Vector and axialvector current correlators within the instanton model of QCD vacuum'

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Vector and axial-vector current correlators
within the instanton model of QCD vacuum.

• A.E. Dorokhov (JINR, Dubna)

V and A current-current corelators OPE vs cQCD
V-A and V correlator and ALEPH data Hadronic
contribution to muon AMM in Instanton
model Topological susceptibility Conclusions
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Vector and Axial-Vector correlators.

• V and A correlators are fundamental quantities of
  the strong-interaction physics, sensitive to
  small- and large-distance dynamics. In the limit
  of exact isospin symmetry they are

where the QCD currents are
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• The two-point correlation functions obey
  (suitably subtracted) dispersion relations

where the imaginary parts of the correlators
determine the spectral functions
ALEPH and OPAL measured V and A spectral
functions separately and with high precision
from the hadronic t-lepton decays
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ALEPH data
r
a1
r
a1
pQCD
pQCD
Inclusive v-a spectral function, measured by the
ALEPH collaboration

• Inclusive va spectral function,
• measured by the ALEPH collaboration

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Vector Adler Function (pQCD and c QCD).
Adler function is defined as
At short distances pQCD predicts (MSbar)
where
and
At large distances pQCD predicts only
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pQCD and cQCD predictions for Adler function
D(Q)
pQCD
?
cQCD
Constituent Quarks Current Quarks
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Adler Function and ALEPH data
Take an ansatz for the spectral function
where
and find the continuum threshold s0 from duality
condition
Using the experimental input corresponding to the
t--decay data and the perturbative expression
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One find matching from duality condition
pQCD
pQCD
ALEPH
ALEPH
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Adler function and ALEPH data
(N)NLO LO
Asymptot Freedom
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Nonlocal Chiral Quark model (cNQM)
SU(2) nonlocal chirally invariant action
describing the interaction of soft quarks
I
Spin-flavor structure of the interaction is given
by matrix products
(Instanton interaction G'-G)
For gauge invariance with respect to external
fields V and A the delocalized quark fields are
defined (with straight line path)
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Quark and Meson Propagators
The dressed quark propagator is defined as
The Gap equation
I
has solution
Mconst.
Mcurr.
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qq scattering matrix
with polarization operator
has poles at posiitons of mesonic bound states
The pion vertex
with the quark-pion constant gpqq satisfying the
Goldberger-Treiman realtion
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Conserved Vector and Axial-Vector currents.
The Vector vertex
as in pQCD
AF
NonLocal part
WTI
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The iso-triplet Axial-Vector vertex has a pole at
Pion pole
AWTI
The iso-singlet Axial-Vector vertex has a pole at
1-GJPP(q2)
h meson pole
anomalous AWTI
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Current-current correlators
Current-current correlators are sum of dispersive
and contact terms
The transverse and longitudinal part of the
correlators are extracted by projectors
I
Contact term
Dispersive term
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Model parameters and Local matrix elements
Profiles for dynamical quark mass in the
Insatanton model
and for the Constrained Instanton it is
approximated by Gassian form
Parameters of the profile are fixed by the pion
weak decay constant
and the quark condensate
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With the model parameters fixed as
one obtains
The couplings Gr,wV and Ga1A are fixed by
requiring that scattering matrix poles coincide
with physical meson masses
The (instanton) contribution to the gluon
condensate appears through using the gap
equation and estimated as
Other condensates are
Averaged quark virtuality in QCD vacuum
ltA2gt condensate
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Current-current correlators in cNQM
V correlator
and the difference of the V and A correlators
One may explicitly varify that the Witten
inequality is fullfiled and that at Q20 one
gets the results consistent with the first
Weinberg sum rule
Above we used definitions
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V-A cNQM vs ALEPH
cQCD
cNQM
OPE pQCD
ALEPH
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Low-energy observables and ALEPH-OPAL data

• E.m.p mass difference. By using DGMLY sum rule

one has
which is in remarkable agreement with the
experimental number (after subtracting md-mu
effect)
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2. Electric polarizability of the charged pion is
defined as
with help of the DMO sum rule
Model calculations provides
and
While from experiment one has
and
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NcQM Adler function and ALEPH data
M(p)
Quark loop
ALEPH
NcQM
AS
r,w
Quark loop
Mesons
NJL
Meson loop
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LO Hadronic contribution to gm-2
The calculations are based on the spectral
representation
which is rewritten via Adler fucntion as
Phenomenological estimates give
and from NcQM one gets
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Other model approaches
Phenomenological estimate
NcQM
Extended Nambu-Iona-Lasinio (Bijnens, de Rafael,
Zheng)
Minimal hadronic approximation (Local
duality) (Peris, Perrottet, de Rafael)
Lattice simulations (Blum Goeckler et.al. QCDSF
Coll.)
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Light-by-Light contribution to muon AMM
Vector Meson Dominance like model (Knecht,
Nyffeler)
VMD OPE (Melnikov, Vainshtein)
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LO vs NLO corrections
1 from phenomenology, 10 from the model
NO phenomenology, 50 from the existing
model, The aim to get 10 accuracy
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Singlet axial-vector current correlator and the
topological susceptebility
Due to anomaly singlet axial-vector current is
not conserved
Longitudinal part of singlet correator is
related to topological susceptibility
by
OPE, SVZ
Crewther theorem
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Topological susceptibilty in NcQM
Model predicts
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Topological susceptibility vs Q2 predicted by NcQM
pQCD
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Conclusions

• Non-local chiral quark model NcQM is appropiate
  for the study of vacuum and light meson internal
  structure.
• it is consistent with low-energy theorems and its
  predictions of the local matrix elements
  (low-energy constants Li, form factors slopes,
  etc.) are close to the predictions of the local
  effective models.
• However, the non-locality allows us effectively
  resum infinite number of local matrix elements.
  This property is crucial in attemps to predict
  the form factors in a wide kinematical region and
  to extract asymptotic (light-cone) distributions
  like Distribution Amplitudes, (Generalized)
  Parton Distributions, etc.
• The nonlocality may be naturally attributed to
  existance of QCD instantons
• We shown agreement of the model predictions on
  V-A correlator with ALEPH-OPAL data, pion
  transtion form factor with CLEO, pion e.-m. form
  factor with JLAB.