L ~ T T ~ ~L ~VOVO C~I]~NWO VOL. 30, ~. 16 18 Aprile 1981

Asymptotic-Freedom Variation of ~ and the QQ States.

S. 3~I~A~I and R. NAKAS~I~'~A

Depa~'tment o] Physics, Osaka City Unive~'sity - Sumiyoshi-ku, Osaka 558, Japan

(rieevuto il 23 Febbraio 1981)

The purpose of this paper is to examine how the asymptotic-freedom variations of ~ have a large effect on the predictions for the QQ states in the heavy-quark region.

First we consider ~ and the masses of the QQ states. The v~lue of ~(mr can be determined by tracing the procedure (~) mentioned below. As is well known, the lep- tonic decay width of a vector meson (n3S~) is given by

(1) F(naS1 --~ e+e -) : 16~2e~[~n(0)Q[2(1 - - 16~J3~)/M(n3S1) ~ ,

where ~n(0)q is the origin value of wave function for the n 8S~ state. According to experimental results (2,a), i) for the p, % ~/J ~nd T,

(2) F(VQ -+ e+e-)/e~ _~ 11 KeV,

and ii) for the p', ~' and T', on the other hand, it is not the ~(V~ --~ e+e-)/e~, but the / �89 I + o M(Vq) F(Vq-+ e e-)/e~ that can almost be regarded as ~ constant, that is (1)

I ! 2 a (3) M(VQ)~F(VQ --~ e+e-)/eq ~ 0.301 (MeV)~,

where VQ, V~, ~q and ~ denote, respectively, the l aS~, 23Sx, 1~So and 21So states in th~ QQ system. Then, the I~(0)u] ~ in eq. (1) can be expressed as follows:

(4)

where

(5)

a n d

(6)

I~l(0)ql ~ : ~I[M(VQ)IM=] ~ 2V1~ and [~2(0)q] 2 : ~[M(V~)/M=]# Ma=,

$1(1 - - 16~,/3~) --~ 0.03

~ ~ ~2(1 - - 16~J3~) _ 0.068.

(1) S. MINA)~I a n d I t . NAKASHIMA: ] ) r ep r in t s OCU-70 a n d OCU-71 (1980). (3) F o r e x a m p l e , see: REVIEW OF PARTICLE PROPERTIES: ~e~). Mod. Phys., 52, S1 (1980); C. QUIGG: F E R M I L A B - C o n f - 79 ] 7 4 - T t t Y (1979). (a) G. WOLF: D E S Y 80/13 (1980).

493

~94 S. MINAI~II and ~. lqAKASttI3IA

W h e n the spin-spin in te rac t ion (8/9)z~58(r)(ala2)/m~ in the nonrela t iv is t ic po- t en t i a l model for the QQ states is t r ea ted as a small per turba t ion , the mass difference (AE~)Q be tween the n3S~ and n~So states is expressed by

(7) (AE~)Q ~ M(n 3S1) Q - - M(n ~ So) Q = (32/9)~]~~ ~ ,

where o 9n(0)q is the unpe r tu rbed wave funct ion for the s ta te specified by the pr incipal o 0 q u a n t u m n u m b e r n and the orbi ta l angular m o m e n t u m L ~ 0. Since y~(0)q _~ 9~( )q,

i t follows f rom eqs. (4) and (7) t h a t

(8)

and

(9)

(AEx) Q ~ M(VQ) - - M(~Q) = (32/9) m e j ~ 2 ~I[M(VQ)/M=] 2 Ma~,

(32/9)z~mQ ~2[M(VQ)/ ~n]' M~

The observed masses of t he ~/J(3100) and ~ (2980) give us i m p o r t a n t in format ion about (AE1) r t h a t is (AE~)o _ 120 MeV. B y mak ing use of this va lue and a re la t ion 2mQ --~ M(VQ), an in teres t ing result for the coupling cons tant

(10) q~(mr --~ 0.31

is ob ta ined by e l iminat ing ~1 f rom eqs. (5) and (8). I n order to see the mQ-dependence of a~(mQ), we pay a t t en t ion to ~,(Q2) expressed by

12z 1 12z (11) ~(Q~) -- 3 3 - - 2nf In (Q2/A2) 25 In . ~ , - 2 . ' for n~ : 4 . ( y / ~ l )

The va lue of A can be es t imated on the basis of the result (10), because as(ran) cor- responds to c~(Q 2) for Q2 = 4m~ _~ M(VQ) e. Thus we get A _~ 272 MeV. This is con- s is tent w i th the resul t (A--~ 300MeV) der ived f rom the analysis of deep inelast ic sca t te r ing (4). F r o m rela t ion (11), for example,

(12) ~(mb) ----- 0.21,

where we h a v e adopted m c - - 1550MeV and m b - - 4730MeV as the quark mass parameters .

F o r t he t -qua rk mass, var ious kinds of predict ions have been made by m a n y authors (5.7). Some of t h e m said tha t m e ~ 26 GeV (5) and others m t _~ 114 GeV (e) or 20 GeV (7). Accord ing to the recent da t a for e+e - collisions up to 31 GeV (s), how- ever, the re seems to be no evidence for t he Vt. I n v iew of this fact we here consider

(4) A. 5. BURAS a n d K . J . F . GAE~ERS: Nucl. Phys. B, 132, 219 (1978). (5) J . D. BJORKEN: S L A C - P U B - 2 1 9 5 (1978); S. PAKVASA a n d H . SUGAWAR&: Phys. Lett. B, 82, 105 (1979) ; S. MINAMI a n d R . NAKASttIMA: Lett. Nuovo Cimento, 26, 341 (1979). (6) I t . HARARI, H . HAUT a n d J . WEYERS" Phys. Lett. B, 78, 459 (1978); H . GEORGI a n d D. V. NANOPOULOS: Phys. Lett. B, 82, 392 (1979); C. E . VAYOI~'AKIS : Phys. Lett. B, 82, 224 (1979); M . A . DE CRO~BRUGGHE: Phys. Left. B, 80, 365 (1979). (~) G. PREPARATA: Phys. Left. B, 82, 398 (1979) ; G . SEGRR, H . A. ~VVELDON a n d ft. WEYERS: Phys, Lett. B , 83, 351 (1979). (s) F o r e x a m p l e , see : G. W o w : D E S Y 80/13 (1980).

ASTMPTOTIC-FREEDO~ VARIATION OF a s AND THE QQ STATES 4~

the two cases m~ = 20 GeV and m t = 26 GeV. Then, eq. (11) leads to

(13) / 0.15 for mt = 20 GeV ,

~s(mt )

t 0.14 for m~ ~ 26 G e V .

I n order to examine how the mQ-dependence of ~s(mQ) has a large effect on the proper t ies of the QQ states, let us consider the fol lowing cases: I) a, = const (_~ 0.31) and I I ) ~,(mq) has the mQ-dependence g iven by eq. (11). I n wha t follows the former and the la t te r are referred to as cases I) and I I ) , respect ively.

W e now s tudy the masses of the QQ states. I n case I), needless to say, no t only ~1, b u t also (AE1) Q is f lavour independent (cf. eqs. (5) and (8)). I n case I I ) , on the o ther hand, (AE1) Q is no longer f lavour independent . As is seen f rom eqs. (5) and (6), bo th ~1 and ~ decrease wi th decreasing ~,. Therefore, the values of (hE1) q and (AEs)Q in case I I ) are smaller t h a n those in case I), respect ively. I n o ther words, the mass of the ~Q(~) in case I f ) is larger t han tha t in case I) when the VQ(VQ) mass is known.

I n table I are shown the pred ic ted masses of the ~q and ~ which are ob ta ined by using eqs. (8) and (9) under the assumpt ion tha t M(Vt) -- 2m~ and M(V~) - - M(Vt) ----- _ ~ M ( Y ' ) - - M ( T ) ~ 558MeV. Fo r the ~b('O~), the difference be tween the pred ic ted masses in cases I) and I I ) is abou t 60 MeV (80 MeV). F o r the ~ (-oft), on the o ther hand, the difference is p r e t t y small, about 20 MeV (10 MeV).

! TABLE I. -- Predicted masses o] the ~qQ and ~TQ.

Case I) Case II)

~,(mb) 0.31 0.21

M(~}b) MeV 9 340 9 398

M ( ~ ) MeV 9 980 10 001

m~ 20 GeV 26 GeV 20 GeV 26 GeV

~(mt) 0.31 0.31 0.15 0.14

M ( ~ ) MeV 39 880 51 880 39 962 51 964

M(~!t) MeV 40 541 52 543 40 553 52 554

! ! N e x t we t ry to examine decays of the 7~Q, 7~Q, VQ and Vq. W h e n the decay w id th

of a pseudoscalar meson (P) in to the channel 2y is ca lcula ted wi thou t any considera- t ions about the effects due to v i r tua l gluons (*), i t can be expressed by (1)

(14) F(P --~ 2y) ~-- 12z~e~I~v(O)QI2m~(Mp/Mv) 3 ~(V, p)2.

I n order to get the expressions for r(~Q -+ 2y) and F ( ~ -+ 2y), we h a v e only to rewri te ]~v(0)Qr: in (14) as 0.03 [M(VQ)/M=]2M~ and 0.068 []I(V+Q)/Mn]~M~, respect ively, (cf.

(*) If the leptonie width /~(V-+e+e -) is calculated similarly, needless to say, its expression has a form without the factor (t--16~s/3~) in eq. (1). In this ease, ~1 and ~2 in eq. (4) must be equal to 0:03 and 0.068, respectively, in order to explain the observed leptonie widths for the VQ and V S. This will be used in our calculation for _P(~lq-+2~) and /'07~-+2y).

496 s. MINAMI and ~. NAKASHIlV[A

eq. (4)). Since the decay width for V--~,/P is g iven b y (~)

(15) / ' (V -+ yP) = (c~/6)(e~/m~)[(M~ - - M~)/Mv] 3/2(V, P)e ,

we can easily ob ta in the expressions for F(VQ --+ y~qq), F(V~--+ ~ ) and F(V~ --+ u (*). r

A n d the decay wid th for "~Q-+yVq can be expressed by (~)

! 2 2 ! 2 f ) 2 (16) r(~Q -~ ~VQ) = (~/2)(eQ/~Q)[{;~(~) - - ~ ( V ~ ) ~ } / ~ ( ~ ) ] ~ ~(VQ, ~Q .

For the Q(V, P), the following results have been given by several authors (9) in the i r analysis of the rad ia t ive decays of vector and pseudoscalar mesons:

(17) [2(haS1, m LSo) 2 ~_ 0.6, for m = n ,

0 .004, for m = q~ 4- 1 .

B y using these values of ~ , the rad ia t ive decay widths of vector and pseudoscalar s tates in the b~ and ~ systems are est imated.

! ! TA~L]~ I I . - Predicted decay widths o] the ~1Q, ~Q, VQ and VQ.

Case I) Case I I )

/~(~b --~ 27) eV 240 245

F ( ~ --~ 2y) eV 74 75

P ( ~ -~ y~') eV 75 85

F(]~' -->Y~b) eV 54.8 42.3

T(Y -+~'~b) eV 49.0 6.8

r ( ~ ' -~ y ~ ) eV 1.85 o.17

ms 20 GeV 26 GeV 20 GeV 26 GeV

/ '(~t -+27) eV 3960 3970 3990 3990

F ( ~ -+ 2y) eV 547 478 547 478

/ , ( t --~ yV~) cV 20.1 12.1 21.5 12.9

T(V[ --~ y~t) eV 13.1 7.8 9.0 5.3

/~(Vt -+ Y~t) eV 11.2 6.6 0.36 0.18

ff(V~ --~ Y~t) eV 3.2.10 -2 1.3- 10 -2 8.1.10 -a 2 .5.10 -~

The results are shown in table I I . Here we wish to emphasize t ha t there is no t a large difference be tween the predicted decay widths in cases I) and I I ) except f o r / ' ( V a --~ y~Q)

(*) Note that the Q(V~, ~q) should be employe4 as the overlap integral for the V~-->-y~Q process. (*) N. ISGUR: Phys. Rev. Left., 36, 1262 (1976); H. FmTZSCH and J. D. JACKSON: Phys. Lett. •, 66, 365 (1977); J. BORENSTEIN and R. SHANKAR: Phys. Rev. Lett., 34, 619 (1975).

ASY~cIPTOTIC-FREI~DOM VARIATIO;N OF ~s AND THE QQ STATES 497

and F(V~ ' ) ' ) - + 7 ~ - The theoret ica l F(V~--~7"~) and F ( V ~ - + 7 ~ in case I I ) are, respect ively, much smaller t han those in case I). This comes f rom the difference be- tween the vo lume e lements of the phase space in these two cases, t ha t is the values of (AEx) q and (AEs) a in case I I ) are smaller than those in case I), respect ively.

P rev ious ly we h a v e po in ted out t ha t in t he h e a v y - q u a r k region (~o),

(A)

and

(B)

F(-qq -+ 2y)/e~ ~ const

=~/(Vq)~F(VQ -+ 7~) / e~ ~ coas t .

I n case I I ) , re la t ion (A) does hold, while re la t ion (B) does not, because the f lavour independence of (AE~)Q has no th ing to do wi th the fo rmer (A) (*), b u t i t is an essential condi t ion for the l a t t e r (B).

F ina l ly the hadronie decay wid ths of ~q are e s t ima ted by mak ing use of our results for F(~Q--+ 2y) and the re la t ion

(lS) F(~q --~ gg -+ hadrons)/F(-~Q -+ 2u = (2/9)[o~(mQ)/oce~] 2 ,

which was po in ted out by some authors (~). The results for case I I ) are as follows:

and

F(~r --~ hadrons) ~ 7.6 M e V , F(~ b -+ hadrons) _~ 3.5 M e V ,

(19) F(~ t -+ hadrons) _~ l 1.9 MeV for m t ~ '20 G c V ,

1.7 MeV for m t ~ 26 G e V .

Since /~(Bq ~-~ 2y)/e~ does no t depend on f lavour as was emphas ized in ref. (~0), F(~Q -+hadrons ) ought to be ahnost the same for the ~r ~b and ~t, so long as ~ is regarded app rox ima te ly as a cons tan t (see eq. (18)). However , the resul ts shown in (19) indica te t ha t the effects of asympto t ic - f reedom var ia t ions of ~ on the proper t ies of the Q ~ states are no t so small even in the upsi lon system.

(~o) S. MZNA~II and R. NAKASHI~IA: Let~. Nuovo Cimento, 29, 140 (1980). (*) Under the assumption t ha t ~r ~= 1, we have Predicted t h a t -P(~Q--+2~{)/e~ is roughly the same, of the order of 30 It:eV, for the ~]c,~b and ~]s (lo). When ~'~(VQ, ~Q)2__~ 0,6, however, the prediction must be modified as fol lows: /~(~Q-->2~f)/eQ_ 18 KeV. (n) For example, see: ~I. KRAMMER and H. KRASEMAhrN: DESY 79/20 (1979), and ref. (3).