the required conﬁdence interval.We use the form of the maximum likeli-hood statistic introduced by Cash(1979),such that changes in the statistic have the same signiﬁcance as changes in chi-squared.Hence,for each parameter,an increase in the Cash statistic of1corresponds to a68%conﬁdence interval,and an increase of2.71to90%conﬁdence.The above process is repeated for each model parameter for which an error esti-mate is required.Errors in physical quantities which are functions of ra-dius(such as mass or temperature)are generally aﬀected by several model parameters.We derive error envelopes for such quantities by taking the outer envelope of all of the curves generated by perturbing each free parameter to its upper and lower error bounds.Because these envelopes are derived using every parameter combination,each oﬀset to their error bounds,the result is a conservative estimate of the statistical uncertainty.Once the total gravitating mass distribution has been determined(using Equation1,if the mass has not been mod-elled directly)the various mass components in the cluster can be separated.The gas mass proﬁle is calculated from theﬁtted parameters.The galaxy mass proﬁle can be con-structed from the observed luminosity proﬁle for the cluster, assuming a constant mass-to-light ratio.Subtraction of these components from the total mass proﬁle then yields the dark matter proﬁle.c 1998RAS,MNRAS000,1–164 D.B.Cannon,T.J.Ponman &I.S.HobbsFigure 1.Gas temperatures derived in projected annuli.The barred crosses are taken from ROSAT PSPC analysis and the di-amonds from ASCA GIS analysis.The results suggest a temper-ature decline with radius,with weak evidence (from the PSPC)for central cooling.2.2ROSAT PSPC reduction The aim of the ROSAT analysis is to obtain well constrained gas density parameters,which can then be utilised in the ASCA analysis.The raw data,obtained on May 25th 1991,were re-duced using the Starlink ASTERIX X-ray analysis package.Periods of high background were removed from the data,reducing the eﬀective exposure time to 42ksec but making the background subtraction substantially more reliable.Subtraction of the X-ray background was accomplished by selecting data from an annulus (27′−33′),ignoring pix-els covered by the detector support structure or contain-ing point source emission.This background sample was then extrapolated to cover the whole ﬁeld,using the PSPC energy-dependent vignetting function.Since the X-ray sur-face brightness proﬁle for A2218can be traced to a maxi-mum radius of 12′,the chosen background annulus is free of source emission.In the following analysis,the data are re-stricted to lie within 9′to avoid possible systematic eﬀects from uncertainties in the background subtraction at large radius.The exposure-corrected,background-subtracted PSPC data were summed to provide an integrated cluster spec-trum and split into concentric annuli,centred on the clus-ter core,within which spectra were extracted.Fitting these spectra gives an indication of the depth of the cluster poten-tial well and the radial structure of the cluster gas density and temperature parameters (see Fig 1).However,any gra-dients present in these distributions will tend to be underes-timated,due to the smoothing eﬀects of the instrument psf and projection along the ing annuli of width greater than the instrument psf minimizes the former eﬀect.In order to create a spectral image dataset,allowing a full spatial and spectral analysis,the data were formed into images of channel width 10,over the channel range 11-230(approximately 0.2-2.3keV).This results in a data cube,from which speciﬁc regions can be selected and analysed.Within the cluster emission there is one bright source contaminating the data,located at a radius of 11.1′from the cluster centre.In the PSPC analysis,the point source can be eliminated by ignoring the data collected in that region.However,this is not possible with ASCA,since the extended psf ensures that it is not discernable as a discrete source.To determine whether the source signiﬁcantly aﬀects our analysis,we model it using the PSPC data.The cluster emission is ﬁrst ﬁtted with the point source pixels removed.This model of the cluster emission is then subtracted from the original data,leaving behind just the point source,which is ﬁtted with a power-law spectral model.The ﬁtted index, 1.25,indicates the softness of the source -the majority of the ﬂux is emitted below 0.5keV.This is consistent with identiﬁcation of the source as SAO17151,a bright star with a soft spectrum (Markevitch 1997).If the PSPC-determined model for the point source is subtracted from the ASCA data,the cluster ﬁts are modiﬁed to the extent that a 1.5%diﬀerence in the total gravitating mass at 2Mpc results.This eﬀect is negligible compared to other errors,so no attempt has been made to remove the source from the ASCA data.2.3ASCA GIS reductionThe ASCA analysis aims to constrain the gas temperature and metallicity proﬁles,using the PSPC derived gas density proﬁle.A2218was observed by the ASCA X-ray telescope on April 30th 1993.In this paper we use only data from the two gas imaging spectrometers (GIS2and GIS3),since these have a wider ﬁeld of view (∼50′diameter)and greater high-energy detector eﬃciency (up to 10keV)than the CCD detectors (SIS0and SIS1).An additional reason is that an accurate model for the large,asymmetrical and energy de-pendent psf is available for the GIS detectors,constructed from Cyg X-1observations.These restrict analysis to a max-imum radius of 18′and an energy-range of 1.5-11keV (Taka-hashi et al.1994),which is not a signiﬁcant limitation in the case of A2218.Standard procedures for ASCA analysis,followed in this paper,are given by Day et al.(1995).The recommended screening criteria are applied to the raw data,removing data taken during times of high background ﬂux.Subtraction of the X-ray background is complicated by the telescope psf.This has the eﬀect of ensuring that no region of the detector is free from source ﬂux.Hence,we extract an “average”back-ground dataset from the publicly distributed set of blank-sky pointings (Day et al.1995).These datasets suﬀer from mild point-source contamination,as sources are not completely averaged out,but are currently the best available solution.The results of a naive annular spectral analysis of the ASCA data are shown in Fig 1.However,it is important to bear in mind the limitations of this approach.Cross-talk between annuli is signiﬁcant (Takahashi et al.1994),and the energy-dependent spreading of ﬂux results in a dis-torted temperature proﬁle,such that analysis of a simulated isothermal cluster would give a temperature which rises withc1998RAS,MNRAS 000,1–16The mass and dynamical state of Abell22185 radius.In practice,the temperature appears to decline withradius,indicating that a real gradient is present.For3D analysis,spectral-image datasets with contigu-ous energy bands of width50raw channels are created.Sincethe cluster centre is oﬀset from the detector centre,data be-yond a radius of9′are notﬁtted(the oﬀset added to theradial extent of the source is similar to the maximum radiuswhere psf calibrations apply).This restriction minimises theeﬀects of poor calibration and high background near the de-tector edge.As both GIS instruments behave similarly,thedatasets areﬁtted simultaneously.The Cash statistic hasbeen used to identify best-ﬁt models,but similar results forbest-ﬁt parameters and for comparison between the qualityofﬁt of diﬀerent models,is obtained using theχ2statistic.3RESULTSWeﬁrst compare the results of our analysis with publishedstudies,to investigate whether theﬁtted models are consis-tent with earlier work on A2218.Integrated spectral analyses of clusters produce“mean”quantities which are representative of the entire object.As-suming isothermality,McHardy et al.(1990)derived a gastemperature of6.72+0.5−0.4keV together with an iron abun-dance of0.2+0.2−0.2Z⊙(using an RS emission code,where all other heavy element abundances wereﬁxed at0.5Z⊙). This is in agreement with an earlier,much less well con-strained examination(Perrenod&Henry1981).More re-cently,Mushotzky&Loewenstein(1997)have derived T= 7.2keV and Z=0.18±0.07Z⊙,from an integrated ASCA spectrum.Fitting an RS model to our ROSAT data results in agas temperature of4.7+1.1−0.9keV and a hydrogen absorptioncolumn of2.6+0.2−0.1x1020cm−2(with metallicityﬁxed at the Ginga value).This absorption agrees with the Stark level of2.58+0.18−0.18x1020cm−2(Stark et al.1992).The temperatureis lower than the Ginga result of McHardy et al.(1990),but the energy range of the PSPC is not very suitable for determining the temperature of such hot gas.It has been found previously(Markevitch&Vikhlinin1997)that PSPC results tend to be biased low for high temperature clusters. Fitting an integrated spectrum simultaneously to the GIS2 and GIS3data gives a gas temperature of6.73+0.46−0.44keV anda metallicity of0.20+0.08−0.08Z⊙(with the hydrogen columnﬁxed at the PSPC value),in good agreement with the Ginga and Mushotzky&Loewenstein(1997)results.The ability to extract and analyse spectra from inde-pendent regions of the cluster represents a signiﬁcant ad-vance over analysis of the integrated emission.Fig1shows the derived annular temperature proﬁles from both ROSAT and simultaneous GIS2/GIS3analysis.The PSPC results are consistent with Siddiqui(1995),with a possible temper-ature drop visible in the central bin.However,the evidence for central cooling is statistically rather weak,and the de-rived central cooling time of∼1.5x1010yr is comparable with the Hubble time,so a strong steady-state coolingﬂow appears to be ruled out.If the hydrogen column isﬁtted,using the PSPC data, it is found to be consistent with the Stark value(Stark et al. 1992)throughout the cluster,apart from a slight rise in the centre.This may be due to matter deposited by an ear-lier,disrupted coolingﬂow(as noted by Siddiqui1995).The ASCA analysis suggests that the metallicity may be slightlylower than the integrated value in the cluster centre with a shallow radial rise.However,all points are consistent with the McHardy et al.(1990)value of0.2Z⊙.Analysis using spectral-image datasets allows extrac-tion of the3D gas density and temperature distributions within A2218.In the subsequent cluster analysis,both‘tem-perature models’(ﬁtting forρgas(r)and T gas(r))and‘mass models’(ﬁtting forρgas(r)and M tot(r))are used.Parame-ters representing the metallicity and cluster position are also ﬁtted.The best constrained parameters derived from PSPC analysis pertain to the shape of the gas density distribution. Comparable analyses have been carried out using Einstein (Perrenod&Henry1981;Boynton et al.1982;Birkinshaw &Hughes1994),and ROSAT(Siddiqui1995)data.These studies agree that when a King proﬁle is assumed,A2218is well modeled with a core radius,r c,of∼1′and an index,αρ,of∼1(equivalent to aβ-value of0.67).Higher resolu-tion analysis,using the ROSAT HRI,has been performed by Squires et al.(1996)and Markevitch(1997).These studies detect the presence of smaller scale structure,with three sur-face brightness peaks visible within the central arcminute, none of which coincides with the central cD galaxy.This structure cannot be resolved with either the PSPC or GIS detectors,and indicates that the core of A2218departs from a fully relaxed state.Since ASCA is poor at constraining the gas density dis-tribution for such a compact source,the PSPCﬁt values for core radius and index are carried over into the ASCA spectral-image analysis.To obtain the appropriate density parameters,a linear temperature ramp model isﬁtted to the entire PSPC data within a radius of9arcmin,with the temperature parametersﬁxed at those derived from an initialﬁt to the GIS data.This model is then re-ﬁtted to the GIS data,complete to a radius of9arcmin,with the gas density core radius and indexﬁxed,allowing aﬁt of the temperature parameters.In an iterative process,this model is alternatelyﬁtted to the PSPC and GIS datasets until no further change is observed in the parameter values. With consistency achieved,the“standard”gas distribution for A2218is determined to be r c=0.91+0.03−0.03arcmin,αρ=0.96+0.02−0.01,in good agreement with the comparable results discussed above.If an NFW proﬁle is assumed,the requiredscale radius,r s,is10.27+0.21−0.20arcmin.However,the NFW parameterisation is neither preferred nor disallowed by the PSPC data,so we only use a King parameterisation for the gas density distribution in the following analysis.3.1All cluster dataFixing the gas density shape parameters at these King val-ues,a range of models wereﬁtted to the ASCA spectral-image data.On the basis of their Cash statistic,a set of models best-ﬁtting the observed cluster data within a ra-dius of9arcmin is selected(Table1lists the main parame-ters for each model)as representative of A2218.As can be seen from the Table,the isothermal model is a signiﬁcantly poorerﬁt to the data.It is included for comparison with the other models.All of the temperature proﬁles obtained for this set(apart from that for the isothermal model)arec 1998RAS,MNRAS000,1–166 D.B.Cannon,T.J.Ponman&I.S.Hobbs plotted in Fig2.Beyond the central arcminute(∼230kpc) the proﬁles are in good agreement and quite non-isothermal. The typical90%conﬁdence envelope for an individual model (remember that these are conservative envelopes)generally encompasses the spread of these best-ﬁt proﬁles(a single error envelope is included to illustrate this point).Within the central region,which lies within a single ASCA psf,a greater spread in temperature is allowed by the data.How-ever,despite this divergence in temperature at small radius, all of the best-ﬁt models,bar the isothermal model,have similar Cash statistics(see Table1).The abundance of heavy elements has been assumed,in all of the above models,to be constant over the cluster. However,the spectral capabilities of ASCA allow us to test this assumption.Allowing a linear metallicity gradient with the best-ﬁt linear temperature ramp model gives a slope of2.8+4.5−6.5x10−2Z⊙arcmin−1,a value consistent with uni-form metallicity.The eﬀect of this best-ﬁt slope upon other model parameters is negligible,hence freezing the metallic-ity gradient at zero does not bias our analysis.3.2Central cluster dataWe now examine the claim that lensing analyses require a larger cluster mass than is consistent with the X-ray data. The model which provides the greatest gravitating mass at the critical radius(∼85kpc)is therefore selected for exam-ination.This model,the DHF best-ﬁt,using a Hernquist description for the DM distribution and a King description for the gas density proﬁle(see Table1),implies a high cen-tral gas temperature.In the following comparisons,we refer to this model as the“maximum-mass model”(MMM).The MMM temperature proﬁle features a factor of2rise within the central1′.This raises two questions:does such a steep temperature gradient raise physical problems(would it be convectively unstable?),and is it consistent with the X-ray spectral data observed within the central region?We will return to theﬁrst question in Section5.To address the latter question,GIS3spectra integrated within r=1′,where the MMM temperature rises steeply, are compared with the predictions of the MMM and isother-mal models in Fig3.For clarity,only the energy range of 5-10keV is displayed,since this is where the impact of very hot gas will be most apparent.Although the MMM provides a reasonableﬁt to the spectral imaging data as a whole,it does not follow that the data in the central regions need be consistent with the high model temperature.In practice, for both instruments(while the GIS2spectra are not shown in Fig3,they behave similarly to the GIS3spectra)the MMM is a good match to the data.The isothermal model is,however,also consistent with the restricted dataset.The conclusion from this is that while the data do not rule out a central temperature rise,they do not require one either.The reason for this is that within the ASCA bandpass,plasmas with temperatures of8and18keV do not have substantially diﬀerent spectral signatures and are thus diﬃcult to diﬀer-entiate(this is analogous to the diﬃculty that the PSPC has in dealing with clusters hotter than2-3keV).This problem is compounded by the limited spatial resolution of ASCA.LTF9.640.63-636.08LTF(ISO)7.90zero-626.85TTF19.01 1.20-634.53KTF8.98 3.18-635.96DNF15.59n/a-636.12DMF9.24n/a-635.20DHF(MMM)20.42n/a-635.33Table1.Temperature and statistic parameters for a variety of ﬁtted3D models are shown.Note that the MMMC is not a best-ﬁt model in the same sense as the others,as is described in the text.For all models,the gas density core radius,r c,and index,αρ,areﬁxed at the values determined from the ROSAT data. Explanations for the model acronyms are given in the text.4COMPARISON OF X-RAY AND LENSING MASSESDeep optical observations of A2218reveal a number of major arcs and a wealth of minor arclets.These features are the result of gravitational lensing,an eﬀect which is independent of the physical state of the cluster gas.Instead,uncertainty lies in the characteristics of the background galaxies and the possibility of matter sub-clumps along the line-of-sight.Using the modelsﬁtted to A2218,we can derive pro-jected gravitating mass proﬁles,suitable for comparison with the results of lensing analysis.This involves assuming a max-imum outer radius for the cluster mass distribution.In the following analysis we take this to be the maximum radius of the data used in clusterﬁtting,which is9′(∼2Mpc).The choice of projection radius has a minor impact upon the derived projected mass proﬁles(compared to other uncer-tainties)so long as the chosen radius is suﬃciently large,≥2Mpc.For example,the diﬀerence in projected mass within2Mpc,between models with maximum radii of2.0 and3.0Mpc,is9%.Since lensing analysis measures the matter distribution on both small(strong lensing occurs where the surface mass density is high)and large(weak lensing is theoretically ob-servable to the edge of the cluster)scales,comparison with X-ray results is extremely informative.In Fig4the X-ray derived projected mass proﬁles for the MMM and isother-mal model are plotted,together with the lensing results of Kneib et al.(1995),Loeb&Mao(1994)and Squires et al. (1996).4.1Strong lensingStrong lensing analysis has been performed,using spectro-scopically observed arcs,by a variety of groups(Loeb& Mao1994;Kneib et al.1995;Kneib et al.1996;Saraniti, Petrosian&Lynds1996;Natarajan&Kneib1996).These analyses provide the gravitating mass within the critical ra-dius associated with the formation of giant arcs,which in A2218is22.1′′(∼85kpc).The two strong lensing points,plotted in Fig4at this radius,are consistent with a projected mass of∼6x1013M⊙,c 1998RAS,MNRAS000,1–16。

Skyrmions and domain walls

(9)
where f, Θ and Ω are the usual polar angles on S 3 . If we take for Θ and Ω any constant value, then f is the only remaining ﬁeld and the equation for the Skyrme model reduces to 2 fµµ + m2 π sin(2f ) = 0, (10)
2 2 1 1 2 2 fπ mπ 1 2 ( T r(U )2 − 1) = − m2 π sin |π | ≃ − mπ |π | . 8 4 2 2
(6)
We see thus that the two choices of potentials are identical in the limit of small pion ﬁelds. On the other hand, the second choice has the property of having 2 vacua: U = ±1.
∗ e-mail † e-mail
(2)
address: B.M.A.G.Piette@ address: W.J.Zakrzewski@
1
This potential term is responsible for making the pions of the model massive. The pion ﬁeld π is related to the unitary ﬁeld U as follows: U = exp 2i π.τ , fπ (3)
showing that the two vacua are given by φ0 = ±1. The real vector φ = (φ0 , φ) can be parametrised as follows cos f sin f cos Θ φ = sin f sin Θ cos Ω sin f slutions.

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1
It is well known that in SU(Nc ) Quantum Chromodynamics in the limit of Nc going to inﬁnity the baryons behave as solitons in an eﬀective meson ﬁeld theory [1]. A popular candidate for such an eﬀective ﬁeld theory is the topological Skyrme Model [2]. It has been extensively studied for two or more ﬂavours [3] and it has been shown that the resemblance of the topological soliton to the baryon in the quark model in the large Nc limit is very strong [4,5]. It’s baryon number and the fermionic character is also well understood [3,6]. Theoretically the most well studied and experimentally the best established model of particle physics is the Standard Model ( SM ) based on the group SU (3c ) ⊗ SU (2)L ⊗ U (1)Y [3]. The model consists of a priori several disparate concepts which are brought together to give the SM its structure as a whole. The successes of the SM are many however, it is believed to have a few shortcomings.It has been a folklore in particle physics that the electric charge is not quantized in the SM. It was felt that one has to go to the Grand Uniﬁed Theories to obtain quantization of the electric charge. It turned out to be a false accusation against the SM. It was clearly and convincingly demostarted in 1989/1990 that the electric charge is actually quantized in the SM [7,8]. The author showed [7,8] that the property of charge quantization in the SM requires the complete machinery which goes in to make it. The SM property of having anomaly cancellation generation by generation, the breaking of symmetry spontaneously through a Higgs doublet which also generates all the masses etc., all go into bringing in quantization of the electric charge in SM. These facts are important as there were several attempts to demonstrate charge quantization in SM using only part of the whole scheme, eg. using only anomaly cancelation [3]. The ﬂaws in such logic have been pointed out by the author [8]. Also analytically the author obtained the color dependence of the electric charge in the SM as [7] 1 1 Q(u) = Q(c) = Q(t) = (1 + ) (1) 2 Nc 1 1 Q(d) = Q(s) = Q(b) = (−1 + ) (2) 2 Nc for Nc = 3 this gives the correct charges. A short derivation of the result is given in the Appendix. It was also demonstrated by the author [7] that these were the correct charges to use in studies for QCD for arbitrary Nc . 2
Is topological Skyrme Model consistent with the Standard Model?
arXiv:hep-ph/9902246v2 9 Nov 2000
Afsar Abbas Institute of Physics Bhubaneswar-751005, India e-mail: afsar@iopb.res.in
where Lµ = U † ∂µ U . The U ﬁeld for the three ﬂavour case for example is a φa (x) ] U (x) = exp[ iλ f π with φa the pseudoscalar octet of π , K and η mesons. In the full topological Skyrme this is supplemented with a Wess-Zumino eﬀective action
Abstract
The topological Skyrme model is known to give a successful description of baryons. As a consistency check, here it is shown that in view of the recent discovery of charge quantization as an intrinsic and basic property of the Standard Model and the color dependence arising therein, the Skyrme Model is indeed completely consistent with the Standard Model.
Hale Waihona Puke This was contrary to many who had been using static ( ie. independent of color ) charges 2/3 and -1/3 [1,4,5,6]. Hence in addition to the other well known properties of the SM, I would like to stress that the quantization of the electric charge and the structure of the electric charge arising therein, especially its color dependence, should be treated as an intrinsic property of the SM. A consistency with the SM should be an essential requirement for phenomenological models which are supposed to work at low energies and for any extensions of the SM which should be relevant at high temperatures especially in the context of the early universe. The color dependence of the electric charge shown above should be viewed in two independent but complementary ways. Firstly for Nc = 3 it is diﬀerent from the static charges Q(u)=2/3 and Q(d)=-1/3. Secondly even for Nc = 3 it should be viewed as providing an anatomic view of the internal structure of the electric charge, meaning as to how are 2/3 and -1/3 built up and in what way the three colors contribute to it. For example the SM is making the statement that in 1/3 the 3 is not entirely due to the 3 of the QCD group SU (3c ). However this is what the SU(5) Grand Uniﬁed Theory says [9,10,11], 1 wherein Q(d) =-1/3 = N− . This is conﬂict with the SM expression where c =3 1 1 Q(d)= -1/3 = 2 (−1 + Nc =3 ). Hence the expression for the electric charge can be a very discriminating and restrictive tool for extensions beyond the SM. This has been used to check consistency of various models in a fruitful manner [9,10,11]. Quite clearly low energy phenomenological models of hadrons should be consistent with the SM in all respects. Is it true for the topological Skyrme Model? It shall be demonstrated below that the answer to the question in the title of the paper is in the aﬃrmative. To do so let us start with the Skyrme Lagrangian [6] LS = fπ 2 1 T r (Lµ Lµ ) + T r [Lµ , Lν ]2 2 4 32e (3)