Helmholtz-Instituts für Strahlen- und Kernphysik
Precise dispersive analysis of
the f0(500) and f0(980) resonances
J. Ruiz de Elvira
R. García Martín, R. Kaminski, J. R. Peláez, JRE,
Phys.Rev. Lett. 107, 072001 (2011)
R. García Martín, R. Kaminski, J. R. Peláez, JRE, F. J. Yndurain.
PRD83,074004 (2011)
Motivation: The f0(500)/σ and the f0(980)
I=0, J=0  exchange very important for nucleon-nucleon attraction
Scalar multiplet identification still controversial
Too many scalar resonances below 2 GeV.
Glueball search: Characteristic feature of non-abelian QCD nature
Possible exotic nature: tetraquarks,molecules,glueballs…
All these states do mix
EFT: Chiral symmetry breaking. Vacuum quantum numbers.
Role on values of chiral parameters.
Similarities and differences with EW-Higgs boson. Strongly interacting EWSBS.
Motivation: The f0(500) controversy until 2012
Very controversial since the 60’s.
“not well established” 0+ state in PDG until 1974
Removed from 1976 until 1994.
Back in PDG in 1996
The reason: The f0(500) is a EXTREMELY WIDE.
Usually refereed to its pole:
s pole  M  i  / 2
Mostly “observed” in  scattering, but no “resonance peak”.
After 2000 also observed in Dalitz plots in production process
PDG2002: “σ well established”
However, since 1996 still quoted as
Mass= 400-1200 MeV
Width= 600-1000 MeV
Most confusion
due to using
MODELS
(with questionable
analytic properties)
1973
1987
1972
1979
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 is irrelevant once
it is used in the integral.
A precise  scattering analysis can help determining the
 and f0(980) parameters
OUR AIM
Precise DETERMINATION of f0(500) 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
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)
Essential
for
f0(980)
Roy Eqs. vs. Forward Dispersion Relations
They both cover the complete isospin basis
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 F00 and F0+ FDR
No subtraction for the It=1FDR.
Roy Eqs. vs. Forward Dispersion Relations
They both cover the complete isospin basis
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 F00 and F0+ FDR
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.
Limited to ~ 1.1 GeV.
Twice substracted.
Good at low energies, interesting for ChPT.
When combined with ChPT precise for f0(500) pole determinations. (Caprini et al)
But we here do NOT use ChPT, our results are just a data analysis
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.
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….
2
1
Re t( I ) ( s )  ST( I ) ( s )   PP
I ' 0  ' 0
SUBTRACTION
TERMS
(polynomials)
Partial wave
on
real axis
“OUT”
s max
I I'
(I )
(I )
ds
'
K
(
s
'
)
Im
t
(
s
'
)

DT

 (s)
  '
4 M 2
KERNEL TERMS
known
DRIVING
TERMS
(truncation)
Higher waves
and High energy
ROY: 2nd order
More energy suppressed
Very small
GKPY: 1st order
Less energy suppressed
small
=?
“IN (from our data parametrizations)”
Similar
Procedure
for FDRs
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
ROY vs. GKPY Eqs.
Roy Eqs. Require HUGE cancellations
between terms above 400 MeV
Both KT and ST
are FAR LARGER than
UNITARITY BOUNDS
GKPY do not
Note the difference
In scale!!
ROY vs. GKPY Eqs.
This the real proportion
Our series of works: 2005-2011
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 2011, PRD83,074004 (2011)
Independent and simple fits
to data in different channels.
“Unconstrained Data Fits UDF”
Check Dispersion Relations
Impose FDRs, Roy Eqs and Sum Rules
on data fits
“Constrained Data Fits CDF”
Describe data and are
consistent with Dispersion relations
All waves uncorrelated.
Easy to change or add
new data when available
Some data sets
inconsistent with FDRs
Some data fits
fair agreement with FDRs
Correlated fit to all waves
satisfying FDRs.
precise and reliable predictions.
from DATA unitarity and analyticity
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
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
including the NEWEST:
NA48/2 results
Get rid of K → 2
Isospin corrections from
Gasser to NA48/2
It does NOT HAVE
A BREIT-WIGNER
SHAPE
Tiny uncertainties
due to NA48/2 data
S0 wave above 850 MeV
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)
Similar Initial UNconstrained FIts for all other waves and High energies
From older works:
R. Kaminski, J.R.Pelaez, F.J. Ynduráin.
Phys. Rev.
D77:054015,2008.
J.R.Pelaez , F.J. Ynduráin.
Eur.Phys.J.A31:479-484,2007,
PRD74:014001,2006
PRD71, 074016 (2005),
UNconstrained Fits for High energies
In principle any parametrization of data is fine. For simplicity we use
UDF from older works and Regge parametrizations of data J.R. Pelaez, F.J.Ynduráin. PRD69,114001 (2004)
Factorization
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
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
2 over these points, that we call d2
d2 close to 1 means that the relation is well satisfied
d2>> 1 means the data set is inconsistent with the relation.
Forward Dispersion Relations for UNCONSTRAINED fits
FDRs averaged d2
<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
Roy Eqs. for UNCONSTRAINED fits
Roy Eqs. averaged d2
<932MeV
<1100MeV
S0wave
0.64
0.56
P wave
0.79
0.69
S2 wave
1.35
1.37
GOOD! But room for improvement
GKPY Eqs. for UNCONSTRAINED fits
Roy Eqs. averaged d2
<932MeV
<1100MeV
S0wave
1.78
2.42
P wave
2.44
2.13
S2 wave
1.19
1.14
GKPYBAD!.
Eqs Need
are much
stricter
PRETTY
improvement.
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. and GKPY Eqs.
We obtain CONSTRAINED FITS TO DATA (CFD) by minimizing:
2
00
  {d  d
2
2
0
d
3 FDR’s
d
2
SR1
d
2
SR 2
2
It 1
d
2
S 0 roy
d
2
Proy
d
2
S 2 roy
3 Roy Eqs.
d
2
S 0GKPY
d
2
P GKPY
d
2
S 2GKPY
}W 
3 GKPY Eqs.
( pk  pkexp ) 2
 k
pk
Parameters of the
Sum Rules for
unconstrained data fits
crossing
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
Forward Dispersion Relations for CONSTRAINED fits
FDRs averaged d2
<932MeV <1400MeV
00
0.32
0.51
0+
0.33
0.43
It=1
0.06
0.25
Roy Eqs. for CONSTRAINED fits
Roy Eqs. averaged d2
<932MeV
<1100MeV
S0wave
0.02
0.04
P wave
0.04
0.12
S2 wave
0.21
0.26
GKPY Eqs. for CONSTRAINED fits
Roy Eqs. averaged d2
<932MeV
<1100MeV
S0wave
0.23
0.24
P wave
0.68
0.60
S2 wave
0.12
0.11
Despite the remarkable improvement
the CFD are not far from the UFD and the data
is still welll described…
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
Some of them prefer a “dip” structure…
... whereas the other one does not
DIP vs NO DIP inelasticity scenarios
Now we find large differences in GKPY S0 wave d2
UFD
CFD
992MeV< e <1100MeV
850MeV< e <1050MeV
Dip
1.02
No dip
3.49
Dip
No dip
6.15
23.68
Improvement possible?
But becomes
the “Dip” solution
No dip (enlarged errors) 1.66
No dip (forced)
2.06
Other waves
worse
and data
on phase
NOT described
Analytic continuation to the complex plane
Now, good description up to 1100 MeV.
We can calculate in the f0(980) region.
Effect of the f0(980) on the f0(500) under control.
We do NOT obtain the poles directly from the constrained parametrizations,
which are used only as an input for the dispersion relations.
The σ and f0(980) poles and residues are obtained from the
DISPERSION RELATIONS extended to the complex plane.
Residues from:
This is parametrization and model independent.
or residue theorem
Final Result: Analytic continuation to the complex plane
s pole
f0(980)
f0(500)
Roy Eqs. Pole:
445  25  i278
Residue:
MeV
1003  i21 MeV
5
 27
g  2.5
g  3.4  0.5
Residue:
GKPY Eqs. pole:
22
18
14
15
11
7
(457 )  i(279 )MeV
0.11
g  3.590.13 GeV
10
8
0.2
 0.6
(996  7)  i(25106 )MeV
g  2.3  0.2 GeV
Fairly consistent with other ChPT+dispersive results
Caprini, Colangelo, Leutwyler 2006
16
9
f 0(980) pole  1001  i14MeV


441

i
272
1 overlap with
pole
8
12.5 MeV
We also obtain the ρ pole:
 pole  763
1.7
1.5
 i73.2
1.0
1.1
MeV
g  6.0100..0407
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)
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
“Dip scenario” for inelasticity favored
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, but are obtained
within a model independent precise analysis of the latest data
Epilogue
Actually, after our work was published, the PDG 2012
edition made a major revision of the σ and f0(980).
PDG σ estimate until 2012
PDG 2012 revision for the σ
53
PDG f0(980) estimate until 2012
PDG 2012 revision for the f0(980)
THANK YOU
SPARE SLIDES
Epilogue
“One might also take the more radical point of view and just
average the most advanced dispersive analyses, for they
provide a determination of the pole positions with minimal bias.
This procedure leads to the much more restricted range of
f0(500) parameters”
“Note on scalar mesons PDG2012”
Motivation
Properties
Nature
SSB
Conclusions
Poles
The results from the GKPY Eqs. with the CONSTRAINED Data Fit input
Poles: PDG 2012 revision for the σ
Jacobo Ruiz de Elvira Carrascal
Doctoral Dissertation
55
Epilogue
“In this issue we extended the allowed range of the f0(980) mass
to include the mass value derived in Ref. 10. We now quote for
the mass”
“Note on scalar mesons PDG2012”
Jacobo Ruiz de Elvira Carrascal
Doctoral Dissertation
56
Final Result: discussion
Fairly consistent with other ChPT+dispersive results:
1 overlap with
 pole  441168  i272129 .5 MeV
f 0 (980) pole  1001 i14MeV
Caprini, Colangelo, Leutwyler 2006
and in general with every other dispersive result.
The existence of two kaon thresholds is relevant for the f0(980). We
have repeated the UFD to CFD process for the two extreme cases and
added half the difference as a systematic uncertainty. It is only relevant
for the f0 width and amounts to 4 MeV
Threshold parameters
We can now use sum rules to obtain threshold parameters:
We use the Froissart Gribov representantion, Olsson sum rule, and a couple of other sum rules we
have derived
We START by parametrizing the data
To avoid model dependences we only require analyticity and unitarity
For the integrals any data parametrization could do.
We use something SIMPLE at low energies (usually <850 MeV)
We use an
effective range formalism:
+a conformal expansion
 ( s) 
s  s0  s
s  s0  s
f L (s) 
2 s
1
 k 2 s k 2 L1 L ( s ) i
 L (s) 
If needed we explicitly factorize a
value where f(s) is imaginary
or has an Adler zero:
 L (s) 
s0=1450

Bn  ( s ) n
M2
s  z A2

Bn  ( s ) n
ON THE REAL ELASTIC AXIS
this function coincides with cot δ
One has to get used to thinking in terms of the w variable,
that deforms considerably the complex plane, and recall
that the expansion is convergent in w, not in s.
1)The left cut lies rigth on |w|=1.
2)The KK cut lies rigth on |w|=1.
Both cannot be described
with the truncated
conformal expansion
For example, s=0 is
outside the inner disk
These points may look close to threshold, but they are not in terms of w.
Our expresions cannot be used in any of those places.
If one does, one very likely gets nonsense
Of course, we cannot use the full series. So, we have to truncate it .
How many parameters we need? The 2 will tell us.
Again one has the systematic uncertainty of the term one is dropping.
Where do we expect it to converge with few parameters?
As before, we have to stay far
from the borders of the circle.
For instance, the Adler zero
comes out right since it is
put there by hand, but w=-0.82,
Beyond that we are too close to
the border, and a truncated
expansion may be bad.
In particular one can get spurious
poles with that particular
parametrization.
as noted by Caprini, Colangelo, Gasser Leutwyler
But the NA48/2 data falls very much inside the
circle:
barycenters between w=-0.537 and -0.401
We can for instance include compatible data points
here, with large uncertainties
S0 wave parametrization: details
To avoid coming close
to the edges of the circle
s<(0.85 GeV)2
and we use FOUR terms
in the expansion
k2 and k3
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 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 pipi-pipi physics.
That is VERY WRONG.
Dispersion relations include ALL contributions to
elasticity (compatible with data)
above 2Mk
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, SolutionC
and the global fits
Other fits do not
have the shoulder
and are separated
from pure Kl4
Kaminski et al.
lies in between
with huge errors
Solution E
deviates strongly
from the rest but has
huge error bars
Note size of
uncertainty
in data
at 800 MeV!!
Regge parameters of N and NN
JRP, F.J. Ynduráin. PRD69,114001 (2004)
Fit to 270 data points of N , KN and NN total cross sections
for kinetic energy between 1 and 16.5 GeV. The Pomeron is very precise!!
 pp  ppˆ
2

4 2
1
 ( s , m 2p , m 2p ) 2
  P(s)  P' (s)
f NP
f P
2


N

4 2
 ( s , m , m 2p )
2

f NP
P
f
1
6
( P ( s )  P ' ( s ))   ( s )

Regge fits:  total cross sections
R.Kaminski, JRP, F.J. Ynduráin PRD74:014001,2006
We have allowed both for degenerate and non degenerate P’ and .
No drammatic difference but non-degeneracy preferred
t dependence needed in Roy Eqs. (up to -0.43 GeV2)
Large errors to cover fits of Rarita et al. and Froggat Petersen
Not very relevant.
In contact with I.Caprini to understand wheter we actually agree on this input
UFD
When fitting also Zakharov data
The effective range: A model independent and SIMPLE parametrization of S0 wave data
The effective range formalism
ensures unitarity
Is related to the phase shift
f L ( s) 
2 s
1
 k 2 s k 2 L1 L ( s ) i
2 s1 / 2
cot  0 ( s ) 
 L (s)
k
0
The effective range function  is analytic with cuts from 0 to – on the left
and also the INELASTIC cuts. (KK in practice)
Thus, it does not have the pion-pion righth hand cut,
and thus can be expanded in that region
However, the usual expansion in momenta has very small convergence radius

4M 
Re f 0 ( s )  al  bl k  ...
k 0
P wave
THIS IS A NICE BREIT-WIGNER !!
Up to 1 GeV This NOT a fit to  scattering
but to the FORM FACTOR
de Troconiz, Yndurain, PRD65,093001 (2002), PRD71,073008,(2005)
Above 1 GeV, polynomial fit
to CERN-Munich & Berkeley
phase and inelasticity
2/dof=1 .01
D2 and S2 waves
Very poor data sets
Phase shift should go to n at 
Elasticity above 1.25 GeV not measured
assumed compatible with 1
-The less reliable. EXPECT LARGEST CHANGE
We have increased the systematic error
For S2 we include an Adler zero at M
- Inelasticity small but fitted
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 lower energies: CERN-Munich
and Berkeley data (is ZERO below 800 !!)
plus threshold psrameters from
Froissart-Gribov Sum rules
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
UNconstrained Fits for High energies
In principle any parametrization of data is fine. For simplicity we use
UDF from older works and Regge parametrizations of data J.R. Pelaez, F.J.Ynduráin. PRD69,114001 (2004)
Factorization
SUM RULES
J.R.Pelaez, F.J. Yndurain Phys
Rev. D71 (2005)
They relate high energy parameters to low energy P and D waves