γ*π0γ Transition Form Fatctor: QCD & BaBar
Adnan Bashir
Michoacán University, Mexico
Argonne National Laboratory, USA
Kent State University, USA
March 15, 2012
Jefferson Laboratory, USA
Contents
• Introduction
• Schwinger-Dyson Equations
Quark Propagator: Quark Mass Function
Pion Bethe-Salpeter Amplitude
Quark-Photon Vertex
•
Pion Electromagnetic Form Factor
•
Pion Transition Form Factor
• Conclusions
Introduction
The
transition form factor is measured
through the process:
Introduction
The
transition form factor:
CELLO H.J. Behrend et.al., Z. Phys C49 401 (1991).
0.7 – 2.2 GeV2
CLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998).
1.7 – 8.0 GeV
The leading twist pQDC calculation was carried out
in:
2
G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980).
BaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2
Introduction
Transition form factor is the correlator of two currents :
Collinear factorization:
T: hard scattering amplitude with quark gluon sub-processes.
is the pion distribution amplitude:
In asymptotic QCD:
Schwinger-Dyson Equations
• Schwinger-Dyson Equations (SDE) are the fundamental
equations of a field theory.
• SDE for QCD have been extensively applied to meson
spectra and interactions below the masses ~ 1 GeV.
• SDE have been employed to calculate:
the masses, charge radii and decays of mesons
elastic pion and kaon form factors
P. Maris, C.D. Roberts, Phys. Rev. C56 3369 (1997).
pion and kaon valence quark-distribution functions
P. Maris, P.C. Tandy, Phys. Rev. C62 055204 (2000).
nucleon
form factors
D. Jarecke, P. Maris, P.C. Tandy, Phys. Rev.
C67 Rev.
035202
(2003). (2011).
T. Nguyen, AB, C.D. Roberts, P.C. Tandy, Phys.
C83062201
G.
“Collective
Eichmann,
Perspective
et. al., Phys.
on advances
Rev. C79
in 012202
DSE QCD”,AB
(2009)., L. Chang, I.C. Cloet, B.
D.
El Bennich,
Wilson, L.
Y. Chang
Liu, C.D.
and Roberts,
C.D. Roberts,
P.C. Tandy,
Phys. Rev.
arXiv:1201.3366[nucl-th]
C85 025205 (2012).
Schwinger-Dyson Equations
Rainbow-ladder truncation
SDE: Full quark propagator:
Schwinger-Dyson Equations
Contact interaction:
Schwinger-Dyson Equations
Bethe-Salpeter amplitude for the pion:
Goldberger-Triemann
relations:
Schwinger-Dyson Equations
Pion Form Factor:
Thus the pseudo-vector
component of the BSamplitude dictates the
transition of the pion
form factor to the
perturbative limit.
P. Maris and C.D. Roberts, Phys. Rev. C58 3659-3665 (1998).
Schwinger-Dyson Equations
For the contact interaction:
Employing a proper time regularization scheme, one can
ensure (i) confinement, (ii) axial vector Ward
Takahashi identity is satisfied and (iii) the corresponding
Goldberger-Triemann relations are obeyed:
Schwinger-Dyson Equations
D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990)
AB, M.R. Pennington Phys. Rev. Phenomenology
D50 7679 (1994)
A. Kizilersu and
M.R. Pennington Phys. Rev. D79 125020 (2009)
Gauge
Lattice
Covariance
L. Chang, C.D.
Roberts, Phys. Rev. Lett. 103 081601 (2009)
AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D83 033003 (2011)
AB, R. Bermudez, L. Chang, C.D. Roberts, arXiv:1112.4847 [nucl-th].
Significantly, this last ansatz
contains nontrivial factors
Quark-photon/
Perturbation
Multiplicative
quark-gluon
associated
is solely
Theorywith those tensors whose appearance
Renormalization
vertex
driven by dynamical chiral symmetry
breaking.
It yields gauge independent critical coupling in QED.
Quark-photon Vertex
It also reproduces large anomalous magnetic moment for
electrons in the infrared.
Pion Electromagnetic Form Factor
Within
The pattern
the rainbow
of chiral
ladder
symmetry
truncation,
breaking
the dictates
elastic the
electromagnetic
momentum dependence
pion form
of the
factor:
elastic pion form factor.
L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C81 065202 (2010).
Pion Electromagnetic Form Factor
When do we expect perturbation theory to set in?
Perturbative
2<9isGeV
2 electromagnetic
Momentum
transfer
primarily
shared equally
Jlab 12GeV:
2<QQ
and (Q/2)
among
quarks as
BSA
is peaked
transition
pion
form
factors.at zero relative momentum.
Pion Electromagnetic Form Factor
• We can dress quark-photon vertex:
• The corresponding IBS-equation thus yields:
H.L.L. Robertes, C.D. Roberts, A. Bashir, L.X. Gutiérrez and P.C. Tandy,
Phys. Rev. C82, (065202:1-11) 2010.
Pion Electromagnetic Form factor
• Dressed quark-photon vertex with ρ-pole on the timelike axis does not alter the asymptotic behaviour of the
form factor for large space-like momenta.
H.L.L. Robertes, C.D. Roberts, A. Bashir, L.X. Gutiérrez and P.C. Tandy,
Phys. Rev. C82, (065202:1-11) 2010.
Pattern of DCSB & Experimental Signatures
Pattern of DCSB & Experimental Signatures
Quark propagator: quark mass function
Pion Bethe-Salpeter amplitude
Dressed quark-photon vertex
QCD based prediction
through SDE
Contact interaction
PQCD prediction
H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys.
Rev. C82, (065202:1-11) 2010.
Pattern of DCSB & Experimental Signatures
Pattern of DCSB & Experimental Signatures
• Other transition form factors are in accordance with
asymptotic QCD:
Pattern of DCSB & Experimental Signatures
Could anything conceivably go astray in the experiment?
A possible erroneous way to extract the pion transition
form factor from the data could be the problem of
π0-π0 subtraction.
There is another channel
which is the crossed
channel of virtual Compton scattering on a pion.
The misinterpretation of some events (where the second
π0 is not seen) may be different at different Q2.
Diehl et. al. Phys. Rev. D62 073014 (2000).
Conclusions
Dynamical chiral symmetry breaking and the momentum
dependence of the quark mass function in QCD have
experimental signals which enable us to differentiate its
predictions from others.
A fully consistent treatment of the contact interaction
model produces results for pion elastic and transition
form factors that are in striking disagreement with
experiment.
In a fully consistent treatment of pion in SDE QCD (static
properties, elastic and transition form factors), the
asymptotic limit of the product Q2G(Q2), is not exceeded
at any finite value of spacelike momentum transfer and is
in disagreement with BaBar data for large momenta.