1. Pion Transition Form Factor University of Michoacán, Mexico
Adnan Bashir
University of Michoacán, Mexico
Collaborators:
F. Akram, University of Punjab, Pakistan
J. Aslam, Quaid-i-Azam University, Pakistan
B. El-Bennish, Cruzeiro do sul, Brazil
Y.X. Liu, Peking University, China
M.R. Pennington, Durham University, UK
J.R. Quintero, Huelva University, Spain
Raya, Michoacán University, Mexico
C.D. Roberts, Argonne National Laboratory, USA
P.C. Tandy, Kent State University, USA
Collaborators:
L. Albino, University of Michoacán, Mexico
M.A. Bedolla, University of Michoacán, Mexico
R. Bermudez, University of Sonora, Mexico
J. Cobos, University of Michoacán, Mexico
L. Chang, University of Adelaide, Australia
L.X. Gutiérrez, University of Michoacán, Mexico
K. Raya, University of Michoacán, Mexico

2. MRP

3. MRP

4. Contents Facts and Challenges Pions and Chiral Symmetry Breaking
Schwinger-Dyson Equations (SDE)
The Quark Propagator
The Quark-Photon Vertex
Bethe Salpeter Amplitude
Pion Electromagnetic Form Factor
Pion Transition Form Factor
SDE - Scope

5. Facts and Challenges
Color degrees of freedom (quarks and gluons) are not
observable (confinement).
Dynamical mass generation for massless quarks;
(dynamical chiral symmetry breaking).
Both these phenomena are emergent and owe themselves
to large coupling strength in the infrared.
How do we study physics beyond perturbation theory?
Studying QCD: lattice, Schwinger-Dyson and Bethe-
Salpeter equations, chiral perturbation theory,
effective quark models.

6. Schwinger-Dyson Equations
Through SDEs, we can study the structure of hadrons
through first principles in the continuum.
SDE for QCD have been extensively applied to the study
of quark and gluon propagators, their interactions, meson
spectra and interactions below the masses ~ 1 GeV.
They have been employed to study:
The gluon propagator
The quark propagator
A.C. Aguilar, A.A. Natale, JHEP 08, 057 (2004).
A. Ayala, AB, D. Binosi, M, Crisoforetti, J. Rodríguez, Phys. Rev. D (2012).
The quark/gluon-photon interactions:
P. Maris, C.D. Roberts, P. Tandy, Phys. Lett. B (1998).
The masses, charge radii, decays of light/heavy mesons
A. Kizilersu and M.R. Pennington Phys. Rev. D (2009)
P. Maris, P. Tandy, Phys. Rev. C (1999).
Pion and kaon valence quark-distribution functions
P. Maris, C.D. Roberts, Phys. Rev. C (1997).
P. Maris, P.C. Tandy, Phys. Rev. C (2000).
M.A. Bedolla, J.J. Cobos Martínez, AB, Phys. Rev. D (2015).
M.A. Bedolla, K. Raya, J.J. Cobos Martínez, AB, Phys. Rev. D (2016).
L. Chang, C.D. Roberts, Phys. Rev. Lett (2009)
AB, R. Bermudez, L. Chang, C.D. Roberts, Phys. Rev. C85, (2012).
R. Bermudez, L. Albino-Fernández, L.X. Gutiérrez, M.E. Tejeda, AB (in progress).
L. Albino-Fernández, AB, L.X. Gutiérrez, Y. Concha, Phys. Rev. D (2016).
Elastic and transition pion form factors
T. Nguyen, AB, C.D. Roberts, P.C. Tandy, Phys. Rev. C (2011).
Nucleon elastic and transition form factors
L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C (2010).
L. Chang et. al., Phys. Rev. Lett. 111, (2013).
K. Raya et. al., Phys. Rev. D93, (2016).
L.X. Gutiérrez, K. Raya, AB, C. Roberts, D. Wilson (in progress)
“Collective Perspective on advances in DSE QCD”, AB , L. Chang, I.C. Cloet, B. El Bennich, Y. Liu, C.D. Roberts, P.C. Tandy, Commun. Theor. Phys (2012)

7. Chiral Symmetry Breaking
Pions
&
Chiral Symmetry Breaking

8. Pions and Chiral Symmetry Breaking
In October 1934, Hideki Yukawa predicted the existence of
a “heavy quantum”, meson, exchanging nuclear force between
neutrons and protons.
1949
1950
1969
2008
It was discovered by Cecil Powel in 1949 in cosmic ray
tracks in a photographic emulsion.
Pion was nicely accommodated in The Eight Fold way of
Murray Gell –Mann in 1961.
Yoichiro Nambu associated it with CSB in 1960.

9. Chiral Symmetry and Its Breaking
The connection of the smallness of pions with chiral
symmetry breaking is reflected in the long known
following relations:
GellMann-Oakes-
Renner relation
Gell-Mann-Okubo
mass formulae
~500 MeV
Parity Partners &
Chiral Symmetry Breaking
~500 MeV

10. Chiral Symmetry and Its Breaking
Nucleon
And its
Parity
Partner

11. Schwinger-Dyson Equations: From Infrared to Ultraviolet

12. Schwinger-Dyson Equtions
Schwinger-Dyson equations are the fundamental equations
of QCD and combine its UV and IR behaviour.
Observing the transition of the pion (form factors, such
as ) from quarks and gluons to one with valence
quarks alone can be studied naturally through SDE.

13. The Quark Propagator
The quark propagator:

14. The Quark Propagator
“Bridging a gap between continuum-QCD and ab initio predictions of hadron observables” , Binosi,, Chang, Papavassiliou, Roberts,  Phys.Lett. B (2015).
This solution violates the axiom of reflection positivity
and the corresponding excitation is confined.
Thus chiral symmetry breaking and confinement are
intimately connected to each other.

15. The Quark-Photon Vertex
For current conservation and Ward identities, a proper quark-photon vertex is essential.
“Truncating the Schwinger-Dyson equations: How multiplicative renormalizability and the Ward identity restrict the three point vertex in QED”
D.C. Curtis and M.R. Pennington, Phys. Rev. D (1990)
“Nonperturbative study of the fermion propagator in quenched QED in covariant gauges using a renormalizable truncation of the Schwinger- Dyson equation”
D.C. Curtis and M.R. Pennington, Phys. Rev. D (1993)
“Gauge independent chiral symmetry breaking in quenched QED”
AB, M.R. Pennington Phys. Rev. D (1994)
“The Nonperturbative three point vertex in massless quenched QED and perturbation theory constraints”
AB, A. Kizilersu, M.R. Pennington Phys. Rev. D (1998)

16. The Quark-Photon Vertex

17. The Quark-Photon/Gluon Vertex
For current conservation and Ward identities, a proper quark-photon vertex is essential.
AB, M.R. Pennington Phys. Rev. D (1994)
D.C. Curtis and M.R. Pennington Phys. Rev. D (1990)
A. Kizilersu and M.R. Pennington Phys. Rev. D (2009)
L. Chang, C.D. Roberts, Phys. Rev. Lett (2009)
AB, R. Bermudez, L. Chang, C.D. Roberts, Phys. Rev. C (2012).
LA Fernández, AB, L.X. Gutiérrez, Y. Concha, Phys. Rev. C (2016).
R. Delbourgo, A. Salam, Phys. Rev (1964).
It respects the separation of scales involved.
It ensures chiral anomaly at zero momentum transfer.
It reproduces one-loop vertex for asymptotic momenta.
It satisfies multiplicative renormalizability condition.
AB, A. Raya, S. Sanchez-Madrigal, C.D. Roberts Few Body Syst (2009).
M. J. Aslam, AB, L.X. Gutiérrez, Phys. Rev (2016).

18. The Bethe-Salpeter Amplitudes
Bethe-Salpeter Amplitude for the pion:
Goldberger-Triemann
relations:
Nakanishi-like representation

19. Pion Transition Form Factor

20. Pion to * Transition Form Factor
The transition
is studied through process:

21. Pion to * Transition Form Factor
The transition form factor:
CELLO H.J. Behrend et.al., Z. Phys C (1991) – 2.2 GeV2
The leading twist pQDC calculation was carried out in:
CLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998) – 8.0 GeV2
G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980).
BaBar R. Aubert et. al., Phys. Rev. D (2009). 4.0 – 40.0 GeV2

22. Pion to * Transition Form Factor
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:

23. Valence Quark Parton Distribution Amplitude for Pion

24. Valence Quark Parton Distribution Amplitude
Pion’s PDA – (x,Q2): is a probability amplitude that describes the momentum distribution of a quark and anti-quark in the bound-state’s valence Fock state.
x is the light-cone momentum fraction:
Among other processes, it enters the calculation of both the pion electromagnetic and transition form factors.
(x, Q2): is an essentially non-perturbative quantity whose asymptotic form is known.
What we can know is the evolution of this function with the momentum scale Q2. ERBL evolution equations.

25. Valence Quark Parton Distribution Amplitude

26. Valence Quark Parton Distribution Amplitude

27. Valence Quark Parton Distribution Amplitude
I.C. Cloet, QCD TNT4, Ilhabela, 2015

28. Pion Electromagnetic Form Factor

29. Pion Electromagnetic Form Factor

30. Pion Electromagnetic Form Factor
1980’s
2001
2006
2017?

31. Pion Electromagnetic Form Factor
Within the rainbow ladder truncation, the elastic
electromagnetic pion form factor:
The pattern of chiral symmetry breaking dictates the
momentum dependence of the elastic pion form factor.
Experiments on pions indicate a contact like interaction?
L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C (2010).

32. Pion Electromagnetic Form Factor
L. Chang, I.C. Cloët, C.D. Roberts, S.M. Schmidt, P.C. Tandy,
Phys. Rev. Lett. 111, (2013)

33. Pion Transition Form Factor

34. Pion to * Transition Form Factor

35. Pion to * Transition Form Factor
The transition form factor:
H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 2010.
CELLO H.J. Behrend et.al., Z. Phys C (1991) – 2.2 GeV2
Lowest order in perturbation theory and the leading twist asymptotic QCD calculation:
CLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998) – 8.0 GeV2
BaBar R. Aubert et. al., Phys. Rev. D (2009). 4.0 – 40.0 GeV2
G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980).
Belle S. Uehara et. al., Phys. Rev. D (2012). 4.0 – 40.0 GeV2

36. Pion to * Transition Form Factor
The transition form factor:
K. Raya, L. Chang, AB, J.J. Cobos-Martinez, L.X. Gutiérrez-Guerrero, 
C.D. Roberts, P.C. Tandy, Phys. Rev. D (2016)

37. Pion to * Transition Form Factor
The transition form factor:
Belle II will have 40 times more luminosity.
Vladimir Savinov:
5th Workshop of the APS
Topical Group on Hadronic
Physics, 2013.
Precise measurements at large Q2 will provide a stringent
constraint on the pattern of chiral symmetry breaking.

38. Schwinger-Dyson Equations: The Scope

39. Schwinger-Dyson Equations: The Scope
Faddeev Equation
Masses, Decays
Form Factors
Bethe Salpeter
Equation
Schwinger-
Dyson
Equations
Quark Propagator
Theory Vs.
Experiment

40. Schwinger-Dyson Equations: The Scope
QCD Phase Diagram
Magnetic Catalysis
Hadron Physics
Chiral Symmetry Breaking
Condensed Matter
Dynamical
Masses
Schwinger-Dyson Equations

41. Happy Birthday Mike!