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IL NUOVO CIMENTO Vor.. 40 A, N. 2 21 Luglio 1977
Charmed Mesons in the Two-Step Potential Model.
1:~. S. KAUSHAL
Department o/ Physics, Ram]as College (University Enclave) - Delhi-7, India
D. S. KvLsn~R~,SRTH_~
Department o] Physics and Astrophysics, University o] Delhi - Delhi-7, India
(ricevuto l ' l l Marzo 1977)
Summary. - - We study the mass spectra, decay widths and electro- m,~gnetic mass splittings of the charmed mesons in the two-step potential model. The results are found to be in satisfactory agreement with the experiment and in accordance with the predictions made by other authors.
1. - I n t r o d u c t i o n .
Many a t tempts have been made so far to explain the heavy-meson reso- nances ~(3.1) and ~'(3.7) in terms of some basic quark-ant iquark interaction
and by postulat ing a new quan tum number for the quarks called (( charm i).
I n spite of the fact tha t a quark-confining linear potential for this purpose is
found to be useful (1), others have emphasized the importance of the Coulomb-
type force at short distances (3). A piecewise smooth potential of the form
(la) V(r) = - - E / r , r < B ,
(lb) V(r) ~-- - - Vo + K r , r > B ,
(1) B . J . HARINGTON, S. Y. PARK and A. YILDIZ: Phys. t~ev. Lett., 34, 168 (1975); J. 17. CxUNION and R. S. WILL,Y: Phys. Rev. D, 12, 174 (1975); J. S. KANG and H. J. SCHNI~rZ]~R: Phys. Rev. D, 12, 841 (1975). (2) T. APP:ELQVIST and H. D. POLITZER: Phys. Rev. Lett., 34, 43 (1975).
163
164 R.S. KAUSHAL and D. S. KULSItRESHTHA
has been used earlier (3) to explain the pion form factor da ta und the heavy-
meson resonances. While the two-s tep potent ia l (1) (henceforth t e rmed as TS potent ial) wi th a possible quark-confining mechan ism satisfactori ly pre- dicts (4) the mass spectra of various meson sys tems (start ing f rom the l ightest
pion to, probably , the heavies t tb), the first br~nch of this potent ia l (la) through
the quark-d iquark model has been used (~) in explaining the gTound-state prop-
erties of the composi te sys tems hke neutron, p ro ton and deuteron.
Recent observat ions a t S P E A R have s t rongly suggested (e.,) the existence
of a pseudoscalar charmed meson D at (1.865 ~: 0 .015)GeV and a vector charmed meson D* a t (2.01 • 0.02) GeV. Al though charge states and various other proper t ies of these charmed sys tems are ye t to be confirmed experi- mental ly , several a t t e m p t s (8-~o) have been made in recent t imes to s tudy their
decay ra tes and electromagnet ic mass sphtt ings. I n the present work, we exa- mine the predict ions of the TS potent ia l model wi th reference to the mass
spectra , pionic and rad ia t ive decay ra tes and electromagnet ic mass split t ings
of the charmed mesons. I t is worthwhile to note here t ha t we do not need any fresh input to our model except for the values of the pa ramete r s obta ined
earlier (') for various meson systems. Fur ther , we avoid the use of the conven-
t ional te rminology of u-, d-, s - type quarks and ins tead deal wi th the quark
entities, which are found to be r a the r convenient to describe various meson
(3) R. S. KAUSHAL: Phys..Lett., 57B, 354 (1975); 60B, 81 (1975). (4) R. S. KAUSHAL, D. S. KULSHRESHTHA and D. PARASHAR: -~UOVO Cimento, 37 A, 55 (1977). (5) R. S. KAVSHAL and D. S. KULSHRESHTHA: to be published in Ann. o] Phys. (s) G. GOLI)HABEH, F. M. PIERR]~, G. S. ABRAMS, M. S. AT.A~, A. M. BOYARSKI, M. BREIDEN]3ACH, W. C. CARITHERS, W. CHNOWSKY, S. C. COOPER, R. G. DE VOE, J. M. DORFAN, G. L. F]ELI)MAN, C. E. FRIEDBERG, D. FRYBERGER, G. HANSON, J. JAROS, A. D. JOHNSON, J. A. KADYK, R. R. LAHSEN, D. LOKE, V. Li)TH, H. L. LYNECH, R. J. MADARAS, C. C. MOREHOUSE, H. K. NGUYEN, J. M. PATERSON, M. L. PERL, I. PERVZZI, M. PICCOLO, T. P. PUN, P. RAPIDIS, B. RICHTER, B. SADOULET, R. H. SCHINDLER, R. F. SCHWITTERS, J. SIEGRIST, W. TANENBAUM, G. H. TRILLUNG, F. VAN- NUCCI, J. S. WHITA~:ER and J. E. WIss: Phys. Rev. Left., 37, 255 (1976). (7) I. P~RuzzI, M. PIccoLo, G. J. F~L])~AN, H. K. NOUY~N, J. E. WISS, G. S. A~RA~S, ~/L S. ALAM, A. M. BOYARSKI, M. BREIDENBACH, W. C. CARITHERS, W. CHINOWSKY, R. G. DEVoE, J. M. DORFAN, G. E. FISCHER, C. E. FRIEDBERG, D. FRYBERGER, G. GOL- DHABER, G. HANSON, J. A. JAROS, A. D. JOHNSON, J. A. KADYK, R. R. LARSEN, D. LiJKE, V. LIJTH, H. L. LYNCH, R. J. MADARAS, C. C. MOREHOUSE, J. M. PATERSON, M. L. PERL, F. 1VI. PIERRE, T. P. PUN, P. RAPIDIS, B. RICHTER, R. H. SCHINDLER, R. F. SCHWITTERS, J. JIEGRIST, W. TANENBAUM, G. H. TRILLUNO, F. VANNUCCI and J. S. WHITA~ZER: Phys. Per. Lett., 37, 569 (1976). (s) A. DE RVJULA, H. GEORGI and S. L. GLASHOW: Phys. Per. D, 12, 147 (1975); A. DE RUJULA, H. GEORGI and S. L. GLASHOW: Phys. I~ev. I~ett., 37, 398 (1976); S. 0No: Phys. Rev. Lett., 37, 655 (1976). (9) W. C]~LMAST]~R: Phys. Rev. Zett., 37, 1042 (1976); A. N. MITRA: Phys. Rev. Lett., 37, 1647 (1976). (lo) K. LAN~ and E. EICHTEN: Phys. Bey. JSett., 37, 477 (1976).
CHARMED MESONS IN THE T W 0 - S T E P P O T E N T I A L MODEL 1 6 5
systems in a consistent manner . For this purpose, we consider the composite systems made up of one charmed quark (having (4) the mass me as 1.95 GeV, the coupling strength parameter flo as 1.28 and carrying the character of a ~ system) and one (noncharmed) ant iquark (carrying the character of ei ther 7:, p, K or K* systems). The composite structures thus obtained with the con- s t i tuent quarks of 7:, p, K and K* would, respectively: be identified b y the D, D*, F and F* charmed mesons. The quark masses m~ (coupling s t rength parameters flq) obtained (4) for 7:, p, K and K* systems and used here are 0.252 GeV (2.41), 0.938 G e u (2.17), 0.824 G e u (2.37) and 1.024 GeV (2.13), respectively. I t should be noted here tha t the value of the s t rength parameter fl itself is a measure of the gluon charge on the corresponding quark specie.
2. - Mass spectra and decay rates.
Having identified the charmed mesons D, D*, F and F* as above, now we cMculate their ground-state masses and radial excitations in the f ramework of the TS potent ial model on the lines followed in ref. ('). I n this case, as one of the quarks (anti or vice versa) inside the charmed meson is heavier t han the other counterpar t , i t is ve ry likely tha t the Coulomb-like par t of potent ia l (1) acts up to a little longer distance. For this reason, we assume tha t (la) remains effective up to the second Bohr radius B1, and beyond tha t , the linear pa r t (lb) comes into p lay for all the four systems. I n this way, the first two states are obtained in the Coulomb-like potent ia l b y using
~ 2
(2) M (')~- m o ~ mq 2n ~
for n = 1 and 2. Here # is the reduced mass and fl is the s t rength parameter obta ined f rom fl = (flqflo)i. The subscript q on m and fl stands for the non- charmed quark in the different systems. Fur ther , the spectrum corresponding
to the linear par t (lb) is given by (4)
where - - a . are the zeros of the Airy function. The results of the calculations
for the four charmed mesons are shown in table I , for n ---- 1 to 10. The values of the parameters obtained for various mesons are given in table I I .
I t can be seen from table I t ha t the ground-state masses obtained for D and D* are in ra ther excellent agreement with the experimental values (e,7). The masses obtained for F and F* as 1.896 and 2.059 GeV are also, respectively, in accordance with other predicted values (lo) of 2.00 GeV and 2.15 GeV. Fur the r , i t can be noticed t ha t MD.-- M D ---- 0.150 GeV, which is p r e t t y close
]66 R.S . KAUSHAL and D. s. KULSItRESHTIIA
TABLE I. -- Mass spectra ]or various charmed mesons in the TS potential model.
u D (GeV) D* (GeV) F (GeV) F* (GeV)
1 1.858 2.008 1.896 2.059
2 2.116 2.668 2.554 2.745
1 2.474 3.586 3.468 3.697
2 2.579 3.856 3.736 3.977
3 2.665 4.077 3.956 4.205
4 2.741 4.274 4.151 4.407
5 2.811 4.452 4.327 4.592
6 2.875 4.620 4.492 4.765
7 2.937 4.777 4.648 4.927
8 2.995 4.927 4.798 5.082
9 3.051 5.071 4.940 5.230
10 3.104 5.209 5.076 5.373
TABLE I I . - Potential parameters ]or various charmed-meson systems.
System fl B o ((GeV) -1) B 1 ((GeV) -1) g ((GeV) ~)
D 1.756 2.552 13.36 9.84. l 0 -3
D* 1.666 0.948 4.962 6.77.10 -2
F 1.742 0.992 5.192 6.46.10 -2
F* 1.651 0.902 4.724 7.40-10 -2
to t h e p i o n mass as e x p e c t e d b y DE RVZUL~. et al. (3). A c o m p a r i s o n of t h e
r e su l t s of t a b l e I w i t h t h o s e o b t a i n e d for n o n c h a r m e d m e s o n s in ref . (4) shows
t h a t t h e spac ing b e t w e e n t h e success ive levels decreases , b u t a t a r e l a t i v e l y
s lower r a t e .
The m o s t p r o m i n e n t d e c a y s which a re e x p e c t e d for t h e c h a r m e d mesons a r e
D * - + D ~ a n d D * - + D 7. U s i n g a h a r m o n i c - o s c i l l a t o r c h a r m e d - q u a r k mode l ,
O~0 (8) has s t u d i e d t h e s e p rocesses b y t r e a t i n g A ( ~ M n . - - M , ) as a p a r a -
m e t e r . H i s ca l cu la t ions p r e d i c t a r a t h e r v e r y sma l l b r a n c h i n g r a t i o for t h e
p rocess D* -+ DT. F o l l o w i n g O~0, he re we ca l cu l a t e t h e d e c a y w i d t h s for t he se
p rocesses in t h e TS p o t e n t i a l mode l .
I n t h e r e s t f r a m e of D*, t h e d e c a y w i d t h s in t h e p r e s e n t case a re g i v e n b y
(4a)
(4b)
F(D*~ -+ D~176 = 5 ~ I2(k) '
F(D*~ : 3 27~ ~m~/
CHARMED MESONS IN THE TWO-STEP POTENTIAL MODEL
(4c) F(D*+ -> D+r:~ : 3 2--~ \m, d
(4d) F (D*+-> D~ +)
(4e)
a n d
(41)
where
w i t h
f ( D * o - + D~ =
~ ~,~I ~,(k), 4
4 F ( D * + - ~ D+~) = ~ (#Qq- yo)~kaI2(k),
k ~ [ ( M ~ . - - ~ 2 z 2 ~ M , - - mi) - - 4 M , m ~]/4MI),
3 3 s = -~ lnx = -5 V6.08-~ x 4 ~ ,
I(k) = 82/[(Bo.Bo.,)J(k2--~ ~2)~]
2 = (Bo,~- Bo,.)/BoDBo,..
167
(for i - ~ 7: or ~.),
H e r e BeD a n d B e , . a re t h e f irs t B o h r r ad i i for D a n d D* sys tems , r e s p e c t i v e l y ,
#o a n d #q are t h e m a g n e t i c m o m e n t s of t he c h a r m e d a n d n o n c h a r m e d qua rks
in D. The va lues of /zq are o b t a i n e d f r o m
M P (5) ~ ,= ~ # o n m .
T h e resu l t s o b t a i n e d for t h e d e c a y wid ths are shown in t ab l e I I I a long
TABLE I I I . - Final-particle momenta, decay rates and branching ratios /or charmed-meson decays in the TS potential model.
Decay Momentum k (McV) Rate (keV) Branching ratio
pre- LE pre- LE pre- LE S U4 sent sent sent
D *+ ~ D~ + 66 53 289 182 0.45 0.67 0.66
D *+--> D+r: ~ 79 75 269 84 0.42 0.31 0.31
D*+ --> D+~ �9 151 149 79 4.5 0.12 0.02 0.03
D *~ -+ D+~ - 45 53 87 64 0.30 0.32 0.27
D *~ --> DOrr176 65 73 137 84 0.47 0.42 0.36
D *~ --> Do~( 141 149 67 54 0.23 0.27 0.37
F* --)-F~ 156 - - 178 1.6(~) - - - - - -
(a) Resul t of SU6 quoted in LE,
168 R. 8. KAUSHAL and D. S. KULSHRESHTHA.
with the ones predic ted b y LA~E and ~EICHTE]N (~o) (referred to as LE). I t can be noticed f rom this table t ha t the present model predicts branching
rat ios for the processes D *+-+ D~ + and D *+--> D+T: ~ of comparable magni- tudes (unlike those obta ined b y LE) , while the branching ra t io for the pro-
cess D *+--> D+y turns out to be much larger (by a fac tor of six) t han the one
predic ted b y LE.
3. - Electromagnetic mass splittings.
l~ext we calculate the e lectromagnet ic mass split t ings in the present model for various meson systems. The interact ions responsible for Coulomb and magnet ic hyperf ine split t ings are (1~)
(6a) H ~ - - qlq2e~ ,
?'12
87~ (6b) H'~ - - 3 qsq~/~l~(a~ "a2)(~(r~- r~),
where ql and q~ are the electric-charge fract ions on the quark and on the anti- quark a n d / ~ and ~2 are thei r corresponding magnet ic moments . The ground-
s ta te wave funct ion for the cq (or ~q) sys tem in the present model is given b y
which leads to
(7a) M~ -- o 2 e 2 16 ~ul ~u2 mq M s : 3 B o ~ -~ 3 B~M me'
for the mass spli t t ings in the case of scalar charmed mesons, and to
(7b) Mv ~ -- M~ -~ . . . . . 2 eo 16 .--#1#~ __m~ 3Bo~ 9 B ~ m o '
for the mass split t ings of vec tor charmed mesons. The results obta ined for
var ious charmed mesons are shown in table IV. Here , we have assumed F
and F* to be the vec tor mesons. Fo r uncha rmed mesons, however, the results
are found to be much larger t h a n the exper imenta l ones.
Since the magnet ic in terac t ion H ' m in eq. (6) is of zero range, therefore, its use is more appropr ia te for the charmed mesons (in which one of the constit-
uents is heavier t han the other) t han t h a t for the mesons hav ing equal-mass
(11) See for example, S. ONO: Nucl..Phys., 107B, 522 (1976).
CHARMED MESONS IN THE TWO-STEP POTENTIAL MODEL 169
TABL~: IV. - Electromagnetic mass splittings ]or various meson systems in the TS poten- tial model.
Meson Point Empirical CELMAS- Experi- interaction (present) TER (~) mental (present) (MeV) (MeV) (12) (MeV) (MeV)
Noncharmed ~ _ ~ o 34.6 4.6 (*) 3.2 4.6
~o - - ~ 27.15 2.6 (*) 1.6 2.6 -t- 2.2
K ~ - - K • 29.03 0.75 4.07 3.99
K * ~ *• 26.05 4.17 0.27 6.1 • 1.5
Charmed D • - - D O 3.01 6.89 1.02 - -
D * ~ D** 2.72 3.62 0.74 - -
F ~ - - F ~ 1.55 1.62 - - - -
F *~ - - .F *+ 3.58 4.85 - - - -
~o - - ~ 6.97 10.87 - - - -
(*) Used as i npu t .
cons t i t uen t s . F o r mesons h a v i n g equa l -mass cons t i t uen t s , we, however , cal-
cu la t e e l ec t romagne t i c mass sp l i t t ings r a t h e r empir ica l ly , i n s t e a d of u s ing a
f in i t e - range i n t e r a c t i o n . F o r th is purpose , we use the e x p e r i m e n t a l va lues for
t h e 7:- a n d 0-meson mass sp l i t t ings as i n p u t a n d o b t a i n the va lues of t he
p a r a m e t e r s a a n d b f rom the equa t ions
a b ~•176 ~'~ Bo~ "~-B--~ ----- 4"6"10-3 G e V ,
a b ~• __ Qo __ - - 2.6 "10 -~ GeV
Bop 3Bo~p
as a -~ 0.0116 (GeV) -2 a n d b ~ 0.039 (GeV) -~. B y us ing these va lues of a a n d
b a n d the va l ue of Bo ( in (GeV) -1) f rom ref. (4) a n d t ab le I I , t he mass spl i t -
t ings are ca l cu la t ed for b o t h c h a r m e d a n d n o n c h a r m e d mesons . The resu l t s
o b t a i n e d wi th th is empi r i ca l fit a re shown i n t ab l e IV . The e x p e r i m e n t a l re-
sul ts (~2) a n d the resu l t s o b t a i n e d b y CELMASTn~ (~) are also g iven for com-
par i son .
4 . - C o n c l u s i o n s .
Whi l e the mass differences b e t w e e n charged s ta tes of D a n d D* have been
t he m a t t e r of d i spu te i n these days , t he p r e s e n t mode l p red ic t s these values ,
(12) V. CHALOUPKA: Phys. Ldt. , 50 B, 1 (1974).
170 R. 8. KAUSHAL &lld D. S. KULSH_RESI-ITI-IA
respect ively, as 3.01 MeV and 2.72 MeV with the point magnet ic in teract ion and as 6.89 MeV and 3.62 MeV with an empirical fit (el. table IV). I n fact , m a n y a rguments have been pu t forward (9) in favour of the value of the mass difference D = ~ - - D o < 5 MeV. However , much has not ye t been known
abou t the mass difference of D* charge s ta tes for which the present model
predicts a value somewhat smaller t han t ha t for D. I n any case, the mass
spl i t t ing for D and D* in the present model is found to be less t h a n 15 MeV, which, though con t ra ry to the predict ions of De Ru ju la et al. (s) is, however, in agrceraent with the conclusions drawn b y C~M~ST]~R (~) and ]V[ITRA (~).
I n spite of the fact t ha t the contr ibut ion of magnet ic hyperf ine in teract ion is found to be as i m p o r t a n t as the Coulomb contr ibut ion, our results are com- pat ible with the predict ions of Celmaster (~) as regards the mass splittings.
I n cognizance of the present predictions, we claim tha t the TS potent ia l provides a consistent model for analysing the propert ies of the charmed mesons.
The main advan tage of using this potent ia l is t h a t the Sehr6dinger equat ion is analyt ical ly solvable with bo th the pieces (cf. eq. (1)), which is not otherwise possible, if one uses the potent ia l of the t y p e V ( r ) : - - ( ~ / r ) ( 1 - - r 2 / A 2 ) .
I n fact , the use of the la t ter potent ia l would demand several approx imat ions a t the computa t iona l s tage (~0).
* * *
The authors wish to thank Prof. A. 5Y. MITRA and Drs. 1~. P. SAX]~A and K. C. TRIPATttY for their interest in the present work. One of us (D.S.K.)
grateful ly acknowledges the financial assistance f rom CAI~PA, Univers i ty of Delhi.
�9 R I A S S U N T O (*)
Si studiano gli spcttri di massa, le ampiezze di decadimento e gli splitting di massa elet- tromagnetiei dci mesoni con charm nel modello di potenziale a due stadi. Si trova che i risultati sono in aeeordo soddisfaeente con l'esperienza e anche con le predizioni fatte da altri autori.
(*) Traduzione a cura della Redazione.
OqapoBaHHMe Me3OHM B M O ~ e J 1 H ~ B y X e T y f l e H q a T o r o HOTeHHHa~a.
Pe3mMe (*). - - Mbr rtccne~tyeM Maccoabte crterTpl, i, mrtpi4Hr~ pacna~oa H anegTpO- MarHnTnbte pacmertneHrm Macc oqaposam~b~x Me3OHOB B MO~eJ'IH )iByxcTyneriqaToro nOTeHUnana. O6napy~eHo, nTO nony,fertrmre pe3ynbTaT~,t y~oBneTBoprITeJIbrIO corna- cyrorcn c aKcrmpr~MeHroM H COOTBeTCTByrOT npe~cKa3am~aM ~pyrnx aBTOpOB.
(*) l-lepeaec)eno pe~)amtue?t.
