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Baryon structure in the global color symmetry model of QCD
Nuclear Physics A 790 (2007) 593c–597c
Baryon structure in the global color symmetry model of QCD∗ Bin Wang a , Yu-xin Liu a
a†
Department of Physics, Peking University, Beijing 100871, China
Baryons are considered as solitons in the global color symmetry model (GCM) of QCD. With both the scalar and pseudoscalar chiral fields being included and the Hedgehog approximation being taken, we obtain a GCM soliton profile whose mass and radius are quite close to the experimental data of a nucleon. We then quantize the angular momentum and isospin with the collective quantization method and obtain a nucleonDelta mass split comparable with experimental data. PACS numbers: 12.40.Yx, 03.75.Lm, 12.39.-x, 11.10.Lm 1. INTRODUCTION How to describe hadron states directly from the fundamental theory of strong interaction, i.e. QCD, is still an open problem. One promising approach may be the lattice QCD because of the advancements in numerical calculation. Another approach is to construct models to describe hadrons. Such models are designed to successfully fit experiment data and to pursue for the QCD foundation, then the Nambu–Jona-Lasinio (NJL) model and the golobal color symmetry model (GCM) have been developed. People have devoted to these models to mimic baryons as solitons [1–15]. Theoretically, the GCM can imbed not only the chiral symmetry spontaneous breaking but also the confinement and the nonlocal property of the low energy QCD. Practically, the GCM have described many meson data. One then expects the model can be a successful tool for the research of baryon structure. The paper is organized as following. In section II, we describe briefly the main points of the GCM and the classical soliton at the mean field level and its quantization with a semiclassical method. In section III, we describe the numerical calculation and result and some discussion. 2. GCM SOLITON AND ITS QUANTIZATION After bosonizing the lagrangian [11,12], one have the GCM as an effective field theoretical model of QCD with the quark and gluon degrees of freedom being changed to quark ∗ This work was supported by the National Natural Science Foundation of China (NSFC) under contract Nos. 10425521 and 10575004, the Key Grant Project of Chinese Ministry of Education (CMOE) under contact No. 305001 and the Research Fund for the Doctoral Program of Higher Education of China under grant No. 20040001010. One of the authors (Y.X. Liu) would acknowledge the support of the Foundation for University Key Teacher by the CMOE, too. The authors are also indebted to Professor Xiao-fu L¨ u for his stimulating discussions. † Corresponding author.
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.100
594c
B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
and meson fields. As one keeps only the scalar, pseudoscalar and vector meson fields, the QCM action reads S[U] = −T r ln G−1 [U] + I[U],
(1)
where the inverse of quark propagator can be conventionally written as x+y )B(x − y) . (2) 2 The configuration V corresponds to the scalar field and pseudoscalar field, and reads √ 1 − γ5 + 1 + γ5 2 a a U+ U , V= with U = exp i π F , (3) 2 2 fπ G−1 [U] = γ · ∂x A(x − y) + V(
where F a is the generator of U(2) group defined in [6]. The vacuum is set to U˜ = 1 (π a = 0) according to the chiral symmetry spontaneous breaking. The function A and B are related to the vacuum configuration of the above mentioned truncated fields, and can be determined by the truncated Dyson-Schwinger equation 2 ˜ Σ(p) = iγ · p[A(p2 ) − 1] + VB(p ) = g2
λa d4 q 1 λa γ γ , D(p − q) μ μ (2π)4 2 iγ · q + Σ(q) 2
(4)
where D(p − q) is the effective gluon propagator. The I[U] is just a functional of the ˜ which has been given explicitly in Ref. [6]. vacuum configuration U, Within the GCM, the nucleon current correlation function is given as {f } {g}∗
ΠN (T ) = ΓN ΓN
1 Z
DU
Nc i=1
γ4 G(x,
T T ; y , − , [U])fi ,gi e−S[U ] , 2 2
(5)
{f }
where ΓJJ3 ,T T3 is a matrix to show the spin and flavor structure of the nucleon, fi denotes the flavor and spin structure. To calculate the correlation function, it is convenient to make the spectral decomposition [15] of the quark propagator G(x, y; [U]) =
j
1 ¯ dω ψj,ω (x) ψj,ω (y) , 2π iλj (ω)
(6)
with ψj,ω (x) = uj (x)eiωx4 , λj (ω) = ω − i j (ω). When people take the limit T → ∞, the quark propagator reduces to the contribution of the lowest poles of λj 1(ω) . One usually refers to these poles as the valence quarks with energy val . Therefore, lim
T →+∞
Nc i=1
γ4 G(x,
T T ; y, − , [U])fi ,gi ∼ e−Nc T val . 2 2
With a static meson field U, one can get
˜ = −Nc T r ln G−1 [U]/G−1 [U˜ ] = Nc T S[U] − S[U]
(7)
( i − ˜ i ) = T Esea [U] ,
(8)
i <0,˜ i <0
which is usually denoted as the sea quarks’ contribution because it comes from the negative Dirac sea.
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B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
Up to now, we obtain a factor e−T (Nc val+Esea ) in the nucleon correlation function which can be related to a soliton mass Mcl [U] = Nc val [U]+Esea [U]. Here the subscript cl means that this mass is the classical mean field mass because there is no quantum fluctuation of meson fields U. The meson fields can be determined with the stability principle δMcl [Ucl ] =0, δUcl
(9)
This equation can be explicitly written as a set of coupled equations of quark and meson fields, which have been given in Refs. [8,15]. It should be noted that in contrast to the chiral quark model[1] (based on NJL Lagrangian), the classical mass here is not divergent because of the nonlocal property of quark-quark interaction. The above soliton equation (9) is difficult to solve because of the four independent π a fileds and the high non-linearity. We take the usual Hedeghog form solution as the profile, V (r) = χ(r)eiγ5
τ a π a (r) fπ
,
π a (r) = rˆa π(r) ,
(10)
where a = 1, 2, 3 corresponds to the iso-triplet part of the field and χ(r) stands for the iso-singlet part of the field. The Hedehog ansatz has the symmetry of simultaneous coordinate-space-rotation and iso-rotation, G = J + τ . In practical calculation, we take the valence quarks as the states with G = 0 as one often did. The Hedgehog soliton state does not have good quantum numbers of angular momentum and isospin. To identify the GCM solitons with baryons, one should restore them the quantum numbers with angular momentum projection or collective quantization. We here take the collective quantization scheme [3,16] and introduce time dependent fields y V (x, y) = R(tx )Vcl ( x+ )R+ (ty ) , where Vcl is the classical soliton solution, R(t) is a time 2 dependent SU(2) matrix which we will quantize. For the non-local model, we take this ˙ ansatz to make the quark field locally rotate with an angular frequency Ω = iR+ (t)R(t). Taking the adiabatic low angular frequency approximation and discarding all the terms with higher orders of Ω2 , one can get an exponential factor exp{−T I[Ucl ]/2Ω2a } in the current correlation function, where I[Ucl ] = I1 [Ucl ] + I2 [Ucl ], with I1 [Ucl ] =
Nc T r{G[Ucl ](x, y)γ4A(y − z)τ 3 G[Ucl ](z, w)γ4 A(w − x)τ 3 } − {Ucl → U˜ } , 4T
(11)
I2 [Ucl ] =
Nc (val|τ 3 A|j)(j|τ 3 A|val) . 2 j =val −Zval λj (i val )
(12)
The note 1/Zj is the corresponding residue of the spectral function 1/λj (ω), and (i|τ 3 A|j) =
d3 xd3 yd(x4 − y4 )u+ x)τ 3 A(x − y)e−iω(x4 −y4 ) uj (y ) i (
ω=ij
.
(13)
As a result of the collective quantization, the presently derived exponential factor corresponds to a Hamiltonian of a rotator Hrot , and I[Ucl ] corresponds to the moment of inertia of the rotator. Quantizing the Hrot , one can get the good quantum numbers of iso-spin and angular momentum. We have thus quantized the rotation and iso-rotation modes and obtained a mass of the quantum system M[Ucl ] = Mcl [Ucl ] + J(J + 1)/2I[Ucl ] , and also the nucleon and Delta mass split MΔ − MN = 2I3 .
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B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
3. NUMERICAL RESULTS AND SUMMARY With a simple effective gluon propagator, the Munczek-Nomirovsky model [17] 3 2 4 η δ (q) , (14) 16 we have solved the Dyson-Schwinger equation, and in turn the quark and meson equations. We obtained then some properties of nucleon [15], for instance, MN = 916 MeV, rN = 0.56 fm. We have also found that the pion field strengthens the attractive interaction and then reduces the mass of soliton much more close to the experiment data and leaves room for the contribution of the collective quantization. We will also take the fitted data η = 1.04 in the calculation of the moment of inertia. With the classical solution, we can make the spectral decomposition of the quark propagator in Eq. (6). We take the common eigenstate of the grand spin G = l + s + τ and the parity P , |K, M, P , as the basis of the spectral decomposition, g 2 D(q) = (2π)4
f1 (q)|(K, K + 12 )KM + f2 (q)|(K, K − 12 )KM f3 (q)|(K + 1, K + 12 )KM + f4 (q)|(K − 1, K − 12 )KM f3 (q)|(K + 1, K + 12 )KM + f4 (q)|(K − 1, K − 12 )KM f1 (q)|(K, K + 12 )KM + f2 (q)|(K, K − 12 )KM
,
for P = (−1)K ,
(15)
,
for P = (−1)K+1 .
(16)
The notation |(l, j)KM stands for the coupling of orbital angular momentum l, spin s and isospin τ to a function with good quantum number G = K, G3 = M. With the derived soliton profile in [15] and the spectral information, we can evaluate the contribution of the valence quarks to the moment of inertia (i.e., I2 ) and obtain I2 = 1.04 GeV−1 . As for the contribution of the sea quarks to the moment of inertia (i.e., I1 ), since both of the two terms of I1 have only ultraviolet divergence
∞ 1 T rG0 γ4 Aτ 3 G0 γ4 Aτ 3 ∝ V (3) ds, (17) T 0 which is just the effect of the asymptotic behavior of the function A(p) and can be exactly counteracted in I1 , it is then definitely convergent. However, it is difficult to get the value of I1 . Usually, one can calculate the corresponding term in the NJL based chiral quark soliton model and turn the trace to a imaginary ω residual summation. Nevertheless, because of the confinement in GCM, the integrand in the second term of I1 has no poles in the complex ω plane and thus it can not be calculated with such a method. Then one can not find a systematic cutoff scheme to reduce the two terms to a finite quantity in numerical evaluation. Alternatively, one can calculate the two terms both on the real ω axes and it is much easy to take a systematic cutoff scheme for the numerical calculation, however it needs too much calculation power. There exists also the third scheme, where one expands the terms of I1 as a series of y+z ) − 1] , (18) G0 B Vˆ (x, z) := d4 yG0 (x, y)B(y − z)[V ( 2 and the terms to the second order of G0 B Vˆ are −2T r[G0 B Vˆ G0 γ4 Aτ 3 G0 γ4 Aτ 3 ] and 2T r[G0 B Vˆ G0 B Vˆ γ4 Aτ 3 G0 γ4 Aτ 3 ] + T r[G0 B Vˆ G0 γ4 Aτ 3 G0 B Vˆ G0 γ4 Aτ 3 ]. Here one still encounters a difficulty that the expansion is not convergent which we numerically proved. In
B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
597c
fact, this expansion is equivalent to the expansion in terms of the derivative of the chiral field V . The soliton field V is not very smooth and then the expansion is not convergent. Taking only the lowest order of G0 B Vˆ in I1 , we obtain I1 = 3.06 GeV−1 . Then the total moment of inertia is I = 4.1 GeV−1 and the corresponding nucleon-Delta mass split is 366 MeV, which is comparable with experimental data. In summary, we have quantized the GCM solitons with the collective quantization scheme so that they take the quantum numbers of low-lying baryons. Then the lowlying baryons can be described by the GCM solitons. We also numerically evaluated the moment of inertia. The obtained mass and radius of nucleon and the nucleon-Delta mass split are comparable with experimental data. However, it is still necessary to handle the numerical calculation of the moment of inertia more seriously. We are now seeking for new scheme to determine the moment of inertia more rigorously. REFERENCES 1. Y. Nambu, and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; ibid, 124 (1961) 246; U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 91; S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 649; T. Hatsuda and T. Kunihiro, Phys. Rept. 247 (1994) 241; R. Alkofer, H. Reinhardt, and H. Weigel, Phys. Rept. 265 (1996) 139. 2. A. Manohar, and H. Georgi, Nucl. Phys. B 234 (1984) 189. 3. C.V. Christov, A. Blotz, H. C. Kim, P. Pobylitsa, T. Watabe, T. Meissner, E. R. Arriola, and K. Goeke, Prog. Part. Nucl. Phys. 37 (1996) 91. 4. C.D. Roberts, and R.T. Cahill, Phys. Rev. D 32 (1985) 2419. 5. J. Praschifka, C. D. Roberts, and R. T. Cahill, Phys. Rev. D 36 (1987) 209. 6. C.D. Roberts, R.T. Cahill, J. Praschifka, Ann. Phys. 188 (1988) 20. 7. R.T. Cahill, Aust. J. Phys. 42 (1989) 171. 8. M.R. Frank, P.C. Tandy, G. Fai, Phys. Rev. C 43 (1991) 2808. 9. M.R. Frank, P.C. Tandy, Phys. Rev. C 46 (1992) 338. 10. C.D. Roberts, R.T. Cahill, M.E. Sevior, N. Iannella, Phys. Rev. D 49 (1994) 125 . 11. P.C. Tandy, Prog. Part. Nucl. Phys. 39 (1997) 117. 12. R.T. Cahill, and S.M. Gunner, Fizika B 7 (1998) 171. 13. X.F. L¨ u, Y.X. Liu, H.S. Zong and E.G. Zhao, Phys. Rev. C 58 (1998) 1195. 14. Y.X. Liu, D.F. Gao, and H. Guo, Nucl. Phys. A 695 (2001) 353; Y.X. Liu, D.F. Gao, J.H. Zhou, and H. Guo, Nucl. Phys. A 725 (2003) 127; L. Chang, Y.X. Liu, and H. Guo, Nucl. Phys. A 750 (2005) 324. 15. Bin Wang, Hui-chao Song, Lei Chang, Huan Chen, and Yu-xin Liu, Phys. Rev. C 73 (2006) 015206. 16. J.L. Gervais, B. Sakita, Phys. Rev. D 11 (1975) 2943. 17. H.J. Munczek, and A.M. Nemirovsky, Phys. Rev. D 28 (1983) 181.
Baryon structure in the global color symmetry model of QCD∗ Bin Wang a , Yu-xin Liu a
a†
Department of Physics, Peking University, Beijing 100871, China
Baryons are considered as solitons in the global color symmetry model (GCM) of QCD. With both the scalar and pseudoscalar chiral fields being included and the Hedgehog approximation being taken, we obtain a GCM soliton profile whose mass and radius are quite close to the experimental data of a nucleon. We then quantize the angular momentum and isospin with the collective quantization method and obtain a nucleonDelta mass split comparable with experimental data. PACS numbers: 12.40.Yx, 03.75.Lm, 12.39.-x, 11.10.Lm 1. INTRODUCTION How to describe hadron states directly from the fundamental theory of strong interaction, i.e. QCD, is still an open problem. One promising approach may be the lattice QCD because of the advancements in numerical calculation. Another approach is to construct models to describe hadrons. Such models are designed to successfully fit experiment data and to pursue for the QCD foundation, then the Nambu–Jona-Lasinio (NJL) model and the golobal color symmetry model (GCM) have been developed. People have devoted to these models to mimic baryons as solitons [1–15]. Theoretically, the GCM can imbed not only the chiral symmetry spontaneous breaking but also the confinement and the nonlocal property of the low energy QCD. Practically, the GCM have described many meson data. One then expects the model can be a successful tool for the research of baryon structure. The paper is organized as following. In section II, we describe briefly the main points of the GCM and the classical soliton at the mean field level and its quantization with a semiclassical method. In section III, we describe the numerical calculation and result and some discussion. 2. GCM SOLITON AND ITS QUANTIZATION After bosonizing the lagrangian [11,12], one have the GCM as an effective field theoretical model of QCD with the quark and gluon degrees of freedom being changed to quark ∗ This work was supported by the National Natural Science Foundation of China (NSFC) under contract Nos. 10425521 and 10575004, the Key Grant Project of Chinese Ministry of Education (CMOE) under contact No. 305001 and the Research Fund for the Doctoral Program of Higher Education of China under grant No. 20040001010. One of the authors (Y.X. Liu) would acknowledge the support of the Foundation for University Key Teacher by the CMOE, too. The authors are also indebted to Professor Xiao-fu L¨ u for his stimulating discussions. † Corresponding author.
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.100
594c
B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
and meson fields. As one keeps only the scalar, pseudoscalar and vector meson fields, the QCM action reads S[U] = −T r ln G−1 [U] + I[U],
(1)
where the inverse of quark propagator can be conventionally written as x+y )B(x − y) . (2) 2 The configuration V corresponds to the scalar field and pseudoscalar field, and reads √ 1 − γ5 + 1 + γ5 2 a a U+ U , V= with U = exp i π F , (3) 2 2 fπ G−1 [U] = γ · ∂x A(x − y) + V(
where F a is the generator of U(2) group defined in [6]. The vacuum is set to U˜ = 1 (π a = 0) according to the chiral symmetry spontaneous breaking. The function A and B are related to the vacuum configuration of the above mentioned truncated fields, and can be determined by the truncated Dyson-Schwinger equation 2 ˜ Σ(p) = iγ · p[A(p2 ) − 1] + VB(p ) = g2
λa d4 q 1 λa γ γ , D(p − q) μ μ (2π)4 2 iγ · q + Σ(q) 2
(4)
where D(p − q) is the effective gluon propagator. The I[U] is just a functional of the ˜ which has been given explicitly in Ref. [6]. vacuum configuration U, Within the GCM, the nucleon current correlation function is given as {f } {g}∗
ΠN (T ) = ΓN ΓN
1 Z
DU
Nc i=1
γ4 G(x,
T T ; y , − , [U])fi ,gi e−S[U ] , 2 2
(5)
{f }
where ΓJJ3 ,T T3 is a matrix to show the spin and flavor structure of the nucleon, fi denotes the flavor and spin structure. To calculate the correlation function, it is convenient to make the spectral decomposition [15] of the quark propagator G(x, y; [U]) =
j
1 ¯ dω ψj,ω (x) ψj,ω (y) , 2π iλj (ω)
(6)
with ψj,ω (x) = uj (x)eiωx4 , λj (ω) = ω − i j (ω). When people take the limit T → ∞, the quark propagator reduces to the contribution of the lowest poles of λj 1(ω) . One usually refers to these poles as the valence quarks with energy val . Therefore, lim
T →+∞
Nc i=1
γ4 G(x,
T T ; y, − , [U])fi ,gi ∼ e−Nc T val . 2 2
With a static meson field U, one can get
˜ = −Nc T r ln G−1 [U]/G−1 [U˜ ] = Nc T S[U] − S[U]
(7)
( i − ˜ i ) = T Esea [U] ,
(8)
i <0,˜ i <0
which is usually denoted as the sea quarks’ contribution because it comes from the negative Dirac sea.
595c
B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
Up to now, we obtain a factor e−T (Nc val+Esea ) in the nucleon correlation function which can be related to a soliton mass Mcl [U] = Nc val [U]+Esea [U]. Here the subscript cl means that this mass is the classical mean field mass because there is no quantum fluctuation of meson fields U. The meson fields can be determined with the stability principle δMcl [Ucl ] =0, δUcl
(9)
This equation can be explicitly written as a set of coupled equations of quark and meson fields, which have been given in Refs. [8,15]. It should be noted that in contrast to the chiral quark model[1] (based on NJL Lagrangian), the classical mass here is not divergent because of the nonlocal property of quark-quark interaction. The above soliton equation (9) is difficult to solve because of the four independent π a fileds and the high non-linearity. We take the usual Hedeghog form solution as the profile, V (r) = χ(r)eiγ5
τ a π a (r) fπ
,
π a (r) = rˆa π(r) ,
(10)
where a = 1, 2, 3 corresponds to the iso-triplet part of the field and χ(r) stands for the iso-singlet part of the field. The Hedehog ansatz has the symmetry of simultaneous coordinate-space-rotation and iso-rotation, G = J + τ . In practical calculation, we take the valence quarks as the states with G = 0 as one often did. The Hedgehog soliton state does not have good quantum numbers of angular momentum and isospin. To identify the GCM solitons with baryons, one should restore them the quantum numbers with angular momentum projection or collective quantization. We here take the collective quantization scheme [3,16] and introduce time dependent fields y V (x, y) = R(tx )Vcl ( x+ )R+ (ty ) , where Vcl is the classical soliton solution, R(t) is a time 2 dependent SU(2) matrix which we will quantize. For the non-local model, we take this ˙ ansatz to make the quark field locally rotate with an angular frequency Ω = iR+ (t)R(t). Taking the adiabatic low angular frequency approximation and discarding all the terms with higher orders of Ω2 , one can get an exponential factor exp{−T I[Ucl ]/2Ω2a } in the current correlation function, where I[Ucl ] = I1 [Ucl ] + I2 [Ucl ], with I1 [Ucl ] =
Nc T r{G[Ucl ](x, y)γ4A(y − z)τ 3 G[Ucl ](z, w)γ4 A(w − x)τ 3 } − {Ucl → U˜ } , 4T
(11)
I2 [Ucl ] =
Nc (val|τ 3 A|j)(j|τ 3 A|val) . 2 j =val −Zval λj (i val )
(12)
The note 1/Zj is the corresponding residue of the spectral function 1/λj (ω), and (i|τ 3 A|j) =
d3 xd3 yd(x4 − y4 )u+ x)τ 3 A(x − y)e−iω(x4 −y4 ) uj (y ) i (
ω=ij
.
(13)
As a result of the collective quantization, the presently derived exponential factor corresponds to a Hamiltonian of a rotator Hrot , and I[Ucl ] corresponds to the moment of inertia of the rotator. Quantizing the Hrot , one can get the good quantum numbers of iso-spin and angular momentum. We have thus quantized the rotation and iso-rotation modes and obtained a mass of the quantum system M[Ucl ] = Mcl [Ucl ] + J(J + 1)/2I[Ucl ] , and also the nucleon and Delta mass split MΔ − MN = 2I3 .
596c
B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
3. NUMERICAL RESULTS AND SUMMARY With a simple effective gluon propagator, the Munczek-Nomirovsky model [17] 3 2 4 η δ (q) , (14) 16 we have solved the Dyson-Schwinger equation, and in turn the quark and meson equations. We obtained then some properties of nucleon [15], for instance, MN = 916 MeV, rN = 0.56 fm. We have also found that the pion field strengthens the attractive interaction and then reduces the mass of soliton much more close to the experiment data and leaves room for the contribution of the collective quantization. We will also take the fitted data η = 1.04 in the calculation of the moment of inertia. With the classical solution, we can make the spectral decomposition of the quark propagator in Eq. (6). We take the common eigenstate of the grand spin G = l + s + τ and the parity P , |K, M, P , as the basis of the spectral decomposition, g 2 D(q) = (2π)4
f1 (q)|(K, K + 12 )KM + f2 (q)|(K, K − 12 )KM f3 (q)|(K + 1, K + 12 )KM + f4 (q)|(K − 1, K − 12 )KM f3 (q)|(K + 1, K + 12 )KM + f4 (q)|(K − 1, K − 12 )KM f1 (q)|(K, K + 12 )KM + f2 (q)|(K, K − 12 )KM
,
for P = (−1)K ,
(15)
,
for P = (−1)K+1 .
(16)
The notation |(l, j)KM stands for the coupling of orbital angular momentum l, spin s and isospin τ to a function with good quantum number G = K, G3 = M. With the derived soliton profile in [15] and the spectral information, we can evaluate the contribution of the valence quarks to the moment of inertia (i.e., I2 ) and obtain I2 = 1.04 GeV−1 . As for the contribution of the sea quarks to the moment of inertia (i.e., I1 ), since both of the two terms of I1 have only ultraviolet divergence
∞ 1 T rG0 γ4 Aτ 3 G0 γ4 Aτ 3 ∝ V (3) ds, (17) T 0 which is just the effect of the asymptotic behavior of the function A(p) and can be exactly counteracted in I1 , it is then definitely convergent. However, it is difficult to get the value of I1 . Usually, one can calculate the corresponding term in the NJL based chiral quark soliton model and turn the trace to a imaginary ω residual summation. Nevertheless, because of the confinement in GCM, the integrand in the second term of I1 has no poles in the complex ω plane and thus it can not be calculated with such a method. Then one can not find a systematic cutoff scheme to reduce the two terms to a finite quantity in numerical evaluation. Alternatively, one can calculate the two terms both on the real ω axes and it is much easy to take a systematic cutoff scheme for the numerical calculation, however it needs too much calculation power. There exists also the third scheme, where one expands the terms of I1 as a series of y+z ) − 1] , (18) G0 B Vˆ (x, z) := d4 yG0 (x, y)B(y − z)[V ( 2 and the terms to the second order of G0 B Vˆ are −2T r[G0 B Vˆ G0 γ4 Aτ 3 G0 γ4 Aτ 3 ] and 2T r[G0 B Vˆ G0 B Vˆ γ4 Aτ 3 G0 γ4 Aτ 3 ] + T r[G0 B Vˆ G0 γ4 Aτ 3 G0 B Vˆ G0 γ4 Aτ 3 ]. Here one still encounters a difficulty that the expansion is not convergent which we numerically proved. In
B. Wang, Y.-x. Liu / Nuclear Physics A 790 (2007) 593c–597c
597c
fact, this expansion is equivalent to the expansion in terms of the derivative of the chiral field V . The soliton field V is not very smooth and then the expansion is not convergent. Taking only the lowest order of G0 B Vˆ in I1 , we obtain I1 = 3.06 GeV−1 . Then the total moment of inertia is I = 4.1 GeV−1 and the corresponding nucleon-Delta mass split is 366 MeV, which is comparable with experimental data. In summary, we have quantized the GCM solitons with the collective quantization scheme so that they take the quantum numbers of low-lying baryons. Then the lowlying baryons can be described by the GCM solitons. We also numerically evaluated the moment of inertia. The obtained mass and radius of nucleon and the nucleon-Delta mass split are comparable with experimental data. However, it is still necessary to handle the numerical calculation of the moment of inertia more seriously. We are now seeking for new scheme to determine the moment of inertia more rigorously. REFERENCES 1. Y. Nambu, and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; ibid, 124 (1961) 246; U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 91; S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 649; T. Hatsuda and T. Kunihiro, Phys. Rept. 247 (1994) 241; R. Alkofer, H. Reinhardt, and H. Weigel, Phys. Rept. 265 (1996) 139. 2. A. Manohar, and H. Georgi, Nucl. Phys. B 234 (1984) 189. 3. C.V. Christov, A. Blotz, H. C. Kim, P. Pobylitsa, T. Watabe, T. Meissner, E. R. Arriola, and K. Goeke, Prog. Part. Nucl. Phys. 37 (1996) 91. 4. C.D. Roberts, and R.T. Cahill, Phys. Rev. D 32 (1985) 2419. 5. J. Praschifka, C. D. Roberts, and R. T. Cahill, Phys. Rev. D 36 (1987) 209. 6. C.D. Roberts, R.T. Cahill, J. Praschifka, Ann. Phys. 188 (1988) 20. 7. R.T. Cahill, Aust. J. Phys. 42 (1989) 171. 8. M.R. Frank, P.C. Tandy, G. Fai, Phys. Rev. C 43 (1991) 2808. 9. M.R. Frank, P.C. Tandy, Phys. Rev. C 46 (1992) 338. 10. C.D. Roberts, R.T. Cahill, M.E. Sevior, N. Iannella, Phys. Rev. D 49 (1994) 125 . 11. P.C. Tandy, Prog. Part. Nucl. Phys. 39 (1997) 117. 12. R.T. Cahill, and S.M. Gunner, Fizika B 7 (1998) 171. 13. X.F. L¨ u, Y.X. Liu, H.S. Zong and E.G. Zhao, Phys. Rev. C 58 (1998) 1195. 14. Y.X. Liu, D.F. Gao, and H. Guo, Nucl. Phys. A 695 (2001) 353; Y.X. Liu, D.F. Gao, J.H. Zhou, and H. Guo, Nucl. Phys. A 725 (2003) 127; L. Chang, Y.X. Liu, and H. Guo, Nucl. Phys. A 750 (2005) 324. 15. Bin Wang, Hui-chao Song, Lei Chang, Huan Chen, and Yu-xin Liu, Phys. Rev. C 73 (2006) 015206. 16. J.L. Gervais, B. Sakita, Phys. Rev. D 11 (1975) 2943. 17. H.J. Munczek, and A.M. Nemirovsky, Phys. Rev. D 28 (1983) 181.