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

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 inside the integral. A precise  scattering analysis can help determining the  and f0(980) parameters

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 F 0+   0  +  0  +, F 00  0  0  0  0 No subtraction for the It=1FDR. They both cover the complete isospin basis

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 F 0+   0  +  0  +, F 00  0  0  0  0 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

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 (Pomeron) growth cancels. if antisymmetric dominated by rho exchange (softer). ONE SUBTRACTION also allowed GKPY Eqs. But no need for it!

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…. Partial wave on real axis SUBTRACTION TERMS (polynomials) KERNEL TERMS known 2nd order 1st order More energy suppressed Less energy suppressed Very small small ROY: GKPY: DRIVING TERMS (truncation) Higher waves and High energy “IN (from our data parametrizations)” “OUT” =? Similar Procedure for FDRs

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

Our series of works: Independent and simple fits to data in different channels. “Unconstrained Data Fits=UDF” Check Dispersion Relations Impose FDRs, Roy & GKPY Eqs on data fits “Constrained Data Fits CDF” Describe data and are consistent with Dispersion relations R. Kaminski, JRP, F.J. Ynduráin Eur.Phys.J.A31: ,2007. PRD74:014001,2006 J. R. P,F.J. Ynduráin. PRD71, (2005), PRD69, (2004), R. García Martín, R. Kaminski, JRP, J. Ruiz de Elvira, F.J. Yduráin. PRD83, (2011) Continuation to complex plane USING THE DISPERIVE INTEGRALS: resonance poles

The fits 1)Unconstrained data fits (UDF) All waves uncorrelated. Easy to change or add new data when available The particular choice of parametrization is almost IRRELEVANT once inside the integrals we use SIMPLE and easy to implement PARAMETRIZATIONS.

S0 wave below 850 MeV R. Garcia Martin, JR.Pelaez and F.J. Ynduráin PR D74:014001,2006 Conformal expansion, 4 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 MeV, where they are consistent within 10 o to 15 o error. Tiny uncertainties due to NA48/2 data It does NOT HAVE A BREIT-WIGNER SHAPE

S0 wave above 850 MeV R. Kaminski, J.R.Pelaez and F.J. Ynduráin PR D74:014001,2006 Paticular care on the f0(980) region : Continuous and differentiable matching between parametrizations Above1 GeV, all sources of inelasticity included (consistently with data) Two scenarios studied CERN-Munich phases with and without polarized beams Inelasticity from several   ,   KK experiments

S0 wave: Unconstrained fit to data (UFD)

P wave Up to 1 GeV This NOT a fit to  scattering but to the FORM FACTOR de Troconiz, Yndurain, PRD65, (2002), PRD71,073008,(2005) Above 1 GeV, polynomial fit to CERN-Munich & Berkeley phase and inelasticity  2 /dof=1.01 THIS IS A NICE BREIT-WIGNER !!

For S2 we include an Adler zero at M  - Inelasticity small but fitted D2 and S2 waves Very poor data sets Elasticity above 1.25 GeV not measured assumed compatible with 1 Phase shift should go to n  at  -The less reliable. EXPECT LARGEST CHANGE We have increased the systematic error

D0 wave D0 DATA sets incompatible We fit f 2 (1250) mass and width Matching at lower energies: CERN-Munich and Berkeley data (is ZERO below 800 !!) plus threshold psrameters from Froissart-Gribov Sum rules Inelasticity fitted empirically: CERN-MUnich + Berkeley data The F wave contribution is very small Errors increased by effect of including one or two incompatible data sets NEW: Ghost removed but negligible effect. The G wave contribution negligible THIS IS A NICE BREIT-WIGNER !!

UNconstrained Fits for High energies JRP, F.J.Ynduráin. PRD69, (2004) UDF from older works and Regge parametrizations of data Factorization In principle any parametrization of data is fine. For simplicity we use

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.

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!!.

Forward Dispersion Relations for UNCONSTRAINED fits FDRs averaged d 2  0   0  I t = <932MeV <1400MeV NOT GOOD! In the intermediate region. Need improvement

Roy Eqs. for UNCONSTRAINED fits Roy Eqs. averaged d 2 GOOD! But room for improvement S0wave P wave S2 wave <932MeV <1100MeV

GKPY Eqs. for UNCONSTRAINED fits Roy Eqs. averaged d 2 PRETTY BAD!. Need improvement. S0wave P wave S2 wave <932MeV <1100MeV GKPY Eqs are much stricter Lots of room for improvement

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)

Imposing FDR’s, Roy Eqs and GKPY as constraints To improve our fits, we can IMPOSE FDR’s, Roy Eqs. 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 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.

Forward Dispersion Relations for CONSTRAINED fits FDRs averaged d 2  0   0  I t = <932MeV <1400MeV VERY GOOD!!!

Roy Eqs. for CONSTRAINED fits Roy Eqs. averaged d 2 S0wave P wave S2 wave <932MeV <1100MeV VERY GOOD!!!

GKPY Eqs. for CONSTRAINED fits Roy Eqs. averaged d 2 S0wave P wave S2 wave <932MeV <1100MeV VERY GOOD!!!

S0 wave: from UFD to CFD Only sizable change in f0(980) region

S0 wave: from UFD to CFD As expected, the wave suffering the largest change is the D2

DIP vs NO DIP inelasticity scenarios Longstanding controversy for inelasticity : (Pennington, Bugg, Zou, Achasov….) There are inconsistent data sets for the inelasticity... whereas the other one does notSome of them prefer a “dip” structure…

DIP vs NO DIP inelasticity scenarios Dip 6.15 No dip MeV< e <1100MeV UFD Dip 1.02 No dip MeV< e <1050MeV CFD GKPY S0 wave d 2 Now we find large differences in No dip ( forced) 2.06 Improvement possible? No dip (enlarged errors) 1.66 But becomes the “Dip” solution Other waves worse and data on phase NOT described

Final Result: Analytic continuation to the complex plane Roy Eqs. Pole: Residue: GKPY Eqs. pole: Residue: f0(600) f0(980) We also obtain the ρ pole:

Comparison with other results:The f 0 (600) or σ

Only second Riemann sheet reachable with this approach Comparison with other results:The f 0 (600) or σ

Summary GKPY Eqs. pole: Residue: Simple and easy to use parametrizations fitted to  scattering DATA CONSTRAINED by FDR’s+ Roy Eqs+ 3 GKPY Eqs σ and f0(980) poles obtained from DISPERSIVE INTEGRALS MODEL INDEPENDENT DETERMINATION FROM DATA (and NO ChPT). “Dip scenario” for inelasticity favored

SPARE SLIDES

We START by parametrizing the data We could have use ANYTHING that fits the data to feed the integrals. We use an effective range formalism: +a conformal expansion If needed we explicitly factorize a value where f(s) is imaginary or has an Adler zero: We use something SIMPLE at low energies (usually <850 MeV) But for convenience we will impose unitarity and analyticity ON THE REAL ELASTIC AXIS this function coincides with cot δ

Truncated conformal expansion for s<(0.85 GeV) 2 Simple polynomial beyond that k 2 and k 3 are kaon and eta CM momenta Imposing continuous derivative matching at 0.85 GeV, two parameters fixed In terms of δ and δ’ at the matching point S0 wave parametrization: details

s>(2 M k ) 2 Thus, we are neglecting multipion states but ONLY below KK threshold But the elasticity is independent of the phase, so… it is not necessarily only due to KKbar, (contrary to a 2 channel K matrix formalism) Actually it contains any inelastic physics compatible with the data. A common misunderstanding is that Roy eqs. only include ππ->ππ physics. That is VERY WRONG. Dispersion relations include ALL contributions to elasticity (compatible with data) above 2M k

Fairly consistent with other ChPT+dispersive results: Caprini, Colangelo, Leutwyler  overlap with Only second Riemann sheet pole reachable within this approach for f0(980). Width now consistent with lowest bound of PDG band, which was not the case for most scattering analysis of the f0(980) region Final Result: discussion and in general with every other dispersive result. To be compared wih what one obtains by using directly the UFD without using disp.relations : Not too far because the parametrization was analytic, unitary,etc…

Fairly consistent with other ChPT+dispersive results: Caprini, Colangelo, Leutwyler  overlap with Final Result: discussion Falls in te ballpark of every other dispersive result. To be compared wih what one obtains by using directly the UFD without using disp.relations : Not too far because the parametrization was analytic, unitary,etc…

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 and residues 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. Remember this is an isospin symmetric formalism. We have added a systematic uncertainty as the difference of using M K+ or M K-. It is only relevant for th f0(980) with yielding an additional ±4 MeV uncertainty Residues from: or residue theorem

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. Essential for f 0 (980)

Forward dispersion relations Used to check the consistency of each set with the other waves Contrary to Roy. eqs. no large unknown t behavior needed Complete set of 3 forward dispersion relations: Two symmetric amplitudes. F 0+   0  +  0  +, F 00  0  0  0  0 Only depend on two isospin states. Positivity of imaginary part Can also be evaluated at s=2M  2 (to fix Adler zeros later) The I t =1 antisymmetric amplitude At threshold is the Olsson sum rule Below 1450 MeV we use our partial wave fits to data. Above 1450 MeV we use Regge fits to data.