Departamento de Física Teórica II. Universidad Complutense de Madrid José R. Peláez ON THE NATURE OF THE LIGHT SCALAR NONET FROM UNITARIZED CHIRAL PERTURBATION THEORY AND LARGE N C Phys.Rev.Lett.92:102001,2004

OUTLOOK OVERVIEW of Unitarization of ChPT amplitudes: Complete one-loop meson-meson scattering Nature of resonances: Link with QCD through the large Nc limit Some light on the hadron spectroscopy has come from unitarized Chiral Lagrangians The paradigmatic example is meson-meson scattering and unitarized Chiral Perturbation Theory Poles in the amplitudes with the complete calculation. Multiplets.

OUTLOOK INTRODUCTION: Unitarization of ChPT amplitudes: Complete one-loop meson meson scattering

Chiral Perturbation Theory Weinberg, Gasser & Leutwyler , K,  Goldstone Bosons of the spontaneous chiral symmetry breaking SU(N f ) V  SU(N f ) A  SU(N f ) V QCD degrees of freedom at low energies  4  f  1.2 GeV  ChPT  The most general expansion in energies compatible with the QCD symmetry breaking Leading order parameters: breaking scale f 0 and masses At 1-loop, QCD dynamics encoded in measurable chiral parameters: L 1...L 8 also their large N c behavior known from QCD Limited to low energies

Partial wave unitarity (On the real axis above threshold) exactly unitary !! ChPT = series in p 2 perturbative unitarity provides.... If t 4 defined to satisfy IAM exact Unitarity and the Inverse Amplitude Method: One channel

Truong, Dobado, Herrero, Pelaez Unitarity + Chiral Low energy expansion Justified within dispersion Theory (But left cut approximated !!) Dynamically Generates Poles  Resonances: , K*,  The Inverse Amplitude Method: Results for one channel   (770) K * (890) Width/2 Mass

Partial wave unitarity (On the real axis above all thresholds) ChPT = series in p 2 perturbative unitarity provides.... exactly unitary !! exact If T 4 defined to satisfy Coupled channel IAM Oller, Oset, Peláez, PRL80(1998)3452, PRD59,(1999) Unitarity and the Inverse Amplitude Method: Multiple channels

  SU(2): Gasser & Leutwyler, Ann. Phys.158,142 (1984) SU(3): Bernard,Kaiser,Mei  ner, Nuc.Phys B357 (1991)129 K  K  : Bernard,Kaiser,Mei  ner, Phys.Rev. D43 (1991) 2757 Bernard,Kaiser,Mei  ner, Phys.Rev. D44 (1991) 2398   K  K  : K  K     8 Amplitudes +chiral symmetry +crossing Complete meson-meson scattering below 1.2 GeV Guerrero, Oller, Nuc.Phys.B537 (1999) 459 K + K -  K + K - K + K -  K 0 K 0 Gómez-Nicola, Peláez Phys. Rev. D65:054009, (2002) this 3 new amplitudes at one-loop in MS-1 scheme + All the others recalculated to satisfy “exact” perturbative unitarity Meson-meson Scattering at one loop in Chiral Perturbation Theory

IAM fit to meson-meson scattering (+3% syst.)  f 0 (980) a0a0  (900) K * (892)  low-energy  00 -  11   00 d   /dE lab   20      3/2 0 K  K   1/2 1 K  K   1/2 0   11    00

First, IAM with previous ChPT parameters (we could not compare before)  resonant features Incompatible sets of Data. Customarily add systematic error: MINUIT fit : 3%, Final error: MINUIT error combined with syst. error Identical curves but variation in parameters 1%, 5% IAM fit parameters compatible with ChPT !! Complete Meson-meson Scattering in Unitarized Chiral Perturbation Theory

OUTLOOK INTRODUCTION: Unitarization of ChPT amplitudes: generation of resonances Complete one-loop meson-meson scattering Poles in the amplitudes with the complete calculation. Multiplets. Simultaneous description of low energy and resonances Fully renormalized and with parameters compatible with ChPT.

Coupled channel IAM : A. Gomez Nicola and J.R.P. PRL80(1998)3452, PRD59,(1999)  and f 0  a0a0 K*  (octet)  With the full one-loop SU(3) unitarized ChPT, we GENERATE, the following resonances, not present in the ChPT Lagrangian, as poles in the second Rieamann sheet J.R.Pelaez, hep-ph/ AIP Conf.Proc.660: ,2003 Brief review: Mod.Phys.Lett.A19: ,2004

a0a0    f0f0 Re  s  M (MeV)-Im  s  /2 (MeV) 754       Resonance POLE POSITIONS... Complete IAM Complete IAM cusp? K* 889   4 Light scalar nonet

OUTLOOK INTRODUCTION Unitarization of ChPT amplitudes: Complete one-loop meson meson scattering Nature of resonances: Link with QCD through the large N c limit Poles in the amplitudes with the complete calculation. Multiplets. Scalar poles found with complete and fully renormalized UNITARIZED amplitudes with standard ChPT parameters

Large N c Limit But the QCD L i large Nc SCALING, is known and Model Independent We do not know how to obtain the L i from QCD in a model independent way Pions, kaons and etas states: The qqbar meson masses M=O(1) and their decay constants f=O(  N c )

M N /M 3 N/3N/3 NcNc N/3N/3 NcNc VECTOR MESONS qqbar states: The  (770) The K*(892) M N /M 3 N/3N/3

Subtlety: The large N c does not see the scale of L i (  ) but is expected to correspond to  MeV Since now we use the completely renormalized expresions of ChPT we can choose any  for the large N c scaling and we get the uncertainty NcNc Resonance poles in the Large N Limit

The IAM generates the expected large N c behavior of established qq states What about the  and the  ? NcNc M N /M 3 N/3N/3 The  (  =770MeV) The  (  =500MeV) NcNc M N /M 3 N/3N/3

Large dependence on  choice for the large Nc scaling The  and the  pole movement is NOT like that of qq states Resonance poles in the Large N Limit

Another way of seeing it is with the modulus of the amplitude Resonance poles in the Large N Limit N c =3 N c =5 N c =10

What about the f 0 and the a 0 ? Following the pole large Nc behavior is complicated by the nearby thresholds, at least for relatively low Nc. In addition, the a 0 (980) amplitude on the real axis is more robust than the pole position For sufficienly high Nc they also disappear in the continuum like the  and  N c =3 N c =5 N c =10 N c =25  =770MeV

What are they? Resonance poles in the Large N Limit Possibility: Some “four quark” or “two meson” states are predicted to become the Meson- meson continuum in the large N c (R. Jaffe) The main component of the  and the , does not behave as qqbar state in the large Nc limit. qqbar resonance Im t  O(1) at peak cannot be present for  and  4 quark state or glueball Im t  O(1/N c 2 ) qqbar resonance t channel contributes to Re t  O(1/Nc) but Im t = 0 For the  and f 0 glueball interpretation also possible. Likely some mixing. Not for  or a 0 four quark=two meson =glueball ( if I=0 J=0 ) in this counting

The imaginary part of amplitudes vanish consistently with Imt~1/N c 2 in one corner of  space, the a 0 (980) still behaves as qqbar  =0.55 to 1 GeV a 0 (980): for almost all   =0.5 to 1 GeV f 0 (980): for all   =0.50 to 0.55 GeV

Complete one loop ChPT meson-meson scattering With unitarized meson-meson scattering amplitudes: Good fit with errors, simultaneously of resonances and low-energy Fit parameters compatible with previous ChPT values Unitarized meson-meson scattering results robust. , K *, , f 0 and a 0 poles appear simultaneously allowing for low energy description Suggests nonet interpretation for light scalars Pole positions of resonances robust. (Except a 0. but cusp disfavored) Vector meson large N c behavior is OK with qqbar nature , f 0 and a 0 large N c QCD behavior at odds with dominant qqbar nature Summary

Warning for practitioners... We use the large Nc behavior, but it is only relevant not far from reality  N c =3 Only qqbars survive at Nc= ... For sufficiently large Nc, even the tiniest admixture of qqbar could be made dominant. The mathematics could be correct, but gives no information about the DOMINANT component of physical states at N c =3. Corner of parameter space  =1000 MeV  ( H. Zheng et al. hep-ph/ hep-ph/ hep-ph/ )