no ordinary mesons: light scalars nc and quark mass dependence
TRANSCRIPT
Departamento de Física Teórica II. Universidad Complutense de Madrid
José R. Peláez
NO ORDINARY MESONS:Light scalars Nc and quark mass
dependence from Unitarized
Chiral Perturbation Theory:
INT.Seattle.
November
20th
2009
Outline
Scalar
Mesons:
A long story
in short
Unitarized
Chiral Perturbation
TheoryThe
Inverse
Amplitude
Method
(IAM)
Nc
behavior
of
scalars
Chiral Extrapolation: the
quark mass
dependence
One
loop
SU(3) ChPT: scalar
nonet
Two
loop
SU(2) ChPT: the
JRP, Phys.Rev.Lett. 92:102001,2004
G. Ríos and
JRP, Phys.Rev.Lett.97:242002,2006.
C. Hanhart, G. Ríos and
JRP, Phys.Rev.Lett.100:152001,2008.
Scalar Mesons:
A long story in short
Motivation: The
f0
(600)
or
σ, half
a century
around
I=0, J=0
exchange
very
important
for
nucleon-nucleon
attraction!!
Crude
Sketch
of
NN potential:
From
C.N. Booth
Scalar-isoscalar
field
already
proposed
by Johnson
& Teller
in 1955
Soon
interpreted
within
“Linear sigma model”
(Gell-Mann) or
Nambu
Jona Lasinio
-
like
models, in the
60’s. Exchange in t-channel, not sensitive to
details in NN
Other
scalar
mesons
narrower
like
f0(980), a0(980) well
established
but not
well
understood
Is there a light scalar nonet as chiral partner
of the light pseudoscalar nonet?
The problem with the data. The S0 wave
From Ke (Kl4
)Pions on-shell. Very precise, but 00
-11. Recent data below mK.
Geneva-Saclay (77), E865
(01)
Example:CERN-Munich5 different analysis of same pn data !!
from Grayer et al. NPB (1974)
Systematic errors of 10o !!
NN
Initial state not well defined, model dependent off-shell
extrapolations (OPE, absorption, A2
exchange...) Phase shift ambiguities, etc...
From N scattering
Motivation: The
f0
(600) controversy
Very
controversial since
the
60’s.
The
reason: The
f0
(600) (and
kappa) is
a EXTREMELY WIDE.
Usually
refereed
to
its
pole:
Mostly
“observed”
in
scattering, but
no “resonance
peak”.
“not
well
established”
0+ state
in PDG until
1974
Removed from
1976 until
1994.
Back in PDG in 1996
2/ iMspole
1987
1979
1973
1972
The
light
scalar
controversy. The theory side...
Scalar
SU(3) multiplets
identification
controversial
f 0
a0
Some
people
claim/claimed
some
of
these
did/do not
exist(Pennington, Minkowski, Ochs, Narison…σ
as glueball
supportersin
general)
Possible
exotic
nature: tetraquarks
(Jaffe
’77) (issues
with
chiral symmerty
and
excess
of
states),
Too
many
resonances?Two
multiplets
+
glueball?..
Glueball
search: Characteristic
feature
of
non-abelian
QCD nature
f0
(980)
(800)
a0
(980)
Light scalar
nonet:
σ(500)Non-strange
heavier!!Inverted
hierarchy
Molecules (Achasov, Jülich-Bonn),…
Modified quark-antiquark models with meson interactionsVan Beveren, Rupp
Why the renewed interest in recent years?
New experimental data
Theoretical progress
“Recent“
progress: The
experiment
scattering
data from
the
70’s poor. Was
main
source
for
sigma discussions
Remarkable
improvement
from
Kl4 decays
in NA/48 (2008)
Different experiments:
(the
ones
that
most
convinced
PDG)Fermilab E791, Focus, Belle, KLOE, BES“Production” from J/Ψ, B-
and
D-
mesons, and
Φ
radiative
decays.
Very
good
statisticsClear
initial
states
and
different
systematic
uncertainties.
Strong experimental claims for wide and light
around 500 MeV
Personal Caveat: not really model independent analysis (Isobar-
Kmatrix models, etc…)
“Strong”
experimental claims for wide and light
around 800 MeV
Lots of data on decays, couplings, etc... for all light scalars
Motivation: The
f0
(600) and recent data
Very
controversial since
the
60’s.
The
reason: The
f0
(600) (and
kappa) is
a EXTREMELY WIDE.
Usually
refereed
to
its
pole:
Mostly
“observed”
in
scattering, but
no “resonance
peak”.
“not
well
established”
0+ state
in PDG until
1974
Removed from
1976 until
1994.
Back in PDG in 1996
PDG2002: “σ
well
established”
However, since
1996 still
quoted
as
Mass= 400-1200 MeV
Width= 600-1000 MeV
2/ iMspole
After 2000 also observed in production process
’95 model
by one
of
the
authors
of
PDG review
1987
1979
1973
1972
“Recent”
progress: The
theory
Many
old an new theoretical
studies
based
on
crude/simple models,
Issues
with
chiral symmetry, Adler
zeros
Often
ignoring
mixing: arguing
about
decay
modes
too simple
Often
neglecting
widths!!
Spurious
parameters
At
best…. “QCD inspired”.
Lattice
still
far, but
maybe
in the
not
too distant
future…
Hadronization missing
Suspicion: What you put in is what you get out??Improvements: With the new data this is no longer possibleModels are more respectful with chiral symmetry and include some
mixing
Dominant quark-antiquark component very disfavouredGlueball very unlikely
Tetraquark-molecule preferred
The real improvement:
Analyticity and Effective Lagrangians
The 60’s and early 70’s: Strong constraints on amplitudes from ANALYTICITY
in the form of
dispersion relations
But poor input on some parts of the integrals and poor knowledge/understanding of subtraction constants = amplitudes at
low
energy values
The 80’s and early 90’s: Development of Chiral Perturbation Theory (ChPT).(Weinberg, Gasser, Leutwyler)
It is the effective low energy theory of QCD. Provides information/understanding on low energy amplitudes
The 90’s and early 2000’s: Combination of Analyticity and ChPT(Truong, Dobado, Herrero, Donoghe, JRP, Gasser, Leutwyler, Bijnens, Colangelo, Caprini, Zheng, Zhou, Pennington...)
QCD: quarks, gluons, dynamics at high energy. Symmetries
Analyticity and Effective Lagrangians: two approaches
1st approach.
(Roy-like equations)
Use data on all waves + high energy + ChPT numerical predictions to determine amplitudes. The most precise
and model independent way of
getting poles. (Colangelo Gasser, Letwyler group, Moussallam, Paris group, Lesniak Krakow group,
UCM-UAM-Madrid Group)
2nd approach.
(Unitarized ChPT)
Use ChPT amplitudes inside dispersive integrals.
Not so precise, but all input controlled with the effective Lagrangian and QCD.
Reliable for understanding and connecting with QCD
sigma and kappa existence , mass and
width
firmly established
Data after
2000 both
scattering
and
productionDispersive-
model
independent
approaches
Chiral symmetry
correct
Outline
Scalar
Mesons:
A long story
in short
Unitarized
Chiral Perturbation
TheoryThe
Inverse
Amplitude
Method
(IAM)
Nc
behavior
of
scalars
Chiral Extrapolation: the
quark mass
dependence
One
loop
SU(3) ChPT: scalar
nonet
Two
loop
SU(2) ChPT: the
JRP, Phys.Rev.Lett. 92:102001,2004
G. Ríos and
JRP, Phys.Rev.Lett.97:242002,2006.
C. Hanhart, G. Ríos and
JRP, Phys.Rev.Lett.100:152001,2008.
Unitarized Chiral Perturbation TheoryThe Inverse Amplitude Method (IAM)
Chiral Perturbation Theory in the meson sector Weinberg, Gasser & Leutwyler
ChPT
is the most general expansion
in energies of a lagrangian made only of pions, kaons and etas
compatible with the QCD symmetry breaking
Leading order parameters: breaking scale f0
and masses
At 1-loop, QCD dynamics encoded inchiral parameters:
L1
...L8Determined from
EXPERIMENT
leading 1/Nc
behavior known from QCD
ChPT is the QCD Effective Theorybut is limited to low energies
, K ,
Goldstone Bosons of the spontaneous
chiral symmetry breakingSU(Nf
)V
SU(Nf
)A
SU(Nf
)V
QCD degrees of freedomat low energies << 4f~1.2 GeV
Partial wave unitarity(On the real axis above threshold)
t1Im
itt
1Re
1
2Im tt
exactly unitary !!
ChPT = series in p2
perturbative unitarity 2
24Im tt
...42 ttt
...)Re(Re 422
21 tttt
provides ....
2242
22
Re titttt
42
22
tttt
IAM
Unitarity and the Inverse Amplitude Method: One channel –one loop NAIVE DERIVATION
Truong
‘89, Truong,Dobado,Herrero,’90, Dobado, JRP,‘93,‘96
The
Inverse
Amplitude
Method: Results
for
one
channel
Fit
and K ELASTIC scattering
data
Preliminary Update: J. Nebrera and JRP ‘09
Truong ‘89, Truong,Dobado,Herrero,’90, Dobado JRP,‘93,‘96
EXTREMELY SIMPLE
Originally obtained from dispersion relation This allows us to go to the complex plane.
Dynamically Generates Poles: Resonances: f0
(600) or “sigma”, K*, ρ
The Inverse Amplitude Method: Results for one channel
f
(770) K*(890)
Width/2
Mass
f
pole: 440-i245 MeVDobado, Pelaez ‘96
Systematic extension to higher orders
Unitarity + Chiral Low energy expansion
The
Inverse
Amplitude
Method: Dispersive
Derivation: THE REAL THING
Write
dispersion
relations
for
G and
t4
t1Im 2
24Im tt
We
have
just
seen
that, for
physical
s
and
,2
2
ttG Define
Gtt ImIm 224
PHYSICAL cutEXACTLY Opposite
to
each
other
Subtraction
Constantsfrom
ChPT expansion
OK since
s=0G(0)=t2
(0)-t4
(0)
Up
to
NLO ChPTOpposite
to
each
other
42
22
tttt
IAM
All
together…we find AGAIN
PC is
O(p6) andwe
neglect
it
or use ChPT
Partial wave unitarity(On the real axis above all thresholds)
1ImT
11 )(Re iTT
*Im TTT
ChPT = series in p2
perturbative unitarity
224Im TTT
...42 TTT
1242
12
1 ...)Re(Re TTTTTprovides
....
exactly unitary !!
21
22422 )Re( TTTiTTTT
21
422 )( TTTTT Coupled
channel
IAM
Oller,
Oset,
JRP, PRL80(1998)3452, PRD59,(1999)074001A. Gomez Nicola and J.R.P.
PRD65:054009, (2002)
Unitarity and the Inverse Amplitude Method: Multiple channels–one loop
To the DATA !!
One-loop ChPT IAM fit to meson-meson scattering (+3% syst.)
f
f0
(980)
a0
κ(900)K* (892)ρ
low-energy 00
-1100
d
/dElab
KK
3/2 0
KK
1/2 1
KK
1/2 0
KK
00
Gómez-Nicola, JRP, Phys. Rev. D65:054009, (2002)
20
11
Incompatible sets of Data. Customarily add systematic error:
MINUIT fit :
3%,
Final error: MINUIT error + Systematic error
Identical curvesbut variation in
parameters1%, 5%
ChPT(=M) IAM fit (+3%) IAM fits L1 0.4 0.3 0.561 0.008 0.6 0.1 L2 1.35 0.3 1.211 0.001 1.2 0.1 L3 -3.51.1 -2.79 0.02 -2.79 0.14L4 -0.3 0.5 -0.36 0.02 -0.36 0.17L5 1.4 0.5 1.39 0.02 1.4 0.5 L6 -0.2 0.3 0.07 0.03 0.07 0.08 L7 -0.4 0.2 -0.444 0.03 -0.44 0.15L8 0.9 0.3 0.78 0.02 0.8 0.2
Complete Meson-meson Scattering in Unitarized Chiral Perturbation Theory
Gómez-Nicola, JRP, Phys. Rev. D65:054009, (2002)
Simultaneous description of low energy and resonances
Fully renormalized and with parameters compatible with ChPT.
Coupled channel IAM Oller, Oset, JRP, PRL80(1998)3452, PRD59,(1999)074001A. Gomez Nicola and J.R.P.
PRD65:054009, (2002)
400 600 800 1000 1200-300
-200
-100
0
100
400 600 800 1000 1200-300
-200
-100
0
100
900 950 1000 1050 1100 1150 1200-300
-200
-100
0
100
1000 1020 1040 1060 1080 1100-100
-80
-60
-40
-20
0
700 800 900 1000 1100 1200-300
-200
-100
0
100
700 800 900 1000 1100-300
-200
-100
0
100
f0(600) and f0
(980)
κ a0
K* (octet)
With the full one-loop SU(3) unitarized ChPT, we GENERATE,the following resonances, not present in the ChPT Lagrangian,as poles in the second Riemann sheet
J.R.P, hep-ph/0301049. AIP Conf.Proc.660:102-115,2003
Brief review: Mod.Phys.Lett.A19:2879-2894,2004
without a priori assumptions on on their existence or nature
a0 (980)
f0
(600)
f0 (980)
754 18
440
8
753
52
74
10
212
15
235
33
973 +39-127 11+189
-11
12 +43-12
Resonance POLE POSITIONS...
CompleteIAM
CompleteIAM
1117+24-320
cusp?
K* 889
13 24
4
Light scalar nonet(Only statistical errors)
(MeV)MRe poles (MeV)2/Im poles
Intro -
Summary
Unitarized Chiral Perturbation Theory
The Inverse Amplitude Method (IAM)Description of meson-meson scattering compatible with ChPTNo model dependence in one-channel (not in coupled channels)Both LIGHT VECTORS and SCALARS as polesNLO (1 loop) and NNLO(2 loops) IAM availableNo supurious parameter dependence in cutoffs or other parameters
Outline
Scalar
Mesons:
A long story
in short
Unitarized
Chiral Perturbation
TheoryThe
Inverse
Amplitude
Method
(IAM)
Nc
behavior
of
scalars
Chiral Extrapolation: the
quark mass
dependence
One
loop
SU(3) ChPT: scalar
nonet
Two
loop
SU(2) ChPT: the
JRP, Phys.Rev.Lett. 92:102001,2004
G. Ríos and
JRP, Phys.Rev.Lett.97:242002,2006.
C. Hanhart, G. Ríos and
JRP, Phys.Rev.Lett.100:152001,2008.
Nc
behavior of scalars
(x10-3) ChPT (=M)
IAM fits Large Nc SCALING
2L1- L2 -0.6 0.6 0.0 0.2 O(1) L2 1.4 0.3 1.2 0.1 O(Nc) L3 -3.5 1.1 -2.79 0.14 O(Nc) L4 -0.3 0.5 -0.36 0.17 O(1) L5 1.4 0.5 1.4 0.5 O(Nc) L6 -0.2 0.3 0.07 0.08 O(1) L7 -0.4 0.2 -0.44 0.15 O(1) L8 0.9 0.3 0.8 0.2 O(Nc)
Leading 1/Nc
expansion
We cannot obtain the Li from QCD, BUT their 1/Nc expansion, is known and Model Independent
Pions, kaons and etas states: )/1(),1( cNOOM The qqbar meson masses M=O(1)
and their decay constants f=O(Nc
)
Our IAM ChPT amplitudes do not
have
any other parameter hiding Nc dependencelike cutoffs, subtractions, etc...
We can thus study
the Nc scaling of the resonances
The IAM generates the expected1/Nc
scaling of established qq statesJRP, Phys.Rev.Lett. 92:102001,2004
0 5 10 15 20
0.20.40.60.8
11.21.4
MN
/M3
N
/3
Nc 0 5 10 15 20
0.20.40.60.8
11.21.4
MN
/M3
N
/3
Nc
LIGHT VECTOR MESONSqqbar states:
)/1(),1( cNOOM
0 5 10 15 20
0.20.40.60.8
11.21.4
MN
/M3
N
/3
The (770)
400 600 800 1000 1200M
- 70
- 60
- 50
- 40
- 30
- 20
- 10
0
-iG
2
Nc=3
Nc=5
Nc=10
Nc=20
-i
/2
The K*(892)
700 800 900 1000 1100 1200M
- 20
- 15
- 10
- 5
0
-iG
2
Nc=3
Nc=5
Nc=10
Nc=20
-i
/2
0 5 10 15 20
0.250.5
0.751
1.251.5
1.752
What about scalars
?
Nc
MN
/M3
N
/3
0 5 10 15 20
0.250.5
0.751
1.251.5
1.752
Nc
MN
/M3
N
/3
Similar results follow for the f0
(980) and a0
(980) Complicated by the presence of THRESHOLDS and except in a corner
of parameter space for the a0
(980)
JRP, Phys.Rev.Lett. 92:102001,2004
The
(=770MeV)
400 600 800 1000 1200M
- 1000
- 800
- 600
- 400
- 200
0
-iG
2
Nc=3Nc=5
Nc=10
Nc=20
-i
/2
The
(=500MeV)
400 600 800 1000 1200M
- 1000
- 800
- 600
- 400
- 200
0
-iG
2
Nc=3Nc=5
Nc=10
-i
/2
Since for quark antiquark states )/1(),1( cNOOM
qqNOTisqqis 1,1 22
G. Ríos and JRP, Phys.Rev.Lett.97:242002,2006.
2-like function to measure how close a resonance is to a quark antiquark
One loop
Two loops
This
2-like can be used to:
1) check
if a resonance behaves as quark-antiquark
2) Try to force
a resonance to behaves as quark-antiquark by tuning parameters
The rho always comes out naturally as a quark-antiquark
The sigma cannot be made to behave as a quark-antiquarkeven by forcing it
G. Ríos and JRP, Phys.Rev.Lett.97:242002,2006.
Large Nc behavior of UNITARIZED
TWO LOOP
ChPT
The f0
(600)
still does NOT
behave DOMINANTLY
as quark-antiquark
BUT, from Nc>8 or 10, the f0
(600) we might
be seeing a
quark-antiquark subdominant
component whose large Nc mass is
1 GeV
quark-antiquark mixingmay emerge at larger Ncat M1 GeV
The sigma:
MN
/M3
N
/ 3
These results suggest what LIGHT SCALARS ARE NOT predominantly made of...
The light scalar nonet DOMINANT component
is NOT a quark-antiquark
state
Results consistent at higher orders.The
cannot
be forced to behave as a
quark-antiquark.
At two loops, a subdominant quark-antiquark component emerges
above
1 GeV!!
(consistent with two nonet picture, the lightestnon-quark antiquark)
In agreement with similar conclusions of quark models by Rupp-Van Beveren et al.or unitary chiral approach by Oset-Oller. Or report of Tornqvist & Close J.PhysG28:R249(2002)
Outline
Scalar
Mesons:
A long story
in short
Unitarized
Chiral Perturbation
TheoryThe
Inverse
Amplitude
Method
(IAM)
Nc
behavior
of
scalars
Chiral Extrapolation: the
quark mass
dependence
One
loop
SU(3) ChPT: scalar
nonet
Two
loop
SU(2) ChPT: the
JRP, Phys.Rev.Lett. 92:102001,2004
G. Ríos and
JRP, Phys.Rev.Lett.97:242002,2006.
C. Hanhart, G. Ríos and
JRP, Phys.Rev.Lett.100:152001,2008.
Chiral Extrapolation: the quark mass dependence
ChPT provides the correct QCD dependence of quark masses as an expansion...
The LATTICE provides rigorous and systematic QCD results in termsof quarks and gluons with growing interest in scattering and the
scalar
sector.
Caveat:
small, realistic, quark masses are hard to implement.
We can study the scalars in Unitarized ChPT for larger quark masses(chiral extrapolation)
and provide a reference for lattice studies
Motivation for Chiral extrapolation
Anthropic considerations...
Truong
‘89, Truong,Dobado,Herrero,’90, Dobado, JRP,‘93,‘96
The
Inverse
Amplitude
Method: Results
for
one
channel
Fit
and K ELASTIC scattering
data + LATTICE
Preliminary J. Nebreda, JRP 2009
The (770) The f0
(600) or
Becomes narrower,and gets closer to the
new thresholdAt some points enters the first
sheet (bound state)
Becomes narrower.When in real axis
“splits” in two real poles
Pole movements with
increasing m
UChPT
SU(2) single channel
calculation
(770) versus f0(600)
Pole movements with
increasing m
Pole movements with
increasing m
To follow the position relative to threshold: normalize to m
units
-
ρ
pole-
σ
pole
Conjugate poles reach the real axis AT THRESHOLD: The rho:
- one pole in the 1st
sheet (bound state).
-
another in the 2nd
sheet in almost the same position
Pole movements with
increasing m
The sigma: 1) Conjugate poles reach the real axis
BELOW threshold:
2)
TWO real POLES on the 2nd sheet: “Splitting”
typical of scalars.
-
ρ
pole-
σ
pole
3) One moves towards threshold until it jumps to the 1st
sheet.The other remains on the 2nd
sheet in ASYMMETRIC position
If very asymmetric: sizable “molecular” componentMorgan, Pennington. PRD48 (1993) 1185
The rho mass grows
slower
than sigma
There is a “non-analyticity”
in the sigma mπ
dependence.
Resonane mass
m
dependence
Resonance width
m
dependence
Resonance width
m
dependence vs. phase space
For
a narrow
vector particle
(like
the
rho) the
decay
width
is
given
by
Phase
space
Coupling
topions
We
can calculate
the
width
variation
due
to
phase
space
reductionand
compare with
our
results. The
difference
gives
the
dependence
of
the
coupling
constant
on
the
pion
mass
Rho width
m
dependence vs. phase space
couplingalmost
independent
of
mπ(assumption in somelattice calculations)
Width behaviorexplained by phase space
Γρ
/ Γρphys
phase spaceΓρ
/ Γρphys
IAM
It
does not follow
the phase
space decrease of a Breit-Wigner:
The
dynamics
of
the
sigma decay
depends
strongly
on
the
pion(quark)
mass(Recall
that
some
pion-pion
vertices
in ChPT depend
on
the
pion
mass).
Very badapproximation
for
a wide
resonanceas the
sigma
g dependence
on
mπ
Γσ
/ Γσphys
phase spaceΓσ
/ Γσphys
IAM
Rho width
m
dependence vs. phase space
CAUTION!!! We
give
POLE MASS
in complex plane
Lattice caveats:Improved actions,Lattice spacing...Finite volume...WIDTHLESS rho
The best would be to use ChPT on the lattice....future work
Comparison with
lattice
results
for
the
rho
Comparison with
lattice
results
for
the
sigma
AGAIN CAUTION!!! We
give
POLE MASS
in complex plane + usuallattice caveats
IMPORTANT REMARKExtrapolations should take careof known scalar mass “splitting”
non-analyticity
Summary:Scalars in Unitarized Chiral Perturbation Theory
Simultaneously
resonances and low energy meson-meson scattering
with parameters compatible with ChPT
Generates Light Scalar
Nonet and vector octet
SUBDOMINANT quark-antiquark component around 1.1 GeV.(Suggests mixing with heavier ordinary scalar nonet)
SCALARS predominantly NOT
quark-antiquark states
quark-antiquark remarkably good for vectors
Nc
behavior of light resonances
M
behavior
qualitatively similar to lattice,
Caution: lattice non-zero width
and that we use pole masses
We
predict
the
parameters
giving
the
M
dependence
on M.
Consistent with chiral fits.
The
sigma mass
dependence is stronger than for the rho
In progress...
SU(3), strange resonances.... DONE !!. Surprises in KSFR.
Γ
just obeys the
BW phase
space reduction whereas Γ
does
not.Dynamical M
effects through 2 pion couplings!
Summary: 2nd part
Pion mass dependence of (770)
and f0(600) mass and width
IDEA FROM THIS WORKSHOP @ INT !!!Calculate also phase shift chiral extrapolations. Within ChPT and IAM. May be feasible in lattice in the not too distant future