Precise Dispersive Analysis of f0(600) and f0(980) Resonances

1. Departamento de Física Teórica II. Universidad Complutense de Madrid Precise dispersive analysis of the f0(600) and f0(980) resonances J. R. Peláez R. García Martín, R. Kaminski, JRP, J. Ruiz de Elvira, Phys.Rev. Lett. 107, 072001 (2011) R. García Martín, R. Kaminski, JRP, J. Ruiz de Elvira,F. J. Yndurain. PRD83,074004 (2011)

2. A precise  scattering analysiscan help determining the  and f0(980) parameters Motivation: Why a dispersive approach? It is model independent. Just analyticity and crossing properties 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 isirrelevant once itisusedinsidethe integral.

3. Roy Eqs. vs. Forward Dispersion Relations They both cover the complete isospin basis FORWARD DISPERSION RELATIONS (FDRs). (Kaminski, Pelaez and Yndurain) Oneequation per amplitude. Positivity in theintegrandcontributions, goodforprecision. Calculated up to 1400 MeV OnesubtractionforF0+ 0+0+, F00 00 00 No subtractionfortheIt=1FDR.

4. Roy Eqs. vs. Forward Dispersion Relations They both cover the complete isospin basis FORWARD DISPERSION RELATIONS (FDRs). (Kaminski, Pelaez and Yndurain) Oneequation per amplitude. Positivity in theintegrandcontributions, goodforprecision. Calculated up to 1400 MeV OnesubtractionforF0+ 0+0+, F00 00 00 No subtractionfortheIt=1FDR. ROY EQS (1972) (Roy, M. Pennington, Caprini et al. , Ananthanarayan et al. Gasser et al.,Stern et al. , Kaminski . Pelaez,,Yndurain). Coupledequationsforallpartialwaves. Twicesubstracted. Limitedto~ 1.1 GeV. Good at lowenergies, interestingforChPT. Whencombinedwith ChPT precise for f0(600) pole determinations. (Caprini et al) Butwehere do NOT use ChPT,ourresults are just a DATA analysis

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: But no need for it! In fixed-t dispersion relations at high energies: if symmetric the u and s cut (Pomeron) growth cancels. if antisymmetric dominated by rho exchange (softer). ONE SUBTRACTION also allowed GKPY Eqs.

6. Structure of calculation: Example Roy and GKPY Eqs. Both are coupled channel equations for the infinite partial waves: I=isospin 0,1,2 , l =angular momentum 1,2,3…. DRIVING TERMS (truncation) Higher waves and High energy SUBTRACTION TERMS (polynomials) KERNEL TERMS known Partial wave on real axis ROY: 2nd order More energy suppressed Very small GKPY: 1st order Less energy suppressed small Similar Procedure forFDRs “OUT” “IN (from our data parametrizations)” =?

7. UNCERTAINTIES IN Standard ROY EQS. vs GKPY Eqs Why are GKPY Eqs. relevant? One subtraction yields better accuracy in √s > 400 MeV region Roy Eqs. GKPY Eqs, smaller uncertainty below ~ 400 MeV smaller uncertainty above ~400 MeV

8. Our series of works: 2005-2011 Check Dispersion Relations ImposeFDRs, Roy & GKPY Eqs ondata fits “Constrained Data Fits CDF” Describe data and are consistentwithDispersionrelations R. Kaminski, JRP, 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), R. García Martín, R. Kaminski, JRP, J. Ruiz de Elvira, F.J. Yduráin. PRD83,074004 (2011) Independent and simple fits to data in differentchannels. “Unconstrained Data Fits=UDF” Continuationtocomplexplane USING THE DISPERIVE INTEGRALS: resonancepoles

9. The fits • Unconstrained data fits (UDF) • All waves uncorrelated. Easy to change or add new data when available The particular choice of parametrization isalmost IRRELEVANT once insidetheintegrals we use SIMPLE and easytoimplement PARAMETRIZATIONS.

10. S0 wave below 850 MeV R. Garcia Martin, JR.Pelaez and F.J. Ynduráin PRD74:014001,2006 Conformal expansion, 4 terms are enough. First, Adler zero at m2/2 Average of N->N data sets with enlarged errors, at 870- 970 MeV, where they are consistent within 10o to 15o error. We use data on Kl4 includingthe NEWEST: NA48/2 results Getrid of K → 2 Isospin correctionsfrom GassertoNA48/2 ItdoesNOT HAVE A BREIT-WIGNER SHAPE Tinyuncertainties dueto NA48/2 data

11. S0 wave above 850 MeV R. Kaminski, J.R.Pelaez and F.J. Ynduráin PRD74:014001,2006 • Paticularcareonthe f0(980) region : • Continuous and differentiablematchingbetweenparametrizations • Above1 GeV, allsources of inelasticityincluded (consistentlywith data) • Twoscenariosstudied Inelasticity from several   ,   KK experiments CERN-Munich phases with and without polarized beams

12. S0 wave: Unconstrainedfitto data (UFD)

13. P wave THIS IS A NICE BREIT-WIGNER !! Above 1 GeV, polynomialfit to CERN-Munich & Berkeley phase and inelasticity 2/dof=1 .01 Up to 1 GeV This NOT a fitto scattering buttothe FORM FACTOR de Troconiz, Yndurain, PRD65,093001 (2002), PRD71,073008,(2005)

14. D2 and S2 waves For S2 we include an Adler zero at M Verypoor data sets Phaseshiftshouldgoto n at  - Inelasticitysmallbutfitted Elasticityabove 1.25 GeV notmeasured assumed compatible with 1 • Thelessreliable. EXPECT LARGEST CHANGE • Wehaveincreasedthesystematic error

15. D0 wave D0 DATA sets incompatible We fit f2(1250) mass and width Inelasticity fitted empirically: CERN-MUnich + Berkeley data THIS IS A NICE BREIT-WIGNER !! Matching at lowerenergies: CERN-Munich and Berkeley data (is ZERO below 800 !!) plus thresholdpsrametersfrom Froissart-GribovSum rules The F wave contributionisverysmall Errorsincreasedbyeffect of includingoneortwo incompatible data sets NEW: Ghost removed butnegligibleeffect. The G wave contributionnegligible

16. UNconstrainedFitsforHighenergies In principleanyparametrization of data is fine. Forsimplicitywe use JRP, F.J.Ynduráin. PRD69,114001 (2004) UDF fromolderworks and Reggeparametrizations of data Factorization

17. The fits • 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.

18. How well the Dispersion Relations are satisfied by unconstrained fits d2close to 1 means that the relation is well satisfied d2>> 1 means the data set is inconsistent with the relation. There are 3 independent FDR’s, 3 Roy Eqs and 3 GKPY Eqs. For each 25 MeV we look at the difference between both sides of the FDR, Roy or GKPY that should be ZERO within errors. We define an averaged2over these points, that we call d2 This is NOT a fit to the relation, just a check of the fits!!.

19. Forward Dispersion Relations for UNCONSTRAINED fits FDRs averagedd2 <932MeV <1400MeV 00 0.31 2.13 0+ 1.03 1.11 It=1 1.62 2.69 NOT GOOD! In the intermediate region. Need improvement

20. Roy Eqs. for UNCONSTRAINED fits Roy Eqs. averagedd2 <932MeV <1100MeV S0wave 0.64 0.56 P wave 0.79 0.69 S2 wave 1.35 1.37 GOOD! But room for improvement

21. GKPY Eqs. for UNCONSTRAINED fits Roy Eqs. averagedd2 <932MeV <1100MeV S0wave 1.78 2.42 P wave 2.44 2.13 S2 wave 1.19 1.14 GKPY Eqs are much stricter Lots of room for improvement PRETTY BAD!. Need improvement.

22. The fits • 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)

23. Imposing FDR’s , Roy Eqs and GKPY as constraints 3 GKPY Eqs. 3 Roy Eqs. 3 FDR’s Sum Rules for crossing Parameters of the unconstrained data fits and GKPY Eqs. To improve our fits, we can IMPOSE FDR’s, Roy Eqs. We obtain CONSTRAINED FITS TO DATA (CFD) by minimizing: W roughly counts the number of effective degrees of freedom (sometimes we add weight on certain energy regions) 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

24. Forward Dispersion Relations for CONSTRAINED fits FDRs averagedd2 <932MeV <1400MeV 00 0.32 0.51 0+ 0.33 0.43 It=1 0.06 0.25 VERY GOOD!!!

25. Roy Eqs. for CONSTRAINED fits Roy Eqs. averagedd2 <932MeV <1100MeV S0wave 0.02 0.04 P wave 0.04 0.12 S2 wave 0.21 0.26 VERY GOOD!!!

26. GKPY Eqs. for CONSTRAINED fits Roy Eqs. averagedd2 <932MeV <1100MeV S0wave 0.23 0.24 P wave 0.68 0.60 S2 wave 0.12 0.11 VERY GOOD!!!

27. S0 wave: from UFD to CFD Onlysizablechange in f0(980) region

28. S0 wave: from UFD to CFD As expected, the wave sufferingthelargestchangeisthe D2

29. DIP vs NO DIP inelasticity scenarios Longstanding controversy for inelasticity : (Pennington, Bugg, Zou, Achasov….) There are inconsistent data sets fortheinelasticity Some of them prefer a “dip” structure… ... whereas the other one does not

30. DIP vs NO DIP inelasticity scenarios GKPY S0 waved2 Now we find large differences in CFD UFD 850MeV< e <1050MeV 992MeV< e <1100MeV Dip 1.02 No dip 3.49 Other waves worse and data on phase NOT described Dip 6.15 No dip 23.68 Improvement possible? No dip (enlarged errors) 1.66 But becomes the “Dip” solution No dip (forced) 2.06

31. Final Result: Analyticcontinuationtothecomplexplane f0(980) f0(600) Roy Eqs. Pole: Residue: GKPY Eqs. pole: Residue: Wealsoobtaintheρ pole:

32. Comparisonwithotherresults:The f0(600) orσ

33. Comparisonwithotherresults:The f0(600) orσ OnlysecondRiemann sheetreachable withthisapproach

34. Summary “Dipscenario” forinelasticityfavored Simple and easy to use parametrizations fitted to  scattering DATA CONSTRAINED by FDR’s+ Roy Eqs+ 3 GKPY Eqs GKPY Eqs. pole: Residue: σand f0(980) polesobtainedfromDISPERSIVE INTEGRALS MODEL INDEPENDENT DETERMINATION FROM DATA (and NO ChPT).

35. SPARE SLIDES

36. We START by parametrizing the data Wecouldhave use ANYTHING thatfitsthe data tofeedtheintegrals. We use something SIMPLE at lowenergies(usually <850 MeV) Butforconveniencewewillimpose unitarity and analyticity We use an effective range formalism: +a conformalexpansion Ifneededweexplicitlyfactorize a valuewhere f(s) isimaginary or has an Adler zero: ON THE REAL ELASTIC AXIS thisfunction coincides withcotδ

37. S0 wave parametrization: details Truncatedconformalexpansionfor s<(0.85 GeV)2 Simple polynomialbeyondthat k2 and k3 are kaon and eta CM momenta Imposingcontinuousderivativematching at 0.85 GeV, twoparametersfixed In terms of δ and δ’ at thematchingpoint

38. S0 wave parametrization: details s>(2 Mk)2 Thus, we are neglectingmultipionstatesbut ONLY below KK threshold Buttheelasticityisindependent of thephase, so… itisnotnecessarilyonlyduetoKKbar, (contraryto a 2 channel K matrixformalism) Actuallyitcontainsanyinelasticphysics compatible withthe data. A commonmisunderstandingisthat Roy eqs. onlyincludeππ->ππphysics. Thatis VERY WRONG. Dispersionrelationsinclude ALL contributionstoelasticity (compatible with data) above 2Mk

39. Final Result: discussion 1 overlapwith Caprini, Colangelo, Leutwyler 2006 and in general witheveryotherdispersiveresult. Nottoofarbecausethe parametrizationwasanalytic, unitary,etc… OnlysecondRiemannsheet pole reachablewithinthisapproachfor f0(980). Widthnowconsistentwithlowestbound of PDG band, whichwasnotthe case formostscatteringanalysis of the f0(980) region Tobecomparedwihwhatoneobtainsbyusingdirectlythe UFD withoutusingdisp.relations: FairlyconsistentwithotherChPT+dispersiveresults:

40. Final Result: discussion 1 overlapwith Caprini, Colangelo, Leutwyler 2006 Falls in te ballpark of everyotherdispersiveresult. Nottoofarbecausethe parametrizationwasanalytic, unitary,etc… Tobecomparedwihwhatoneobtainsbyusingdirectlythe UFD withoutusingdisp.relations: FairlyconsistentwithotherChPT+dispersiveresults:

41. 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. We can calculate in the f0(980) region. Effect of the f0(980) on the f0(600) under control. Now, good description up to 1100 MeV. Rememberthisisan isospin symmetricformalism. Wehaveadded a systematicuncertainty as thedifference of using MK+ or MK-. Itisonlyrelevantforth f0(980) withyieldinganadditional ±4 MeVuncertainty In previous works dispersion relations well satisfied below 932 MeV Thisisparametrization and modelindependent. Residuesfrom: orresiduetheorem Theσ and f0(980) polesand residues are obtainedfromthe DISPERSION RELATIONS extended tothecomplexplane.

42. OUR AIM Precise DETERMINATION of f0(600) and f0(980) pole FROM DATA ANALYSIS We do not use the ChPT predictions. Our result is independent of ChPT results. 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 Essential for f0(980) 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 Use of Roy and GKPY dispersion relations for the analytic continuation to the complex plane. (Model independent approach)

43. Forward dispersionrelations Twosymmetric amplitudes. F0+ 0+0+, F00 00 00 Onlydependontwo isospin states. Positivity of imaginarypart Can alsobeevaluated at s=2M2 (tofix Adler zeroslater) Below 1450 MeVwe use ourpartial wave fitsto data. TheIt=1 antisymmetricamplitude Above 1450 MeV we use Regge fits to data. At thresholdistheOlssonsum rule Usedtochecktheconsistency of each set withtheotherwaves Contraryto Roy. eqs. no largeunknown t behaviorneeded Complete set of 3 forward dispersionrelations: