Departamento de Física Teórica II. Universidad Complutense de Madrid

J. Ruiz de Elvira

in collaboration withR. García Martín

R. KaminskiJose R. PeláezF. J. Yndurain.

Precise dispersive analysis of the f0(600) and f0(980)resonances from pion-pion scattering.

Motivation: The f0(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.

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

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 MeVOne subtraction for F00 and F0+ FDRNo 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 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!

UNCERTAINTIES IN Standard ROY EQS. vs 1s Roy like GKPY EqsGarcia Martin, R. Kaminski, J.R.Pelaez, F.J. Yndurain

smaller uncertainty below ~ 400 MeV smaller uncertainty above ~400 MeV

Why are GKPY Eq. relevant?

One subtraction yields better accuracy in √s > 400 MeV region

Roy Eqs. GKPY Eqs,

OUR AIM

Precise DETERMINATION of f0(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.

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.

We START by parametrizing the data

To avoid model dependences we only require analyticity and unitarity

We use an effective range formalism:

sss

ssss

0

0)(

s0=1450

+a conformal expansion

isksks

LLL sf

)(2

1212)(

nnL

sBs )()(

If needed we explicitly factorize a value where f(s) is imaginary

or has a zero:

nnL

sBzs

Mss )()( 2

2

For the integrals any data parametrization could do. We use something SIMPLE at low energies (usually <850 MeV)

S0 wave below 850 MeV R. Garcia Martin, JR.Pelaez and F.J. Ynduráin PRD74:014001,2006

Conformal expansion, three terms are enough. First, Adler zero at m2/2

We use data on Kl4including the NEWEST:

NA48/2 resultsGet rid of K → 2

Isospin corrections fromGasser to NA48/2

Average of N->N data sets with enlarged errors, at 870- 970 MeV, where they are consistent within 10o to 15o error.

Note that it is just used

in the real axisfor physical s

S0 wave above 850 MeV R. Kaminski, J.R.Pelaez and F.J. Ynduráin PRD74:014001,2006

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.

NEW:S0 wave with improved matching

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:

Similar Initial UNconstrained FIts for all other waves and High energies

1.5 2 5 10 15sGeV

10

20

30

40

50

latoT

bm

Regge: PY

Data: Robertson etal.Biswas etal.Abramowicz etal.

PYHooglandetal.

1.5 2 5 10 15sGeV

10

20

30

40

50

latoT

bm

2 4 6 8 10sGeV

10

20

30

40

50

latoT

bm

Data : Robertson et al.

Biswas et al.

Hanlon et al.PY

Regge: PY

Hyams et al.

2 4 6 8 10sGeV

10

20

30

40

50

latoT

bm

1.5 2 2.5 3 3.5 4sGeV

10

20

30

40

50

latoT

0bm Regge: PY

Data: Biswas etal.

PY

1.5 2 2.5 3 3.5 4sGeV

10

20

30

40

50

latoT

0bm

J.R.Pelaez, F.J. Ynduráin. PRD69,114001 (2004)From older works:

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 d2

For each 25 MeV we look at the difference between both sides ofthe FDR, Roy or GKPY that should be ZERO within errors.

d2 close 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.

This is NOT a fit to the relation, just a check of the fits!!.

Forward Dispersion Relations for UNCONSTRAINED fits

FDRs averaged d2

00 0.52 1.84

0+ 1.02 1.11

It=1 0.89 2.50

<932MeV <1400MeV

NOT GOOD! In the intermediate region.Need improvement

Roy Eqs. for UNCONSTRAINED fits

Roy Eqs. averaged d2

GOOD!. But room for improvement

S0wave 0.80 0.70

P wave 0.64 0.56

S2 wave 1.22 1.23

<932MeV <1100MeV

GKPY Eqs. for UNCONSTRAINED fits

Roy Eqs. averaged d2

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 scricterLots 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 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

kk

kkSRSR

SPSSPSIt

p

ppdd

Wddddddddd GKPYGKPYGKPYroyroyroy

2exp22

21

22

220

22

220

21

20

200

2

)(

}{

3 FDR’s 3 GKPY Eqs

Sum Rules forcrossing

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 d2

00 0.34 0.55

0+ 0.31 0.47

It=1 0.12 0.33

<932MeV <1400MeV

GOOD!.

Roy Eqs. for CONSTRAINED fits

Roy Eqs. averaged d2

S0wave 0.17 0.22

P wave 0.07 0.15

S2 wave 0.28 0.32

<932MeV <1100MeV

GOOD!.

GKPY Eqs. for CONSTRAINED fits

Roy Eqs. averaged d2

S0wave 0.45 0.50

P wave 0.85 0.79

S2 wave 0.17 0.28

<932MeV <1100MeV

GOOD!.

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.

Final Result: Analytic continuation to the complex plane

Fairly consistent with other ChPT+dispersive results

Caprini, Colangelo, Leutwyler 2006

MeV272441 95.12

168

ipole1 overlap with

Roy Eqs:

GKPY Eqs: MeV)11266()21460( i

f0(600) f0(980)

MeV)619()6999( i

poles

poles

Results are PRELIMINARY. Still honing the uncertainties, will probably turn out slightly bigger and asymmetric

From

MeV141001)980(0 if pole

MeV1926929449 i MeV4381002 474

i

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

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

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.

SUM RULES

J.R.Pelaez, F.J. Yndurain Phys Rev. D71 (2005)

They relate high energy parameters to low energy P and D waves

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 MeVRoy Eqs. satisfied except S2, but still within 1.3 sigmas

All waves correlated.Differ from Uncorrelatedby less than 1 sigmaExcept D2 wave, that differs1.5 sigma

CONSTRAINED FITSFDRs, Roy Eqs and Sum rules

satisfied remarkably well.Very reliable.

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 setsinconsistent with FDRs

All waves uncorrelated.Easy to change or add

new data when available

Some data fitsfair agreement with FDRs

Correlated fit to all wavessatisfying 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. PRD74:014001,2006J. 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

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, SolutionCand the global fits

Other fits do nothave the shoulder and are separated from pure Kl4

Kaminski et al.lies in betweenwith huge errors

Solution Edeviates stronglyfrom the rest but has huge error bars

Note size ofuncertainty

in dataat 800 MeV!!