departamento de física teórica ii. universidad complutense de madrid
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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!!
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