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Variational study of weakly coupled triply heavy baryons

Yu JiaInstitute of High Energy Physics, CAS, Beijing

[work based on JHEP10 (2006) 073]

5th International Workshop on Heavy Quarkonia,19 October 2007, DESY

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Outline1. Introduction to triply heavy baryons 2. What is a weakly-coupled QQQ state? Coulomb system + ultrasoft gluons3. 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 inequalities5. Summary and Outlook

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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 nonperturbativ

e sector of QCD

Steady progress also made in the conventional sector of spectroscopy ’c hc ’c etc. Talk by Seth

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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

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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

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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)

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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 pT>10 GeV and |y|<1, the amount of produced bcc and ccc can reach 6 108 to 1 108

It seems promising for future identification of these states at LHC with such a large amount of yield.

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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)

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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 determ

ined 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

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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

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Separating the Hamiltonian governing internal motion Our task is then solving Schrödinger equation

Define new variables

CM part

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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

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Sketch of bcc coordinate system

M

m

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Solve Schrödinger equation The Hamiltonian governing internal motion

where mred= m M/(m+M) Baryonic unit: mres2s/3=1 and mres (2s/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

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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 quarkContrary to helium, expecting >1 on physical ground

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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 12 term vanishes as a consequence of spherical symmetry of 1s wave function.

Effect of kinetic energy of b is embodied entirely in reduced mass

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Variational estimate for bcc (cont’)

Variational principle requires dE/d = 0

Indeed > 1 as is expected

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Trial state for ccc ground state

symmetry constraint JP = 3/2+ The Hamiltonian governing internal motion

Again I adopt baryonic unit: mres2s/3=1 and mres (2s/3)2=1 with mres= m/2

Fermi statistics trial wave function is taken asfully symmetric

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Variational estimate for ccc (cont’s)

The ground state energy is thus

where

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Variational estimate for ccc (cont’)

I obtain

Variational principle dE/d = 0

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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

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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

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Sketch of bbc coordinate system

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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)

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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

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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 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

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Two alternative approaches incorporating finite diquark radius effect

2. One step variational estimateIntroduce 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.

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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 2s/3=1 Having defined = mred /Mred

Choosing the trial wave function as

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Variational determination of energy of bbc baryons

I obtain

where

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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

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Variational determination of energy of ground bbc state

The deviation between point-like diquark approx. and one-step variational approach increases as increases

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Optimal values of and

The impact of b on c is more important than the impact of c on b

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“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

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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

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Mass of Lowest-lying bcc state

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Mass of Lowest-lying ccc state

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Mass of Lowest-lying bbc state

With the input

Using variational calculus, I obtain

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Mass of Lowest-lying bbc state (cont’)

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Various QCD mass inequalities

My results are compatible with these inequalities

Nussinov (PRL 83,84)Martin et al (PLB 86)Richard (PLB 84)

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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

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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

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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

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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