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Variational study of weakly coupled triply heavy baryons

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. Outline. 1. Introduction to triply heavy baryons

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Variational study of weakly coupled triply heavy baryons

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

  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, H2+ion, etc. 4. Predictions of masses and QCD inequalities 5. Summary and Outlook

  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 • ’chc’c etc.  Talk by Seth

  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. 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. 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. 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-1data (one year run at LHC design luminosity), with cuts pT>10 GeV and |y|<1, the amount of producedbccand ccccan reach 6108to 1108 • It seems promising for future identification of these states at LHC with such a large amount of yield.

  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. Weakly-coupled QQQ states The state satisfiesm v QCDis 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 , Bc, even J/ as weakly coupled system  talks by Pineda, Garcia Tormo and Vairo

  10. Static potential of QQQ state • Singlet-channel static potential (familiar Coulomb potential) • Lamb’s shift (color octet effect) Ultrasoft gluon (p ~m v2)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. Separating the Hamiltonian governing internal motion • Our task is then solving Schrödinger equation • Define new variables CM part

  12. Solve Schrödinger equation • The Hamiltonian governing internal motion • Where r12 = |r1-r2|, and mij is the reduced mass of mi and mj • 3-body problem not exactly solvable. • Must resort to approximation • Variatonal method is a simple and economic way

  13. Sketch of bcc coordinate system m M

  14. Solve Schrödinger equation • The Hamiltonian governing internal motion where mred= m M/(m+M) • Baryonic unit: mres2s/3=1 and mres (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. 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. 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. Variational estimate for bcc (cont’) • Variational principle requires dE/d = 0 •  Indeed  > 1 as is expected

  18. Trial state for ccc ground state • symmetry constraint JP = 3/2+ • The Hamiltonian governing internal motion • Again I adopt baryonic unit: mres2s/3=1 and mres (2s/3)2=1 with mres= m/2 • Fermi statistics  trial wave function is taken as fully symmetric

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

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

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

  23. Sketch of bbc coordinate system

  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<r>  <R>, • 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. Ground state energy in point-like diquark approximation • In the limit <r>  <R>, approximate r1  r2  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. Two alternative approaches incorporating finite diquark radius effect 1. Born-Oppenheimer (adiabatic) approximation Well motivated for diatomic molecule like H2+ ion ion. Justified by strong separation of time scales between electronic and vibrational nuclear motion [ N/e~ (MN/me)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 ofb is much (~M/m) shorter than that of c. Exhibiting completely anti-adiabatic behavior Conceptually inappropriate to use adiabatic approx. to bbc state

  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. bbc Baryons • Symmetry between two b quarks JP = 3/2+ or 1/2+ • The Hamiltonian governing internal motion • I adopt heavier baryonic unit: Mred =1 and 2s/3=1 Having defined  = mred /Mred • Choosing the trial wave function as

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

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

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

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

  35. Mass of Lowest-lying bcc state

  36. Mass of Lowest-lying ccc state

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

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

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

  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. 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. 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. 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 <s G2>) Also aims to determine the wave function at the origin of QQQ states, which will be needed for more reliably estimating the fragmentation function

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