LETTERE AL NUOYO CIMENTO VOL. 43, N. 4 16 Giugno 1985

Hadronic Spectroscopy: Light and Heavy Mesons (*).

A . S. DE CASTRO and H. P. I)E CARVALHO

Universidade Federal do R io de Jane i ro Ins t i tu to de F i s i ca I l ha do ~Wundgo - R io de Janeiro, I~S, C E P 21.941, B f a s i l

A. B. d'OLIVEIF, A

.Fac,ddade de Engenharia , Depa?~ de Yf,~sica e Q u i ~ i c a , ~ . E G - U . N E S P Guaratinvuetd, SSo Paulo , S P - C E P , C E P 12.500, Bras i l

(rieevuto il 7 Gennaio 1985)

PACS. 12.85. - Composite models of particles.

S u m m a r y . - We calculate the spectrum of the masses of light and heavy mesons, using a potential that shows confinement and asymptotic freedom for quark-antiquark pairs. From the analysis of the results we est imate the masses of mesons with bot tom not yet found experimentally. We discuss the behaviour of the confinement potent ial under Lorentz transformation and calculate the spin-dependent corrections. We also calculate the electromagnetic decay rates and the leptonie and hadronic decay widths for some m e s o n s .

Hadrou spectroscopy has given us, in these last few years, many clues to unravel the structure of matter. In this article, we want to study a li t t le more the meson spectrum. Much data are being obtained in high-energy machines, and this furnishes us with data that can be used to study hadron fenomenology. The theoretical framework is given by QCD which may be the right theory to describe strong interactions. To test QCD predictions is one of the objectives of hadron spectroscopy. Hadrons are interpreted as being composed of quarks. Unfor tunate ly up to now we have been unable to ex- tract from theory the quark bound states. However, great success has been achieved in the description of heavy mesons, bound states of Q~ using a potential model to describe the interaction between quarks (1,2). This interaction is given, in a non- relativistic approximation by a potential tha t shows asymptot ic freedom at small distances and confinement at large distances.

(*) ~Vork p a r t i a l l y s u p p o r t e d b y C N P q a n d F I N E P . (i) W. APP:ELQIYIST &tld I{. D. POLITZER: Phys . Rev. Lett. , 34, 43 (1975). (3) E. EICttTEN, ~J~. GOTTFI~IED, T. KINOSHITA, J . ]~OGUT, K . D. I~AN]q; a n d T. M. YAN: Phys . Rev. JLett., 34, 369 (1975).

161

I n a p r e v i o u s w o r k (3), we h a v e s h o w n t h a t t h e p o t e n t i a l

V ( r ) = K r ~ - - - ~ + c 3r

- 0 . 3

used t o g e t h e r w i t h t h e Sch r 5d i uge r e q u a t i o n descr ibes well t h e J / ~ a n d Y famil ies , g iv ing exce l l en t a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a . The m e t h o d of n u m e r i c a l so lu t ion h a s b e e n p r e v i o u s l y desc r ibed b y one of us (4).

M o t i v a t e d b y t h e success of t h e n o n r e l a t i v i s t i c de sc r ip t i on of t he famil ies of h e a v y mesons , we e x t e n d e d t h e ana lys i s for t h e l i g h t mesons , w h e r e t h i s mode l is less re l iab le ( re la t iv i s t i c effects m a y be i m p o r t a n t ) . W i t h t h i s ob jec t ive , we cons ider K un ive r sa l , i.e. t h e s a m e lor all q u a r k - a n t i q u a r k pa i rs . T h e ef fec t ive coup l ing c o n s t a n t % depends on t h e r e l a t i v e m o m e n t u m , however , we a s s u m e i t to h a v e t he s ame v a l u e for all famil ies , a n d C is t h e u n i q u e p a r a m e t e r to be d e t e r m i n e d for e ach fami ly . W e t a k e ~ = 0.187 (ca lcu la t ion in f i rs t o rde r of t h e r a t i o Fr247 a n d we fix K b y f i t t i ng t h e f i rs t four S - s t a t e s of e h a r m o n i n m . W e cons ider t h e 4 8 s t a t e to h a v e t h e mass of 4.415 GeV. W e d e t e r m i n e C for al l k n o w n v e c t o r m e s o n s b y f i t t i ng t h e 1S s ta tes , a n d ca l cu la t e t h e nmss s p e c t r u m . Our r e su l t s a re s h o w n in t a b l e I .

Now we t r y to e s t a b l i s h t h e d e p e n d e n c e of t h e masses of t h e mesons on t h e v a l u e of t h e p a r a m e t e r C. T h r o u g h a f i t t i ng p r o c e d u r e , we ge t C = A 2 x 2 + A~x + Ao, w h e r e x = In [ (m~m6b + mq m~o) (GeV) -3] a n d a a n d b are f l avour indices. T h e coef- f ic ients d e t e r m i n e d b y l e a s t - s q u a r e fi ts a re A 2 = 0 . 0 1 0 G e V , A~ = 0.146 GeV a n d A 0 = -- 1.412 GeV. T h e g r e a t i n t e r e s t in t h e d e t e r m i n a t i o n of th i s f u n c t i o n a l depend- ence lies in t h e poss ib i l i ty of o b t a i n i n g b y i n t e r p o l a t i o n v a l u e s o f C for t he r e s o n a n c e s

- 0 . 5

- 0 . 7

- 0 .9

o '~ -1.~

-1.3

-1.5

-1.')

/ r I t I i I I I I

-1.6 0 1.6 3.2 4.8 X

162 A . s . DE CASTRO, t I . F. :DE CARVALttO and A. B. D'OLIV:EIRA

Fig. 1. - Parameter C(QaQb) in terms of the constituent masses x, given by x = ln[(m~amQb 2 + ?~Qa ?nQb) (GoV)-3] .

(s) H.F . DE CARVALHO, I~. CHANDA and A. B. D'OLIVEIR.~.: Left. Nuovo Uimento, 22, 679 (1978); Ho F. ])E CARVA~ttO ~nd_ _A. ]3. ~D'OLIvEIRA: Lett. Nuovo Cimento, 33, 572 (1982). (') E. GERCK, A. ]3. D'OLIVEIRA and J. A. C. GALLES: Phys. Rev. A, 26, 662 (1982); Rev. Bras. Fis., 13, 183 (1983).

HADRONIC SPECTROSCOPY: L I G H T AND HEAVY MESONS ] 6 ~

p r e d i c t e d theore t i ca l ly , b u t n o t y e t f o u n d e x p e r i m e n t a l l y . Th i s is t h e case for t h e mesons s5 a n d cb. T he masses t h u s d e t e r m i n e d are (fig. 1): M ~ = 5.338 GeV a n d Mc~ = 6.329 GeV.

I t is i n t e r e s t i n g to obse rve t h a t t h e masses of t h e r e s o n a n c e s v a r y l i nea r ly w i t h t h e sum of t h e masses of t h e c o n s t i t u e n t s , as s h o w n in fig. 2. Th i s serves as a m e t h o d to fix t h e ef fec t ive masses of t h e quarks . T h e coefficients of t h e l i nea r f u n c t i o n ~l fQ.~( l~q) : = a,(mQ= ~- m ~ ) ~- a 0 are a , = 1.058 a n d a 0 = -- 0.013 GeV. Th i s w a y we d e t e r m i n e t h e masses of r e sonances sb a n d eb d i r ec t ly w i t h o u t u s i n g a d y n a m i c a l p r o c e d u r e : M~(1S) = 5.277 GeV a n d Mo~(1S) = 6.335 GeV. U p to now, we h a v e es sen t i a l ly ana - lysed t h e s p e c t r u m of l igh t a n d h e a v y mesons . W e n o w ana ly se t h e l ep ton ic a n d h a d r o n i e decay, r e l a t iv i s t i c co r rec t ions d e p e n d i n g on sp in a n d e l e c t r o m a g n e t i c t r a n s i t i o n s for all v e c t o r mesons f o r m e d b y pa i r s ( Q ~ ) .

9.1

7.7

3.5

6.3

,~ 4.9

2.1

0.70 1.8 3.6 5.4 7.2 9.0

(rn Q=+ mob ) (GeV)

10.5

Fig. 2. - Masses of the resonances M a~b(1S) in terms of the sum of the constituent masses: mQ~ + m~b.

T h e express ions for t h e l ep ton ic a n d h a d r o n i c decays of n381 s ta tes , i n c l u d i n g per- t u r b a t i v e co r rec t ions f rom QCD are g i v e n b y (5)

/'~-~a~a .... = F~ [1 + (4.9 =}=0.5)cQ(M~)] ,

Fr->a~a .. . . = F~ [1 ~ - ( 3 . 8 :J: 0.5) ~ ' (MT)]

(5) :a . BARBIERI, l~. GATTO, P~. K~GERTEN and Z. KOMSZT: Phys. Lett. B, 57, 455 (1975); P. ]3. ~VIACKENZIE and G . P . LEPAGE: Phys. Rev. Lett., 47, 124~: (1981).

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I IADl lONIC SPECTlgOSCOPY: L I G I l T AND I IEAVY MFSONS 1 ~ 5

TABLE I I . - S-states (nlSo) and P-states ]or bottomonium and charmonium, using line and hyper/ine splitting, in GeV.

0.0 0.4 0.5 0.6 1.0 E x p e r - m e n t a l (6)

llSo b b 9,396 9.408 9.411 9.414 9.427 cc 2.917 2.965 2.977 2.989 3.036 2.981

2 ' 8 o b b 9,976 9.982 9.983 9.985 9.990 cc 3.598 3. 622 3.628 3. 634 3.659 3.592

13P2 b[) 9.896 9.888 9.887 9.885 9.877 9.912 cc 3,592 3.559 3.551 3.543 3.511 3.551

13Pi bE 9.857 9.863 9.865 9.867 9.873 9,892 c~ 3,453 3.482 3.489 3.496 3.525 3.505

laPo bE 9.818 9.835 9.840 9.844 9.861 9.870 cc 3.327 3.401 3.419 3.438 3.512 3,414

111)1 bE 9.865 9.868 9.869 9.870 9.874 c~ 3.475 3.491 3.495 3.499 3.516

A~ b ~ 1.03 0.90 0.86 0.80 0.32 0,91 cc 1.09 0.95 0.89 0.80 - - 1.11 0.50

2~P2 bE 10.260 10.255 10.254 10.253 10.248 10,266 cc 4.006 3.984 3.979 3.973 3,952

2aP1 b ~ 10.233 10.237 10.238 10.239 10.244 10.251 c~ 3.909 3.928 3.933 3.938 3.958

23Po bE 10.205 10.217 10.219 10.222 10.234 10.234 c~ 3.818 3.869 3.882 3.894 3.945

21p~ bE 10.238 10.241 10.242 10.242 10.244 cc 3.925 3.936 3.939 3.942 3.953

J2 bi) 1.01 0.89 0.85 0.80 0.41 0.88 cc 1.08 0,94 0.88 0.80 -- 0.50

TABLE 1II . -- S-states (nlSo) and P-states ]or light mesons, in GeV.

s~ u ~

f = 0.5 ] = 0.6 E xpe r i - ] --~ 0.5 ] = 0.6 Expe r i - m e n t (6) m e n t (~)

P S o 0.666 0.716 0.285 0.357 ~ (0.139) 21So 1.521 1.548 1,228 1,267 ~ ' (1.240) 13P2 1.621 1.585 f ' (1.515) 1.439 1.386 f (1.273)

A2(1.318) 13P1 1.406 1.438 E (1.420) 1.138 1.185 D (1.285) 13Po 1.171 1.254 g~(1.240) 0.812 0.933 ~ (0.983)

3*(0.975) 11.1)1 1.407 1.425 1.135 1.161 /I 1 0.92 0.80 0.92 0.80 2aP2 2.118 2.093 gT(2.160) 1.949 1.913 23P1 1.961 1.983 1.731 1.764 23Po 1.789 1.846 1.494 1.578 21./)1 1.965 1.978 1.733 1.571 A 2 0.91 0.80 0.92 0.80

f* (1.799)

S*(1.770)

1 6 6 A . S . DE CASTRO, H. F. DE CARVALHO and A. B. D'OLIV]~IRA

with

F o = 160 ~R(0) [2

Our results arc shown in table I. For the light mesons, the hadronic decay width has been calculated without per turbat ivc corrections from QCD.

The spin-dependent relativistic corrections also depend upon the behaviour of the potential under Lorentz transformations. The Coulomb term is due to the exchange of one gluon and has relativistic corrections analogous to the case of QED (positronium), i .e . , behaves as a vector. The nature of the confinement potential is not very clear and different hypotheses give different contributions to the spin-dependent forces (~). We analyse this question from a purely phenomenological point of view, considering a mixture of scalar and vector couplings, i .e.

and

with

and

V(r) = Veon~(r) -1- Vcoul(r) -I- G ( Q a ~ a )

Y ( r ) = Vv( r ) - ] Vs( r ) -~- C(Qa-Qa)

V~(r) = (1 -- ]) V~o~(r) ~- Vco.l(r)

V~(r) =]Vooof(r),

where 0 < ] ~< 1. For ] = 0 the confining potential will be a pure vector and for ] = 1 a pure scalar. The spin-dependent Hamil tonian is writ ten as follows: H~D~= = Hso ~-

Hss § HT, where

H s ~ r \ dr

2 H s s = 3m~Q V2VvS1" S 2 ,

1 [d2V,~ 1 dr,~\ S S HT = 3 m ~ \ d r 2 r dr J( 1' 2--3S1'r

A complete analysis for charmonium and bot tomonium is given in table II, where we define A n = [M(n3P2-- M ( n 3 P 1 ) ] / [ M ( n 3 p 1 - M(n3P0)]. From this analysis we con- elude that the scalar and vector mixture in the confinement potential should include between 50% to 60% scalar coupling. We made this analysis for the vector mesons (Qa~a) more lights with ] = 0.5 and ] = 0.6 as shown in table I I I .

(~) ]=[. J . SCttNITZER; Phys. Rev. D, 18, 3~:82 (1978); A. B. I:fENRIQUES, B. ]:I. ]~ELLET and It . G. ~r Phys. Lett. B, 64, 85 (1976); 5. D. JACKSON: T t t -2351-CERN (1977).

l l A D R O N I C S P E C T R O S C O P Y : L I G H T A N D H E A V Y M ] ~ S O N S 167

The expressions for the e lec t romagne t ic t rans i t ions in the electr ic-dipole approxi- mat ions are g iven by (s)

co

4 (2J @ 1)~e2Q(o ~ fR~.(r)raS~s(r)d r F ~ ( 2 ~ S ~ + Y@laP+) -- 3 9

o

and

co

FE~(laP3 =~ Y-F- 13S1) = ~ zce~w 3 ls(r)r3R1p(r) dr ,

o

where ~o is the pho ton energy and R(r) is t he normal ized radia l w a v e funct ion. Our resul ts are shown in tab le IV. These calcula t ions h a v e been done by means of the the- oret ical and the expe r imen ta l va lues for oJ.

TABL]~ IV. - Radia t ive transi t ions i n charraonium and bottomonium, in KeV. The co lumn indicate~ by * shows resul ts using the expe r imen ta l va lues of the P-s ta tes .

cc b]~

] = 0 . 5 ] = 0 . 6 * experi- ] = 0 . 5 ] = 0 . 6 * experi- m e n t (~) m e n t (~)

T'~ (23S1 -+ ~'laP2) 48.84 57.48 40.0 17 3.318 3.469 2.0 FE~ (2aS1 ~ ylaP1) 85.66 76.97 57.8 19 3.202 3.094 2.0 F ~ (2aS1 -> y13Po) 68.43 55.52 64.7 21 1.714 1.588 1.1 F~I (1~/)2 -+ y13S1) 616.1 584.2 601.7 330 39.60 39.07 49.5 FEl(13Pl-+ ylaS1) 398.0 420.3 436.7 -< 700 33.79 34.22 43.2 /r'E1 (laPo --> ~flaS1) 222.4 262.6 206.7 97 27.77 28.75 36.9

0.7 1.6 1.0

Our resul ts shown in table I, a l though we use a nonre la t iv i s t ic approx ima t ion , reproduce the spec t rum of l ight mesons re la t ive ly well. These resul ts are equ iva l en t to the resul ts of o ther authors (9), where t h e y use a re la t iv i s t ic model .

The behav iour of the p a r a m e t e r C ( Q ~ ) shown in fig. 1 was useful in the e s t ima te of the masses of the mesons wi th b o t t o m Ms~ (1S) = 5.338 GeV and Mr = 6.329 GeV. Since the meson B(u~) has recen t ly been found (6) we hope t h a t in a near fu tu re i t wil l be possible to tes t our predict ions . F r o m fig. 2 M~(1S) = 5.269 GeV and Mc~(1S ) =

6.324 GeV poin ts to an excel lent ag reemen t be tween the two methods . Recen t ly J ] ~ A (10) m a d e a ca lcula t ion analogous to ours, using a confining po ten t i a l

of the t y p e V(r) = -- 8.0248 + 6.757r~ In our case, besides t he confining po ten t ia l , we took into considerat ions t he Coulomb p a r t of t he po ten t i a l and we de t e rmined C(Qa-~b), t ak ing as i npu t the masses of t he known resonances. Thus we ob ta ined C(Q~b) . The quark masses h a v e been t aken arbi t rar i ly . W e m a d e a choice of masses t h a t resu l ted in the l inear dependence shown in fig. 2. I t wil l be in te res t ing to app ly this m e t h o d

(s) V . A . N o v i K o v , L . B . OKUM, M . A . SmF~iAN, A . I . VAI~'SHTEIh r, ]V[. B. •OI, OSItIg a n d V . I . ZAKHAROV; Phys. Rep. C, 41, 1 (1978) ; J . D. JACKSON: S L A C r e p o r t No . 198 (1976). (~) B. I~LI5IA a n d I t . ]V[AOR: D E S Y 84-029 ; S . N . JENA a n d T. TRIPATI: Phys. Rev. D, 28, 1780 (1983). (lo) S . N . JENNY.: Phys. Rev. D, 27, 244 (1983); ')8, 244, 2326 (1983).

168 A . S . D]~ CASTRO, I I . F . D E CAI~VALttO and A. B. D~OLIY]~]I>A

to possible s ta tes i nvo lv ing a rb i t r a ry pairs (Q~i)-b). In the work by JENA, he e s t ima ted the fol lowing masses : Mb~(1S ) = 5.390 GeV and Mb~(1S ) = 6.307 GeV, in full agree- m e n t wi th our es t imates . Our resul ts disagree for h igher states. E v e n for cha rmonium, in our model , the 4.414 GeV s ta te is the 4aS1 and no t the 5aS1 . We th ink t h a t the dis- covery of the tg sys tem, will al low us to improve this phenomenolog ica l analysis, since this sys tem possesses m a n y S s ta tes wi th energy below t h a t of p roduc t ion of mesons wi th t -f lavours . The ca lcu la t ion of leptonic decay wid th using pe r tu rba t ion correc- t ions in QCD, shows a good ag reemen t for the sys tems c~ and bg when compared wi th the expe r imen ta l values . The same cannot be said about the hadronic decay wid th , where the theore t i ca l va lue is too high. This calculat ion, however , depends s t rongly on the effect ive coupl ing cons t an t e~. W e assume the same va lue for ~ for all systems. The sp in -dependen t t e rms such as hyperf ine spl i t t ing and spin-orbi t separa- t ion give us a s t rong ind ica t ion t h a t the confining po ten t ia l is composed of a scalar and vec to r pa r t unde r Loren tz t r ans fo rmat ion . Accord ing to tables I I and I I I , i t is suggested tha t the scalar p a r t represen ts a t least 50~o of the cont r ibut ion . This is conven ien t for appl icat ions to re la t iv i s t ic models , to avo id the Klein paradox . W e show the radia- r ive t rans i t ions in tab le I V w i t h o u t correct ions such as re la t iv is t ic correct ions to the w a v e funct ions or over lap in tegra l (1P[E~[nS} (n). These resul ts are still grea ter t han the expe r imen ta l values. I n Leipzig conference (12) two new S-s ta tes for b o t t o n i u m were p resen ted : Tss(10.866 ) and T6s( l l .020 ). I n our mode l the theore t ica l va lues are Tss(10.831 ) and T6s( l l .021 ), which are in comple te accordance wi th the exper imenta l values.

(11) R. MCGLARu a n d I t . BYERS: Phys. Rev. D, 28, 1692 (1983). (1~) A. SILVERI~IAN: X X I I Internat ional Conference on High E n e , g y Phys ics , Leipzig, J u l y 1984.