Departamento de Física Teórica II. Universidad Complutense de Madrid

1. Departamento de Física Teórica II. Universidad Complutense de Madrid The nature of the lightest scalar meson, its Nc behavior and semi-local duality J.R. Peláez In collaboration with: J. Ruiz de Elvira, M. Pennigton and D. Wilson arXiv:1009.6204 [hep-ph]

2. Outline ●Introduction ●UChPT andthe 1/Nc expansion. ●FESR and local duality. ●Results

3. The ρ becomes narrower with Nc, as expected for a meson. Introduction and motivation Light scalars, and particularly the sigma are of interest for nuleon-nucleon attraction, glueballs, chiral symmetry breaking, Chiral Perturbation Theory etc… In general they are hard to accommodate as ordinary mesons Actually, there is mounting evidence that these states may not be ordinary quark-antiquark states Jaffe, van Beveren,, Rupp, Tornqvist, Roos, Close, Schecter, Sannino, Fariborz, Black, Oset, Oller, JRP, Hanhart, Achasov, Kalashnikova, Maiani Polosa, Riquer and many others… The scalar nonet may appear above 1 GeV Actually, NLO ChPT+ dispersion relations findsdifferent Nc behaviours JRP, Phys.Rev.Lett. 92:102001,2004, The σ becomes broader and its contribution to the amplitude decreases

4. At NNLO a subdominant component suggested for the σ around >1 GeV. (probably related to the ordinary nonet around that region ) G. Ríos and JRP Phys.Rev.Lett.97:242002,2006, Here we show that this >1 GeV subdominant component ensuresthat local duality is still satisfied. Introduction and motivation In general, non- states have DIFFERENT Nc dependence than the ρ PROBLEM: Local duality requires cancellation between the σandρ. IF SIGMA “DISAPPEARS AT LARGER Nc Possible contradiction with local duality?

5. Outline ●Introduction ●UChPT andthe 1/Nc expansion.

6. ’sGoldstone Bosons of the spontaneous chiral symmetry breaking SU(2)V SU(2)ASU(2)V QCD degrees of freedom at low energies << 4f~1 GeV , : : ChPT limited to low energies Chiral Perturbation Theory Weinberg, Gasser & Leutwyler ChPTis the low energy EFFECTIVE THEORY OF QCD most general low-energy expansion of a pion lagrangian with the QCD symmetries Leading order parameters: At higher orders, QCD dynamics encoded in Low Energy Constants determined fromexperiment ππscattering leading 1/Nc behavior known from QCD !!!

7. Elastic two-body Unitarity Constraints: One channel Partial wave UNITARITY (On the real axis above threshold) KNOWN EXACTLY (kinematics) Unitarity bound We only need the Real part of 1/t (dynamics) exactly unitary !! EXACT unitarity not satisfied by ChPT series (or any other series) Badly violated if ChPT series extrapolated to high energies or resonance region How to fix that? We can use ChPT for Re 1/t But it is better to use this info inside a dispersion relation

8. The Inverse Amplitude Method: Dispersive Derivation: THE REAL THING Define Write dispersion relations for G and t4 IAM Up to NLO ChPT Opposite to each other Subtraction Constants from ChPT expansion OK since s=0 G(0)=t2(0)-t4(0) PHYSICAL cut EXACTLY Opposite to each other All together…we find AGAIN We have just seen that, for physical s and PC is O(p6) and we neglect it or use ChPT

9. The Inverse Amplitude Method: Results for one channel Dobado, JRP ‘96 f0(600) pole: 440-i245 MeV Mass Width/2 K*(890) (770) =f0(600) Truong ‘89, Truong,Dobado,Herrero,’90, Dobado JRP,‘93,‘96 Very simple. Systematic extension to higher orders Simultaneously: Unitarity + Chiral expansion ChPT used ONLY at low energies: subtraction constants and left cut, NOT in resonance region Dispersion relation allows us to go to complex plane. Generates Poles of Resonances: f0(600) or “”, ρ(770), (800), K*(892),

10. The 1/Nc expansion The 1/Nc expansion provides a clear definition of states ChPT parameters: Leading 1/Nc behavior known and model Independent UChPT predicts 1/Nc Behavior of resonances The IAM reliable for Nc < 15 – 30 at most beyond that, just a qualitative model (since QCD weakly interacting for large Nc)

11. LIGHT VECTOR MESONS The IAM generates the expected Nc scaling of established qq states JRP, Phys.Rev.Lett. 92:102001,2004 The (770) The K*(892) MN/M3 MN/M3 MN/M3 N/3 N/3 N/3 Nc qqbar states: Nc

12. JRP, Phys.Rev.Lett. 92:102001,2004 What about scalars ? The (=770MeV) The (=500MeV) MN/M3 N/3 Nc MN/M3 N/3 Nc Similar conclusions for the f0(980) and a0(980) Complicated by the presence of THRESHOLDS and except in a corner of parameter space for the a0(980) Requires coupled channel formalism

13. Results O(p6): the sigma Near Nc = 3 similar results to those at O(p4): Robust Non qqbar dominant component M becomes constant ~ 1GeV Γ starts decreasing G. Ríos and JRPelaez, Phys.Rev.Lett.97:242002,2006 For Nc ~ 10 tor 12 Mixing? The O(p6) calculation suggests a subdominantqqbar component for the σ with a LARGER MASS ~ 2.5 Mσ ~ 1 to 1.2GeV This subdominant qqbar component can fix the duality problem of a non-qqbar interpretattion for the sigma

14. Outline ●Introduction ●UChPT andthe 1/Nc expansion. ●FESR and local duality.

15. Introduction. Local Duality Local duality implies that a large number of s-channel resonances are, “on the average“, dual to t-channel Regge exchanges. No resonances exchanged in repulsive I = 2 ππ scatterings-channel I = 2 t-channel exchange should be suppressed respect to other isospin Crossing relates t-channel I=2 amplitude to s-channel amplitudes: σ ρ T Very small The I=2 suppression requires strong σ-ρcancellation

16. Local duality & FESR “On the average-cancellation" properly defined via Finite Energy Sum Rules. Regge theory interpretation is:

17. Local duality vs. non-qqbar sigma The I=2 ππscattering s-channel remains non resonant with Nc. In t-channel suppressed respect to other isospins The Regge parameters don’t depend on Nc. (at LO) The I=2 FESR should be still suppressed for any Nc. σ - ρcancellation needed for all Nc But if σ - ρ behave differently with Nc, this cancellation does not occur!!

18. Outline ●Introduction ●UChPT andthe 1/Nc expansion. ●FESR and local duality. ●Results

19. FESR for Nc = 3. Check with real data For Nc =3, local duality is satisfied. First point: Check the FESR suppression for Nc=3 Using real data parametrizations, we have checked: Kaminski, JRP and Yndurain, PRD77:054015,2008 for t = th

20. FESR and IAM For n= 2, 3, this cancellationoccurs below 1-1.5 GeV. We can use the IAM to study local duality, but only applies for S0, P and S2 waves We calculate the FESR using the IAM and check the influence of those waves. The influence of higher waves is around 10%. The IAM predicts correctly the FESR suppression. We can use the IAM to study the FERS dependence on Nc

21. FESR and Nc. Case with vanishing σ Local duality implies a σ - ρcancellation with Nc. However, the σ and ρ mesons showa different Nc behaviour. If we take a case where the σ amplitude vanishes (typically the NLO IAM) the ρ dominates the FESR. T T Vanish with Nc SMALL Local duality spoilt at larger Nc!!

22. FESR and Nc. Case with vanishing σ At higher Nc The σ amplitude vanishes: there is no σ-ρ cancellation. Local duality fails CONFLICT WITH LOCAL DUALITY IF THE SIGMA DISAPPEARS COMPLETELY This is the expected problem FESR suppression, checked using a real parametrization.

23. But if a subleading component for the σ emerges around 1 GeV, As it happens naturally within two-loop ChPT. FESR and Nc. Case with subdominant quark-antiquark mixture There is still a cancellation between the σ and ρ amplitudes. The FESR are still suppressed with Nc Local duality is still satisfied

24. FESR and Nc. Case with subdominant quark-antiquark mixture FESR remain small with Nc. The subleading qqbar σ component at 1 GeV , ensures local duality. LOCAL DUALITY IS SATISFIED with Nc Two loop UChPT solves the problem naturally FESR suppression, checked using a real parametrization

25. Case with subdominant quark-antiquark mixture. Other states Cancellation occurs only if the subdominant state has a mass below 1.5 GeV Important: a LARGE width when reaching back the real axis around1.2 GeV (FESR are 1/sn suppressed), otherwise no cancellation Most likely this is an ordinary meson component common to other mesons in that region (J.Ruiz de Elvira, F. J. llanes Estrada, JRP in preparation) OTHER mesons or qqbar components in that region are not enough for the cancellation at large Nc. They have a too narrow width for larger Nc

26. Case with subdominant quark-antiquark mixture. Other states In particular f0(980) effect too small We have also added a crude model of the f2(1275). It contributes a littke to the cancellation, but not enough. The effect of the Subdominant component is larger.

27. FESR and Nc: With and without subdominant quark-antiqurk admixture No subdminant component: (typical @NLO) No FESR suppression With subdominant component (natural @NNLO) FESR suppression Local duality is satisfied. Local duality fails

28. We have even extrapolated to (too) large Nc. The cancellation continues The suppresion continues. It is an stable efffect But IAM reliable for Nc < 15 – 30 at most beyond that, just a qualitative model (since QCD weakly interacting for large Nc)

29. Actually, the 1/Nc expansion within UChPT shows that the σ meson is not predominantly a state, while genrating the correct ρ dependence. The σ 1/Nc behavior is predominantly that of a non ordinary meson, but a subdominant component with the 1/Nc behavior naturally suggested by two-loop unitarized ChPT ensuresthat local duality is still satisfied. Summary Light scalars and particularly the σ seem likely non ordinary quark-antiquark mesons All non-qqbar scenarios where the σ completely disapperars from the spectrum (typical @NLO-UChPT, pure tetraquark, pure molecule, etc…), suffer CONFLICT WITH LOCAL DUALITY