We calculate low-energymeson decay processes and pion-pion scattering lengths in a two-flavour linear sigma model with global chiral symmetry, exploring the scenario in which the scalar mesons f0(600) and a0(980) are assumed to be ¯qq states

Giacosa, Francesco

Parganlija, Denis

Rischke, Dirk-Hermann

English

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:0812.2183v1  [hep-ph]  11 Dec 2008 Decay widths of resonances and pion scatteringlengths in a globally invariant sigma model withvector and axial-vector mesonsDenis Parganlija∗Goethe-Universität Frankfurt am MainE-mail: parganlija@th.physik.uni-frankfurt.deFrancesco GiacosaGoethe-Universität Frankfurt am MainE-mail: giacosa@th.physik.uni-frankfurt.deDirk H. RischkeGoethe-Universität Frankfurt am MainE-mail: drischke@th.physik.uni-frankfurt.deWe calculate low-energymeson decayprocesses and pion-pionscattering lengths in a two-ﬂavourlinear sigma model with global chiral symmetry, exploring the scenario in which the scalarmesons f0(600) and a0(980) are assumed to be ¯ qq states.8th Conference Quark Conﬁnement and the Hadron SpectrumSeptember 1-6 2008Mainz, Germany∗Speaker.c ￿ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/A Linear Sigma Model with Vector Mesons and Global Chiral Invariance Denis Parganlija1. IntroductionEffective ﬁeld theories provide a very efﬁcient means to describe Quantum Chromodynam-ics (QCD) at low energies. They possess the same global symmetries as QCD - e.g., the chiralSU(Nf)r ×SU(Nf)l symmetry, where Nf is the number of ﬂavours - and are expressed in terms ofhadronic degrees of freedom rather than in terms of quarks and gluons. Spontaneous breaking ofthe chiral symmetry leads to the emergence of low-mass pseudoscalar Goldstone bosons and theirchiral partners, large-mass scalar states.In this paper we present a linear sigma model with global chiral invariance, similar to the oneof Ref. [1]. The model contains scalar and pseudoscalar as well as vector and axialvector mesons.The global invariance is motivated by the results of Refs. [2, 3] where it has been shown that alocally invariant linear sigma model fails to describe simultaneously pion-pion scattering lengthsand some important decay widths. For the globally invariant model additional terms appear in theLagrangian which introduce new coupling constants that can in principle be adjusted to improvethe agreement with the experimental data. In this paper we show the ﬁrst results from this approachfor the case of Nf = 2.As outlined in Ref. [3], there are two possibilities to interpret the scalar sand a0 ﬁelds con-tained in the model where they are ¯ qq states [s = 1 √2(¯ uu+ ¯ dd), a00 = 1 √2(¯ uu− ¯ dd)]: (i) they areidentiﬁed with f0(600) and a0(980) which form a part of a larger nonet that consists of f0(980),a0(980), k(800) and f0(600) (resonances below 1 GeV); (ii) they are identiﬁed with the f0(1370)and a0(1450) resonances forming a part of a nonet that consists of f0(1370), f0(1500), f0(1710),a0(1450), K0(1430) - i.e., resonances above 1 GeV (see Ref. [4]). In the assignment (ii), scalarmesons below 1 GeV, whose spectroscopic wave functions possibly contain a dominant tetraquarkor mesonic molecular contribution [5], may be introduced as additional scalar ﬁelds.In this paper, we describe brieﬂy the consequences of assignment (i); the consequences ofassumption (ii) as well as more detailed results in assignment (i) may be found in Ref. [6].2. The Linear Sigma Model with Global Chiral SymmetryThe Lagrangian of the globally invariant linear sigma model with U(2)R ×U(2)L symmetryreads [2, 3, 7]:L = Tr[(DmF)†(DmF)]−m20Tr(F†F)−l1[Tr(F†F)]2−l2Tr(F†F)2−14Tr[(Lmn)2+(Rmn)2]+m212Tr[(Lm)2+(Rm)2]+Tr[H(F+F†)]+c(detF+detF†)−2ig2(Tr{Lmn[Lm,Ln]}+Tr{Rmn[Rm,Rn]})−2g3{Tr[(¶mLn +¶nLm){Lm,Ln}]+Tr[(¶mRn +¶nRm){Rm,Rn}]}+L4+h12Tr(F†F)Tr[(Lm)2+(Rm)2]+h2Tr[(FRm)2+(LmF)2]+2h3Tr(FRmF†Lm), (2.1)with F=(s +ihN)t0+(  a0+i  p)   t (scalar and pseudoscalar mesons; our model is valid for Nf = 2and thus our eta meson hN contains only non-strange degrees of freedom); Lm = (wm − fm1 )t0 +(  rm −  am1)   t and Rm = (wm + fm1 )t0+(  rm +  am1)   t (vector and axialvector mesons), where t0,  t arethe generators ofU(2); DmF=¶mF+ig1(FLm −RmF), Lmn =¶mLn−¶nLm, Rmn =¶mRn −¶nRm2A Linear Sigma Model with Vector Mesons and Global Chiral Invariance Denis Parganlijaand L4 containing all the vertices with four vector and axialvector mesons. Explicit breaking of theglobal symmetry is described by the term Tr[H(F+F†)] ≡ hs(h = const.). The chiral anomaly isdescribed by the term c(detF+detF†) [8].Irrespective of L4, the model contains 13 parameters - 12 parameters from the Lagrangian(2.1), plus the wave function renormalisation constant of the pseudoscalar mesons [9], Z. However,only seven of those (Z, g1,2, h1,2,3, l2) are relevant for the decays that will be considered in thefollowing. The parameters g1, h3 and l2 are expressed in terms of Z:g1 = g1(Z) =ma1Z fpr1−1Z2, h3 = h3(Z) =1Z2f2p m2r −m2a1Z2!, l2 = l2(Z) =Z2m2a0 −m2hNZ4f2p,and thus the number of independent relevant parameters is decreased to four. Additionally, ms(which is a function of m0, l1, l2, c and Z) is taken as a parameter that can be determined from thepion-pion scattering lengths yielding ﬁve independent parameters for the meson decay modes andscattering lengths described below.2.1 Relevant Decay Modes and pp Scattering LengthsThe following decay modes of two-ﬂavour low-energy mesons have been taken into account[parameter dependence in brackets]: r → pp [Z, g2], f1 → a0p [Z, h2], a1 → pg [Z], a0 → hp[Z, h2], s → pp [Z, h1, h2], a1 → sp [h1, h2, Z], a1 → rp [g2, Z]. We have also considered thepion-pion scattering lengths a00(h1, h2, Z, ms) and a20(h1, h2, Z, ms).Given that the decay widths for the channels s → pp [Z, h1, h2], a1 → sp [h1, h2, Z] anda1 → rp [g2, Z] are poorly known, we have not taken any numerical values for these decay widthsto ﬁt our parameters - these decay widths will be calculated as a consistency check on the basis ofthe results obtained from the other decay widths and the scattering lengths.Here, we will present formulas that have been used to ﬁt the parameters; for all other formulas,see Ref. [6].Decay width for r → pp. The decay width readsGr→pp =m5r48pm4a1"1−￿2mpmr￿2# 32 h￿g1−g22￿Z2+g22i2.The experimental value is (149.4±1.0) MeV [10].Decay width for f1 → a0p. The following formula for the decay width is obtained:Gf1→a0p =g21Z22pk3m2f1m4a1￿m2r −12(h2+h3)f2￿2, k =12sm2f1 −2(m2p +m2a0)+(m2a0 −m2p)2m2f1where f ≡ Z fp is the vacuum expectation value of the s ﬁeld.Decay width for a1 → pg. The Lagrangian leading to the formula for the decay width Ga1→pgis obtained from the Lagrangian (2.1) by coupling the photon to the relevant part of the axial3A Linear Sigma Model with Vector Mesons and Global Chiral Invariance Denis Parganlijacurrent Jm =−ig1Z2fp(a+1mp−−a−1mp+)−Zw(a+1mn¶np−−a−1mn¶np+), where w=g1fm2a1, and readsLa1pg = eJmAm.The decay width readsGa1→pg =e296p(Z2−1)ma1"1−￿mpma1￿2#3.Note that the sole dependence of the a1 → pg decay width on the parameter Z may in prin-ciple lead to an accurate determination of this parameter. However, the experimental value of thea1 → pg decay width is not very precise (Ga1→pg = 0.640±0.246 MeV [10]) and thus we haveused the c2 method to determine all the parameters from the decay widths and scattering lengths.Decay amplitude for a0 → hNp. The mass of the hN meson can be calculated using thewell-known mixing of strange and non-strange contributions in the physical ﬁelds h and h′(958)yielding h = hNcosj +hSsinj; h′ = −hNsinj +hScosj, where hS denotes a pure ¯ ss state andj ≃ −36◦ [11]. Then we obtain mh = 716 MeV.Note that we have used the decay amplitude for the a0 →hNp decay instead of the decay widthas quoted by the PDG [10] in order to ﬁt the parameters of the model. The experimental value ofthe decay amplitude is known from the Crystal Barrel data: Aa0hp =(3330±150) MeV [12] whichfor our purposes has to be divided by cosj; the formula for the decay amplitude obtained from Eq.(2.1) isAa0hp =m2h −Z2m2a0f+g21fm2a1￿￿1−12Z2f2m2a1(h2−h3)￿(m2a0 −m2p −m2h)+Z2m2a0￿.Scattering length a00. The formula for a00 is calculated using the partial wave decomposition[13] which leads toa00 =14p￿2g21Z4 m2pm4a1￿m2r +f216(12g21 −2(h1+h2)−14h3)￿−58Z2m2s −m2pf2p−32"2g21Z2fm2pm2a1 1+m2r −f2(h1+h2+h3)/22m2a1!−Z2m2s −m2p2f#214m2p −m2s+￿g21Z2fm2pm4a1￿m2r −f22(h1+h2+h3)￿+Z2m2s −m2p2f￿2 1m2s).We are using the result a00 = 0.233±0.023 (normalised to the pion mass) in accordance withdata published by the NA48/2 collaboration [14].Scattering length a20. An analogous calculation as in the case of the scattering length a00 leadstoa20 = −14p￿14Z2m2s −m2pf2p+g21Z4 m2pm4a1￿m2r −f22(h1+h2+h3)￿4A Linear Sigma Model with Vector Mesons and Global Chiral Invariance Denis Parganlija−￿g21Z2fm2pm4a1￿m2r −f22(h1+h2+h3)￿+Z2m2s −m2p2f￿2 1m2s).The result for a20 from the NA48/2 collaboration [14] is a20 = −0.0471±0.015.3. ResultsIn order to ﬁt the relevant parameters of our model (Z, g2, h1,2, ms) to experimental data (forthe aforementioned decay widths, a0 → hp decay amplitude and scattering lengths) we have usedthe c2 method. The error for the mixing angle j is neglected in this ﬁrst case study.Our best ﬁt yields the minimal value of c2min. = 0.752516 per degree of freedom which leads to thefollowing values of parameters: Z = 1.5217, g1 = 6.59, g2 = 0.3365, h1 = −100.7, h2 = 106.045,h3 = −2.63, ms = 330 MeV.It is interesting to note that, although new parameters have been introduced in the globally invariantmodel, the values of Z = 1.5217, g1 = 6.59, and ms = 330 MeV are virtually the same as thoseobtained in the locally invariant model where the corresponding values were Z = 1.586, g1 = 6.51,and ms ≃ (315−345) MeV [3]. Note also that the value of h1 does not appear to be large-NCsuppressed, although the parameter h1/2 is the prefactor to a term consisting of a product of twotraces Tr(F†F)Tr[(Lm)2+(Rm)2] - in fact, the modulus of the corresponding prefactor h1/2 is byabout a factor of ten larger than the prefactor to the term Tr(FRmF†Lm) (i.e., 2h3 = −5.26).Using the parameters above leads to the following consequences: (i) Ga1→sp = 90.163 MeV;(ii) given that in the globally invariant model the r mass term is m2r = m21 +f2(h1 +h2+h3)/2,it is possible to calculate the contribution of the bare mass (m21) to the total mass m2r - the resultm1 ≃ 758 MeV is obtained, leading to a very small contribution of the quark condensate to the rmass; (iii) the s → pp decay width has a value of less than 10 MeV - it is thus too small - and thea1 → rp decay width has the value of 1.4 GeV - it is thus too large.Hence, in the light of our results we conclude that the ¯ qq assignment of the light scalar mesonsleads to contradictions to experiment. For a deﬁnite conclusion, the errors of the parameters in themodel should be evaluated (see Ref. [6]), but it is already clear from our current results that theassignment of f0(600) and a0(980) as ¯ qq states may be problematic.A possible way to resolve the aforementioned problem is to redeﬁne s and a0 mesons in themodel as f0(1370) and a0(1450), respectively, and hence assign the scalar meson states to theenergy region above 1 GeV [6]. Then, the mixing of quarkonia and tetraquark states [16] needs tobe examined.4. Conclusions and OutlookA globally invariant linear sigma model with vector and axial vector mesons and its conse-quences for low-energy meson decay channels and pion-pion scattering lengths have been pre-sented. Results obtained in the assignment in which scalar mesons are identiﬁed as states under1 GeV indicate contradictions to experimental data, hence raising questions about the justiﬁcationof the mentioned assignment. Thus, a detailed study of the other possible assignment for scalarmesons (in which those states are located in the energy region above 1 GeV) is necessary. In the5A Linear Sigma Model with Vector Mesons and Global Chiral Invariance Denis Parganlijafuture, other relevant issues in connection with vacuum phenomenology will be addressed such asthe inclusion of the nucleon ﬁeld together with its chiral partner [15] as well as extending the workof Refs. [9, 16] to consider chiral symmetry restoration at nonzero temperature.AcknowledgmentsThe authors thank H. Leutwyler and S. Strüber for valuable discussions during the preparationof this work.References[1] M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002) [arXiv:hep-ph/0102260].[2] S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. 41, 531 (1969); P. Ko and S. Rudaz, Phys. Rev. D50, 6877 (1994)[3] D. Parganlija, F. Giacosa and D. H. Rischke, AIP Conf. Proc. 1030, 160 (2008) [arXiv:0804.3949[hep-ph]].[4] C. Amsler and F. E. Close, Phys. Lett. B 353, 385 (1995) [arXiv:hep-ph/9505219];W. J. Lee andD. Weingarten, Phys. Rev. D 61, 014015 (2000) [arXiv:hep-lat/9910008];F. E. Close and A. Kirk,Eur. Phys. J. C 21, 531 (2001) [arXiv:hep-ph/0103173];F. Giacosa, T. Gutsche, V. E. Lyubovitskijand A. Faessler, Phys. Rev. D 72, 094006 (2005) [arXiv:hep-ph/0509247];F. Giacosa, T. Gutsche,V. E. Lyubovitskij and A. Faessler, Phys. Lett. B 622, 277 (2005) [arXiv:hep-ph/0504033].[5] R. L. Jaffe, Phys. Rev. D 15, 267 (1977); R. L. Jaffe, Phys. Rev. D 15, 281 (1977); L. Maiani,F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. Lett. 93, 212002 (2004) [arXiv:hep-ph/0407017];A. H. Fariborz, R. Jora and J. Schechter, Phys. Rev. D 72, 034001 (2005) [arXiv:hep-ph/0506170];F. Giacosa, Phys. Rev. D 74, 014028 (2006) [arXiv:hep-ph/0605191];A. H. Fariborz, R. Jora andJ. Schechter, Phys. Rev. D 76, 114001 (2007) [arXiv:0708.3402[hep-ph]]; F. Giacosa, Phys. Rev. D75, 054007 (2007) [arXiv:hep-ph/0611388].[6] D. Parganlija, F. Giacosa and D. H. Rischke, in preparation.[7] J. Boguta, Phys. Lett. B 120, 34 (1983); O. Kaymakcalan and J. Schechter, Phys. Rev. D 31, 1109(1985); R. D. Pisarski, arXiv:hep-ph/9503330.[8] G. ’t Hooft, Phys. Rept. 142, 357 (1986).[9] S. Strüber and D. H. Rischke, Phys. Rev. D 77, 085004 (2008) [arXiv:0708.2389[hep-th]].[10] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 1 (2008)[11] F. Giacosa, arXiv:0712.0186[hep-ph].[12] D. V. Bugg, V. V. Anisovich, A. Sarantsev and B. S. Zou, Phys. Rev. D 50, 4412 (1994).[13] B. Ananthanarayan,G. Colangelo, J. Gasser and H. Leutwyler, Phys. Rept. 353, 207 (2001)[arXiv:hep-ph/0005297].[14] J. R. Batley et al. [NA48/2 Collaboration], Eur. Phys. J. C 54, 411 (2008).[15] S. Wilms, F. Giacosa and D. H. Rischke, arXiv:nucl-th/0702076.[16] A. Heinz, S. Strüber, F. Giacosa and D. H. Rischke, arXiv:0805.1134[hep-ph].6

Decay widths of resonances and pion scattering lengths in a globally invariant sigma model with vector and axial-vector mesons

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