We apply the linear sigma model to analyze pion-nucleon sigma-term. Analytic expresions for these quantities are obtained in terms of the parameters of low-energy physics (pion decay constant and the pion mass). Good results have been obtained in comparison with author\u27s previous work and with other models.Primijenili smo linearni sigma model u analizi pionsko-nukleonskog člana sigma. Analitički izrazi za te veličine izvode se na osnovi parametara nisko-energijske fizike (konstanta raspada i masa piona). Postigli smo dobre ishode u usporedbi s autorovim ranijim radovima i s drugim modelima

Ali, T. S. T.

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7PION-NUCLEON SIGMA TERM IN THE LINEAR SIGMA MODELT. S. T. ALIDepartment of Mathematical Sciences, UAE University, P. O. Box. 17551,AL-AIN U.A.E. College of Science, Helwan University, Cairo, EgyptReceived 9 September 2009; Revised manuscript received 23 February 2010Accepted 3 March 2010 Online 19 March 2010We apply the linear sigma model to analyze pion-nucleon sigma-term. Analytic ex-presions for these quantities are obtained in terms of the parameters of low-energyphysics (pion decay constant and the pion mass). Good results have been obtainedin comparison with author’s previous work and with other models.PACS numbers: 14.20.Dh, 13.75.Gx, 11.30.Rd UDC 539.125, 539.141Keywords: pion-nucleon interaction, sigma term, mean field approximation1. IntroductionIn recent years there has been a growing interest in studying the sigma-termσ (πN). Many groups made a significant progress towards understanding pion-nucleon sigma-term σ (πN) using such models. One of these models is the MeanField Approximation (MFA). In this approach the meson fields are treated as time-independent classical fields.In a MFA which has the hedgehog property, a similar model has been consid-ered by Kälbermann and Eisenberg [1] and Birse and Banerjee [2], while high-ordermesonic interactions in the linear sigma model was considered Sahu and Ohanishi[3] and by M. Rashdan et al. [4, 5]. They used the mean field theory to get betterdescription of the nucleon properties. Ali et al. [6, 7] also used the coherent pairapproximation to study the nucleon properties. Hadron models set up to under-stand the structure of the nucleon should respect the constraints imposed by chiralsymmetry. In the first approaches in chiral perturbation theory [8, 9] and latticeQCD [10, 11], also, for example, in the chiral effective theories [12, 13], similarlylarge values for σ (πN) were obtained in the (light-cone) QCD sum rule approaches[14, 15]. Spontaneous and explicit chiral symmetry breaking require the existenceof the pion whose mass vanishes in the limit of zero current mass. Few solutions forFIZIKA B (Zagreb) 19 (2010) 1, 9–16 9ali: pion-nucleon sigma term in the linear sigma modelthe lagrangian of chiral linear soliton models, when applied to the nucleon, havebeen suggested. The sigma term is an important quantity in that it measures thedegree of chiral symmetry breaking. To achieve these goals, one should not, as webelieve, replace the whole system by a boson field (as Skyrme does), nor replace theDirac sea by a boson field (as in the dispersion-relation techniques). For a systemof the size of a nucleon, gradient expansions do not converge well and, hence, onehas to treat the Dirac sea “exactly”. No experimental method is known to directlymeasure the pion-nucleon sigma term. In an earlier analysis, Gasser et al. [16] ob-tained a value of about 45 ± 8 MeV, which was generally accepted until the late1990’s. Recent analyses, however, tend to yield higher values in the range (80 – 90)MeV (Kaufmann and Hite [17], Pavan et al. [18]) and also using the pion-nucleonscattering data of Mekterović et al. [19].The mean-field equations are a straightforward extension of that outlined byGoldflam and Wilets [20]. We consider a model based on the idea of strong QCDforces.The aim of the present work is to investigate and analyze the sigma-term σ (πN)depending on different values of the quark and sigma masses, with fixed prameterslike the pion decay constant fπ = 91.9 MeV and the pion mass mπ = 138.04 MeV,introduced by Struber and Rischke [21]. Description of the model is given in Sec. 2.The results and discussion are given in Sec. 3.2. Chiral quark-sigma modelWe describe the interactions of quarks with σ− mesons and pions by Birse andBanerjee [2]. The Lagrangian density isL = iΨγµ∂µΨ+12(∂µσ∂µσ + ∂µ−→π .∂µ−→π ) + gΨ(σ + iγ5τ.−→π )Ψ− U (σ,−→π ) , (1)whereU (σ, π) =λ24(σ2 + π2 − ν2)2+m2πfπσ, (2)is the meson-meson interaction potential where the Ψ, σ and π are the quark ,(scalar, isoscalar) sigma and (pseudoscalar, isovector) pion fields, respectively.The meson-meson interactions in Eq. (2) lead to hidden chiral SU(2) × SU(2)symmetry with σ (r) taking on a vacuum expectation value〈σ〉 = −fπ, (3)where fπ is the pion decay constant. The final term in Eq. (2) is included to breakthe chiral symmetry. It leads to partial conservation of axial-vector isospin current(PCAC). The parameters λ2, ν2 can be expressed in terms of fπ ,the mass of the σand mass of the π mesonλ2 =m2σ −m2π2f2π, (4)10 FIZIKA B (Zagreb) 19 (2010) 1, 9–16ali: pion-nucleon sigma term in the linear sigma modelν2 = f2π −m2πλ2. (5)Now we expand the extremum, with the shifted field defined asσ = σ′ − fπ, (6)substituting by equation(6) into equation (1) we getL = iΨγµ∂µΨ+12(∂µσ′∂µσ′ + ∂µ−→π .∂µ−→π )− gΨfπΨ+ gΨσ′Ψ+ igΨγ5τ.−→π Ψ−U (σ′, π) , (7)withU (σ′, π) =λ24(σ′2 + π2)2 − λ2fπσ′(σ′2 + π2)+m2π2(σ′2 + π2)+ λ2f2πσ′2, (8)the time-independent fields σ′ (r) and π (r) are to satisfy the Euler-Lagrangianequations, and the quark wave function satisfies the Dirac eigenvalue equation.Substituting by equation (7) in Euler-Lagrangian equation we get(+m2σ)σ′ = gΨΨ− λ2σ′(σ′2 + π2)+ λ2fπ(3σ′2 + π2), (9)(+m2π)π = igΨγ5 · τΨ− λ2(σ′2 + π2 − 2fπσ′)π . (10)where τ refers to Pauli (iso)spin-matrices, γ5 =(0 11 0).By using the hedgehog Ansatz [2], whereπ(r) = ṙπ(r) . (11)The Dirac wave functions areΨ (r) =1√4π[u (r)iw (r)]and Ψ (r) =1√4π[u (r) iw (r)]. (12)The chiral Dirac equation for the quarks isdudr= −P (r)u+ (E −mq + S(r))w, (13)where S(r) = g 〈σ′〉 , P (r) = 〈π · r̂〉 , E are the scalar potential, the pseudoscalarpotential and the eigenvalue of the quarks spinor Ψ, anddwdr= − (E −mq + S(r))u+(2r− P (r))w. (14)FIZIKA B (Zagreb) 19 (2010) 1, 9–16 11ali: pion-nucleon sigma term in the linear sigma modelThe set of equations( 9, 10, 13, 14) is solved following the method used by Goldflamand Wilets [20] and by Birse and Banarjee [2] for the soliton bag model (for moredetails see Refs. [22, 4]). Including the color degree of freedom, one has gΨΨ →NcgΨΨ, where Nc = 3 colors and g is the coupling constant. Thusρs = NcgΨΨ =3g4π(u2 − w2), (15)ρp = iNcgΨγ5τΨ =34πg (−2uw) , (16)ρv =3g4π(u2 + w2), (17)where ρs, ρp and ρv are the sigma density, pion density, and vector density, respec-tively. These equations are subject to the boundary conditions that asymptoticallythe fields approach their vacuum values,σ (r) ∼ −fπ π (r) ∼ 0 at r → ∞. (18)Finally, we have solved the Eqs. (13, 14) using fourth-order Rung-Kutta method.Due to the implicit nonlinearly of our Eqs. (9, 10), it is necessary to iterate thesolution until self-consisteny is achieved. To start this iteration process, we use thechiral circle form for the meson fieldsS(r) = mq(1− cos θ) , (19)P (r) = −mq sin θ , (20)whereθ = π tanh r . (21)3. Results and conclusionsThe field equations (9,10,13,14) have been solved by iteration for different valuesof the quark and sigma masses. The dependence of the nucleon properties on thesigma and quark masses is listed in Tables 1 and 2. Tables 1 and 2 show the nucleonobservables calculated for mq ≥ 370 − 440 MeV, mσ 1 900 − 980 MeV. As seenfrom Table 1, increasing the sigma mass decreases the sigma-term σ (πN) and thengets closer to the experimental data. The experimental values of the pion decaycoupling constant and the pion mass are of the order of 93 MeV and 139.6 MeV,respectively, but by decreasing these values (Hemmert et al. [23]) and choosing thevalues 91.9 MeV and 138.04 MeV decreases the sigma-term σ (πN). I thus suggesta value of the quark mass to be in the range of 370− 440 MeV, depending on themodel parameters.12 FIZIKA B (Zagreb) 19 (2010) 1, 9–16ali: pion-nucleon sigma term in the linear sigma modelTABLE 1. Values of σ (πN) (MeV), gπNN (0)mπ2MBand gA(0), at mq = 400 MeV.mσ MeV 900 940 960 980σ (πN) 83.03 81.61 80.98 76.99gπNN (0) mπ/(2MB) 1.35 1.35 1.35 1.32gA(0) 1.77 1.77 1.77 1.76TABLE 2. Values of σ (πN) (MeV), and gA(0), at mq = 1000 MeV.mqMeV 440 420 400 380 370σ (πN) 88.48 88.81 79.86 72.46 65.96gA(0) 1.82 1.80 1.78 1.74 1.72In fact, neither mσ nor mq have been determined by experiment, since the quarkcannot be isolated (because of the confinement principle), and we believe mq is ofthe order of one third of the nucleon mass, and from the Nambu and Jona-Lasiniomodel [24], mσ is twice mq. Birse and Banerjee [2] suggested a value 500 MeV formq, which is very large. Decreasing this value has the effect of decreasing sigma-term σ (πN) as seen from Table 2. A better result for the sigma term σ (πN) =65.96 MeV is obtained when mq = 370 MeV and mσ = 1000 MeV as seen fromTable 2. Furthermore, sigma-term σ (πN) is better reproduced than that of Birseand Banerjee [2] who used standard potential as seen from Table 3.TABLE 3. Values of the sigma-term σ (πN)MeV compared with my previousworks.Observable [23] [2] [4] [7] this workσ (πN) 54 92.3 88 88.9 65.96gπNN (0) mπ/(2MB) 0.61 1.29 1.09 1.35 1.25gA(0) 0.78 1.85 1.80 1.46 1.72In particular, σ (πN) is improved because following Refs. [23, 17, 18], recentupdated analyzes of pion-nucleon scattering data lead to a dramatic increase ofthe value of σ (πN). For the sigma form factor σ(t) at the Cheng-Dashen pointt = 2m2π it was found σ(2m2π) = (85 ± 15) MeV by Kaufmann and Hite [23] orσ(2m2π) = (79 ± 7) MeV by Pavan et al. [18]. With the value ∆σ = 15 MeVfrom the work by Gasser et al. [16], one concludes for σ (πN) = σ(2m2π)−∆σ theexperimental results σ (πN) : 73± 15 MeV [23] or 64± 7 MeV [17]. In this contextFIZIKA B (Zagreb) 19 (2010) 1, 9–16 13ali: pion-nucleon sigma term in the linear sigma modelone should also mension the recent pion-nucleon scattering data by Mekterović etal. [19], along with the axial-vector coupling constant gA, the pion-nucleon couplingconstant gπNN depending on the quark and sigma masses. I compared the freeskyrmion model by Braghin and Cavalcante [25], Birse and Banerjee [2], Rashdanet al. [4] and Ali et al. [7] (my previous works) with the present work.Finally, in all of my previous work I made a modification in the potential of thelinear sigma model, but in this paper calculations were done differently, using thestandard potential of the linear sigma model without any modifications. The newparameterization describes the nucleon properties that have been introduced.References[1] G. Kälbermann and J. Eisenberg, Phys. Lett. B 139 (1984) 337.[2] M. C. Birse and M. K. Banerjee, Phys. Rev. D 31 (1985) 118.[3] P. K. Sahu and A. Ohnishi, Prog. Theor. Phys. 104 (2000) 6.[4] M. Rashdan, M. Abu-Shady and T. S. T. Ali, Int. J. Mod. Phys. E 15 No.1, (2006)143.[5] M. Rashdan, M. Abu-Shady and T. S. T. Ali, Int. J. Mod. Phys. A 22 (2007) 2673.[6] T. S. T. Ali, M. Rashdan and M. Abu-Shady, Int. J. Theor. Phys. 45 (2006) 1654.[7] T. S. T. Ali, J. A. McNeil and S. Pruess, Phys. Rev. D. 60 (1999) 114022.[8] B. C. Lehnhart, J. Gegelia and S. Scherer, J. Phys. G 31 (2005) 89 [arXiv: hep-ph/0412092].[9] M. Frink, U. G. Meissner and I. Scheller, Eur. Phys. J. A 24 (2005) 395 [arXiv:hep-lat/0501024].[10] D. B. Leinweber, A. W. Thomas and R. D. Young, Phys. Rev. Lett. 92 (2004) 242002[arXiv:hep-lat/0302020].[11] M. Procura, B. U. Musch, T. Wollenweber, T. R. Hemmert and W. Weise, Phys. Rev.D 73 (2006) 114510 [arXiv:hep-lat/0603001].[12] P. Scweitzer, Phys. Rev. D 69 (2004) 034003 [arXiv:hep-ph/0307336].[13] T. Inoue, V. E. Lyubovitskij, T. Gutsche and A. Faessler, Phys. Rev. C 69 ( 2004)035207 [arXiv: hep-ph/0311275].[14] Z. G. Wang, S. L. Wan and W. M. Yang, Phys. Rev. D 73 (2006) 094011 [arXiv:hep-ph/0601025].[15] G. Erkol and M. Oka. Phys. Lett.B 659 (2008) 176 [arXiv:0705.4046 [hep-ph]].[16] J. Gasser, H. Leutwyler and M. E. Sainio, Phys. Lett. B 253 (1991) 252.[17] W. B. Kaufmann and G. E. Hite, Phys. Rev. C 60 (1999) 055204.[18] M. M. Pavan, I. I. Strakovsky , R. L. Workman and R. A. Arnt, πN Newsletter 16(2002) 110.[19] D. Mekterović et al. [The Crystal Ball Collaboration], Phys. Rev. C 80 (2009) 055207;e-print arXiv; 0908.3845[hep-ex].[20] R. Goldflam, L. Wilets, Phys. Rev. D 25 (1982) 1951.14 FIZIKA B (Zagreb) 19 (2010) 1, 9–16ali: pion-nucleon sigma term in the linear sigma model[21] Stefan Struber and Dirk H. Rischke, Phys. Rev. D 77 (2008) 085004;hep-th/0708.2389v2 (2007).[22] M. Abu-Shady, M. Rashdan and T. S. T. Ali, Fizika B 16 (2007) 59.[23] R. Hemmert, M. Procura and W.Weise, Phys. Rev D 68 (2003) 075009.[24] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246.[25] F. L. Braghin and I. P. Cavalcante, Phys. Rev. C 67 (2003) 065207.FIZIKA B (Zagreb) 19 (2010) 1, 9–16 15ali: pion-nucleon sigma term in the linear sigma modelPIONSKO-NUKLEONSKI ČLAN SIGMA U LINEARNOM SIGMA MODELUPrimijenili smo linearni sigma model u analizi pionsko-nukleonskog člana sigma.Analitički izrazi za te veličine izvode se na osnovi parametara nisko-energijskefizike (konstanta raspada i masa piona). Postigli smo dobre ishode u usporedbis autorovim ranijim radovima i s drugim modelima.16 FIZIKA B (Zagreb) 19 (2010) 1, 9–16

Pionsko-nukleonski član sigma u linearnom sigma modelu

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