1. Flavor structure of Lambda baryons from lattice QCD - from strange to charm
P. Gubler, T.T. Takahashi and M. Oka, Phys. Rev. D 94, (2016).
Talk at the ECT* workshop “The Charm and Beauty of Strong Interactions”,
Trento, Italy
Philipp Gubler (Keio U.)
Toru T. Takahashi (Gunma National College of Technology)
Makoto Oka (Tokyo Institute of Technology)

2. Λ baryons Λ Λc I=0 I=0 S=-1 C=1 u u d d s c JP=1/2±, 3/2±,…

3. Motivation Λ(1405) , u u u u d s s d d (JP=1/2-)
Lightest negative parity baryon, even though it contains a strange quark!
Λ(1405) u d s
N*(1535) u d s u d , u
quark masses
→ Difficult to reproduce Λ(1405) in
conventional quark models

4. N Σ Λ(1405) K π ? Λ(1405) (JP=1/2-) Meson-baryon molecule?
Can naturally explain the lightness of Λ(1405) thanks to the strong and attractive KN interaction
R. H. Dalitz, T. C. Wong, and G. Rajasekaran, Phys. Rev. 153, 1617 (1967).
N. Kaiser, P.B. Siegel and W. Weise, Nucl. Phys. A 594, 325 (1995).
E. Oset and A. Ramos, Nucl. Phys. A 635, 99 (1998).
D. Jido et al., Nucl. Phys. A 725, 181 (2003).
T. Hyodo, and W. Weise, Phys. Rev. C 77, (2008).
. . .

5. Octet or singlet?
Σ -
Σ +
Σ 0 Λ
Ξ -
Ξ 0 Λ

6. Replace an s-quark with a c-quarkd s u d c
magnetic moment of quark scales as 1/mq
→ heavy quark spin decouples
heavy quark plays the role of a static color source
→ excitations determined by diquark
degrees of freedom

7. Naïve expectations from the quark model?
Dominates lowest excitation?
ρ-excitation
Diquark interpretation of Λc spectrum
λ-excitation

8. λ-mode and ρ-mode
Consider a simple harmonic oscillator potential model:
Rewrite using:
with
Lowest excitation is a λ-mode
and
T. Yoshida, E. Hiyama, A. Hosaka, M. Oka and K. Sadato, Phys. Rev. D 92, (2015).

9. ? How does the spectrum evolve as a function of the heavy quark mass?
octet ?
ρ-excitation ? u d
singlet ?
λ-excitation s u u d s
octet
Change of internal structure?
Manifestation of diquark degrees of freedom?

10. Interpolated using non-physical quark masses
Our strategy
Investigate masses and flavor structures of Λ-baryons as a function of the heavy quark mass
physical
non-physical
physical
heavy quark mass
Interpolated using non-physical quark masses

11. Our Method: Lattice QCD
→ Reformulate QCD on a lattice
→ Go to imaginary time
→ Perform the path integral statistically
(important sampling)
→ Extract the spectrum

12. How to extract the spectrum
Operators constructed from quark fields
t: large
ground state mass

13. Sample results for baryonic ground states
newest LHCb measurement
arXiv: [hep-ex]
C. Alexandrou et al. (ETM Collaboration), Phys. Rev. D 90, (2014).

14. Method for studying the flavor structure
Use correlation matrix:
Eigenvectors
octet
Couplings to singlet/octet operators provide information about flavor structure of the states
singlet

15. Our interpolating fields
+ parity
- parity
octet
singlet
S. Basak et al. [Lattice Hadron Physics (LHPC) Collaboration], Phys. Rev. D 72, (2005).

16. Recent studies of the Λ spectrum
JP=1/2-
- full 2+1-flavor QCD
- lightest pion mass: ~156 MeV
→ Λ(1405)
reproduced?
J.M.M. Hall et al., Phys. Rev. Lett. 114, (2015).

17. Recent studies of the Λ spectrum (cont`d)
Measurement of the strange magnetic form factor of the Λ(1405) s u
→ Information on u d
internal structure of the state
J.M.M. Hall et al., Phys. Rev. Lett. 114, (2015).

18. Our lattice parameters
- 2+1-flavors of dynamical quarks
(Gauge configurations generated by the PACS-CS Collaboration)
- Renormalization-group improved gauge action at β=1.9 and a= fm
(Iwasaki gauge action + nonperturbatively O(a) improved Wilson quark action)
- Lattice size: 323 x 64 → (2.9 fm)3 x (5.8 fm)
- 4 Hopping parameters for light quark masses
→ Pion masses: 296 MeV ~ 702 MeV
- 5 Hopping parameters for heavy quark masses
→ 2 for physical strange and charm quarks
→ 3 for non-physical quarks interpolating between s and c

19. Effective masses (Λ) mπ = 570 MeV
P. Gubler, T.T. Takahashi and M. Oka, Phys. Rev. D 94, (2016).

20. Effective masses (Λc) mπ = 570 MeV
P. Gubler, T.T. Takahashi and M. Oka, Phys. Rev. D 94, (2016).

21. Chiral extrapolations (Λ)
Extracted masses are a bit too high compared to experiment.
→ too large strange quark mass?
P. Gubler, T.T. Takahashi and M. Oka, Phys. Rev. D 94, (2016).

22. Chiral extrapolations (Λc)
Good agreement of ground state masses with experimental values
P. Gubler, T.T. Takahashi and M. Oka, Phys. Rev. D 94, (2016).

23. Relations to thresholds
JP=1/2- Λ Λc
P. Gubler, T.T. Takahashi and M. Oka, Phys. Rev. D 94, (2016).

24. Letting the Λ evolve into the Λc (Masses)ms mc

25. Definition of flavor components
octet
singlet

26. Chiral extrapolations of flavor components (Λ)
The light quark mass dependence is relatively weak

27. Chiral extrapolations of flavor components (Λc)
[Flavor decomposition, SU(3): (u,d,c)]
The light quark mass dependence is relatively weak

28. Letting the Λ evolve into the Λc
[Flavor decomposition, SU(3): (u,d,Q)] ms mc ms mc

29. Relation to simple quark model picture
→ ρ-mode (diquark p-wave excitation)
ρ-mode
ORBITAL
→ λ-mode (diquark’s CM p-wave excitation) u d
→ diquark with spin 1 (total 1/2)
λ-mode
SPIN
→ diquark with spin 0 (total 1/2) c
→ diquark with spin 1 (total 3/2)
singlet:
λ-mode - ρ-mode
Rewrite SU(3) wavefunctions in terms of diquark wavefunctions:
octet:
λ-mode + ρ-mode
octet:
ρ-mode
T. Yoshida, E. Hiyama, A. Hosaka, M. Oka and K. Sadato, Phys. Rev. D 92, (2015).

30. Possible interpretation of lattice results for Λc(1/2-)
Consistent with λ-mode dominated wavefunction:

31. Summary and Conclusions
- Lattice QCD can be used to study hadrons
(energy levels, internal structure)
- We have studied the spectrum and flavor structure
of Λ baryons with Jπ=1/2±
- In contrast to most other channels, the flavor structure
of the Λc(1/2-) differs strongly from its strange counterpart
Λ(1/2-)

32. Backup slides

33. Λ Λc Experimental spectrum **** *** *** * **** **** *** *** **** ***
1890 MeV
****
1810 MeV
1800 MeV
***
***
1710 MeV
1690 MeV
1670 MeV *
****
1600 MeV
****
***
2625 MeV
1520 MeV
2595 MeV
1405 MeV
***
****
***
****
1116 MeV
2286 MeV
****
****
1/2-
3/2-
1/2-
3/2-
1/2+
3/2+
1/2+
3/2+

34. What about the Λ(1405)? ?
Does this state correspond to the lower (singlet dominated) dynamically generated pole of the chiral unitary approach?
D. Jido et al., Nucl. Phys. A 725, 181 (2003).

35. Recent studies of the Λ spectrum
JP=1/2-
- full 2-flavor QCD
- lightest pion mass: ~ 500 MeV
→ no Λ(1405) ?
T.T. Takahashi and M. Oka, Phys. Rev. D 81, (2010).

36. Recent studies of the Λ spectrum (cont`d)
JP=1/2-
- full 2-flavor QCD
- lightest pion mass: ~ 255 MeV
→ Λ(1405) successfully
reproduced?
G.P. Engel, C.B. Lang and A. Schaefer (BGR Collaboration), Phys. Rev. D 87, (2013).

37. Recent study of the Λc spectrum
- full 2+1-flavor QCD
- lightest pion mass: ~259 MeV
JP=1/2+
JP=1/2-
P. Pérez-Rubio, S. Collins and G.S. Bali, Phys. Rev. D 92, (2015).

38. Change to particle basis
SU(3) Clebsh-Gordan coefficients give:
As however gD and gF are undetermined, the states can at present not be uniquely expressed in the particle basis

39. Chiral extrapolations (Λ,J=3/2)

40. Chiral extrapolations (Λc,J=3/2)