1. Departamento de Física Teórica II. Universidad Complutense de Madrid J. Ruiz de Elvira in collaboration with R. García Martín R. Kaminski Jose R. Peláez F. J. Yndurain. Precise dispersive analysis of the f0(600) and f0(980) resonances from pion-pion scattering.

2. Motivation: The f 0 (600) and f0(980) I=0, J=0  exchanges are very important for nucleon-nucleon attraction Scalar multiplet identification still controversial Chiral symmetry breaking. Vacuum quantum numbers.

3. It is model independent. Just analyticity and crossing properties Motivation: Why a dispersive approach? Determine the amplitude at a given energy even if there were no data precisely at that energy. Relate different processes Increase the precision The actual parametrization of the data is irrelevant once it is used in the integral. A precise  scattering analysis can help determining the  and f0(980) parameters

4. Roy Eqs. vs. Forward Dispersion Relations FORWARD DISPERSION RELATIONS (FDRs). (Kaminski, Pelaez and Yndurain) One equation per amplitude. Positivity in the integrand contributions, good for precision. Calculated up to 1400 MeV One subtraction for F00 and F0+ FDR No subtraction for the It=1FDR. ROY EQS (1972) (Roy, M. Pennington, Caprini et al., Ananthanarayan et al. Gasser et al.,Stern et al., Kaminski. Pelaez,,Yndurain). Coupled equations for all partial waves. Twice substracted. Limited to ~ 1.1 GeV. Good at low energies, interesting for ChPT. When combined with ChPT precise for f0(600) pole determinations. (Caprini et al) But we here do NOT use ChPT, our results are just a data analysis They both cover the complete isospin basis

5. NEW ROY-LIKE EQS. WITH ONE SUBTRACTION, (GKPY EQS) When S.M.Roy derived his equations he used. TWO SUBTRACTIONS. Very good for low energy region: In fixed-t dispersion relations at high energies : if symmetric the u and s cut growth cancels. if antisymmetric dominated by rho exchange. ONE SUBTRACTION also allowed GKPY Eqs R. Garcia Martin, R. Kaminski, J.R.Pelaez, F.J. Yndurain Int.J.Mod.Phys.A24, AIP Conf.Proc.1030, Int.J.Mod.Phys.A24, Nucl.Phys.Proc.Suppl.186 Already introduced here in Montpellier in QCD 08 But no need for it!

6. UNCERTAINTIES IN Standard ROY EQS. vs 1s Roy like GKPY Eqs Garcia Martin, R. Kaminski, J.R.Pelaez, F.J. Yndurain smaller uncertainty below ~ 400 MeVsmaller uncertainty above ~400 MeV Why are GKPY Eq. relevant? One subtraction yields better accuracy in √s > 400 MeV region Roy Eqs.GKPY Eqs,

7. OUR AIM Precise DETERMINATION of f 0 (600) and f0(980) pole FROM DATA ANALYSIS Use of Roy and GKPY dispersion relations for the analytic continuation to the complex plane. (Model independent approach) Use of dispersion relations to constrain the data fits (CFD) Complete isospin set of Forward Dispersion Relations up to 1420 MeV Up to F waves included Standard Roy Eqs up to 1100 MeV, for S0, P and S2 waves Once-subtracted Roy like Eqs (GKPY) up to 1100 MeV for S0, P and S2 We do not use the ChPT predictions. Our result is independent of ChPT results.

8. The fits 1)Unconstrained data fits (UDF) Independent and simple fits to data in different channels. All waves uncorrelated. Easy to change or add new data when available R. Garcia-Martin, R. Kaminski, J.R Pelaez, F. Yndurain Int.J.Mod.Phys.A24, AIP Conf.Proc.1030, Int.J.Mod.Phys.A24, Nucl.Phys.Proc.Suppl.186 This is our starting point. We use for all the waves previous fits except for the S0 in the f(980) region that we improve here.

9. We START by parametrizing the data To avoid model dependences we only require analyticity and unitarity We use an effective range formalism: s 0 =1450 +a conformal expansion If needed we explicitly factorize a value where f(s) is imaginary or has a zero: For the integrals any data parametrization could do. We use something SIMPLE at low energies (usually <850 MeV)

10. S0 wave below 850 MeV R. Garcia Martin, JR.Pelaez and F.J. Ynduráin PR D74:014001,2006 Conformal expansion, three terms are enough. First, Adler zero at m  2 /2 We use data on Kl4 including the NEWEST: NA48/2 results Get rid of K → 2  Isospin corrections from Gasser to NA48/2 Average of  N->  N data sets with enlarged errors, at 870- 970 MeV, where they are consistent within 10 o to 15 o error. Note that it is just used in the real axis for physical s

11. S0 wave above 850 MeV R. Kaminski, J.R.Pelaez and F.J. Ynduráin PR D74:014001,2006 CERN-Munich phases with and without polarized beams Inelasticity from several   ,   KK experiments We have updated the S0 wave using a polynomial fit to improve: the intermediate matching between parametrizations (continuous derivative). the flexibility of the f0(980) region.

12. NEW:S0 wave with improved matching

13. Similar Initial UNconstrained FIts for all other waves and High energies R. Kaminski, J.R.Pelaez, F.J. Ynduráin. Phys. Rev. D77:054015,2008. Eur.Phys.J.A31:479-484,2007, PRD74:014001,2006 J.R.Pelaez, F.J. Ynduráin. PRD71, 074016 (2005), From older works:

14. Similar Initial UNconstrained FIts for all other waves and High energies J.R.Pelaez, F.J. Ynduráin. PRD69,114001 (2004) From older works:

15. The fits 1)Unconstrained data fits (UDF) Independent and simple fits to data in different channels. All waves uncorrelated. Easy to change or add new data when available Check of FDR’s Roy and other sum rules.

16. How well the Dispersion Relations are satisfied by unconstrained fits We define an averaged  2 over these points, that we call d 2 For each 25 MeV we look at the difference between both sides of the FDR, Roy or GKPY that should be ZERO within errors. d 2 close to 1 means that the relation is well satisfied d 2 >> 1 means the data set is inconsistent with the relation. There are 3 independent FDR’s, 3 Roy Eqs and 3 GKPY Eqs. This is NOT a fit to the relation, just a check of the fits!!.

17. Forward Dispersion Relations for UNCONSTRAINED fits FDRs averaged d 2  0  0 0.52 1.84  0  + 1.02 1.11 I t =1 0.89 2.50 <932MeV <1400MeV NOT GOOD! In the intermediate region. Need improvement

18. Roy Eqs. for UNCONSTRAINED fits Roy Eqs. averaged d 2 GOOD!. But room for improvement S0wave 0.80 0.70 P wave 0.64 0.56 S2 wave 1.22 1.23 <932MeV <1100MeV

19. GKPY Eqs. for UNCONSTRAINED fits Roy Eqs. averaged d 2 PRETTY BAD!. Need improvement. S0wave 1.33 4.78!!!! P wave 2.48 2.16 S2 wave 0.59 0.56 <932MeV <1100MeV GKPY Eqs are much scricter Lots of room for improvement

20. The fits 1)Unconstrained data fits (UDF) Independent and simple fits to data in different channels. All waves uncorrelated. Easy to change or add new data when available Check of FDR’s Roy and other sum rules. Room for improvement 2) Constrained data fits (CDF)

21. Imposing FDR’s, Roy Eqs and GKPY as constraints To improve our fits, we can IMPOSE FDR’s, Roy Eqs W counts the number of effective degrees of freedom The resulting fits differ by less than ~1  -1.5  from original unconstrained fits The 3 independent FDR’s, 3 Roy Eqs + 3 GKPY Eqs very well satisfied 3 FDR’s 3 GKPY Eqs Sum Rules for crossing Parameters of the unconstrained data fits 3 Roy Eqs We obtain CONSTRAINED FITS TO DATA (CFD) by minimizing: and GKPY Eqs.

22. Forward Dispersion Relations for CONSTRAINED fits FDRs averaged d 2  0  0 0.34 0.55  0  + 0.31 0.47 I t =1 0.12 0.33 <932MeV <1400MeV GOOD!.

23. Roy Eqs. for CONSTRAINED fits Roy Eqs. averaged d 2 S0wave 0.17 0.22 P wave 0.07 0.15 S2 wave 0.28 0.32 <932MeV <1100MeV GOOD!.

24. GKPY Eqs. for CONSTRAINED fits Roy Eqs. averaged d 2 S0wave 0.45 0.50 P wave 0.85 0.79 S2 wave 0.17 0.28 <932MeV <1100MeV GOOD!.

25. Analytic continuation to the complex plane We do NOT obtain the poles directly from the constrained parametrizations, which are used only as an input for the dispersive relations. The σ and f0(980) poles are obtained from the DISPERSION RELATIONS extended to the complex plane. This is parametrization and model independent. In previous works dispersion relations well satisfied below 932 MeV Now, good description up to 1100 MeV. We can calculate in the f0(980) region. Effect of the f0(980) on the f0(600) under control.

26. Final Result: Analytic continuation to the complex plane Fairly consistent with other ChPT+dispersive results Caprini, Colangelo, Leutwyler 2006 1  overlap with Roy Eqs: GKPY Eqs: f0(600) f0(980) Results are PRELIMINARY. Still honing the uncertainties, will probably turn out slightly bigger and asymmetric From

27. The results from the GKPY Eqs. with the CONSTRAINED Data Fit input

29. Summary Simple and easy to use parametrizations fitted to  scattering DATA for S,P,D,F waves up to 1400 MeV. (Unconstrained data fits) 3 Forward Dispersion relations and the 3 Roy Eqs satisfied fairly well Simple and easy to use parametrizations fitted to  scattering DATA CONSTRAINED by FDR’s+ Roy Eqs+ 3 GKPY Eqs 3 Forward Dispersion relations and the 3 Roy Eqs and 3 GKPY Eqs satisfied remarkably well Remarkable agreement with CGL Roy Eqs+ChPT predictions for S, P waves below 450 MeV We obtain the σ and f0(980) poles from DISPERSION RELATIONS extended to the complex plane, without use ChPT. The poles obtained are fairly consistents whit previous ones.

30. Spare transparencies

31. SUM RULES J.R.Pelaez, F.J. Yndurain Phys Rev. D71 (2005) They relate high energy parameters to low energy P and D waves

32. UNCONSTRAINED vs. CONSTRAINED fits UNCONSTRAINED CONSTRAINED All waves uncorrelated. Easy to update if new data available on one channel FDRs very well below 930 MeV, fairly well up to 1400 MeV Roy Eqs. satisfied except S2, but still within 1.3 sigmas All waves correlated. Differ from Uncorrelated by less than 1 sigma Except D2 wave, that differs 1.5 sigma CONSTRAINED FITS FDRs, Roy Eqs and Sum rules satisfied remarkably well. Very reliable.

33. Our series of works: 2005-2010 Independent and simple fits to data in different channels. “Unconstrained Data Fits UDF” Check with FDR Impose FDRs and Sum Rules on data fits “Constrained Data Fits CDF” Some data sets inconsistent with FDRs All waves uncorrelated. Easy to change or add new data when available Some data fits fair agreement with FDRs Correlated fit to all waves satisfying FDRs. precise and reliable predictions. from DATA unitarity and analyticity R. Kaminski, J.R.Pelaez, F.J. Ynduráin Eur.Phys.J.A31:479-484,2007. PRD74:014001,2006 J. R. P,F.J. Ynduráin. PRD71, 074016 (2005), PRD69,114001 (2004) + Roy +GKPY Eqs + New Kl4 decay data !! Phys. Rev. D77:054015,2008 We do not include ChPT (we want to test it), we include data in the whole energy region it used to be called an ENERGY DEPENDENT DATA ANALYSIS

34. The S0 wave. Different sets The fits to different sets follow two behaviors compared with that to Kl4 data only Those close to the pure Kl4 fit display a "shoulder" in the 500 to 800 MeV region These are: pure Kl4, SolutionC and the global fits Other fits do not have the shoulder and are separated from pure Kl4 Kaminski et al. lies in between with huge errors Solution E deviates strongly from the rest but has huge error bars Note size of uncertainty in data at 800 MeV!!