1. 高崇文 Chung-Wen Kao Chung-Yuan Christian University, Taiwan
Dihadron fragmentation functions
In chiral models
高崇文 Chung-Wen Kao
Chung-Yuan Christian University, Taiwan
KEK theory center workshop on Hadron and Nuclear Physics in 2017
(KEK-HN-2017), January 7, 2017

2. Collaboration and Reference
In Collaboration with
Nam Seung-il (KIAS) Dong-Jing Yang, Fu-Jiun Jiang (NTNU, Taiwan)
Wen-Chen Chang (AS, Taiwan)
This talk is based on the following works:
(1) Nam Seung-il, CWK, Rev.Rec .D85, (2012)
(2) Nam Seung-il, CWK, : Phys.Rev.D85, (2012)
(3) Dong Jing Yang, Fu-Jiun Jaing, CWK, Nam Seung-il:
Phys. Rev D87, (2013)
(4) Dong Jing Yang, Fu-Jiun Jaing, CWK, Nam Seung-il: arXiv
(5) Dong Jing Yang, Fu-Jiun Jaing, Wen-Chen Chang, CWK,
Nam Seung-il: Phys. Lett. B 755(2016) 393

3. Motivation
Q: Why should one care about di-hadron fragmentations? A: Because they are important quantities of extracting the transversity from SIDIS with two hadrons in the final states.

4. Transversity: least known leading twist structure function

5. How to measure Transversity?

7. Collins and Transversity

9. Extended dihadron fragmentation function

10. Extended dihadron fragmentation function

11. Extended dihadron fragmentation function

13. A. Courtoy, A.Bacchetta, M.Radici and A.Bianconi,
PRD 85, (2012)

14. How to extract transversity?
Artru-Collins
Asymmetry
input
input
Assume
D1q(z,Mh,Q2)
SIDIS
Asymmetry
input

15. A. Courtoy, A.Bacchetta, M.Radici and A.Bianconi,
PRD 85, (2012)

16. Extraction of transversity
A. Bacchetta, A.Courtoy and M.Radici:
First extraction of valence transversities in a collinear framework,
JHEP 1303, 119 (2013)

17. From SiFF to DiFF
Fitted from “virtual data” does not sound too reliable, at least for me…..
Actually there is a way to connect single hadron FFs to dihadron FFs.
However one cannot obtain DiFF from SiFF directly, but needs rely on the
so-called elementary FFs.

18. The path leading somewhere…

19. Elementary Fragmentation Functions
One step fragmentation process
q(k) →h(p)+Q(k-p)
With a vertex of quark-quark-PS meson, one can calculate the elementary FFs.

20. From elementary FFs to SiFFs

21. Monte-Carlo simulationor
coupled channel integral equations

22. How to model Fragmentation functions?
Fragmentation functions are essentially related to long distance physics, which cannot be calculated by perturbative QCD.
However, chiral physics should be related with them. Hence it is desirable to calculate them by some chiral models.
So is it realistic or just a dream?

23. Dream, Dream, Dream…. All I have to do is dream?
Not really, you need something
more practical…..at least
something you can calculate…

24. How to model elementary FF?
1993, Ji and Zhu suggested to use Georgi-Manohar model to model the coupling between the Goldstone modes and the quarks and model fragmentation functions.
Later people used NJL model to model the same coupling.
We suggest to use nonlocal chiral quark model which is derived from the instanton model.
Ref:Diakonov (1996)

25. Instanton Vacuum model
Effective Chiral Action derived from the instanton configuration in the leading order (LO) of the 1/Nc expansion :
The quarks are moving inside the (anti)instanton ensemble and flipping their helicities. It results in that (anti)quarks acquire the momentum-dependent effective masses dynamically, i.e. constituent-quark masses M(k).

26. Effective quark mass M(k)
Assuming that the zero modes dominate the low-energy phenomena, we can write the Dirac equation for a quark for the (anti)instanton background
By making Fourier transform of
, One obtains ≈
instanton size ≈1/3

27. Nonlocal Chiral quark model
Non-local vertex of
quark-quark-PS meson

28. Inclusion of quark-jet contribution
s → K-
u → K+

29. Inclusion of quark-jet contribution

30. Inclusion of quark-jet contribution
u → barK0
u → K0
u → K-
s → K+

31. Relations between FFs
Consider the isospin symmetry and charge conjugation:
favouredt Fragmentation functions: 16→4
disfavoured Fragmentation functions:26 →7

32. QCD evolution
We employ QCDNUM 17 to evolve our result from Q2=0.36 GeV2 to 4 GeV2.
Compare our result with NJL result and
HKNS and DSS parameterizations.
HKNS
DSS

33. favoured Pion Fragmentation Functions

34. favoured Kaon Fragmentation Functions
u → K+
s → K-

35. Disfavoured pion fragmentation functions

36. Disfavored pion fragmentation functions

37. Disfavoured Kaon Fragmentation Functions
u → barK0
u → K0

38. Disfavoured Kaon Fragmentation Functions
s → K+
u→ K-

39. How to calculate di-hadron FFs?
(1) Monte Carlo simulation
(2) Solving Integral equations with the SiFF as in put:

40. Dihadron Fragmentation Function
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

41. Dihadron Fragmentation Function
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

42. Dihadron Fragmentation Function
s→ π+ π-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

43. Dihadron Fragmentation Function
u→ π+ K-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

44. Dihadron Fragmentation Function
d→ π+ K-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

45. Dihadron Fragmentation Function
s→ π+ K-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

46. Dihadron Fragmentation Function
u→ K+ K-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

47. Dihadron Fragmentation Function
d→ K+ K-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

48. Dihadron Fragmentation Function
s→ K+ K-
Q2=4 GeV2 Left: z1=0.1, Right z1=0.5

49. Dihadron fragmentation functions

50. Extended dihadron fragmentation functions
Previous calculation shows the MC computation works well.
To extract the transversity, one needs another another type of DiFFs.
It is called extended dihadron fragmentation functions. But there is no integral equations for them so one needs rely on MC computation.

51. Extended Dihadron Fragmentation function
Fragmentation functions

52. Inclusion of vector mesons
Vector mesons are also included.
They decay into PS mesons. π π π π ρ π ρ ω
H.H.Matevosyan, A.W.Thomas and W.Bentz,  PRD (2013)

53. u quark case (π+π-)
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

54. u quark case (π0π+ ) ( π0π- ) ( π0π0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

55. s quark case (π0π+ ) ( π0π- ) ( π0π0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

56. u quark case (charged pion pairs)
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

57. u quark case (π+ K0 ) ( π+ K+ ) ( π+ K- ) (π+ K0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

58. u quark case (π- K0 ) ( π- K+ ) ( π- K- ) (π- K0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

59. s quark case (π+ K0 ) ( π+ K+ ) ( π+ K- ) (π+ K0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

60. s quark case (π- K0 ) ( π- K+ ) ( π- K- ) (π- K0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

61. u quark case (K0 K0 ) ( K0 K0 ) ( K0 K- ) (K0 K+ )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

62. u quark case (K- K+ ) ( K0 K0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

63. s quark case (K0 K0 ) ( K0 K0 ) ( K0 K- ) (K0 K+ )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

64. s quark case (K- K+ ) ( K0 K0 )
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

65. Extraction of transversity
Using the extended Di-hadron FFs, one can repeat what Bachetta et. al. have done. Namely we can also obtain some certain combinations of transversity PDFs.
However the results are extremely sensitive
to the unpolarized DiFFs.

66. Extraction of hD1
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

67. Extraction of hP1
Left: NJL-jet model, Right:Nonlocal Chiral quark Model

68. Summary
We use a nonlocal chiral quark model to calculate the unpolarized fragmentation functions of pions an kaons.
We also have calculated the dihadron fragmentation functions.
Our results have been used to extract the transversity.
We have found that the extracted values of transversity are very sensitive to the choices of unpolarized DiFFs.

69. Outlook
Extend to transverse-momentum dependent FF. This is crucial for extraction of Sivers functions.
Extend to Collins FFs which is also important to extract transversity.
Try to find the model-indpenednet elementary FFs to generate reliable DiFFs to extract the transversity.

70. Two hadrons in final state are too many,
But two cups of ice cream are just perfect!