1. 1 Variational study of weakly coupled triply heavy baryons Yu Jia Institute of High Energy Physics, CAS, Beijing [work based on JHEP10 (2006) 073] 5 th International Workshop on Heavy Quarkonia, 19 October 2007, DESY

2. 2 Outline 1. Introduction to triply heavy baryons 2. What is a weakly-coupled QQQ state? Coulomb system + ultrasoft gluons 3. Variational estimate of the binding energy Stimulated by the familiar QM textbook treatment on Helium atom, H 2 + ion, etc. 4. Predictions of masses and QCD inequalities 5. Summary and Outlook

3. 3 Introduction Recent years we have witnessed renaissance of field of heavy hadron spectroscopy which is propelled by emergences of many unexpected XYZ states  Talks by Braaten, Miyabayashi, Prencipe, Yuan, Lü, Mehen, De Fazio, Faustov, Hanhart, Oset and Polosa Enormously enrich our understanding toward nonperturbative sector of QCD Steady progress also made in the conventional sector of spectroscopy  ’ c h c  ’ c etc.  Talk by Seth

4. 4 Why cares about QQQ states? Recent interests in doubly heavy baryons Several tentative ccq candidates  Hu’s talk To complete baryon family, the last missing member is triply heavy baryons (QQQ states) Baryonic analogue of heavy quarkonium, Free from light quark contamination, Clean theoretical laboratory for understanding heavy quark bound state

5. 5 Interests towards QQQ state initiated by Bjorken (85) Estimates masses of various lowest lying QQQ states Discusses discovery potential of the triply charmed baryon at fixed-target experiment Suggest  ccc   + 3  + + 3  may serve as clean trigger for triply charmed baryons

6. 6 Production rate of QQQ states Too low production rate of triply charmed baryon at existing e + e - colliders Baranov and Slad (04) Fragmentation functions of various QQQ states have also been computed Gomshi-Nobary and Sepahvand (05)

7. 7 Production rate of QQQ states at LHC Fragmentation probabilities of c and b to various QQQ states range from 10 -7 to 10 -4 Gomshi-Nobary and Sepahvand (05) For 300 fb -1 data (one year run at LHC design luminosity), with cuts p T >10 GeV and |y|<1, the amount of produced  bcc and  ccc can reach 6  10 8 to 1  10 8 It seems promising for future identification of these states at LHC with such a large amount of yield.

8. 8 Static potential for QQQ states Can be obtained nonperturbatively by measuring Wilson loop from lattice Color tube formed: Y-shape vs.  -shape Takahashi, Matsufuru, Nemoto and Suganuma (PRL 01) Bali (Phys Rept 01)

9. 9 Weakly-coupled QQQ states The state satisfies m v   QCD is called weakly coupled Brambilla, Pineda, Soto and Vairo (RMP 05) Brambilla, Rosch and Vairo (PRD 05) Potential, as the short-distance Wilson coefficient, can be determined via matching from NRQCD, long distance piece of potential is unimportant Integrating out soft and potential gluons, only keeping ultrasoft ones Empirically, one may treat , B c, even J/  as weakly coupled system  talks by Pineda, Garcia Tormo and Vairo

10. 10 Static potential of QQQ state Singlet-channel static potential (familiar Coulomb potential) Lamb’s shift (color octet effect) Ultrasoft gluon ( p  ~m v 2 ) induces color electric-dipole transition between singlet and octets configuration Inter-quark forces in octets and decuplet channels can be repulsive In this work we don’t consider the color-octet effect

11. 11 Separating the Hamiltonian governing internal motion Our task is then solving Schr ö dinger equation Define new variables CM part

12. 12 Solve Schr ö dinger equation The Hamiltonian governing internal motion Where r 12 = |r 1 -r 2 |, and m ij is the reduced mass of m i and m j 3-body problem not exactly solvable. Must resort to approximation Variatonal method is a simple and economic way

13. 13 Sketch of bcc coordinate system M m

14. 14 Solve Schr ö dinger equation The Hamiltonian governing internal motion where m red = m M/(m+M) Baryonic unit: m res 2  s /3=1 and m res (2  s /3) 2 =1 This problem is very much like Helium, except the force between two c quarks is attractive Classical example of application of variational method

15. 15 Variational estimate for bcc The trial wave function assumes the form where is the 1s Coulomb wave function. is a variational parameter: Effective color charge of b perceived by each c quark Contrary to helium, expecting >1 on physical ground

16. 16 Variational estimate for bcc (cont ’ s) The ground state energy is thus where measures average potential energy stored between two c quarks The contribution of the  1  2 term vanishes as a consequence of spherical symmetry of 1s wave function. Effect of kinetic energy of b is embodied entirely in reduced mass

17. 17 Variational estimate for bcc (cont ’ ) Variational principle requires dE/d = 0  Indeed > 1 as is expected

18. 18 Trial state for ccc ground state symmetry constraint  J P = 3/2 + The Hamiltonian governing internal motion Again I adopt baryonic unit: m res 2  s /3=1 and m res (2  s /3) 2 =1 with m res = m/2 Fermi statistics  trial wave function is taken as fully symmetric

19. 19 Variational estimate for ccc (cont ’ s) The ground state energy is thus where

20. 20 Variational estimate for ccc (cont ’ ) I obtain Variational principle  dE/d = 0 

21. 21 Comparing bcc and ccc states Lessons we can learn 1. Symmetrization effects tend to lower the energy, also squeeze the orbital 2. bcc state is more stable than ccc state compatible with the well known fact that electron in hydrogen atom is more stable than in positronium

22. 22 The Hamiltonian for bbc system It is more convenient to adopt a different coordinate The Hamiltonian governing internal motion where M res = M/2, m res = (1/m+1/2M) -1

23. 23 Sketch of bbc coordinate system

24. 24 Diquark picture of ideal bbc state This is more complicated than previous two cases, since the force felt by c is only axially symmetric, no longer spherically symmetric Picture greatly simplifies in the limit M  m  , one can shrink the bb diquark by a point antiquark. The compact diquark picture here is not as good as in doubly heavy baryon states. Savage and Wise (90) Brambilla, Rosch and Vairo (05); Fleming and Mehen (05)

25. 25 Ground state energy in point-like diquark approximation In the limit , approximate r 1  r 2  r, the Hamiltonian collapses into two independent parts One then gets Cannot be accurate if the hierarchy between m and M is not perfect, like in the physical bbc state Finite diquark radius effect should be implemented

26. 26 Two alternative approaches incorporating finite diquark radius effect 1. Born-Oppenheimer (adiabatic) approximation Well motivated for diatomic molecule like H 2 + ion ion. Justified by strong separation of time scales between electronic and vibrational nuclear motion [  N /  e ~ (M N /m e ) 1/2 ] However, it is completely unjustified for an ideal bbc state Both b and c have comparable velocity ( ~  s ), uncertainty principle tells typical time scale of b is much (~M/m) shorter than that of c. Exhibiting completely anti-adiabatic behavior Conceptually inappropriate to use adiabatic approx. to bbc state

27. 27 Two alternative approaches incorporating finite diquark radius effect 2. One step variational estimate Introduce two variational parameters and  : effective charge of b “ seen ” by c  : the impact of c on the bb diquark geometry There is no any other approximation involved, and it is conceptually appropriate to apply to bbc state. Good accuracy can be achieved as long as trial wave function is properly chosen.

28. 28 bbc Baryons Symmetry between two b quarks  J P = 3/2 + or 1/2 + The Hamiltonian governing internal motion I adopt heavier baryonic unit: M red =1 and 2  s /3=1 Having defined  = m red /M red Choosing the trial wave function as

29. 29 Variational determination of energy of bbc baryons I obtain where

30. 30 The limiting case as M  m It is interesting to look at   0 limit Two uncoupled polynomials of and  Using variational principles, one obtain optima =2 and  =1 Recover the point-like diquark picture

31. 31 Variational determination of energy of ground bbc state The deviation between point-like diquark approx. and one-step variational approach increases as  increases

32. 32 Optimal values of and  The impact of b on c is more important than the impact of c on b

33. 33 “ Modified ” diquark picture As  gets large, the naive diquark picture deviates from the true one severely. However, I find numerically the following “modified” diquark approximation renders rather accurate results Only as   0,  2 and   1, corresponding to naive diquark picture

34. 34 Phenomenology Input of charm and bottom mass Most natural to express the m b and m c from masses of  and J/ , by treating them as weakly coupled bound state

35. 35 Mass of Lowest-lying bcc state

36. 36 Mass of Lowest-lying ccc state

37. 37 Mass of Lowest-lying bbc state With the input Using variational calculus, I obtain

38. 38 Mass of Lowest-lying bbc state (cont ’ )

39. 39 Various QCD mass inequalities My results are compatible with these inequalities Nussinov (PRL 83,84) Martin et al (PLB 86) Richard (PLB 84)

40. 40 My numerical predictions Choose renormalization scale  =1.2 GeV (  s =0.43) for bcc, bbb and bbc Choose renormalization scale  =0.9 GeV (  s =0.59) for ccc

41. 41 Indication of my results Bjorken ’ s results are systematically higher than mine. His prediction can be regarded as arising from strongly coupled picture Put another way, ground state QQQ baryons have lower masses if they are weakly coupled. Future experimental findings and accurate lattice measurements will unveil the nature of lowest-lying QQQ states

42. 42 Summary and Outlook We have computed the lowest-order binding energy based on weakly-coupled assumption systematically lower than Bjorken’s predictions One may improve the estimate by including NLO static potential/tree level higher-dimensional potential Other methods are welcome for democratic purpose  Martynenko 0708.2033v2 Relativistic potential model + hyper-spherical expansion

43. 43 Summary and Outlook (cont ’ ) Some non-potential method would also be valuable in providing complementary information  F. K. Guo and Y. J. (work in progress) QCD sum rule (including ) Also aims to determine the wave function at the origin of QQQ states, which will be needed for more reliably estimating the fragmentation function