Transcript Review and Outlook of My Work

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
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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
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Introduction
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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
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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?
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Recent interests in doubly heavy baryons
Several tentative ccq candidates
 Hu’s talk
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To complete baryon family, the last missing
member is triply heavy baryons (QQQ states)
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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)
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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
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Too low production rate of triply charmed
baryon at existing e+ e- colliders
Baranov and Slad (04)
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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
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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
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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
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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
determined via matching from NRQCD, long distance
piece of potential is unimportant
Integrating out soft and potential gluons, only keeping ultrasoft ones
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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
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Singlet-channel static potential (familiar Coulomb potential)
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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
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In this work we don’t consider the color-octet effect
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Separating the Hamiltonian governing
internal motion
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Our task is then solving Schrödinger equation
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Define new variables
CM part
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Solve Schrödinger equation
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The Hamiltonian governing internal motion
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Where r12 = |r1-r2|, and mij is the reduced mass of
mi and mj
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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
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The Hamiltonian governing internal motion
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where mred= m M/(m+M)
Baryonic unit: mres2s/3=1 and mres (2s/3)2=1
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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
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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
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Variational estimate for bcc (cont’s)
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The ground state energy is thus
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where
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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
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Variational estimate for bcc (cont’)
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Variational principle requires dE/d = 0
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Indeed  > 1 as is expected
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Trial state for ccc ground state
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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
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Variational estimate for ccc (cont’s)
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The ground state energy is thus
where
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Variational estimate for ccc (cont’)
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I obtain
Variational principle  dE/d = 0
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
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Comparing bcc and ccc states
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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
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It is more convenient to adopt a different coordinate
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The Hamiltonian governing internal motion
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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
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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
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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 ]
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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 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.
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bbc Baryons
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Symmetry between two b quarks  JP = 3/2+ or 1/2+
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The Hamiltonian governing internal motion
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I adopt heavier baryonic unit: Mred =1 and 2s/3=1
Having defined  = mred /Mred
Choosing the trial wave function as
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Variational determination of energy of bbc
baryons
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I obtain
where
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The limiting case as M  m
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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
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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
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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
Nussinov (PRL 83,84)
Martin et al (PLB 86)
Richard (PLB 84)
My results are compatible with these inequalities
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My numerical predictions
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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
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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
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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’)
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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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