1. Wigner, Husimi and GTMD at small-x
Yoshitaka Hatta
(Yukawa institute, Kyoto U.)
1. Experimentally measuring 5D distributions at small-x
2. Computing 5D distributions in Color Glass Condensate
YH, Bowen Xiao, Feng Yuan, Phys.Rev.Lett. 116 (2016) ,
Yoshikazu Hagiwara, YH, Takahiro Ueda, arXiv:

2. 5D tomography: Wigner distribution— the “mother distribution”
Belitsky, Ji, Yuan (2003);
Lorce, Pasquini (2011)
TMD
GPD
PDF
Form factor
charge

3. GTMD and Husimi GTMD Husimi Hagiwara, YH (2015)
Meissner, Metz, Schlegel (2009)
Hagiwara, YH (2015)
Fourier transform
Gaussian smearing in k, b
PDF
Form factor
charge
TMD
GPD

4. Wigner/Husimi/GTMD model calculations, 1-loop calculations, evolution...
Lorce, Pasquini
Liu, Ma
Mukherjee, Nair, Ojha
Courtoy, Goldstein, Hernandez, Liuti, Rajan
Hagiwara, YH
Echevarria, Idilbi, Kanazawa, Lorce,
Metz, Pasquini, Schlegel
Wigner
Husimi
GTMD

5. Parton orbital angular momentum
Nucleon spin decomposition
Quarks’ helicity
Gluons’ helicity
Canonical Orbital angular momentum
Lorce, Pasquini, (2011);
YH (2011)

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

7. Dipole gluon Wigner distribution at small-x
YH, Xiao, Yuan (2016)
Approximate
``Dipole S-matrix”
Find a process sensitive to both and .

8. Diffractive dijet production in DIS
Proton recoil momentum
Dijet relative momentum
Fourier transform of

9. `Elliptic’ Wigner distribution
correlation between and expected at small-x

10. Wigner/Husimi/GTMD in the Color Glass Condensate
Hagiwara, YH, Ueda (2016)
Small-x evolution of dipole S-matrix including gluon saturation
 Balitsky-Kovchegov (BK) equation
Solve BK keeping the b-dependence

11. Hidden SO(3) symmetry BK equation conformally symmetric.
Gubser (2011)
Assume that a SO(3) subgroup of conformal group survives
Realistic initial conditions break this symmetry…
HOWEVER, the symmetry is
dynamically restored by the equation!
Berger, Stasto (2010)
has a peak at

12. Wigner Peak at the saturation momentum
insert a Gaussian factor to suppress
the large-r region (mimic confinement)
Peak at the saturation momentum

13. Rapidity evolution

14. Elliptic Wigner A few percent effect. Peak does not move with Y
(no geometric scaling)

15. Husimi Positive everywhere  probability distribution of gluons in CGC
Two Gaussian widths inversely related such that
Large-r region naturally suppressed.
Positive everywhere  probability distribution of gluons in CGC

16. GTMD Directly connected to dijet cross section!
peak at , rapidly falls with increasing

17. Conclusions
Let’s get 5 dimensional. Even richer physics than TMD+GPD combined, still largely unexplored.
5D distributions experimentally accessible in DIS
Wigner, GTMD: subject to uncertainties due to confinement effects. A nonperturbative factor needed (like in TMD).
Husimi can avoid this problem.
The elliptic angular correlation—a few percent effect, different rapidity dependence.
 challenging to measure at EIC?