arXiv:1507.00437v2 [hep-ph] 25 Oct 2015 rsn okaesmaie sflos i ydimen- large four- the by the flat in (i) a space to in dimensional theory follows: related Yang-Mills a as points deconstructing summarized properties sionally key are skyrmion Some work the present explored. on widely resonances were some the of (QCD), effects chromodynamics quantum of models graphic such 4]. include extended to [3, models researchers resonances Skyrme-type light to physics, and model nuclear Skyrme meson, the omega in meson, mesons rho res- scalar meson the of as roles freedom essential such of the pro- onances by degree Motivated originally the pion. only The the of include ex- model [2]. been Skyrme physics posed has matter idea and physics, condensed this nuclear also physics, then, particle Since in applied [1]. tensively baryon theory the mesonic with in nontrivial (soliton) certain configuration a field identify topological may one that idea pioneering t trcieeet[–0.I a on httedilaton the that found was It [8–10]. investigate effect to attractive introduced scale a its was the dilaton, (iii) Nambu-Goldstone of the breaking i.e., the 7]; symmetry, spontaneous as [6, with regarded force associated is boson repulsive that strong meson a scalar effective provide chiral to was the term found Wess-Zumino enters homogeneous the that through meson theory next-to-leading sym- omega effective the local the to chiral hidden up order, a the written on approach using (HLS) based by metry mesons (ii) vector of [5]; theory lighter soli- the becomes included, are ton resonances vector isovector more as ∗ ‡ † [email protected] [email protected] [email protected] eety seilyatrteapaac fholo- of appearance the after especially Recently, his proposed Skyrme R. H. T. ago, years 50 than More ffcso clrmsn naSym oe ihhde oa sy local hidden with model Skyrme a in mesons scalar of Effects 2 n h aiso h aynnme est eoe ag,as large, becomes density strengt si number coupling baryon soliton the the the as of radius and on, th the switched smaller between and is four coupling is coupling direct the mass direct no omit soliton when is first the there We mesons, when vector fields. that physica four-quark find the and and model, two- model our omega the In and of meson, fields. states rho meson pion, scalar four-quark including and model mesonic a amining Jilin Physics, of College and Physics Theoretical of Center ASnmes 13.d 23.c 23.e 14.40.Be 12.39.Fe, 12.39.Dc, increases. 11.30.Rd, size t numbers: soliton PACS of the component and two-quark decreases the mass When soliton t the soliton. of the components of four-quark properties and scalar two-quark four-quark the the include between then mixing We strong. becomes meson esuyteeet flgtsaa eoso h krinpro skyrmion the on mesons scalar light of effects the study We .INTRODUCTION I. 1 eateto hsc,Ngy nvriy aoa 464-860 Nagoya, University, Nagoya Physics, of Department igRnHe, Bing-Ran N c ii,i a on that found was it limit, 1, ∗ ogLagMa, Yong-Liang Dtd ac ,2018) 5, March (Dated: eaint hswr.I e.3 ecluaeteskyrmion in the calculate features we typical 3, its sec. In explains and work. this paper to relation this that in model use effective chiral we the construct we how describes nitrcinaogtevco n clrmsn,on skyrmion. mesons, the of scalar of size and strength and con- vector the mass and the as the well among masses as interaction the mesons, an of scalar the effects of the stituents meson study through we model the dynamics, Then, of parameters mesons. me- the scalar omega determining four-quark and the and meson, mass two-quark rho and the the son, effective pion, on chiral the a quarks including constructing first model four by skyrmion and the of quarks size two of made the matter. in in restoration MeV chiral 100 reproduce about can of which force model, attractive an provide can 2, † hsppri raie sflos h etsection next The follows. as organized is paper This u nig a esmaie sfollows: as summarized be can findings Our mesons scalar of effect the explore we paper, this In 2. 1. n aaauHarada Masayasu and n t iebcmslarge. becomes small size becomes mass its soliton and that, the find increased, scalar lighter is we the meson of model, component the two-quark the in when two-quark states the between four-quark mixing and on switch we When a such In state. light that: two-quark find The pure we case, a model. is the meson from four-quark scalar decouples the model, state the scalar two-quark in the states between four-quark mixing and off switch we When (b) (a) nrae,tesltnbcmsheavy, becomes soliton the increases, h nvriy hncu,101,China 130012, Changchun, University, ei agrfrlgtrsaa mesons; scalar lighter for larger is ze eo edi h oe n n that find and model the in field meson clrmsn r enda mixing as defined are mesons scalar l esaa eo ed loaet the affects also fields meson scalar he h eusv oc rsn rmthe from arising force repulsive the elgtrsaa eo sincreased, is meson scalar lighter he est nraewe h opigstrength coupling the increases. when number baryon increase the the density of that size the shows and result mass and coupling soliton The meson the mesons. scalar vector light on the the depends between mass strength in- soliton meson. scalar with The light decrease the size of mass and creasing mass soliton The qaksaa eo edfo the from field meson scalar -quark w-ur clrmsnadthe and meson scalar two-quark e eo ed swl stwo-quark as well as fields meson ete ycntutn n ex- and constructing by perties ,Japan 2, 1, ‡ mmetry ω 2 properties numerically and analyze the effects of scalar and q q , respectively. Therefore, the hadron field R → R mesons on them. The last section discusses the results M(2) transforms as we obtained and presents our conclusions. In the ap- pendixes, we give some details of the calculation. M M . (4) (2) →− (2) We introduce a four-quark field that is schematically II. THE MODEL written at the quark level as

φ q¯ q¯ ǫij qk ql ǫ . (5) We start by constructing a chiral effective model in- ∼ Li Lj R R kl cluding two-quark and four-quark scalar states. Here, In this work, we regard φ as a real field by considering we limit our discussion to the mesons, including up and only the real part of Eq. (5). Under the Z transforma- down quarks. In our model, we introduce a two-quark 2 tion, φ transforms as field M(2) that is schematically written at the quark level as φ φ . (6) j → M(2) q¯R i qL j , (1) i ∼ On the basis of chiral symmetry and Z2 symmetry, the Lagrangian is written in the form where i, j = 1, 2 are the
flavor indices. This, at the hadron level, is rewritten in terms of the isosinglet σ field µ 1 µ = Tr ∂ M ∂ M † + ∂ φ∂ φ and isotriplet π fields as L µ (2) (2) 2 µ  ¯  ¯ 1 V0 V0 VSB VSB , (7) M = (σ + i~π ~τ) , (2) − − − − (2) 2 · where V0 represents the meson
potential
term, and VSB where ~τ are the Pauli matrices. Under chiral transfor- denotes the explicit chiral symmetry breaking term, ¯ ¯ mation, the matrix M(2) transforms as which has values of V0 and VSB, respectively, in vacuum. We note that, in this analysis, we include only the non- M g M g† , (3) derivative interactions. We require that the number of (2) → L (2) R fields in each vertex derived from the potential V0 is less where g SU(2) . Furthermore, we can impose than or equal to four, and the number of quarks included L,R ∈ L,R Z2 symmetry as a remnant of the U(1)A transforma- in each vertex is less than or equal to eight [11]. Then, tion, under which q and q transform as q q the potential is expressed as L R L → − L

2 1 2 2 V = λ Tr M M † M M † m Tr M M † + m φ 0 (2) (2) (2) (2) − 2 (2) (2) 2 4 √    + 2A det M(2) + det M(2)† φ . (8) 
 

We adopt the simplest form of the explicit chiral symme- by the pion fields as try breaking potential VSB: i~π ~τ/(2fπ) ξR = ξ† = e · . (11) 1 L V = f Tr χM † + χ†M , (9) SB − 2 π (2) (2) Then, we write the Lagrangian (7) as   where χ = 2B , and B and are a constant with 1 µ 2 µ 1 µ M M = ∂µσ∂ σ + σ Tr (α µα )+ ∂µφ∂ φ dimension one and the quark mass matrix, respectively. L 2 ⊥ ⊥ 2 In this analysis, we consider an isospin-symmetric case, V0 V¯0 VSB V¯SB + V , (12) that is, = diag(m ¯ , m¯ ), wherem ¯ = (m + m )/2. We − − − − L u d where use theM pion mass as an input to determine the relevant

parameters, so VSB is expressed as 1 α µ = ∂µξR ξ† ∂µξL ξ† , (13) ⊥ 2i · R − · L 1 2   VSB = fπm Tr M † + M(2) . (10) and the newly introduced term represents the La- − 2 π (2) LV   grangian for the vector meson component, which will be In this paper, we introduce the rho and omega mesons explicitly given later. The potentials V0 and VSB are based on the HLS [12, 13]. For this purpose, we take the rewritten as 1 polar decomposition of M(2) as M(2) = 2 ξL† σξR, where 1 4 1 2 2 1 2 2 1 2 V0 = λσ m2σ + m4φ + Aσ φ, ξL and ξR, in the unitary gauge of the HLS, are expressed 8 − 2 2 √2 3

1 2 V = m f σ Tr U + U † , (14) Vµν ∂µVν ∂ν Vµ i[Vµ, Vν ] , (21) SB − 4 π π ≡ − − with i~π ~τ/fπ
where U = ξ† ξ = e . L R · 1 ρ0 + ω ρ+ g √2 µ µ µ The stationary conditions of the potential of the sys- Vµ = 1 0 . (22) √2 ρµ− ρµ + ωµ tem are obtained by taking the first derivative of the
− √2 ! potential V = V0 + VSB with respect to σ and φ: Here g is the gauge coupling constant of the
HLS, ahls ∂V 1 and s0 are real dimensionless parameters, and F is a = λσ3 m2σ + √2Aσφ m2 f , (15) ∂σ 2 − 2 − π π constant of dimension one. When σ has its vacuum ex- ∂V 1 pectation value (VEV), σvac = fπ, the vector meson = m2φ + Aσ2 . (16) 2 2 2 2 4 mass is expressed as mV = ahlsg (s0fπ + (1 s0)F ), ∂φ √2 2 2 − where ahlsg s0fπ represents the mass obtained by spon- After a suitable choice of the model parameters, σ ac- taneous chiral symmetry breaking, and a g2(1 s )F 2 hls − 0 quires its expectation value σvac, which is identified as the is the chiral invariant mass. In this sense, s0 accounts pion decay constant. Through the interaction between for the magnitude of the vector meson mass coming from the two-quark and four-quark scalar states, a nonzero spontaneous chiral symmetry breaking. In this paper, we A leads to a nonzero expectation value of φ, i.e., φvac. take F = fπ, so the vector meson mass is expressed as 2 2 2 The physical scalar meson fields are obtained from the mV = ahlsg fπ, which is the standard form given in the fluctuations with respect to the relevant expectation val- HLS [12, 13]. Note that for 0 , the term linear in the LV ues through σ = fπ +σ ˜ and φ = φvac + φ˜, where σ field is excluded by Z2 symmetry, and s0 can be both 2 Afπ positive and negative. φvac = 2 , as given in Eq. (A4). − √2m4 The intrinsic parity-odd terms of the Lagrangian Taking the second-order derivative, one obtains the anom in Eq. (19) read [13] L mass matrix for σ and φ as 3 m2 m2 2 2 √ 4 Nc σ σφ λfπ + mπ 2Afπ d x anom = ci i , (23) 2 2 = 2 . (17) L 16π2 4 L m m √2Afπ m M i=1  φσ φ   4  Z Z X 4 The physical states f500 and f1370 are the mixing states where M represents the four-dimensional Minkowski of σ and φ through the rotation space, and µνσρ f500 cos θ sin θ σ˜ 1 = iǫ Tr (αLµαLν αLσαRρ αRµαRν αRσαLρ) , = − , (18) L µνσρ − f1370 sin θ cos θ φ˜ = iǫ Tr (α α α α ) ,       L2 Lµ Rν Lσ Rρ = ǫµνσρ Tr [F (α α α α )] , (24) where θ is the mixing angle. In Appendix A, we show L3 V µν Lσ Rρ − Rσ Lρ some relations among the parameters and the physical in terms of the 1-form and 2-form notation with αL = 2 quantities that are useful for this work. From Eq. (18) αˆ α , αR =α ˆ + α , and FV = dV iV . The k − ⊥ k ⊥ − one can see that when cos θ 0, the physical state f500 is low-energy constants c ,c , and c , which are difficult → 1 2 3 dominantly a four-quark state and f1370 is dominantly a to fix from fundamental QCD, are usually estimated two-quark state, whereas when cos θ 1, f500 is almost phenomenologically with respect to the experimental → a two-quark state, but f1370 is almost a four-quark state. data [13]. In this work, we focus on the effect of scalar Now, we are in a position to specify the vector meson mesons on the skyrmion properties; therefore, we choose component of the Lagrangian. Here we use the following the parameters c1 = c2 = 2/3 and c3 = 0, which µ − − form: provides ωµB [14]. = + , (19) LV LV0 Lanom III. NUMERICAL RESULTS FOR THE where V0 is the Lagrangian with only intrinsic parity- SKYRMION even terms,L whereas contains the intrinsic parity- Lanom odd terms relating to the U(2)L U(2)R chiral anomaly. In this section, we make a numerical study of the ef- Explicitly, the intrinsic parity-even× Lagrangian is LV0 fects of scalar mesons on the soliton mass and also the expressed as root-mean-square (RMS) radii of the baryon number den- 2 2 µ sity and energy density. In Table. I, we summarize the V0 = ahls(s0σ + (1 s0)F ) Tr(ˆα µαˆ ) L − k k parameters of the model used in this paper. 1 Tr(V V µν ) , (20) − 2g2 µν µ A. The ansatz whereα ˆ and Vµν are defined as k 1 To study the properties of the soliton obtained from αˆ µ = ∂µξR ξR† + ∂µξL ξL† Vµ , the Lagrangian (12), we take the standard parametriza- k 2i · · −   4

fπ 92.4 MeV baryon number B = 1, the wave functions F (r), G(r), mπ 139.57 MeV and W (r) satisfy the following boundary conditions: Nc 3 F (0) = π, F ( )=0, c1 + c2 0 ∞ − − G(0) = 2, G( )=0, c1 c2 4/3 − ∞ c3 0 W ′(0) = 0, W ( )=0. (27) ∞ g 5.80 ± 0.91 ahls 2.07 ± 0.33 The boundary conditions for the profile functions of the scalar mesons are determined as follows: (I) as r , → ∞ TABLE I: Values of the model parameters. the VEVs of the scalar meson fields should be repro- duced, so bothσ ¯(r) and φ¯(r) must vanish, i.e., tions for the soliton configurations. Following Refs. [1, σ¯( )=0 , φ¯( ) = 0 ; (28) 14], we take the ansatz for the π, ρ and ω fields as ∞ ∞ ¯ iτ rˆF (r) (II) the second-order derivativesσ ¯′′ and φ′′ should be U = e · , (25a) nonsingular as r 0, in view of the last term in Eqs. (B6) G(r) → ρ = (rˆ τ ) , (25b) and (B7); equivalently, gr × ωµ = W (r)δµ0 . (25c) σ¯′(0) = 0 , φ¯′(0) = 0 . (29) We parametrize the scalar meson fields σ and φ as To study the RMS radius of the baryon number den- σ = fπ (1+¯σ(r)) , (26a) sity of the system, we specify the baryon number current

φ = φvac 1+ φ¯(r) , (26b) by introducing the external gauge field µ of the U(1) baryon number. Then, the baryon numberV current is whereσ ¯(r) and φ¯(r) are dimensionless
functions. Sub- obtained by taking a functional derivative of the total stituting Eqs. (25) and (26) into the Lagrangian (12), one Lagrangian (12) with respect to . After an explicit Vµ obtains the equations of motion for the profile functions calculation, the baryon number density B0, which is the F (r), G(r), W (r),σ ¯(r), and φ¯(r). The detailed expres- zeroth component of the baryon number current, is ex- sions are given in Appendix B. For the solutions with the pressed as

2 B = f 2g2r2a W s σ¯2 +2s σ¯ +1 0 − 3gr2 π hls 0 0 2 2  + F ′ α 2G ( α + α + αcos F )+ α cos F 2α cos F + (α α ) G 2 − − 2 3 2 2 − 2 2 − 3 2  2 sin F  2α sin F G′ + α sin F F ′ F ′ , (30) − 3 1 − 2π2r2 where α1, α2, and α3 relate to c1, c2, and c3, respectively, through Eq. (B2). By using the equation of motion for W in Eq. (B5), the baryon number density is simplified as

2 1 d 4α sin F (cos F G 1) 2r W ′ sin(2F ) F B = 3 − − + , (31) 0 r2 dr 3g − 3g 8π2 − 4π2  

2 1/2 which agrees with that obtained in Ref. [15]. Note that, number density, r B , and the RMS radius of the en- by combining Eqs. (31) and (27), one can show that the 2 h1/2i ergy density, r E , which are defined as follows: ∞ 3 h i baryon charge is correctly normalized as 0 dr B0(r)= 1. R 2 1/2 ∞ 3 2 In this analyse, we take c1 + c2 = 0, c1 c2 =3/4, and r B = d rr B0(r) , − h i s 0 c3 = 0; therefore, Eq. (30) is reduced to Z 1/2 1 ∞ 2 2 2 2 2 2 3 2 B = f g r a W s σ¯ +2s σ¯ +1 . (32) r E = d rr Msol(r) , (33) 0 2 π hls 0 0 h i M − 3gr s sol Z0   In our numerical calculation, in addition to the soli- where Msol is the soliton mass, and Msol(r) is the corre- ton mass, we also consider the RMS radius of the baryon sponding energy density given in Eq. (B1). 5

B. Effects of scalar mesons 1.4 1.35

1.3 (GeV)

Now we are in a position to show how the scalar mesons sol 1.25 m affect the skyrmion properties. 0.9 1 1.1 1.2 1.3 1.4 1.5 First, we consider A = 0. In this case, the four- mσ (GeV) 0.73 quark field φ decouples from the other mesons. The La- 0.72 (fm)

grangian (12) is reduced to B 0.71 i 2

r 0.7 1 h µ 2 µ p 0.69 = ∂µσ∂ σ + σ Tr(α µα ) 0.9 1 1.1 1.2 1.3 1.4 1.5 L 2 ⊥ ⊥ mσ (GeV) (Vσ V¯σ) (VSB V¯SB)+ V , (34) 0.62 0.6

− − − − L (fm) 1 4 1 2 2 2 E 0.58 where Vσ = λσ m σ . The parameters λ and m i 8 2 2 2 2 − r 0.56 are related to the masses of σ and π through h

p 0.54 0.9 1 1.1 1.2 1.3 1.4 1.5 3 mσ (GeV) m2 = m2 + λf 2 , σ − 2 2 π 1 FIG. 1: mσ dependence of the soliton mass and radii m2 = m2 + λf 2 . (35) π − 2 2 π with A = s0 = 0.

Here we take the empirical value of mπ as an input and regard mσ as a parameter of the model. In addition, 2 F s0 in V is an unfixed parameter. In the following, we L 0 study the relation between these two parameters and the 0 0.5 1 1.5 r (fm) skyrmion properties. 0

a. The effects of mσ on the skyrmion properties. −1 G We start our discussion from the standard HLS La- −2 0 0.5 1 1.5 grangian [13], which is obtained by taking s0 = 0 in this r (fm) analysis. 0 −0.05

We show the mσ dependence of the soliton mass msol (GeV) 2 2 −0.1 and the radii r B and r E in Fig. 1 and the pro- W 0 0.5 1 1.5 file functions inh Fig.i 2. Fromh i Fig. 1 we find that, in- r (fm) p p 0 creasing mσ, the soliton mass msol increases. This ten- ¯ dency can be understood from the profile function ofσ ¯ σ −0.5 in Fig. 2; when the σ is heavy, the magnitude ofσ ¯ is 0 0.5 1 1.5 r (fm) small. Becauseσ< ¯ 0, as mσ increases, the magnitude of σ = f (1+σ ¯) becomes large; consequently, the contri- π FIG. 2: m dependence of the profile functions with bution of σ2 Tr(α αµ ) to the skyrmion mass becomes σ µ A = s =0 for m = 1 GeV (green solid line) and large. Physically,⊥ for a⊥ heavy σ, the attractive force pro- 0 σ m =1.4 GeV (red dash-dotted line). vided by the σ is small; as a result, the soliton mass is σ large. In addition, Fig. 1 tells us that when mσ is large, the radii r2 and r2 are small. This result can be of the skyrmion by taking a typical value of the scalar h iE h iB understood from the profile functions plotted in Fig. 2; meson mass, mσ = 1.37 GeV. Our numerical results are p p when mσ is large, the profile forσ ¯ becomes narrow. Gen- plotted in Fig. 3. erally, the σ supplies the attractive force, whereas the ω From Fig. 3, we see that, when the value of s0 is in- 2 meson supplies the repulsive force. A narrow σ needs a creased, msol and r B become large. According to narrow ω to balance the attractive and repulsive forces. the equations of motionh fori the vector mesons in Eq. (B5), The shapes of σ and ω dominate the shape of the energy a g2 f 2(1 s )+ps σ2(r) acts as the effective mass of hls ω π − 0 0 distribution. As a result, the narrow σ and ω make the the vector mesons inside the soliton. Because σ(r) gauge boson ofh thei U(1) baryon number symme- 0 p V 6

2 try. The relation between s0 and r B needs a more Eq. (18). Our results are shown in Figs. 5 and 6. Note precise explanation; because the baryonh i number density p is proportional to the effect of the ω meson, as can be 1.35 seen in Eq. (32), the RMS radius of the baryon number 1.3 1.25 2 (GeV)

density r B is large for large s0. sol 1.2

h i m The situation for the RMS radius of the energy density, 1.15 p 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 r2 , is quite complicated. In this analysis, the nu- cos(θ) h iE merical error is about 1-3%, so the third graph in Fig. 4 0.74

p 2 (fm) implies that r E is rather stable against changes in 0.72 B

h i i 2 0.7 s0. This implies that the contribution of ω has the op- r p h posite direction to that of σ, so they cancel each other. p 0.68 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 cos(θ) 1.4 0.75 1.35 0.7 (fm) 1.3 (GeV) E 0.65 i 2 sol 1.25 r 0.6 h m

p 0.55 −1 −0.9 −0.8 −0.7 −0.6 −0.5s −0.4 −0.3 −0.2 −0.1 0 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 cos(θ)

0.65 (fm) FIG. 5: Dependence of skyrmion properties on the B

i 0.6 2 mixing angle between two scalar mesons for s = 0. r 0 h 0.55 p −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 s0

0.65 (fm) 1.3

E 1.28 i 0.6 2

r 1.26 h

0.55 (GeV) 1.24 p

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 sol 1.22

s0 m 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 cos(θ) FIG. 3: s0 dependence of the skyrmion mass and radii 0.7

for mσ =1.37 GeV. (fm) 0.68 B i

2 0.66 r h

p 0.64 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 cos(θ) 2 0.7 F 0.65 0 (fm)

0 0.5 1 1.5 E 0.6 i r (fm) 2 r 0.55 0 h

p 0.5 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 −1 G cos(θ) −2 0 0.5 1 1.5 r (fm) FIG. 6: Dependence of skyrmion properties on the 0 mixing angle between two scalar mesons for s0 = 0.5. −0.05 − (GeV) −0.1

W 0 0.5 1 1.5 r (fm) 0 that we cannot find any stable solution in the regions of ¯ σ cos θ> 0.6 for s = 0 and cos θ > 0.25 for s = 0.5. In −0.5 0 0 other words, the maximal value of cos θ is small− for small 0 0.5 1 1.5 r (fm) s0. This could be understood as follows. The definition of the mixing angle in Eq. (18) implies that, for a large FIG. 4: Profile functions with s0 = 0.5 (green line) value of cos θ, f includes a large amount of the two- − 500 and s0 = 0 (red dotted line). quark componentσ ˜, which generates a strong attractive force. Because the total repulsive force does not change, the attractive force can exceed the total repulsive force, Next, we switch on the mixing between the two scalar and the soliton collapses. A large value of cos θ corre- mesons by setting A = 0 and then study how the mixing sponds to a strong attractive force; therefore, the soliton angle θ affects the skyrmion6 properties. In our calcu- collapses at a certain value of cos θ. On the other hand, a lation, we take two fixed values of s0, i.e., s0 = 0 and decrease in s0 means that the repulsive force mediated by s0 = 0.5, to see the dependence of the mass and the ω meson becomes weak, as we stated above. Because radii of− the skyrmion on the mixing angle θ defined in the soliton is stabilized by the balance between the at- 7 tractive and repulsive forces, the maximal value of cos θ 300 MeV larger than that needed to reproduce the em- is small for small s0. pirical values of the nucleon mass and ∆-N mass split- Note that the decreasing and increasing tendencies in ting. Thus, to reproduce the baryon masses in the real Figs. 5 and 6 are the same, and they are actually consis- world, the present model should be extended. There are tent with those in Fig. 1. As we just stated, increasing several possible extensions, for example, inclusion of the cos θ increases the magnitude of the two-quark compo- effect of the dilaton as the Nambu-Goldstone boson as- nent included in f500. Thus, the large cos θ in Figs. 5 sociated with the scale symmetry breaking of QCD; the and 6 corresponds to the small mass in Fig. 1. effect of the (p4) terms of the HLS, which account for the π-ρ interaction;O and/or the effect of all the homo- geneous Wess-Zumino terms of the HLS, which increase IV. CONCLUSIONS AND DISCUSSIONS the number of π-ρ and/or π-ρ-ω terms. We leave these possibilities as future projects. In addition, even though In this work, we first investigated the effect of a two- the present model cannot reproduce reality, it still merits quark scalar meson on the skyrmion properties when mix- application to dense matter systems to study the effect ing between the two-quark and four-quark states of scalar of the scalar mesons on the phase structure of nuclear mesons is switched off. We found that as the mass of the matter. We will report this progress elsewhere. two-quark scalar meson decreases, the soliton mass de- creases and the RMS radii of the baryon number density, 2 2 r B , and the energy density, r E, increase. We nexth studiedi the effect on the skyrmionh propertiesi of cou- ACKNOWLEDGMENTS ppling between the scalar meson andp vector mesons. It is found that as coupling becomes larger, the soliton mass M. H. was supported in part by the JSPS Grant-in- and radii increase. We finally explored the skyrmion Aid for Scientific Research (S) No. 22224003 and (c) properties by considering the mixing structure of the two No. 24540266. And the work of Y.-L. M. was supported scalar mesons. We showed that, when the percentage of in part by National Science Foundation of China (NSFC) the two-quark component in the lighter scalar meson is under Grant No. 11475071 and the Seeds Funding of Jilin increased, the soliton mass becomes small, and the RMS University. radii ( r2 and r2 ) become large. B E Appendix A: The mixing matrix Our numericalh i resultsh showi that, even though the two- quarkp and four-quarkp scalar mesons are both included in the model, the soliton mass obtained here is still about The masses of f500 and f1370 are

1 m2 = 2f 2 (4A2 λm2 + λm2 )+ f 4λ2 + (m2 m2 ) 2 + f 2λ + m2 + m2 , (A1) f500 2 − π − 4 π π 4 − π π 4 π  q  1 m2 = 2f 2 (4A2 λm2 + λm2 )+ f 4λ2 + (m2 m2 ) 2 + f 2λ + m2 + m2 . (A2) f1370 2 π − 4 π π 4 − π π 4 π q 

The mixing angle θ can be obtained as From Eqs. (15), (16), (A1) and (A2), one gets (m2 m2)(m2 m2 ) 2 f1370 − 4 4 − f500 A = 2 , 2fπ m2 + m2 m2 m2 f500 f1370 − 4 − π λ = 2 , fπ Af 2 φ = π , vac 2 − √2m4 m2 m2 m2 m2 1 m2 m2 f1370 − 4 4 − f500 2 f500 f1370 2 θ = arctan 2 2 = arccos 2 2 . m2 = 2 3mπ , (A4) s m4 mf500 smf1370 mf500 2 m4 − ! − − 2 2 2 (A3) with mf500

Appendix B: Soliton mass and the equations of motion for the profile functions F (r), G(r),W (r), σ¯(r) and φ¯(r)

Substituting the ansatz defined in Eqs. (25) and (26) to the effective Lagrangian (12), we obtain the following expression for the soliton mass:

∞ 2 Msol =4π drr Msol(r) Z0 ∞ 1 2 2 2 2 2 2 2 = 4π dr r 4fπg s0ahlsW σ¯(¯σ + 2) 4fπ (¯σ + 1) F ′ − 0 8 − Z n n 2 2 2 2 2 ¯ 2 m4 m 500 m 1370 m4 (¯σ + 1) φ +1 2 − f f − 2 2 2 2 4 +fπ 2 mf500 + mf1370 m4 mπ (¯σ + 1)   m4 
− − − h 2 2 2 2 2 2 2 2 2 ¯ 2
2 m 500 m 1370 3m4mπ (¯σ + 1) m4 m 500 m4 m 1370 φ +1 f f − − f − f 2 + 2 + 2 +8mπ (¯σ + 1)cos(F )  m4    m4 
i 2 2 2 2 2 ¯ 2 2 2 2 2 2 fπ m4 m 500 m4 m 1370 φ′ fπ m 500 m 1370 +3m4mπ − f − f 2 2 2 2 2 f f 2 + 4 4fπσ¯′ +4fπg ahlsW 2 +4W ′  m4  − −  m4  o 1 F F 8f 2g2a G sin2 s σ¯2 +2s σ¯ +1 4f 2g2σ¯2 sin2 ((s a 1) cos(F ) s a 1) −2g2 π hls 2 0 0 − π 2 0 hls − − 0 hls − n     F
8f 2g2σ¯ sin2 ((s a 1) cos(F ) s a 1)+2f 2g2a G2 s σ¯2 +2s σ¯ +1 − π 2 0 hls − − 0 hls − π hls 0 0  
2 2 4 F 2 2 2 2 2 +8f g a sin f g cos(2F )+ f g +2G′ π hls 2 − π π   2 o α 2G (W F ′ sin(F )W ′)+ G W F ′ + 2 sin(F ) ((cos(F ) 1)W ′ + W G′) − 3 − − 2 2 2 2 G (G + 2) +α W F ′( cos(F )+ G + 1) + α W F ′ sin (F ) ,
(B1) 2 − 1 − 2g2r2 o with 3gN gN gN α = c (c c ) , α = c (c + c ) , α = c c . (B2) 1 16π2 1 − 2 2 16π2 1 2 3 16π2 3 The equations of motion for F (r), G(r), W (r), σ¯(r) and φ¯(r) are expressed as 1 F ′′ = 2 2 2 fπr (1+¯σ) 2G f 2s a σ¯2 sin F +2f 2s a σ¯ sin F + f 2a sin F × π 0 hls π 0 hls π hls n α cos F W ′ α cos F W ′ + α W G′ α W G′ + α W ′ α W ′) − 2 − 3 2 − 3 2 − 3 f 2σ¯ sin F 4s a cos F 4s a 4cos F r2m2 2f 2σ¯2 sin F (s a cos F s a cos F ) − π 0 hls − 0 hls − − π − π 0 hls − 0 hls − 2 2 2 2 2 2 2 2fπr F ′σ¯′ 2fπrσ¯ F ′ 2fπrσF¯ ′ (rσ¯′ +2)+2fπa
hls sin F fπahls sin(2F ) − 2 −2 2 2 − 2 − 2f rF ′ + f m r sin F + f sin(2F ) 2α W cos F G′ +2α W cos F G′ − π π π π − 2 3 1 1 α cos(2F )W ′ 2α cos F W ′ + α cos(2F )W ′ 2α cos F W ′ +2α cos(2F )W ′ +2α W G′ − 2 1 − 2 2 2 − 3 3 2 2 1 3 2α W G′ + (α α ) G W ′ + α W ′ + α W ′ , (B3) − 3 2 − 3 2 1 2 2 2 2 2 2 o G′′ = f g a s σ¯ +2s σ¯ +1 (cos F G 1) α g (W F ′(cos F G 1) + 2 sin F W ′) − π hls 0 0 − − − 3 − − 2 2
G G +3G +2 + α2g W F ′(cos F G 1)+ 2 , (B4) − − r
2 2 2 2W ′ W ′′ = f g a W s σ¯ +2s σ¯ +1 π hls 0 0 − r 1
2 + α F ′ 2G(cos F + 1)+ 2(cos F cos(2F )) + G + 4 sin F G′ r2 − 3 − n 2 2 F

+ α F ′( cos F + G + 1) +2α F ′ sin (cos F + 1) , (B5) 2 − 1 2   o 9

1 2 2 2 2 2 2 2 2 2 2 2 3 σ¯′′ = 2 m4(2¯σ g s0ahlsW + mf500 + mf1370 + F ′ m4 + mf500 + mf1370 m4 mπ σ¯ 2m4 − − − − n 2 2 2 2 2 2 2 2 2 +3 m + m m m σ¯ +2 g s a
W + F ′ m (cos F 1)
f500 f1370 − 4 − π − 0 hls − π − m 2 m2 m2 m2 (¯
σ + 1) φ¯  − 4 − f500 f1370 − 4 2 F 4 oF 2 (¯σ + 1) 8s0ahlsG
sin 8s
0ahls sin 2s0ahlsG + cos(2F ) 1 2¯σ − 2 − 2 − − ′ , (B6) − r2 − r ¯


2 2φ′ φ¯′′ = m φ¯ σ¯ (¯σ + 2) . (B7) 4 − − r

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