1. Accessing the gluon Wigner distribution in ep and pA collisions
Yoshitaka Hatta
(Yukawa inst. Kyoto U.)
YH, B. Xiao, F. Yuan, PRL116 (2016)
Y. Hagiwara, YH, R. Pasechnik, M. Tasevsky, O. Teryaev, PRD96 (2017)

2. Contents Introduction: Phase space distributions in QCD
Wigner distribution at small-x
Diffractive dijet production in ep and pA.

3. Nucleon tomography

4. 1D tomography: Parton distribution function (PDF)
Probability distribution of quarks and gluons with
longitudinal momentum fraction
The nucleon is much more complicated!
Partons also have transverse momentum
and are spread in impact parameter space

5. 3D tomography: Transverse momentum dependent distributions (TMD)
Relevant in semi-inclusive DIS (SIDIS), etc.

6. 3D tomography: Generalized parton distributions (GPD)
distribution of partons in impact parameter space
Fourier transform
Deeply Virtual Compton Scattering (DVCS)

7. 5D tomography: Wigner distribution— the “mother distribution”
Belitsky, Ji, Yuan (2003);
Lorce, Pasquini (2011)
TMD
GPD
PDF
Form factor
charge

8. 5D tomography: Generalized TMD and Husimi
GTMD
Husimi
Meissner, Metz, Schlegel (2009)
Hagiwara, YH (2015)
Gaussian smearing in k, b
Probabilistic interpretation!
PDF
Form factor
charge
TMD
GPD

9. Wigner distribution and orbital angular momentum
Nucleon spin decomposition
Quarks’ helicity
Gluons’ helicity
Canonical Orbital angular momentum
Lorce, Pasquini, (2011);
YH (2011)

10. Electron-Ion Collider (EIC)
 Abhay’s talk
A future (2025~), high-luminosity　 experiment dedicated to
the study of nucleon structure.
arXiv:

11. Wigner distribution: Is it measurable?
In quantum optics, yes!
What about in QCD? Go to small-x!

12. Gluon Wigner distribution
There are two ways to make it gauge invariant
Dominguez, Marquet, Xiao, Yuan (2011)
Dipole distribution
Weizsacker-Williams distribution

13. Dipole gluon Wigner distribution at small-x
YH, Xiao, Yuan (2016)
Approximate
``Dipole S-matrix”
correlation expected
``Elliptic Wigner”
``OAM Wigner”

14. Gluon Wigner from Balitsky-Kovchegov equation
Hagiwara, YH, Ueda (2016)
Peak at the saturation momentum
Small in magnitude (a few percent effect)
Distinct functional dependence on

15. Elliptic Wigner in DVCS
YH, Xiao, Yuan (2017)
Gluon transversity GPD
Elliptic Wigner (GTMD)

16. Elliptic Wigner in high-multiplicity pp and pA
Hagiwara, YH, Xiao, Yuan (2017)
see also, Kopeliovich et al. (2008), Levin Rezaeian (2011),..
Elliptic flow observed in
high-multiplicity pp and pA.
Initial state or final state effect?
Double parton scattering + Elliptic Wigner = Elliptic flow

17. Direct measurement of the Wigner distribution
Find a process sensitive to both and …. Nontrivial!
cf. Vector meson production
One can study the dependence
on Munier, Stasto, Mueller, 2001
but not on

18. Probing Wigner (GTMD) in exclusive diffractive dijet in ep
YH, Xiao, Yuan (2016), see also Altinoluk, Armesto, Beuf, Rezaeian (2015)
NLO calculation by Boussarie et al. (2016)
Jet 1
Jet 2
𝑒 −
Proportional to GTMD

19. Ultra-peripheral pA collisions!
Hagiwara, YH, Pasechnik, Tasevsky, Teryaev, (2017)
preferably small
Use the Weiszacker-Williams photons in UPC!
YH, Xiao, Yuan (2016)
photon flux

20. Analytically invert these relations to get GTMD
BK model
MV model
Can be done also for UPC in AA collisions

21. Nucleon spin puzzle and small-x
Nonzero contribution from `large’-x region
de Florian, Sassot, Stratmann,Vogelsang (2014)
Huge uncertainty in from the small-x region.
Maybe after including the small-x contribution. OAM not needed? Δ𝐺

22. - Longitudinal single spin asymmetry in diffractive dijet 2 2
YH, Nakagawa, Xiao, Yuan, Zhao (2016) 2 2 -
‘Odderon’
at small-x !?

23. Conclusions
Physics of nucleon structure has literally expanded its dimensions, from 1D (PDF) to 3D (GPD/TMD) and 5D (Wigher/Husimi/GTMD). Exciting times ahead at the future EIC.
First ideas to measure 5D gluon distributions: Diffractive dijet events in ep (photoproduction) and pA (UPC). Look for 　　　　 and angular dependences.
Extension to spin-dependent part (OAM), moderate-x, quark Wigner.
YH, Nakagawa, Xiao, Yuan, Zhao (2017)
Ji, Yuan, Zhao (2017)
Bhattacharya, Metz, Zhou (2017)

24. Homodyne Detection
Two electromagnetic fields are incident on a beam splitter
Photon count difference at detectors C and D
: Probability distribution of
(coherent state with phase )
c.f.
Reconstruct via the inverse Radon transformation