Recent developments in the modeling of heavy quarkonia

0.03 MB·English

by  Stanley F Radford

Recent Developments in the Modeling of Heavy
Quarkonia
Stanley F. Radford∗ and Wayne W. Repko†
∗DepartmentofPhysics,MariettaCollege,Marietta,OH45750
6
†DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI48824
0
0
2 Abstract. We examine the spectra and radiative decays of the cc¯ and bb¯ systems using a model

n whichincorporatesthecompleteone-loopspin-dependentperturbativeQCD shortdistancepoten-
a tial, a linearconfiningterm including(spin-dependent)relativistic correctionsto orderv2/c2, and
J a fully relativistic treatment of the kinetic energy. We compare the predictions of this model to

3 experiments,includingstatesanddecaysrecentlymeasuredatBelle,BaBar,CLEOandCDF.

1
v
0 1. INTRODUCTION AND THE POTENTIAL MODEL
2
0
Overthepast25+years,potentialmodelshaveprovenvaluableinanalyzingthespectra
1
0 and characteristics of heavy quarkonium systems [1, 2, 3, 4, 5, 6, 7]. Motivation for
6 revisiting the potential model interpretation of the cc¯ and bb¯ systems is provided by
0
recent experimentalresults:
/
h
p • Thediscoveryofseveralexpectedstates inthecharmoniumspectrum(h C and hC)
-
p • The discovery of new states [X(3872), X(3943)], which could be a interpreted as
e
abovethresholdcharmoniumlevels
h
v: • Thediscoveryofthe13D2 stateoftheupsilonsystem
i • ThedeterminationofvariousE1widthsforcc¯and bb¯.
X
r Ourpurposehere isto examineto what extentasemi-relativisticpotentialmodelwhich
a
includes all v2/c2 and one-loop QCD corrections can fit the below threshold cc¯ and bb¯
dataand accommodatethenewabovethresholdstates.
In ouranalysis,weuseasemi-relativisticHamiltonianoftheform
4a a
H = 2 ~p2+m2+Ar− S 1+ S(12−n )(ln(m r)+g ) +V +V
3r 3p f E S L
p (cid:16) (cid:17)
= H +V +V ,
0 S L
where m is the renormalization scale,V contains the v2/c2 corrections to the confining
L
potentialand theshortdistancepotentialisV =V +V +V +V , with
S HF LS T SI
32pa ~S ·~S a
V = S 1 2 1− S (26+9ln2) d (~r)
HF 9m2 12p
nh i
a lnm r+g 21a lnmr+g
− S (33−2n )(cid:209) 2 E + S(cid:209) 2 E
24p 2 f (cid:20) r (cid:21) 16p 2 (cid:20) r (cid:21)(cid:27)
2a ~L·~S a 11
V = S 1− S −(33−2n )lnm r+12lnmr−(21−2n )(g −1)
LS m2r3 (cid:26) 6p (cid:20) 3 f f E (cid:21)(cid:27)
4a (3S~ ·rˆS~ ·rˆ−S~ ·S~ ) a 4
V = S 1 2 1 2 1+ S 8+(33−2n ) lnm r+g −
T 3m2r3 (cid:26) 6p (cid:20) f (cid:18) E 3(cid:19)
4
−18 lnmr+g −
E
(cid:18) 3(cid:19)(cid:21)(cid:27)
4pa a a lnm r+g 7a m
V = S 1− S(1+ln2) d (~r)− S (33−2n )(cid:209) 2 E − S
SI 3m2 (cid:26) 2p 24p 2 f (cid:20) r (cid:21) 6p r2(cid:27)
h i
UsingthevariationalproceduredescribedinRef.[8],wefittheexperimentaldataforthe
the charm and ¡ systems by varying the parameters A,a ,m,m and f , the fraction of
S V
vectorcouplinginthescalar-vectormixtureoftheconfiningpotential,tofindaminimum
in c 2. Thiswas donein twoways: first by treatingV +V as aperturbation and second
L S
bytreatingtheentireHamiltoniannon-perturbatively.Theresultsare showninTable1.
TABLE1. FittedParameters
cc¯ Pert cc¯ Non-pert bb¯ Pert bb¯ Non-pert
A(GeV2) 0.168 0.175 0.170 0.186
a 0.331 0.361 0.297 0.299
S
m (GeV) 1.41 1.49 5.14 6.33
q
m (GeV) 2.32 1.07 4.79 3.61
f 0.00 0.18 0.00 0.09
V
2. RESULTS AND CONCLUSIONS
Our results1 for the fit to the cc¯ spectrum and the predicted E1 transition rates from
the resulting wave functions are comparable to recent results for charmonium [6, 7],
with the non-perturbative treatment yielding the best fit. The non-perturbative results1
for the bb¯ spectrum and decays are quite reasonable, though not as good as those from
the perturbative treatment. In Table2, we show the fit to the bb¯ spectrum for the case
oftheperturbativetreatment ofV +V , and in Table3, weshow ourpredictionsfor the
L S
observedE1transitionsandthefortheE1 decaysassociated withthe¡ (13D ).
2
Asidefromtheabovethresholdstatesincc¯,wheremixingaswellascontinuumeffects
must be included to describe the X(3872) and the X(3943), both treatments of cc¯ and
bb¯ yield very good overall fits. It is striking that for both systems the perturbative fits
require the confining terms to be pure scalar, while the non-perturbative fits require a
smallamountofvectorexchange.
1 See:http://www.panic05.lanl.gov/sessions_by_date.php#sessions3
TABLE2.
Pert Expt Pert Expt
h (1S) 9411.6 9300±28 h (3S) 10339.5
b b
¡ (1S) 9459.5 9460.3±0.26 ¡ (3S) 10359.5 10355.2±0.5
1c 9862.5 9859.44±0.52 3c 10511.6
b0 b0
1c 9893.2 9892.78±0.40 3c 10534.5
b1 b1
1c 9914.0 9912.21±0.17 3c 10549.8
b2 b2
1h 9902.1 3h 10540.9
b b
h (2S) 9996.5 13D 10149.8.
b 1
¡ (2S) 10020.9 10023.26±0.31 13D 10157.6 10161.1±1.7
2
2c 10228.9 10232.5±0.6 13D 10163.5
b0 3
2c 10254.0 10255.46±0.55 11D 10158.9
b1 2
2c 10270.8 10268.65±0.55
b2
2h 10261.1
b
TABLE3.
G g(E1)(keV) Pert Expt G g(E1)(keV) Pert Expt
¡ (2S)→g 1c 1.12 1.16±0.15 ¡ (3S)→g 2c 1.64 1.30±0.20
b0 b0
¡ (2S)→g 1c 1.79 2.11±0.20 ¡ (3S)→g 2c 2.61 2.78±0.43
b1 b1
¡ (2S)→g 1c 1.76 2.19±0.20 ¡ (3S)→g 2c 2.59 2.89±0.50
b2 b2
2c →g ¡ (13D ) 1.47 2c →g ¡ (13D ) 0.47
b1 2 b2 2
¡ (13D )→g 1c 19.7 ¡ (13D )→g 1c 5.16
2 b1 2 b2
ACKNOWLEDGMENTS
This research was supported in part by the National Science Foundation under Grant
PHY-0244789.
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