Diapositiva 1

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Departamento de Física Teórica II.
Universidad Complutense de Madrid
Nature of the sigma meson from the Nc dependence
of unitarized ChPT
G. Ríos and J. R. Peláez
PRL100152001(2008)
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Introduction
Controversy on the nature of the sigma meson...
The ? and s resonances are model independently
generated with the elastic Inverse Amplitude
method, whose Nc dependence is known from QCD
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Outlook
Chiral Perturbation Theory and the Inverse
Amplitude Method
The 1/Nc expansion. Definition of states.
Nc scaling of the IAM
Results at O(p4)
Results at O(p6)
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Chiral Perturbation Theory
Weinberg, Gasser Leutwyler
ChPT is the most general expansion in energies
of a lagrangian made only of pions compatible
with the QCD symmetry breaking
Leading order parameters
At higher orders, QCD dynamics encoded in Low
Energy Constants determined from experiment
pp scattering
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Inverse Amplitude Method
The (elastic) Inverse Amplitude Method (IAM) uses
elastic unitarity and ChPT to evaluate a
dispersion relation for 1 / t
Imaginary part of 1/t known exactly on RC
Unitarity condition
ChPT, being an expansion satisfies unitarity
only perturbatively
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Inverse Amplitude Method
The analytic structure of 1/t (right cut, left
cut and possible poles) allows us to write a
dispersion relation for
This is exactly t2
Exact dispersion relation for -t4
substraction constants use ChPT
On the right cut we use elastic unitarity
On the left cut we use ChPT
PC is O(p6). We neglect it
Right cut kown exactly (elastic approx.)
Calculated at a low energy point good
approximation
Left cut weighted at low energies, where ChPT
valid
It is small for all s
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Inverse Amplitude Method
The IAM can be systematiclly improved taking into
account higher chiral terms
Dispersion relation for
It has no Right Cut since, on RC
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Inverse Amplitude Method

• The IAM satisfies exact unitarity and matches the
  chiral expansion when reexpanding at low energies
• Describes data up to 1 GeV
• Generates poles on the second Riemann sheet
  associated to resonances. In pp scattering we
  find the ? and the s
• Derived from analyticity, unitarity and ChPT.
• No model dependencies, just aproximations.
  Use of ChPT perfectly justified, always used at
  low energies
• Left cut and substraction constants correct up to
  the given ChPT order used.

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Inverse Amplitude Method
for narrow resonances
M? 750 MeV G? 150 Mev
Im t11
Im t00
we take
as a definition for the wide sigma
Ms 440 MeV Gs 440 MeV
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The 1/Nc expansion
The 1/Nc expansion provides a clear definition of
states
Their masses and widths scale as
The QCD Nc dependence is implemented in ChPT
through the LECs
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The 1/Nc expansion
This is only relevant near Nc 3
This is what gives information about the dominant
component of the Nc 3 physical state
We dont expect the IAM to work well for very
large Nc
Weak interacting limit ? dispersion relation not
dominated by the exactly known right cut
RC 1/Nc suppresed. At Nc3 the IAM describes data
and resonances within 10-20 errors ? 100 error
at Nc 15-30
We estimate the IAM to give reasonable results
for Nc lt 15 30 at most
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Results O(p4)
Fit pion to scattering data
?2data 1.1 (µ770 MeV)
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Results O(p4)
The ? follows the qqbar Nc scaling. The s does
not.
The ?
The s
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With the O(p4) IAM we generate poles in the
amplitudes
This suggests there may be a cancellation which
make the other terms unimportant. Is this
cancellation due to a particular election of LECS?
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Allowing for generous variations of the LECS we
will check the robustness of the O(p4) results
? qqbar, s not qqbar. LECS not fine-tuned
We will check the O(p6) results.
They may reveal a subdominant qqbar component for
the s.
To do so we will present a method to quantify how
much a resonance Nc behavior is like a qqbar
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Taking into account subleading terms, a resonance
can be considered qqbar if
We can calculate the expected M and G at some Nc
from their value at Nc -1(and take the subleading
contributions as uncertainties)
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We constrct the following ?2-like
function If a state is predominantly
qqbar, If it is NOT predominantly qqbar,
We can know from what Nc a resonance starts
behaving as a qqbar By minimizing ?2qq we
can constraint a qqbar behavior on the
resonances
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Results O(p4)
in order to see if our result is due to some
particular LECs election we make a fit minimizing
?2qq for the s
Even if we try we do not manage to constraint the
s to behave as a qqbar ?2qq,s 125 Data is
worse fitted ?2data 1.4
The s not qqbar result is not due to some
fine tuning between the LECs
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Results O(p6)
At O(p6) we find some sets of LECs where the rho
does not behave as a qqbar
We can only trust our results for the sigma if we
correctly reproduce the qqbar behavior of the rho
?2data 1.1 ?2qq.? 0.93 ?2qq.s 15
If we impose qqbar Nc behavior of the ? we get a
not qqbar s (but much more qqbar than at O(p4))
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Results O(p6)
At larger Nc O(p6) terms become more
important than O(p4) loop diagrams
qqbar behavior
The O(p6) calculation seems to reveal a
subdominant qqbar component for the s with a
mass around 2.5M3 1GeV
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Results O(p6)
At O(p6) we can find a qqbar bahavior for the ?
(?2qq.? 0.93) When doing so we find that the
dominant component of the s is not a qqbar
(?2qq,s 15 , M and G rise as Nc increases
near real life Nc 3) A subdominant qqbar
component seems to appear at larger Nc with
mass around 1 GeV
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Results O(p6)
Can we find a LECS set that makes this qqbar
component become dominant?

We get a worse data fit, ?2data 1.4
We try a fit minimizing ?2qq,s
We spoil the qqbar behavior of the ?, ?2qq,? 2.0
Even so we find ?2qq,s 3.5
With the O(p6) calculation we cannot find a
qqbar dominant component for the s
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Results O(p6)
How much of the subdominant qqbar behavior for
the s can we accommodate without spoiling that
of the ??
The ? behavior deteriorates a little ?2qq,? 1.3
We fit minimizing both ?2qq,? and ?2qq,s.
The s does not behave predominantly as
qqbar (?2qq,s 4) but it starts behaving as a
qqbar (?2qq,s 1) from Nc 6
The qqbar component of the s would become
dominant at best at Ncgt6
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Summary
Nc scaling of the resonances genertad in the IAM.
Comparison to qqbar behavior. Method to determine
quantitatively how much close a resonance Nc
depence is to a qqbar behavior
? as qqbar and s not predominantly qqbar.
This result does not depend on the specific
choice of LECS
Extension to O(p6) confirming the the stability
of the O(p4) conclusion and showing that a
possible mixing with a subdominant qqbar
component around 1GeV for the s could exist
This subdominant component would become dominant
at best around Nc gt 6, but our main fit suggest
an even bigger supression
Thank you!