Transcript Folie 1

A chiral model for low-energy
hadrons
Francesco Giacosa
In collaboration with
Achim Heinz, Stefan Strüber, Susanna Gallas,
Denis Parganlija and Dirk Rischke
Frankfurt University
Hirschegg 2011
Francesco Giacosa
Motivation
• Development of a a (chirally symmetric) linear sigma
model for mesons and baryons
• Study of the model for T = μ =0 (spectroscopy in vacuum)
(decay, scattering lengths,…)
Interrelation between
these two aspects!
• Second goal: properties at nonzero T and μ
(Condensates and masses in thermal/matter medium,…)
Francesco Giacosa
Fields of the model:
• Quak-antiquark mesons: scalar, pseudoscalar, vector
and axial-vector quarkonia.
• Additional meson: The scalar glueball
(and evt tetraquarks)
• Baryons: nucleon doublet and its partner
(in the so-called mirror assignment)
How to construct the model:
• (Global) Chiral symmetry: SUR(Nf)xSUL(Nf)xUV(1)
• Retain operators of fourth order (dilatation invariance)
Francesco Giacosa
Mesonic sector (Nf = 2): 16 quark-antiquark fields
4 scalar and 4 pseudoscalar fields
chiral transformation:
  (  i )t 0  (a0  i )  t
  U R U L
J PC  0  
J PC  0  

U R , U L  SU (2)
 
1
(uu  d d )
2
a 0  ud , du,
1
(uu  d d )
2
  ud , du,
1
(uu  d d )
2
1
(uu  d d )
2
First Problem: the scalar mesons!
4 vector and 4 axialvector fields

R  (  f1 )t  (   a1 )  t



0


chiral t ransf.:


R URR U
L  U L LU L

R
0

1
(uu  d d )   ud , du,
2
1
(uu  d d )
2

L  (  f1 )t  (   a1 )  t

J PC  1
J PC  1 
f1 
1
(uu  d d )
2
a1  ud , du,
1
(uu  d d )
2
Francesco Giacosa
Problem of scalars
 is
f 0 (600)
or
f 0 (1370) ???
a0 is a0 (980)
or
a0 (1450) ???
This is an important issue. One shall do the correct assignment.
Many models use  = f0(600) (Lσm, NJL). This has been the usual picture at
nonzero temperature/density.
However, this assignment is found to be incorrect in many studies at zero
temperature (Phenomenology, Large-Nc, Lattice) .
The quantitative effects of scalars both in the vacuum and in a medium are large!
Francesco Giacosa
Lagrangian in the meson sector
In the chiral limit (H=0) two dimensional parameters: Λ (dilatation invariance) and
c (anomaly). Both from the gauge sector.
Here the large Nc counting shows that all the state are quarkonia. (  Nc , 2  Nc-1,... )
Only exception: the glueball.
Francesco Giacosa
Scenario I:   f 0 (600), a0  a0 (980)and G  f 0 (1500)
M  550 MeV
from -scattering.
  f 0 (600)     200 MeV!!!
This is wrong! The experimental value is much larger (500 MeV).
Note, the role of axial-vector mesons is crucial for this result.
We conclude: the assignment is unfavoured!
One should start from:
  f 0 (1370)
and
Details in: Denis Parganlija, F.G., Dirk H. Rischke
Phys.Rev.D82:054024,2010; arXiv:1003.4934 [hep-ph].
a0  a0 (1450)
Francesco Giacosa
Z = 1 corresponds to the (unphysical) decoupling of axial-vector mesons.
The reason for the big change is technical: the a1- mixing.
Details in: Denis Parganlija, F.G., Dirk H. Rischke
Phys.Rev.D82:054024,2010; arXiv:1003.4934 [hep-ph].
Francesco Giacosa
Scenario II: 
 f 0 (1370), a0  a0 (1450)and G  f 0 (1500)
10 free parameters. 6 are fixed through
m , m , mN , ma1 , f , a1 
For the remaining 4: fit to 5 exp quantities:
1


0.25   
(
u
u

d
d
)


2

0.75 
G  gg


a
G
G a ,    (698 39 MeV) 4
 f 0 (1370)   0.75

  

 f 0 (1500)    0.25
C 4   2´ s
to be compared with
a
 2´ s G
G a ,    ( 300 t o 600 MeV) 4
sum rules
lattice
Francesco Giacosa
f 0 (1370) decays mostly in the pion-pion channel. (exp. still ddebated)
Consistent phenomenological picture.
However, the situation can also not be that simple. The
light scalars below 1 GeV should also be included.
What are they? Tetraquarks?
S. Janwoski, D. Parganlija, F.G., D. Rischke, in preparation
Francesco Giacosa
Tetraquark: outlook and small excursus at nonzero T
A possibility is to interpret the light scalar states below 1 GeV
[f0(600), k(800), f0(980) and a0(980)]
as diquark-antidiquark objects: these are the Jaffe’s tetraquarks.
The Nf=3 case is an outlook. Mixing of these tetraquark-quarkonioa takes place.
Black et al, Phys. Rev. D 64 (2001),
F.G., Phys.Rev.D 75,(2007)
For Nf=2 only one tetraquark survives. In this case we studied a simplified system

at nonzero T.
1
T heresonance f 0 (1370)   
2
(uu  d d ) is thechiralpartnerof thepion  .
1
T heresonance f 0 (600)    [u, d ][u, d ] is an extra- scalar state
2
Increasing of mixing:
1) Tc decreases
2) First order softened
3) Cross-over obtained for g large enough
Achim Heinz, Stefan Strube, F.G., Dirk H. Rischke
]
Phys.Rev.D79:037502,2009; arXiv:0805.1134 [hep-ph
Francesco Giacosa
Baryon sector
strategy
Nucleon and its chiral partner : global chiral symmetry and operators of 4-th-order
(Axial)Vector mesons are included
Mirror assignment: (C. De Tar and T. Kunihiro, PRD 39 (1989) 2805)
1, R  U R 1, R
1, L  U L 1, L
2, R  U L 2, R
2, L  U R 2, L
A chirally invariant mass-term is possible!

m0 1, L 2, R  1, R 2, L   2, L 1, R   2, R 1, L

Francesco Giacosa
Mass of the nucleon
N  N (940)
N *  N * (1535)
 M N  M N* 

2
m
0


  ar cosh 
M N ,N*
1

4m02  (...) 2  (...)
2
If m0  0  only thequark condensategeneratesthemasses.
m0 parametrizes the contribution which does not stem from the quark condensate
Crucial also at nonzero temperature and density
also in the so-called quarkyonik phase: L. McLerran, R. Pisarski Nucl.Phys.A796:83-100,2007
Francesco Giacosa
Axial couplings without vector d.o.f
Without vector and axial-vector mesons:
g AN  1
g AN  1
g AN *  1
g AN *   g AN
g AN*  -1
g AN  1.26 (exp), g AN*  0.2 (latt) cannot be described without vm.
However, with (axial-)vector mesons are introduced the axial couplings are modified.
A description of the axial couplings is then possible.
Result for m0:
m0  460 136 MeV
Using g AN  1.26 (exp), g AN*  0.2 (latt) and N*N  67 MeV
Francesco Giacosa
Pion-nulceon scattering lengths
a0  (6.04  0.63) 104 MeV-1
a0(exp)  (6.4  0.1) 104 MeV-1
a0  (from 20 to 20104 ) MeV-1
a0(exp)  (8.8  7.2) 104 MeV-1
Large theoretical uncertainty due to the scalar-isosocalar
Importance of both vector mesons and mirror assignment in order to get these results
Francesco Giacosa

Where does m0 comes from?
m0 1, L 2, R  1, R 2, L   2, L 1, R   2, R 1, L

By requiring dilatationinvarianceoneshould modify themass - termsas :
a  bG1,L 2,R  1,R 2,L   2,L 1,R   2,R 1,L 
tetraquark dilaton
By shifting:   0   , G  G0  G
one has: m0  a0  bG0
m0 originates form the tetraquark and the gluon condensates.
Note, also, a tetraquark exchange naturally arises in nucleon-nucleon interactions
This is an interesting direction for further studies
Francesco Giacosa
Summary and outllok
Chiral model for hadrons based on dilatation invariance and global symmetry
Important role of (axial)vector mesons in all phenomenology
Scalar quarkonium and glueball above 1 GeV (effects in the medium)
Contribution to the nucleon mass which does not stem fro the chiral condensate
Nf=3, additonal tetraquark states, weak decays
Francesco Giacosa
Thank You
for the
attention
Francesco Giacosa
Global chiral symmetry and VMD
Here we follow a different way: we consider –as in QCD- only global chiral
symmetry and we restrict to terms up to order four.
M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)
D. Parganlija, F. G. and D. Rischke, AIP Conf. Proc. 1030, 160 (2008)
Denis Parganlija, F.G., Dirk H. Rischke, Phys.Rev.D82:054024,2010; arXiv:1003.4934 [hep-ph].
We use another realization of VMD: the photon is first introduced via
minimal substitution. The form factors of hadrons receive corrections
from the (gauge-invariant) mixing term   F 
In this realization of VMD no need of universality. Details in:
[H. O’ Connell, B. Pearce, A. Thomas and A. Williams, Prog.Part.Nucl.Phys.39:201-252,1997
Francesco Giacosa
Why only up to 4-th order



Tr    ,
Tr   
 ,
2

Tr   
 ,
4

Tr       

2
One could think that the reason is renormalizability! However, a low-energy theory
of QCD does not need to be renormalizable.
Large-Nc arguments allow to neglect some, but not all higher order diagrams
The reason is: dilatation invariance of the interaction terms and finiteness of potential
G 1

Vdilaton (G)  G 4  log  
 4



Λ only dimensionful param.-which
breas dilatation inv.- in the chiral
limit!
G  G0  .


V (G, )  G2Tr   ... G4Tr       ....
Francesco Giacosa
Local vs global chiral symmetry
In many works: Local Chiral symmetry: SUR(2)xSUL(2).
[S. Gasiorowicz and D. A. Geffen (1969), U. G. Meissner (1988), P. Ko and S. Rudaz (1994)]
In agreement with the VMD-Sakurai:   A
Coupling universality. Small number of parameters
However: ρ decay and other observables are not correct.
To keep ‘local chiral symmetry’ further terms of higher orders
must be included
Francesco Giacosa
Contributions to the rho mass
M 2 
2 (...) m12
M a21  M 2   2 (...)
quark
condensate
The quark condensate decreases with T; the behavior of the rho mass strongly
depends on the composition
In thelocalcase : M 2  m12
(no contribution of the quark condensate)
In general we recognize 3 scenarios
Francesco Giacosa
M 2 
3 scenarios
2 (...) m12
quark
condensate
Ma1
Ma1
Mρ
Mρ
Case A: m1 dominates
T
Mρ
T
Case B: m1 and condensate
have similar contributions
Ma1
Case C: the condensate dominates
Mρ
T
T
Francesco Giacosa
Explicit calculation in the local case
• Calculation in the chirally local case by: S. Strüber and D. Rischke, Phys.Rev.D77:085004,2008
• 2Pi (CJT) formalism: system of coupled Dyson-Schwinger eqs at 2 Pi
• In this work –although with the simplified local case- the first calculation at nonzero T
has been performed.
• Role of vector mesons in the chiral phase transition important
Ma1
M 2  m12
Mρ
As expected –case A. We would like to do it in the global model. Outlook for the future,
Francesco Giacosa
Fixing the free parameters in Scenario I
In order to fix the parameters we have first to specify the assignment
of the scalar mesons
  f 0 (600)
and
a0  a0 (980)
This is the problematic assignment. Nevertheless, as also done in many studies
we test it.
For the remaining 5 parameters use:
  
a1  
f1  a0
a00
a02
One can fix the parameters…BUT then the f0(600) is too narrow.
Francesco Giacosa