1. Chiral Extrapolations of light resonances
Departamento de Física Teórica II Universidad Complutense de Madrid
Chiral Extrapolations of light resonances
from dispersion relations and
Chiral Perturbation Theory
Guillermo Ríos
In colaboration with
C. Hanhart and J. R. Peláez PRL100:152001(2008)
A. Gómez-Nicola and J. R. Peláez PRD77:056006(2008)
C. Hanhart and J. R. Peláez in progress

2. Introduction Why changing the pion mass?
The LATTICE provides rigorous and systematic QCD results in terms
of quarks and gluons.
Caveat: small, realistic, quark masses are hard to implement.
ChPT provides the correct QCD dependence of quark masses as an expansion...
Unitarized ChPT (IAM) generates poles in ππ scattering
amplitudes associated to resonances not present in the lagrangian
without a priori assumptions
We can study the resonances in Unitarized ChPT
for larger pion masses and
provide a reference for lattice studies

3. Inverse Amplitude Method (Dispersive derivation)
Outline
Inverse Amplitude Method (Dispersive derivation)
Results
Pole movements
Resonance mass dependence with pion mass
Resonance width dependence. Comparision with phase space
Comparision with lacttice
Summary

4. Chiral Perturbation Theory
Weinberg, Gasser & Leutwyler
’s Goldstone Bosons
of the spontaneous
chiral symmetry breaking
SU(2)L SU(2)R  SU(2)V
QCD degrees of freedom
at low energies << 4f~1 GeV
ChPT is the most general expansion in energies
of a lagrangian made only of pions
compatible with the QCD symmetry breaking
Leading order parameters: breaking scale fπ and mπ
At 1-loop, QCD dynamics encoded in
Low Energy Constants
ππ scattering: l1,…,l4
Determined from experiment
ChPT is the QCD Effective Theory
but is limited to low energies

5. Elastic Inverse Amplitude Method (Dispersive derivation)
The analytic structure of 1/t (right cut, left cut and possible poles) allows
us to write a dispersion relation for 1/ t substracted at the Adler zero, sA
On the right cut we
use elastic unitarity
and approximate the Adler
zero by its LO approximation
Right cut imaginary part known exactly
LO very good approximation for s’ far from sA

6. Elastic Inverse Amplitude Method (Dispersive derivation)
The analytic structure of 1/t (right cut, left cut and possible poles) allows
us to write a dispersion relation for 1/ t substracted at the Adler zero, sA
On the left cut
we use ChPT
Left cut weighted at low energies where ChPT valid
even more suppresed when s is in the physical and
resonace region (near right cut)

7. Elastic Inverse Amplitude Method (Dispersive derivation)
The analytic structure of 1/t (right cut, left cut and possible poles) allows
us to write a dispersion relation for 1/ t substracted at the Adler zero, sA
This is exactly the dispersion relation
for –t4 / t22 except for the pole contribution

8. Elastic Inverse Amplitude Method (Dispersive derivation)
The pole contributions read
We use ChPT to evaluate the derivatives of t at the Adler zero
This is a low energy point → ChPT perfectly justified

9. physical and resonance
Elastic Inverse Amplitude Method (Dispersive derivation)
In the end
we arrive at
If we set A(s)=0 we get the standard IAM. This is the case of the p wave
Note: A(s) counts O(p6) and is numerically small except near s=sA
- A(s)
The differences with
the standard IAM are
less than 1% in the
physical and resonance
region

10. Elastic Inverse Amplitude Method (Dispersive derivation)
The IAM satisfies exact unitarity and the chiral series is
recovered when reexpanding at low energies
Describes data up to ~ 1 GeV
Generates poles on the second Riemann sheet associated to resonances. In ππ scattering we find the ρ and the σ
Derived from analyticity, unitarity and ChPT.
No model dependencies, just aproximations.
Use of ChPT perfectly justified, always used at low energies
Left cut and Pole Contribution correct up to NLO. Quark mass
dependence in LC un PC correct up to NLO
Has the Adler zero at its NLO approximation and has no spurious
pole

11. Elastic Inverse Amplitude Method fit to data
Fit pion scattering data (μ=770 MeV)
Only fit l1 and l2
lr1 = -3.7 ± 0.2
lr2 = 5.0 ± 0.4
For l3 and l4 we take
the values of
Gasser and Leutwyler
lr3 = 0.8 ± 3.8
lr4 = 6.2 ± 5.7
- lri does not depend on mπ

12. Elastic Inverse Amplitude Method
for narrow resonances
Mρ = 753 MeV Γρ = 150 Mev
Im t11
Im t00
we take
as a definition for the
wide sigma
Mσ = 443 MeV
Γσ = 432 MeV

13. Changing the pion mass (Applicability)
How high can we make the pion mass?
We do not want to spoil the chiral expansion
SU(3) ChPT works
well with kaon
masses ~ 494 MeV
with mπ < 500 MeV
we are OK
But we are working with SU(2) and there are not kaons. We do not
want to reach the kaon threshold
For mπ = 500 MeV we
have mK ~ 600 MeV
Being generous
mπ = 500 MeV is our limit
of applicability

14. Comparison of (770) and f0(600) or  movements with increasing pion mass
Becomes narrower, reaches real axis at threshold (also increasing) and jumps into the 1st sheet (bound state)
The rho
Becomes narrower,reaches real axis below threshold
on 2nd sheet and split in two poles
The sigma

15. Pole Movements (normalized to mπ)
sigma pole
rho pole

16. Pole Movements - ρ pole - σ pole The rho poles reach the real axis
just at threshold and one
of them jumps into the 1st
sheet becoming a bound
state. the other one stays
on the 2nd sheet in almost the
same position
The sigma goes below
threshold with a
siezable width.
Meets its partner on
the real axis and
they split in two poles
on the real axis.
One of them moves toward
threshold and jumps into the
1st sheet becoming a bound state
two asymmetric poles in
different sheets for
scalar channels indicates
siezable molecular component
Morgan, Pennington. PRD48 (1993) 1185
Baru et al. PLB586(2004)53
Similar pole movements
are found with quarks models
E. van Beveren et al. PRD74,037501(2006)
and with Roy equations
G. Colangelo, private communication

17. Simple interpretation of pole movements
These movements are generic for s-wave and p-wave resonances
Simple S-matrix parametrization (in k-space) with resonance pole at k=k0 – iγ :
Partial wave with angular momentum
p-wave
poles
s-wave
poles

18. Simple interpretation of pole movements
In s-plane this reads:
For p-wave:
-Reach the real axis at threshold
For s-wave:
-Can reach threshold with a sizeable width
-Reach the real axis below threshold

19. Simple interpretation of pole movements
ρ pole movement k-plane

20. Simple interpretation of pole movements
σ pole movement k-plane
This behavior is
generic for an
s-wave resonance.
The point where it
occurs is especific
of QCD and is
what we estimate
mπ = (329 ± 4) MeV

21. Mass dependence on mπ
The rho mass grows slower than sigma

22. Width dependence on mπ

23. Width behavior comparison with phase space
For a narrow vector particle (like the rho) the decay width is given by
Phase space
Coupling to
pions
We can calculate the width variation due to phase space reduction
and compare with our results. The difference give the dependence
of the coupling constant on the pion mass

24. Width behavior comparison with phase space
Γρ / Γρphys phase space
Γρ / Γρphys IAM
Width behavior
explained by
phase space
Coupling
constant almost
independent of mπ

25. Γσ / Γσphys phase space Γσ / Γσphys IAM Very bad approximation for
In contrast, the sigma “width” does not follow the decrease in phase space
of a Breit-Wigner resonance:
Γσ / Γσphys phase space
Γσ / Γσphys IAM
Very bad
approximation for
a wide resonance
as the sigma
Differences with only
phase space reduction
g dependence on mπ
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.

26. Rho mass comparison with lattice
We find a cualitative agreement with lattice results

27. Outlook. In progress... Calculation at NNLO
Main diffuculty: More LECs at NNLO LECs and poorly known
We use lattice data to constrain their values
a20 dependence on mπ
fπ dependence on mπ
ETMC hep-lat/
JLQCD hep-lat/
S. Beane et al. PRD77,014505(2008)
S. Beane et al. PRD77,014505(2008)

28. We have calculated the ρ and σ pole mass and width
Summary
We have calculated the ρ and σ pole mass and width
dependence on mπ from the IAM (model independent up to NLO)
Both resonances become bound states for large mπ but they
approach this limit in a very different way
The rho mass grows slower than the sigma mass
The ρππ coupling is almost mπ independent while the σππ coupling
is strongly mπ dependent
We find a qualitative agreement with lattice calculations for the
rho mass
To do so we have presented a modified IAM to account
properly for the Adler zero region

29. Thank you!