J Ruiz de Elvira Precise dispersive analysis of the f0500 and f0980 resonances R García Martín R Kaminski J R Peláez JRE PhysRev Lett 107 072001 2011 ID: 802032
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Slide1
Helmholtz-Instituts für Strahlen- und Kernphysik
J. Ruiz de Elvira
Precise dispersive analysis of the f0(500) and f0(980) resonances
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)
Slide2Motivation
: The f0
(500)/σ and the f0(980)
I=0, J=0 exchange very important for nucleon-nucleon attraction
Scalar multiplet identification still controversial
EFT:
Chiral
symmetry
breaking
.
Vacuum
quantum
numbers.Role on values of chiral parameters.Similarities and differences with EW-Higgs boson. Strongly interacting EWSBS.
f
0
a0
All these states do mix
T
oo
many scalar resonances below 2 GeV.
Possible
exotic nature: tetraquarks,molecules,glueballs…
Glueball search: Characteristic feature of non-abelian QCD nature
Slide3Motivation
: The f0
(500) controversy until 2012
Very controversial since the 60’s.The reason: The f0(500)
is a EXTREMELY WIDE. Usually
refereed
to
its
pole:
Mostly
“
observed” in scattering, but no “resonance peak”.After 2000 also observed in Dalitz plots in production process“not well established” 0+ state in PDG until 1974Removed from 1976 until 1994.Back in PDG in 1996
PDG2002: “σ well established”
However, since 1996 still quoted as
Mass= 400-1200 MeVWidth= 600-1000 MeV
?
Slide41987
1979
1973
1972Most confusion due to usingMODELS(with questionableanalytic properties)
Slide5It
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
Slide6Data after 2000 both scattering
and production
Dispersive- model independent approachesChiral symmetry correct
Slide7OUR AIM
Precise DETERMINATION of
f0(500) 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)
Slide8Roy 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
F00 and F0+ FDR
No
subtraction for the It=1FDR. They both cover the complete isospin basis
Slide9Forward dispersion
relationsUsed to check
the consistency of each set with the other wavesContrary
to
Roy.
eqs
. no
large
unknown
t
behavior
neededComplete set of 3 forward dispersion relations: Two symmetric amplitudes. F0+ 0+0+, F00
0
0 0
0Only depend
on two isospin states
. Positivity
of imaginary
part
Can
also
be evaluated at s=2M2 (to
fix
Adler zeros
later)
The
It=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.
Slide10Roy 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
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
They both cover the complete isospin basis
Slide11NEW 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 allowedGKPY Eqs.
But no need for it!
Slide12Structure
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 waveonreal 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
Slide13UNCERTAINTIES 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,
Slide14ROY vs. GKPY Eqs.
Roy Eqs. Require HUGE cancellations
between terms above 400 MeVBoth KT and STare FAR LARGER thanUNITARITY BOUNDS
GKPY do notNote the differenceIn scale!!
Slide15ROY vs. GKPY Eqs.
This the real proportion
Slide16Our series of works: 2005-2011
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
Some data sets
inconsistent with FDRs
All waves uncorrelated.
Easy to change or add
new data when available
Some data fits
fair agreement with FDRs
Correlated fit to all waves
satisfying FDRs.
precise and reliable predictions.from DATA unitarity and analyticityR. Kaminski, JRP, F.J. Ynduráin
Eur.Phys.J.A31:479-484,2007. PRD74:014001,2006
J. R. P ,F.J. Ynduráin
. PRD
71, 074016 (2005) ,
PRD69,114001 (2004),
R. García Martín, R.
Kaminski
, JRP, J. Ruiz de Elvira, F.J.
Yduráin
2011,
PRD83,074004 (2011)
Continuation
to
complex
plane
USING THE DISPERIVE INTEGRALS:
resonance
poles
Slide17The fits
Unconstrained
data fits (UDF)All waves uncorrelated. Easy to change or add new data when availableThe particular choice of parametrization
is almost IRRELEVANT once inside the integralswe use SIMPLE and easy to implement PARAMETRIZATIONS.
Slide18S0 wave below 850 MeV
Conformal expansion, 4 terms are enough. First, Adler zero at m
2/2We use data on Kl4
including the NEWEST:NA48/2 resultsGet rid of K → 2Isospin corrections fromGasser to
NA48/2
Average of N->N data sets with enlarged errors, at 870- 970 MeV,
where they are consistent within 10
o
to 15
o
error.
Tiny
uncertainties
due to NA48/2 data It does NOT HAVEA BREIT-WIGNERSHAPE
Slide19S0 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
Slide20S0 wave: Unconstrained
fit to data (UFD)
Slide21Similar 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:
Slide22UNconstrained
Fits for High
energies
J.R. Pelaez, 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
Slide23The fits
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.
Slide24How 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
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
.
Slide25Forward Dispersion Relations for UNCONSTRAINED fits
FDRs averaged
d200
0.31 2.130+ 1.03 1.11It=1 1.62 2.69
<932MeV <1400MeV
NOT GOOD! In the intermediate region.
Need improvement
Slide26Roy Eqs. for UNCONSTRAINED fits
Roy Eqs. averaged
d2GOOD! But room for improvement
S0wave 0.64 0.56P wave 0.79 0.69S2 wave 1.35 1.37 <932MeV <1100MeV
Slide27GKPY Eqs. for UNCONSTRAINED fits
Roy Eqs. averaged
d2
PRETTY BAD!. Need improvement. S0wave 1.78 2.42P wave 2.44 2.13
S2 wave 1.19 1.14
<932MeV <1100MeV
GKPY Eqs are much stricter
Lots of room for improvement
Slide28Lesson to learn:
Despite usingVery
reasonable parametrizations with lots of nice properties, obtaining nice
looking fits to data…They are at odds withFIRST PRINCIPLEScrossing, causality (analyticity)…And of course, to
extrapolate to
the
complex
plane
ANALYTICITY
is
essential
Slide29The fits
Unconstrained data fits
(UDF)Independent and simple fits to data in different channels.All waves uncorrelated. Easy to change or add new data when availableCheck of FDR’s Roy and other sum rules.
Room for improvement2) Constrained data fits (CDF)
Slide30Imposing 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 fitsThe 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
.
Slide31Forward Dispersion Relations for CONSTRAINED fits
FDRs averaged
d200
0.32 0.510+ 0.33 0.43It=1 0.06 0.25
<932MeV <1400MeV
Slide32Roy Eqs. for CONSTRAINED fits
Roy Eqs. averaged
d2 S0wave 0.02 0.04
P wave 0.04 0.12S2 wave 0.21 0.26 <932MeV <1100MeV
Slide33GKPY Eqs. for CONSTRAINED fits
Roy Eqs. averaged
d2 S0wave 0.23 0.24
P wave 0.68 0.60S2 wave 0.12 0.11 <932MeV <1100MeV
Slide34Despite
the remarkable improvement
the CFD are not far from the UFD and the datais
still welll described…
Slide35S0 wave:
from
UFD to CFDOnly sizable
change in f0(980) region
Slide36S0 wave: from
UFD to CFD
As expected, the
wave suffering the largest change is the D2
Slide37DIP 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…
Slide38DIP vs NO DIP inelasticity scenarios
Dip 6.15
No dip 23.68992MeV< e <1100MeV
UFDDip 1.02No dip 3.49850MeV< 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) 1.66But becomesthe “Dip” solution
Other waves
worse
and dataon phaseNOT described
Slide39Analytic 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
dispersion
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
.
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.Residues
from
: or
residue
theorem
Slide40Final
Result: Analytic continuation
to the complex plane
Roy Eqs. Pole:Residue:GKPY Eqs. pole:
Residue:
f0(500
)
f0(980)
We
also
obtain
the
ρ
pole:
Fairly
consistent
with
other ChPT+dispersive results
Caprini
,
Colangelo, Leutwyler 2006
1 overlap with
Slide41Fairly
consistent with
other ChPT+dispersive results:
Caprini, Colangelo, Leutwyler 20061 overlap with
Final
Result
:
discussion
Falls in te
bullpark
of
every
other
dispersive result.
Slide42The results from the GKPY Eqs. with the CONSTRAINED Data Fit input
Slide43The results from the GKPY Eqs. with the CONSTRAINED Data Fit input
Slide44Summary
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 Eqs3 Forward Dispersion relations
and the 3 Roy Eqs
and 3 GKPY Eqs satisfied
remarkably well
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
“
Dip
scenario
”
for
inelasticity
favored
Slide45Epilogue
Actually, after our work was published, the PDG 2012 edition made a major revision of the
σ and f0(980).
Slide46PDG
σ
estimate until 2012
Slide47PDG 2012 revision for the
σ
53
Slide48PDG
f0(980)
estimate until 2012
Slide49PDG 2012 revision for the
f0(980)
Slide50THANK YOU
Slide51SPARE SLIDES
Slide52Epilogue
“One might also take the more radical point of view and just
average the most advanced dispersive analyses, for theyprovide a determination of the pole positions with minimal bias.This procedure leads to the much more restricted range off0(500) parameters”
“Note on scalar mesons PDG2012”
Slide53The
results from the
GKPY Eqs. with the CONSTRAINED Data Fit input
Motivation
Nature
Properties
SSB
Conclusions
Poles
55
Jacobo Ruiz de Elvira Carrascal
Doctoral
Dissertation
Poles: PDG 2012 revision for the
σ
Slide54Epilogue
56
Jacobo Ruiz de Elvira Carrascal
Doctoral Dissertation“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”
Slide55Fairly
consistent with
other ChPT+dispersive results:
Caprini, Colangelo, Leutwyler 20061 overlap with
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
MeVFinal Result: discussion
and in general
with
every other dispersive result
.
Slide56We 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 derivedThreshold parameters
Slide57We START by parametrizing the data
To avoid model dependences we only require
analyticity and unitarityWe use an effective range formalism:
s0=1450+a conformal expansion
If
needed
we
explicitly
factorize
a
value where f(s) is imaginaryor has an Adler zero:For the integrals any data parametrization could do. We use something SIMPLE at low energies (usually <850 MeV)ON THE REAL ELASTIC AXIS this function coincides with cot δ
Slide581)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
One
has
to
get used to thinking in terms of the w variable,that deforms considerably the complex plane, and recallthat the expansion is convergent in w, not
in s.
For example
, s=0 isoutside the
inner diskOur
expresions
cannot be
used in any of those
places.
If one
does, one very likely gets nonsenseThese points
may look
close to
threshold, but they are not in terms of w.
Slide59Where do we expect it to converge with few parameters?
Of course, we cannot use the full series. So, we have to
truncate it .
How many parameters we need? The 2 will tell us.As before, we have to stay far from
the borders
of
the
circle
.
For
instance
,
the Adler zero comes out right since it isput there by hand, but w=-0.82,Beyond that we are too
close to
the border, and a
truncatedexpansion may be
bad.In particular
one can
get spurious
poles with that
particular parametrization
.as
noted by Caprini, Colangelo, Gasser Leutwyler
Again one has the systematic uncertainty of the term one is dropping.
Slide60But the NA48/2 data falls very much inside the circle:
barycenters between w=-0.537 and -0.401
Slide61We can
for instance
include compatible data pointshere, with large uncertainties
Slide62To
avoid coming
close to the edges of the circle
s<(0.85 GeV)2and we use FOUR terms in the expansion
k2
and k
3
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 pointS0 wave parametrization
: details
Slide63S0 wave
parametrization: details
s>(2 Mk)
2Thus, 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
Slide64The 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 betweenwith huge errorsSolution Edeviates stronglyfrom the rest but has huge error bars
Note size of
uncertainty
in data
at 800 MeV!!
Slide65Regge
parameters of N and NN
Fit to 270 data points
of N , KN and NN total cross sectionsfor kinetic energy between 1 and 16.5 GeV. The Pomeron
is
very
precise!!
JRP, F.J.
Ynduráin
. PRD69,114001 (2004)
Slide66We
have allowed both for
degenerate and non degenerate P’ and .No drammatic difference but non-degeneracy preferred
Regge
fits
:
total
cross
sections
R.Kaminski
, JRP, F.J.
Ynduráin PRD74:014001,2006t 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
Slide67UFD
Slide68When
fitting also
Zakharov data
Slide69The
effective range: A model
independent and SIMPLE parametrization of S0 wave dataThe effective range
formalism ensures unitarityThe 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 regionHowever, the usual expansion in momenta has very small convergence radiusIs related to the phase shift
Slide70P wave
Up
to 1 GeV This NOT a fit to scattering
but to the FORM FACTORde Troconiz, Yndurain, PRD65,093001 (2002), PRD71,073008,(2005)
Above 1 GeV, polynomial
fit
to
CERN-
Munich
& Berkeley
phase
and
inelasticity
2/dof=1 .01THIS IS A NICE BREIT-WIGNER !!
Slide71For S2 we include an Adler zero at M
- Inelasticity
small but fittedD2 and S2 waves
Very poor data setsElasticity 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
Slide72D0 wave
D0 DATA sets incompatible
We fit f2(1250) mass and widthMatching at lower energies: CERN-Munich
and Berkeley data (is ZERO below 800 !!)plus threshold psrameters from Froissart-Gribov Sum rulesInelasticity fitted empirically:
CERN-MUnich + Berkeley data
The
F wave
contribution
is
very
small
Errors increased by effect of including one or two incompatible data setsNEW: Ghost removed but negligible effect. The G wave contribution negligible
THIS IS A NICE BREIT-WIGNER !!
Slide73UNconstrained
Fits for High
energies
J.R. Pelaez, 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
Slide74SUM RULES
J.R.Pelaez, F.J. Yndurain Phys
Rev. D71 (2005)They relate high energy parameters to low energy P and D waves