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, 072001 (2011)

R. García Martín, R. Kaminski, JRP, J. Ruiz de Elvira,F. J. Yndurain. PRD83,074004 (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 MeVOne subtraction for F0+ 0+0+, F00 00 00

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 MeVOne subtraction for F0+ 0+0+, F00 00 00

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!

)()'(Im)'(')()(Re )()(''

max

4

2

0'

1

0'

)()(

2

sDTstsKdsPPsSTst IIIIs

MI

II

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 waveon

real axis

SUBTRACTIONTERMS

(polynomials)

KERNEL TERMSknown

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 MeV smaller 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: 2005-2011

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.A31:479-484,2007. PRD74:014001,2006J. 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)

Continuation to complex planeUSING 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 parametrizationis 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 PRD74:014001,2006

Conformal expansion, 4 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.

Tiny uncertaintiesdue to NA48/2 data

It does NOT HAVEA BREIT-WIGNER

SHAPE

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

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 scatteringbut to the FORM FACTOR

de Troconiz, Yndurain, PRD65,093001 (2002), PRD71,073008,(2005)

Above 1 GeV, polynomial fitto CERN-Munich & Berkeley

phase and inelasticity2/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 measuredassumed compatible with 1

Phase shift should go to n at

- The less reliable. EXPECT LARGEST CHANGEWe have increased the systematic error

D0 waveD0 DATA sets incompatible

We fit f2(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 smallErrors 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. PRD69,114001 (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 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 0.31 2.13

0+ 1.03 1.11

It=1 1.62 2.69

<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 0.64 0.56

P wave 0.79 0.69

S2 wave 1.35 1.37

<932MeV <1100MeV

GKPY Eqs. for UNCONSTRAINED fits

Roy Eqs. averaged d2

PRETTY BAD!. Need improvement.

S0wave 1.78 2.42

P wave 2.44 2.13

S2 wave 1.19 1.14

<932MeV <1100MeV

GKPY Eqs are much stricterLots 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

kk

kkSRSR

SPSSPSIt

pppdd

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 0.32 0.51

0+ 0.33 0.43

It=1 0.06 0.25

<932MeV <1400MeV

VERY GOOD!!!

Roy Eqs. for CONSTRAINED fits

Roy Eqs. averaged d2

S0wave 0.02 0.04

P wave 0.04 0.12

S2 wave 0.21 0.26

<932MeV <1100MeV

VERY GOOD!!!

GKPY Eqs. for CONSTRAINED fits

Roy Eqs. averaged d2

S0wave 0.23 0.24

P wave 0.68 0.60

S2 wave 0.12 0.11

<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 6.15No dip 23.68

992MeV< e <1100MeV

UFD

Dip 1.02No dip 3.49

850MeV< e <1050MeV

CFDGKPY S0 wave d2Now we find large differences in

No dip (forced) 2.06

Improvement possible?No dip (enlarged errors) 1.66

But becomesthe “Dip” solution

Other waves worse

and dataon phase

NOT described

Final Result: Analytic continuation to the complex plane

Roy Eqs. Pole:

Residue:

GKPY Eqs. pole:

Residue:

MeV)279()457( 117

1415

i

f0(600) f0(980)

MeV)25()7996( 106

i

poles

MeV27825445 2218

i MeV211003 10

8527

i

GeV59.3 11.013.0

g GeV2.03.2 g

5.04.3 g 2.06.05.2

g

We also obtain the ρ pole:

MeV2.73763 0.11.1

7.15.1

ipole 04.0

07.001.6 g

Comparison with other results:The f0(600) or σ

Only second Riemann sheet reachable

with this approach

Comparison with other results:The f0(600) or σ

Summary

GKPY Eqs. pole:

Residue:

MeV)279()457( 117

1415

i MeV)25()7996( 10

6 i

GeV59.3 11.013.0

g GeV2.03.2 g

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:

sss

ssss

0

0)(

+a conformal expansion

isksks

LLL sf

)(2

1212)(

nnL sBs )()(

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

or has an Adler zero:

n

nA

L sBzsM

s )()( 2

2

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 fors<(0.85 GeV)2

Simple polynomial beyond that

k2 and k3

are kaon and eta CM momenta

Imposing continuous derivative matching at 0.85 GeV, two parameters fixedIn terms of δ and δ’ at the matching point

S0 wave parametrization: details

S0 wave parametrization: details

s>(2 Mk)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 2Mk

Fairly consistent with other ChPT+dispersive results:

Caprini, Colangelo, Leutwyler 2006

MeV272441 95.12

168

ipole1 overlap with MeV141001)980(0 if pole

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 :

MeV)10255()17484( iNot too far because the

parametrization was analytic, unitary,etc…

Fairly consistent with other ChPT+dispersive results:

Caprini, Colangelo, Leutwyler 2006

MeV272441 95.12

168

ipole1 overlap with MeV141001)980(0 if pole

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 :

MeV)10255()17484( iNot 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 MK+ or MK-. 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 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.

Essentialfor

f0(980)

Forward dispersion relations

Used to check the consistency of each set with the other wavesContrary to Roy. eqs. no large unknown t behavior needed

Complete set of 3 forward dispersion relations:

Two symmetric amplitudes. F0+ 0+0+, F00 00 00

Only depend on two isospin states. Positivity of imaginary part

)4')(4')('(')'(Im)4'2(')4()4(Re)(Re 22

2

4

22

2

MssMssss

sFMsdsPPMssMFsFM

Can also be evaluated at s=2M2 (to fix Adler zeros later)

The It=1 antisymmetric amplitude

)4')('()'(Im')42()(Re 2

4

2

2

Mssss

sFdsPPMssFM

At threshold is the Olsson sum rule

Below 1450 MeV we useour partial wave fits to data.

Above 1450 MeV we useRegge fits to data.