We review two approaches to studying pseudoscalar meson-meson scattering
amplitudes to beyond 1 GeV using non-linear and linear chiral Lagrangians.
These approaches use two different unitarisation techniques - a generalised
Breit Wigner prescription and K-matrix unitarization respectively. We also
report some preliminary findings on K-matrix unitarisation of the I=J=0
pion-pion scattering amplitude in the non-linear chiral Lagrangian approach and
make some remarks about the light scalar mesons.Comment: Contribution to XII Conference on Hadron Spectroscopy, Frascati 2007.
  8 pages, 2 figure

Black, Deirdre

Gaunt, Jonathan R

English

arXiv

Apollo (Cambridge)

Frascati Physics Series Vol. XLVI (2007), pp. 000-000HADRON07: XII Int. Conf. on Hadron Spectroscopy – Frascati, October 8-13, 2007Parallel SessionStudy of scalar mesons in chiral Lagrangian frameworksDeirdre Black and Jonathan GauntDepartment of Physics, Cavendish Laboratory, Cambridge CB3 OHE, UKAbstractWe review two previous approaches to studying pseudoscalar meson-mesonscattering amplitudes to beyond 1 GeV using non-linear and linear chiral La-grangians. In these approaches we use two different unitarisation techniques- a generalised Breit Wigner prescription and K-matrix unitarization respec-tively. We also report some preliminary findings on K-matrix unitarisation ofthe I=J=0 pipi scattering in a non-linear chiral Lagrangian approach and makesome remarks about the light scalar mesons.1 IntroductionPseudoscalar meson-meson scattering up to the 1-2GeV energy range is ofinterest for several related reasons. On the one hand this region is beyond thatwhere chiral perturbation theory has traditionally been applied and below thatwhere we can use perturbative QCD, so it is a challenge to develop a frameworkto calculate these amplitudes from first principles. At the same time thereare many resonances in this region, some of which are controversial from thepoint of view of establishing their properties experimentally and their quarksubstructure. In particular, the scalar mesons are a long-standing puzzle inmeson spectroscopy because, for example, there are too many states to fit intoa single SU(3) nonet and the masses and decay patterns of some of the scalarresonances are not what would would expect for quark-antiquark scalar states.This talk is based on approaches developed by the Syracuse group. Many otherinteresting approaches are given in the proceedings of this conference and alsocited in the references given in the bibliography.2 Non-linear chiral Lagrangian approach to meson-meson scatter-ingWe begin 1), 2) with the conventional chiral Lagrangian, including only pseu-doscalars:L1 = −F2pi8Tr(∂µU∂µU†)+ Tr [B (U + U †)] , (1)in which U = e2iφFpi , with φ the 3 × 3 matrix of pseudoscalar fields and Fpi =132 MeV the pion decay constant. B is a diagonal matrix (B1, B1, B3) withB1 = m2piF2pi/8 = B2 and B3 = F2pi (m2K −m2pi/2)/4.We add a nonet of scalar mesons, which transform like external fields un-der chiral transformations. It turns out 3) that the trilinear scalar-pseudoscalar-pseudoscalar interaction that follows from the general chiral invariant extensionof L1 to include a scalar meson nonet is given byLNφφ = AabcdefNda∂µφeb∂µφfc + BTr (N) Tr (∂µφ∂µφ)+ CTr (N∂µφ) Tr (∂µφ) + DTr (N) Tr (∂µφ) Tr (∂µφ) (2)The first term of (2) may be eliminated in favor of the more standard formTr (N∂µφ∂µφ), but is interesting because it is the OZI rule conserving term fora dual diquark-antidiquark type nonet mentioned below.The scalar particles with non-trivial quantum numbers are given by:N = N11 a+0 κ+a−0 N22 κ0κ− κ¯0 N33 (3)with a00 = (N11 − N22 )/√2. There are two iso-singlet states: the combination(N11 + N22 + N33 )/√3 is an SU(3) singlet while (N11 + N22 − 2N33 )/√6 belongsto an SU(3) octet. These will in general mix with each other when SU(3) isbroken. We can write the general mass term 3)Lmass = −aTr(NN)− bTr(NNM)− cTr(N)Tr(N)− dTr(N)Tr(NM), (4)where a, b, c and d are real constants. M is the “spurion matrix” M =diag(1, 1, x) , x being the ratio of strange to non-strange quark masses in theusual interpretation.We take a convention where the physical particles, σ and f0, which diag-onalize the mass matrix obtained from (4) are related to the basis states N 33and (N11 + N22 )/√2 by(σf0)=(cosθs −sinθssinθs cosθs)(N33N11+N22√2), (5)For a given set of inputs for the masses of the four scalar mesons σ, f0(980),a0(980) and κ the constants a, b, c and d are fixed and there are two possiblesolutions for the mixing angle θs.We note that there are different possibilities, in addition to quark-antiquarkconfigurations, for the underlying quark substructure of N which all give riseto the same SU(3) transformation properties. For example, forming diquarkobjectsTa = abcq¯bq¯c, T¯ a = abcqbqc, (6)where the antisymmeterisation of the quark fields is implicit, we can form apure tetraquark scalar nonet as follows:N ba ∼ TaT¯ b ∼ s¯d¯ds s¯d¯us s¯d¯uds¯u¯ds s¯u¯us s¯u¯udu¯d¯ds u¯d¯us u¯d¯ud (7)or construct linear combinations of qq¯ and qqq¯q¯ nonets. We studied s-wavepseudoscalar meson scattering in a framework beginning with Eqs.(1) and (2).If we begin with the tree-level scattering amplitudes, which due to chiral sym-metry give good agreement with experiment close to the scattering threshold,we find that they soon deviate from the experimental data. They also violateunitarity. The approach that we took was to add an imaginary piece by handto the tree-level propogator of the s-channel resonance. For pi K scattering wecalled the lightest strange scalar resonance κ and made the substitution:m2κ − s −→ m2κ − s− imκG′κ (8)in the denominator of the s-channel s-wave amplitude. In order to fit to ex-periment the quantity G′κ was left as a free parameter, not necessarily equal tothe perturbative width, Gκ say. Our fit2), shown in Fig. 1 gaveGκG′κ= 0.13showing a substantial deviation from a Breit-Wigner resonance for which thisratio would be exactly equal to 1. Good agreement with experiment was alsofound 1) with this generalised Breit-Wigner prescription for the case of pipi scat-tering. The other fitting parameters are the scalar-pseudoscalar-pseudoscalarcoupling constants, which can all be written in terms of the four coefficients inthe interaction terms in Eq. (2), the scalar meson masses and mixing angle.We note also that, in addition to neatly explaining the mass orderingand general pattern of decays of the scalar states below 1 GeV, a multiquarkinterpretation for these states is also suggested by the value of the scalar mesonoctet-singlet mixing angle θs defined in Eq. (5), which was a parameter fixedby our fits. Our best fit was about −20o which, in our mixing convention,would be close to ideal mixing for a “dual” diquark-antidiquark nonet.3 Pseudoscalar meson-meson scattering in Linear Sigma ModelsIn the three flavor linear sigma model the pseudoscalar and scalar mesonsappear together since the model is constructed from the 3× 3 matrix fieldM = S + iφ, (9)where S = S† represents a scalar nonet and φ = φ† a pseudoscalar nonet.Under a chiral transformation qL → ULqL, qR → URqR of the fundamental leftand right handed light quark fields, M is defined to transform asM −→ ULMU †R. (10)We considered a general non-renormalizable Lagrangian1 of the formL = −12Tr (∂µφ∂µφ)− 12Tr (∂µS∂µS)− V0 − VSB , (11)1See 4) and references therein for more detailwhere V0 is an arbitrary function of the independent SU(3)L×SU(3)R×U(1)Vinvariants Tr(MM †), Tr(MM †MM †), Tr((MM †)3)6(detM + detM †). Ofthese, only I4 is not invariant under U(1)A. The symmetry breaker VSB hasthe minimal formVSB = −2(A1S11 + A2S22 + A3S33), (12)where the Aa are real numbers which turn out to be proportional to the threelight (“current” type) quark masses. In this model there are many constraintsamong the parameters. For example, many of the trilinear scalar-pseudoscalar-pseudoscalar coupling constants are predicted in terms of the pseudoscalar andscalar meson masses. Another difference is that [compare with Eq. (2)] thistrilinear interaction does not involve derivatives. Both models give the “currentalgebra” results in the limit where the scalar mesons are integrated out.If we calculate the tree level s-wave amplitudes they deviate from ex-periment and also violate unitarity as we go beyond the threshold region. Weused 4) the well-known K-matrix procedure to unitarise the linear sigma modelamplitudes and then checked if the resulting unitary amplitudes can give agood fit to data. In the standard parameterization 6) of a given partial waveS-matrix:S =1 + iK1− iK ≡ 1 + 2iT, (13)we identifyK = Ttree. (14)Ttree is the given partial wave T-matrix computed at tree level in the LinearSigma Model and is purely real. This scheme gives exact unitarity for T butviolates the crossing symmetry which Ttree itself obeys. In Fig. 2 we showour best fit to the I=J=0 pipi scattering data. The parameters in this fit arethe “bare” masses of the two I=0 scalar mesons in M and their mixing angle.Using these parameters we can solve for the poles in the unitarised amplitudein the complex s plane. Labelling these poles zσ and zσ′ we can identify thephysical masses and widths as usual from the Real and Imaginary parts, forexample zσ = m2σ − imσΓσ .4 Summary and comparison between modelsWe have found good agreement with scattering data in the approaches basedon the non-linear chiral Lagrangians outlined in Sections 2 and 3. We are0.6 0.8 1.0 1.2 1.4 1.6√s (GeV)−0.6−0.4−0.20.00.20.4R01/2ExperimentTheoryFigure 1: Best fit to the experimental data 5) for the Real part of the I= 12,J=0, pi-K scattering amplitude in our non-linear chiral Lagrangian model 2)with generalised Breit-Wigner prescription.0.2 0.4 0.6 0.8 1 1.2 1.4s−0.6−0.4−0.25.5511e−170.20.40.6R0 0Figure 2: Best fit to the experimental data 7) for the Real part of the I=J=0 pipiscattering amplitude in a linear sigma model with K-matrix unitarisation 4).Table 1: Results for physical I=0 scalar meson parameters from fits to pipiscattering. Comparison between linear and non-linear chiral Lagrangian modelsusing K-matrix and Generalised Breit-Wigner approaches to unitarisation.Scalar Non-linear chiral SU(3) Linear Non-linear chiralparameters approach Sigma Model 4) approach 1)(MeV) K-matrix K-matrix Generalised B-W(preliminary) without/with ρ mesonmσ 444 457 378/559Γσ 604 632 836/370mf 986 993 987Γf 52 51 65currently studying scattering using the non-linear chiral Lagrangian approachoutlined of Section 2, but employing the K-matrix unitarisation as describedin Section 3. This may make it easier to compare the linear and non-linearchiral Lagrangian models more directly and to understand the effects of theunitarisation prescriptions in themselves. This was partly motivated by ourwork on extending the non-linear chiral Lagrangian approach to include vec-tor mesons 8). This enabled us to study the interesting rare radiative decayprocesses φ → pipiγ and φ → piηγ. We found that the shape of the partialbranching fraction depends quite sensitively on whether we use derivative ornon-derivative scalar-pseudoscalar-pseudoscalar coupling as in Section 2 or 3respectively.A summary of our results is shown in Table 1 for the case of pipi scattering.In the third and fourth columns we show the results of the analyses describedin sections 3 and 2 respectively. In the fourth column we give the results withand without the inclusion of the ρ vector meson. In column 2 we show theresults of our current analysis, which are preliminary. However we can seesome trends, namely that the f0(980) parameters are quite stable, whereasthe σ parameters seem to depend more on the model and, even more, onthe unitarisation procedure. These results are preliminary because we haveonly done a fit of pipi scattering data over a limited energy range. Also wehave not included the inelastic channel and so the important KK¯ thresholdregion. These and a similar study of related scattering channels are interestingdirections for future work.5 AcknowledgementsDB would like to thank the conference organisers for a friendly and stimulatingmeeting. The work in much of this presentation is part of an ongoing collabo-ration with Abdou Abdel-Rehim, Amir Fariborz, Masayasu Harada and JosephSchechter. DB is supported by the Royal Society and JG was supported by aSummer Research Studentship from Trinity College, Cambridge.References1. a) M. Harada, F. Sannino and J. Schechter, Phys. Rev. D. 52 1991 (1996),b) M. Harada, F. Sannino and J. Schechter, Phys. Rev. Lett. 78, 1603(1997)2. D. Black, A.H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D 58054012 (1998)3. D. Black, A.H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D 59,074026 (1999)4. D. Black et al, Phys. Rev. D 64 014031 (2001).5. D. Aston et al, Nucl. Phys. B296, 493 (1988).6. S.U. Chung et al, Ann PHys.4 404 (1995).7. E.A. Alekseeva et al, Sov. Phys. JETP 55, 591 (1982), G. Grayer et al,Nucle Physc. B75, 189 (1974).8. D. Black, M. Harada and J. Schechter, Phys. Rev. Lett. 88 181603 (2002),D. Black, M. Harada and J. Schechter, Phys. Rev. D 73 054017 (2006).

Study of scalar mesons in chiral Lagrangian frameworks

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