11001 (2007). 16. F. L. Braghin and I. P. Cavalcante, Phys. Rev. C 67,. 065207 (2003). 17. B. Derek Leinweber, H. Lu Ding, and W. Thomas. Anthony, Phys. Rev.
ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 6, с. 978–982
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
PION MASS DEPENDENCE ON NUCLEON MAGNETIC MOMENTS IN THE EXTENDED LINEAR SIGMA MODEL c 2010
M. Abu-Shady
Department of Mathematics, Faculty of Science, Menofiya University, Egypt Received October 5, 2009
The dependence of the nucleon magnetic moments on the pion mass is studied in the framework of the extended linear sigma model. The field equations have been solved in the mean-field approximation. A remarkable agreement is observed in comparison with other models.
1. INTRODUCTION Low-energy hadronic phenomenon is described quite successfully in terms of effective meson theories. It is therefore important to consider models which consistently combine quark and meson degrees of freedom in respective domains. The Lagrangian for these models should be derivable from quantum chromodynamics (QCD) [1]. The chiral sigma model is a suitable one to study this. The model is formulated on the basis of the chiral symmetry and its spontaneous breakdown. In [1–10], the chiral linear sigma model and its modification have been applied in the nucleon magnetic moments. The model has shown that the proton and neutron magnetic moments can be reproduced with a physical pion mass (mπ ). It is constructed on the basis of the spontaneous symmetry breaking which generates constituent quark mass (mq ) and thus hadron masses. An expected relation between these variables of QCD and hadronic degrees of freedom can be seen through the GellMann–Oakes–Renner relation which, to the lowest order, reads: 2 ¯ q q mq = −fπ2 m2π . The experimental value of the pion decay constant is nearly 93 MeV. Numerical lattice QCD simulations face technical problems such as discretization errors or finite size effects, which are attacked and minimized with increasing success by employing improved versions of discretized actions, or by working on larger lattices available thanks to the steadily growing computer power. Lattices that are still present are too small to accommodate the pion as light as it appears in nature [11–13], therefore there is a need to perform extrapolation from exact calculations in finite lattice to the physical region of observables to ensure credibility, as in [14]. Chiral quark models as Skyrme model and extended Skyrme model [14–16], and
cloudy bag model [17] have been extrapolated lattice data from the region of nowadays for large value of pion mass down to the physical value of the pion mass. The aim of this paper is to estimate the effect of pion mass on nucleon magnetic moments in the framework of extended linear sigma model which is proposed by Rashdan et al. [5] in which the new form of mesonic interaction is presented. The extended chiral sigma model is explained briefly in Section 2. The numerical results and the discussion are presented in Section 3. The nucleon magnetic moments are explained in Section 4. The conclusion of the paper is presented in Section 5. 2. THE EXTENDED LINEAR SIGMA MODEL The extended linear sigma model is described in details in [5]. The following model is a brief summary of the extended linear sigma model with a new version of the meson–meson interaction. The Lagrangian density of the extended linear sigma model, which describes the interactions between quarks via the σ and π mesons, is written as in [5]: ¯ μ∂μΨ + (1) L (r) = iΨγ 1 (∂μ σ∂ μ σ + ∂μ π · ∂ μ π) + 2 ¯ (σ + iγ5 τ · π) Ψ − U1 (σ, π) + gΨ +
with 2 λ2 (2) U1 (σ, π) = 1 σ 2 + π 2 − ν12 + 4 2 2 λ2 + 2 σ 2 + π 2 − ν22 + m2π fπ σ. 4 It is clear that potential also satisfies chiral symmetry. In the original model [1, 2], the extracted term is excluded by the requirement of renormalizability. Since 978
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we are going to use Eq. (1) as an effective model, approximating the underlying quark theory, the model needs not and should not be renormalizable as in [4– 7]. By PCAC and the usual potential minimum conditions, as in [5–7], we get λ21 =
m2σ − m2π , 4fπ2
(3)
m2π , 2λ21
(4)
ν12 = fπ2 − λ22 =
m2σ − 3m2π , 16fπ6
(5)
m2π . 4λ22 fπ2
(6)
ν22 = fπ4 −
Now we can expand the extremum with the shifted field defined as (7) σ = σ − fπ ; substituting Eq. (7) into Eq. (1), we get ¯ μ∂μΨ + (8) L (r) = iΨγ 1 ∂μ σ ∂ μ σ + ∂μ π · ∂ μ π − + 2 ¯ Ψ + igΨγ ¯ 5 · πΨ − U2 σ , π ¯ π Ψ + gΨσ − gΨf with
2 λ2 2 U2 σ , π = 1 σ − fπ + π 2 − ν12 + (9) 4 2 2 2 λ + 2 (σ − fπ )2 + π 2 − ν22 + m2π fπ (σ − fπ ). 4
The time-independent fields σ (r) and π (r) satisfy the Euler−Lagrange equations, and the quark wave function satisfies the Dirac eigenvalue equation. Substituting Eq. (8) in Euler–Lagrange equation, we get: ¯ − λ21 (fπ − σ ) (σ − fπ )2 + (10) σ = gΨΨ 2 2 2 2 2 + π − ν1 − 2λ2 (fπ − σ ) (σ − fπ ) + π × 2 × (σ − fπ )2 + π 2 − ν22 − m2π fπ , ¯ 5 · τ Ψ − λ21 (σ − fπ )2 + (11) π = igΨγ + π 2 − ν12 π − 2λ22 π (σ − fπ )2 + π 2 × 2 × (σ − fπ )2 + π 2 − ν22 , where τ refers to Pauli isospin matrices, ⎛ ⎞ 0 1 ⎠. γ5 = ⎝ 1 0 ЯДЕРНАЯ ФИЗИКА
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Including the color degree of freedom, one has ¯ → Nc gΨΨ, ¯ where Nc = 3 colors. Thus, gΨΨ ⎡ ⎤ u (r) 1 ⎦, (12) Ψ (r) = √ ⎣ 4π iw (r) ¯ (r) = √1 u (r) iw (r) , Ψ 4π then
¯ = 3 u2 − w2 , (13) ρs = Nc ΨΨ 4π ¯ 5 · τ Ψ = 3 g (−2uw), (14) ρp = iNc Ψγ 4π 3 2 u + w2 , (15) ρv = 4π where ρs , ρp , and ρv are sigma, pion, and vector densities, respectively. The field equations satisy the boundary conditions at the vacuum values; σ (r) ∼ − fπ ,
π (r) ∼ 0 at r → ∞.
(16)
By using hedgehog ansatz [5] in which one assumes that the pion field is radially oriented in isospin space, π (r) = ˆrπ (r) . (17) The chiral Dirac equation for the quarks is [5] du = −P (r) u + (W + mq − S(r)) w, (18) dr where the scalar potential S(r) = g σ , the pseudoscalar potential P (r) = π · ˆ r, and W is the eigenvalue of the quark spinor Ψ, dw = − (W − mq + S(r)) u − (19) dr 2 − P (r) w. − r 3. NUMERICAL CALCULATIONS AND DISCUSSIONS
3.1. The Scalar Field σ To solve Eq. (10), we integrate a suitable Green’s function over the source fields [18]. Thus, (20) σ (r) = dr Dσ (r − r ) gρs (r ) − − λ21 (fπ − σ ) (σ − fπ )2 + π 2 − ν12 − − 2λ22 (fπ − σ ) (σ − fπ )2 + π 2 × 2 × (σ − fπ )2 + π 2 − ν22 − m2π fπ , where Dσ (r − r ) =
1 exp(−mσ r − r ), 4π |r − r | 3*
980
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the scalar field is spherical in this model as we only need the l = 0 term exp (−mσ r> ) 1 , (21) sinh (mσ r< ) Dσ r − r = 4π r> therefore σ (r) = ∞ = mσ
r 2 dr
(22)
sinh (mσ r> ) exp (−mσ r> ) × (mσ r> )2
0
× gρs (r ) − λ21 (fπ − σ ) (σ − fπ )2 + π 2 − ν12 − − 2λ22 (fπ − σ ) (σ − fπ )2 + π 2 × 2 × (σ − fπ )2 + π 2 − ν22 − m2π fπ ,
where r< and r> are the smaller and larger of r and r , respectively [18]. We will solve this equation by iterating to self-consistency.
3.2. The Pion Field π To solve Eq. (11), we integrate a suitable Green’s function over the source fields. We use the l = 1 component of the pion Green’s function. Thus, (23) π (r) = mπ × ∞ − sinh (mπ r< ) + mπ r< cosh (mπ r< ) × × r 2 dr (mπ r> )2 0 exp (−mπ r> ) 1 × × 1+ mπ r> mπ r> × gρp − λ21 (σ − fπ )2 + π 2 − ν12 π − − 2λ22 π (σ − fπ )2 + π 2 × 2 . × (σ − fπ )2 + π 2 − ν22 We have solved Dirac Eqs. (18), (19) using fourthorder Rung–Kutta method. Due to the implicit nonlinearly of these Eqs. (10), (11) it is necessary to iterate the solution until self-consistency is achieved. To start this iteration process, we could use the chiral circle form for the meson fields: S(r) = mq (1 − cos θ), P (r) = −mq sin θ, (24) where θ = tanh r (for details, see [5, 6]). 4. NUCLEON MAGNETIC MOMENTS The proton and neutron magnetic moments are given by [1] 1 (25) μp,n = P ↑ dr r × jEM (r) P ↑ , 2
where the electromagnetic current is 1 τ3 ¯ jEM (r) = Ψ(r) γ + Ψ(r) − 6 2 − εαβ3 πα (r) ∇πβ (r), such that (jEM (r))nucleon
¯ = Ψ(r) γ
1 τ3 + 6 2
(26)
Ψ(r),
(jEM (r))meson = −εαβ3 πα (r) ∇πβ (r) .
(27) (28)
We note that the change in μp,n is received from meson fields according to Eq. (28); thus, the mesonic contributions come from the dynamic of fields in Eqs. (10), (11) where the dynamics of fields depends on the form of meson–meson potential (for details, see [8]). 5. RESULTS The field equations (10) and (11) have been solved by iteration method as in [5, 6] for different values of pion masses: 0, 139, 200, 400, 600, 700, 800 MeV. The chiral quark models such as Skyrme model [14], extended Skyrme model [15, 16], and cloudy bag model [17] have predicated successfully the dependence of pion mass on nucleon magnetic moments. The slopes of all curves are the same wherever the magnetic moments of the proton and neutron have decreased by increasing the pion mass (see the figure). The results of the extended linear sigma model are credibility in comparison with the results in [14]. In the same way, we need to examine the effect of pion mass on nucleon magnetic moments in the framework of extended linear sigma model described above in the Section 2. From the figure, the magnetic moments of the nucleon are calculated by cloudy bag model [17] and are related as μ0N , μN (mπ ) = 1 + αmπ + βm2π where μ0N , α, and β are fitted phenomenologically. The magnetic moments μi (i = p, n) of the proton and neutron were calculated as the averaged value of the operator: j 1 k k B + JV (3) , (29) μi = εijk r 2 where B k and JVk (3) are the isoscalar baryonic current and the third component of the isovector current, respectively. The μi was calculated in details for Skyrme model in [14]. For the extended Skyrmion model [16], we calculated the nucleon magnetic moments ЯДЕРНАЯ ФИЗИКА
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Magnetic moments of the proton and neutron (magn.)
4 Skyrme model [14] Extended linear sigma model Cloudy bag model [17] Extended Skyrme model [16]
2
0
–2
–4 0
0.2
0.4 Squared pion mass, GeV2
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0.8
Magnetic moments of the neutron and proton as functions of the squared pion mass, where upper curves represent the proton magnetic moment and lower curves represent neutron magnetic moment.
(μi ) with the observed nucleon mass (nearly MN 940 MeV) by using Eq. (29). We note that the behavior of curves (slopes) is similar in [14]. This may bring a certain credibility to both results but there is no unique desirable possible extrapolation where the results depend on the physical input of the effective model. In particular, we notice that although the behavior of all the curves are similar, for realistic pion mass the resulting values for the magnetic moment may be quite different, mainly for the proton due to the physical input parameters for each model. The extended linear sigma model is given a good values of the magnetic moments of the proton and neutron for physical pion mass: μp = 2.79 (magn.), μn = −2.10 (magn.), which is in a good agreement with experimental data μp = 2.793 (magn.), μn = −1.913 (magn.). 6. CONCLUSION The results of nucleon magnetic moments indicate that the extended linear sigma model has been predicted successfully for the large values of pion mass ЯДЕРНАЯ ФИЗИКА
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in comparison with other models. Also, the extended linear sigma model has predicted successfully nucleon magnetic moments for physical pion mass. REFERENCES 1. M. C. Birse and M. K. Banerjee, Phys. Rev. D 31, 118 (1985). 2. W. Broniowski and M. K. Banerjee, Phys. Lett. B 158, 335 (1985). 3. M. C. Birse, Phys. Rev. D 33, 1934 (1986). 4. W. Broniowski and B. Golli, Nucl. Phys. A 714, 575 (2003). 5. M. Rashdan, M. Abu-Shady, and T. S.T. Ali, Int. J. Mod. Phys. E 15, 143 (2006). 6. M. Rashdan, M. Abu-Shady, and T. S. T. Ali, Int. J. Mod. Phys. A 22, 2673 (2007). 7. M. Abu-Shady, M. Rashdan, and T. S. T. Ali, Fizika B (Zagreb) 16, 59 (2007). 8. M. Abu-Shady, Int. J. Theor. Phys. 48, 115 (2009). 9. M. Abu-Shady, Inter. J. Applied Math. & Info. Sciences (accepted). 10. M. Abu-Shady, Mod. Phys. Lett. A 24, 1617 (2009). 11. J. Gasser, Ann. Phys. 136, 62 (1981). 12. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984).
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13. K. Goeke, J. Ossmann, P. Schweitzer, and A. Silva, hep-lat/0505010v2. 14. F. L. Braghin, Braz. J. Phys. 34, 243 (2004). 15. C. Houghton and S. Magee, Europhys. Lett. 77, 11001 (2007). 16. F. L. Braghin and I. P. Cavalcante, Phys. Rev. C 67, 065207 (2003).
17. B. Derek Leinweber, H. Lu Ding, and W. Thomas Anthony, Phys. Rev. D 60, 034014 (1999). 18. S. E. Koonin and D. C. Meredith, Computational Physics (FORTRAN Version) (AddisonWesley, Reading, 1990).
ЗАВИСИМОСТЬ МАССЫ ПИОНА ОТ МАГНИТНЫХ МОМЕНТОВ НУКЛОНА В РАСШИРЕННОЙ ЛИНЕЙНОЙ СИГМА-МОДЕЛИ М. Абу-Шади В рамках расширенной сигма-модели исследуется зависимость магнитных моментов нуклона от массы пиона. Уравнения поля решаются в приближении среднего поля. Получено очень хорошее согласие с другими моделями.
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