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VOLUME 37, NUMBER 20 PHYSICAL REVIEW LETTERS 1 5 NOVEMBER 1976 Interrelations between Meson Spectra J. Pasupathy Tata Institute of Fundamental Research, Bombay 400005, India (Received 15 April 1976) It is proposed that the leading Regge trajectory on which q(q^ mesons {i,j = u,d,s ,c...) lie satisfies the equation a {j (t) =a + a i /(t -b) with (a^') 2 = c^/a,/. The latter is justified with use of the factorization property of Regge poles. J/tp-like states associated with still heavier quarks are predicted to be not narrow if their mass exceeds' 5.6 GeV. I shall regard all low-lying meson states to be quark-antiquark composites {q { q i9 q i =u 9 d,s,c, .. .) with the lightest vector meson multiplet to be identified as p° = (uu -dcl)/S2, oo = (uu+dd)/yf2, cp =ss, j/^(3095) =cc, etc. The search 1 for p- like excited states has been stimulated by a num- ber of theoretical ideas, especially dual models which predict a large number of such states. In the ip family apart from ^(3684), existence of one more state #(4414) has been established 2 and the presence of more states is indicated in the struc- ture at 4.1 GeV. On the other hand it is known 3 that there are no narrow resonances ine*e~ for center-of-mass energies up to 7.6 GeV which might correspond to additional quarks. This rais- es two questions: (1) Is there a simple connec- tion between spacings of resonances associated with different q i q i composites (p, D*, and ip states, for example)? (2) Are the #-like states associated with additional quarks broad (i.e., lie above the threshold for the production of parti- cles with the corresponding quantum number)? The considerations outlined in this Letter lead to an upper bound of 5.6 GeV for the mass of the ip- like state associated with an additional quark if it is to be found as a narrow resonance. My considerations are based on the following two assumptions: (1) The leading Regge trajec- tory in all q i q j sectors is linear, 4 i.e., ot ij (t)=a + a ij '(t-b), (1) i=u,d 9 s ,c,.. . , and j =u,a r ,'s ,c,.. . , where a and b are constants independent of i and j. (2) The slopes ot { / of the leading trajectory satisfy the factorization property {a u '? = a u 'a„'. (2) I shall provide at the end of this note a justifica- tion of Eq. (2) using the factorization property of Regge pole residues. It follows from Eq. (1) that the square of the mass of a state of angular momentum J (J p = 1", 2 + ,3",... ) is given by (mi/r^b+iJ-aUoii/V 1 . (3) For the leading trajectory states ideal mixing is exact 5 (i.e., cp a n d / ' a r e pure ss states, 6 for ex- ample). It,is worth noting that over the years, the mass of p° has converged towards the mass of co(783) as demanded by ideal mixing 5 and the remaining splitting of a few MeV 7 can be easily attributed to electromagnetism. 8 For the p- and co-like states with J p = 3", I have candidates in #•(1680) and a?'(1675); the spin parity of the latter has been established to be 3" recently. 9 The con- stants a and b are then fixed by using the values (all masses now in GeV units) w(p°)=ra(co) = 0.78, m(g)=m(oo') = 1.68, m{<p) = 1.019, w7(/') = L514, which gives a = -1.93, b = - 2.63 (GeV) 2 , (aj)" 1 = 1.11 (GeV) 2 , and (a,j-')" 1 = 1.25 (GeV) 2 . Using these in Eqs. (3) and (2), I get m(f)=m(A 2 ) = 1.31, m{h) = 1.98, while for the first three K* states, 1", 2 + , and 3", the mass values 0.90, 1.41, and 1.78, respectively, which are in agreement with experiment. 10 Having determined the values of a and b from u, d, and s quark states, I determined the ip tra- jectory by using the j/^(3095) mass in Eq. (1) which gives c ^)- 1 = 4.2 (GeV) 2 . (4) Equation (3) then gives the masses of the 2 + , 3", and 4 + states on the leading trajectory cr (2 + ) = 3.71, m cr (3")=4.23, (5) ra cr (4~)=4.70. Experimentally 11 the 3 P 2 cc~ state has not yet been positively identified. In the radiative decay of ^(3684), there is evidence for at least three states x(3410), x (3500), and x(3550), and per- haps a fourth one x(3450). There is marginal evi- 1336

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VOLUME 37, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 1 5 NOVEMBER 1976

Interrelations between Meson Spectra

J . Pasupathy Tata Institute of Fundamental Research, Bombay 400005, India

(Received 15 April 1976)

It is proposed that the leading Regge trajectory on which q(q^ mesons {i,j = u,d,s ,c...) lie satisfies the equation a{j(t) =a + ai/(t -b) with (a^')2 = c^/a , / . The latter is justified with use of the factorization property of Regge poles. J/tp-like states associated with still heavier quarks are predicted to be not narrow if their mass exceeds' 5.6 GeV.

I shal l r e g a r d al l low-lying meson s t a t e s to be quark-an t iquark compos i tes {q{qi9 qi=u9d,s,c, . . . ) with the l ightest vec tor meson mult iplet to be identified a s p° = (uu -dcl)/S2, oo = (uu+dd)/yf2, cp = s s , j / ^ ( 3 0 9 5 ) =cc, e tc . The sea rch 1 for p -like excited s t a t e s has been s t imulated by a num­b e r of theore t ica l i d e a s , especial ly dual models which predic t a l a rge number of such s t a t e s . In the ip family apa r t from ^(3684), exis tence of one m o r e s ta te #(4414) has been established2 and the p r e s e n c e of m o r e s t a t e s i s indicated in the s t r u c ­t u r e at 4.1 GeV. On the other hand it i s known3

that t h e r e a r e no nar row re sonances ine*e~ for c e n t e r - o f - m a s s energ ies up to 7.6 GeV which might co r re spond to additional qua rks . This r a i s ­es two ques t ions : (1) Is t h e r e a s imple connec­tion between spacings of r e s o n a n c e s assoc ia ted with different qiqi compos i tes (p, D*, and ip s t a t e s , for example)? (2) Are the #-l ike s t a t e s assoc ia ted with additional quarks broad ( i .e . , l ie above the threshold for the production of p a r t i ­c l e s with the corresponding quantum number )? The cons idera t ions outlined in th is Le t te r lead to an upper bound of 5.6 GeV for the m a s s of the ip-like s ta te a s soc ia ted with an additional quark if it i s to be found a s a na r row resonance .

My cons idera t ions a r e based on the following two a s sumpt ions : (1) The leading Regge t r a j e c ­tory in all qiqj s e c t o r s i s l inear , 4 i . e . ,

otij(t)=a + aij'(t-b), (1)

i=u,d9s ,c,.. . , and j =u,ar,'s , c , . . . , whe re a and b a r e cons tants independent of i and j . (2) The s lopes ot{/ of the leading t r a j ec to ry satisfy the factor izat ion proper ty

{au'? = au'a„'. (2)

I shall provide at the end of this note a just i f ica­tion of Eq. (2) using the factor izat ion proper ty of Regge pole r e s i d u e s .

It follows from Eq. (1) that the square of the m a s s of a s ta te of angular momentum J (Jp = 1",

2 + , 3 " , . . . ) i s given by

(mi/r^b+iJ-aUoii/V1. (3)

For the leading t r a j ec to ry s t a t e s ideal mixing is exact5 ( i .e . , cp a n d / ' a r e pure s s s t a t e s , 6 for ex­ample) . I t , is worth noting that over the y e a r s , the m a s s of p° has converged towards the m a s s of co(783) a s demanded by ideal mixing5 and the remain ing split t ing of a few MeV 7 can be easi ly a t t r ibuted to e l ec t romagne t i sm. 8 For the p - and co-like s t a t e s with Jp = 3" , I have candidates in #•(1680) and a?'(1675); the spin par i ty of the la t te r has been es tabl i shed to be 3" recent ly . 9 The con­s tan ts a and b a r e then fixed by using the values (all m a s s e s now in GeV units)

w(p°)=ra(co) = 0.78, m(g)=m(oo') = 1.68,

m{<p) = 1.019, w7(/ ') = L514 ,

which gives a = - 1 . 9 3 , b = - 2.63 (GeV)2, (aj)"1

= 1.11 (GeV)2, and (a , j - ' )" 1 = 1.25 (GeV)2. Using these in Eqs . (3) and (2), I get m(f)=m(A2) = 1.31, m{h) = 1.98, while for the f i rs t t h r e e K* s t a t e s , 1", 2 + , and 3 " , the m a s s va lues 0.90, 1.41, and 1.78, respec t ive ly , which a r e in ag reemen t with exper iment . 1 0

Having de te rmined the values of a and b from u, d, and s quark s t a t e s , I de te rmined the ip t r a ­jec tory by using the j / ^ ( 3 0 9 5 ) m a s s in Eq. (1) which gives

( « c ^ ) - 1 = 4.2 (GeV)2. (4)

Equation (3) then gives the m a s s e s of the 2 + , 3 " , and 4+ s t a t e s on the leading t ra jec tory

™ c r (2 + ) = 3.71, m c r ( 3 " ) = 4 . 2 3 , (5)

racr(4~)=4.70.

Experimental ly 1 1 the 3 P 2 cc~ s t a te has not yet been positively identified. In the radia t ive decay of ^(3684), t he re i s evidence for at leas t t h r ee s t a t e s x(3410), x (3500) , and x(3550), and p e r ­haps a fourth one x(3450). The re i s marg ina l ev i -

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V O L U M E 37, N U M B E R 20 P H Y S I C A L R E V I E W L E T T E R S 15 NOVEMBER 1976

dence for the two pseudoscalar decay modes X(3550)-7T7T and KK; and x(3550) is probably the 3P2 state.12 Equations (2) and (4) give for the charmed vector meson masses13 mD* = 1.92 GeV andwF* =2.02 GeV.

Now suppose there is yet another quark (de­noted X) carrying a quantum number X. The J/ip-like vector meson XX then has the mass

mxx2 = b + {l -a){axx

fYly

while the lightest vector meson with nonvanishing X has the mass [see Eq. (2)],

mXr2 = b + (l-a)(axl>a«r')-1'2.

A necessary condition for the XX state to be sta­ble against the Okubo-Zweig-Iizuka-rule-allowed decays is mxx^ 2mxw. With use of the values a, b, and auU-' determined earlier, this leads to the upper bound for the mass mxx - 5.6 GeV. The bound can be slightly lowered if I take into ac­count the splitting of the 0" and 1", Xu states.

In dual models as typified by the string picture, one has many excited vector meson states which are degenerate in mass with every state (2+, 3 ' , . . .) on the leading trajectory. However there are reasons to believe14 that the string picture is expected to work best for the states on the lead­ing trajectory but require modifications for states on the daughter trajectories. Therefore I expect the mass degeneracy between excited vector mes­ons and leading trajectory states to be only ap­proximate, the approximation getting poorer the higher the mass is. In the ucl sector there are indications1 of p-like states at 1.27 and 1.6 GeV to be identified with the daughters of the 2+ and 3" states. In the us sector the reported15 en­hancement in theKTITT system, with spin parity 1" under the tensor state if* (1420), could be the can­didate for the daughter of the 2+ level. In the cc sector ^(3684) readily tends itself to the identifi­cation as the daughter of the 2+ level [Eq. (5)]. The ip states in the 4.1 structure2 are to be identi­fied as the daughters of the 3" level. If there are two closely spaced ip states in this structure, then it follows that p*(1600) enhancement must also contain two p states. This may very well be the case. Because of their large width and large number of channels [27r,47r {pirn,a>7r), r]7nr,KK, . . . ] into which these can decay, experimental delineation of the properties of these states is difficult, which is of course further compounded by the expected presence of negative G-parity co-like states approximately degenerate in mass with p" .

Justification of Eq. (2).—I shall illustrate this by considering a concrete example, although the arguments have more general validity. Consid­er the following amplitudes (a) IT+TT~ -* 7r°77°, {b)K+Tr~ ~K°ir°, 2ind{c)K+K'-K0K0. Fors-~oo, in all the three cases the p trajectory ap(t) should dominate. Following Lovelace and Shapiro,16 I can write17 the dual amplitude as

T * r[i-of0g)]r[i-a^s)] J*-p* r[l-ap(t)-aR(s)] ' (6)

R=A,B,C,

with aA(s)=af(s)=ap(s), aB(s) = aK* (s),ac(s) = <y(fi(s). The constant j3R is related to the on-shell coupling constants gp7[1[ and gpKjc by evaluating the residue of the p-meson pole at ap(t) = 1,which gives

PA=igpmi fiB=i(apf/aK*')gpmgpKK>

Pc=h(0tpf/0t(p

f)gpKK-

Now letting s -*<*>, I have from Eq. (6)

TR = pR(aBfS)aP{t). (7)

Applying the usual factorization requirement18 of Regge residues I have

(*/<**'= (aK.')2, (8)

identical with Eq. (2). The only aspect of duality [in particular, the f$-function model of Eq. (6)] that I have used in the above discussion is the fact that the asymptotic scale s0, which enters into the usual Regge pole expression (S/S0)

a(t\ is not arbitrary in dual models but is set equal to the slope of the Regge trajectory in the s chan­nel. In fact, all that I need is the validity of the asymptotic expression [Eq. (7)] for a small r e ­gion in t; and this is likely to be true in a more general class of models19 than Eq. (6).

The above discussion of the Regge trajectory slopes is purely phenomenological. There are attempts in the literature20 to compute the t ra ­jectory slopes from more fundamental consider­ations of hadron structure. For example, in the Abelian model considered by Nambu,20 the trajec­tory slope depends on the mass of the vector field responsible for binding the quarks. It is not clear whether more realistic generalization of the model could incorporate varying trajectory slopes as envisioned here.

In conclusion, my proposal that all leading t r a ­jectories are linear but have different slopes sat­isfying the factorization property Eq. (2) has the interesting consequence that, at higher energies

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V O L U M E 37, N U M B E R 20 P H Y S I C A L R E V I E W L E T T E R S 15 NOVEMBER 1976

available at SPEAR, heavier quarks carrying newer quantum numbers besides charm may be present but do not manifest themselves as narrow resonances as does the charmed quark through the j/ip particle.

*In ir'p col l is ion, photoproduction, e*e~ annihilation, e t c . F o r a s u m m a r y of the situtation regard ing p' and p» see D. Bollini et al., P h y s . Let t . 61B t 96 (1976), and r e fe rences cited the re in .

2 J . Siegrist<?£ al., P h y s . Rev. Let t . 36, 700 (1976). 3R, F . Schwi t te rs , in Proceedings of the International

Symposium on Lepton and Photon Interactions, Stanford, California, 1975, edited by W. T. Kirk (Stanford Linear Acce le ra to r Cen te r , Stanford, Calif., 1975).

4The suggestion that Regge t r a j ec to r i e s in different QiWj s e c t o r s have different s lopes has been made e a r l i e r by B. W. Lee and C. Quigg, FNAL Report No. F e r m i -lab 74/110-THY (unpublished), and R. D. Field and C. Quigg, FNAL Report No. F e r m i l a b 75/15-THY (un­published). However thei r c^-' do not satisfy our Eq. (2).

5S. Okubo, P h y s . Let t . j>, 164 (1963). 6W. Beuschef al. [Phys . Le t t . 60B, 101 (1975)] give

the upper l imit for Tiff -»Tnr)/T(ff — all) ^ 1% while A. J . Pawlicki et al. ANL Repor t No. ANL-HEP-PR-76 -26 (unpublished)] have de termined the branching ra t io as (1.40 ±0.55)%. Theore t ica l ly the pionic decays of <pmd

/ ' can be understood to a r i sed from uni tar i ty c o r r e c ­t i ons ; see J . Pasupathy, P h y s . Rev. D 12;, 2929 (1975); and J . Pasupathy and C. A. Singh, P h y s . Let t . 61B, 469 (1976).

7G, Graye r et al. [Nucl. P h y s . m B , 234 (1974)] give the value w(p°) = 778±2 MeV; cf. a lso Ref. 1.

8See J . Pasupathy, Ref. 6. 9 F . Wagner , M. Tabak, and D. M. Chew, Lawrence

Berke ley Labora tory Repor t No. LBL-3595 (1975). 1 0Experience with p - m e s o n m a s s fits sugges ts that it

i s not r a s h to hope that the difference i n / and A2 m a s s ­e s may be nar rowed down in the nea r future. F o r the 4 + , h m e s o n , W. D. Apel et al. [Phys. Let t . 57B, 398 (1975)], give the m a s s value 2020±30 MeV; W. Blum et al. [Phys . Let t . 57B, 402 (1975)] give the m a s s value 2050± 25 MeV. F o r the d iscuss ion of the evidence for the 7 = 1 , JPC = 4 + + meson in the 1940-MeV m a s s region s e e , Ch. D'andlauef al., Phys . Let t . j58B, 223 (1976). Fo r the 3 " , K* meson , G. W. Brandenburg et al. [Phys . L e t t - 60B, 478 (1976)], give the m a s s value 1776±26 MeV.

n W . Braunschweig €?£ al., Phys . Let t . 57B, 407 (1975); G. F e l d m a n ^ al., P h y s . Rev. Let t . 35 ,̂ 821 (1975); W. Tanenbaum et al., Phys . Rev. Let t . 35, 1323 (1975); W. Liith, in P roceed ings of the International Neutr ino Conference, Aachen, West Germany , June 1976 (to be published); F . P i e r r e , in Proceedings of the Eighteenth Internat ional Conference on High Energy P h y s i c s , Tbi l i ­s i , U. S. S. R., July 1976 (to be published).

12On the other hand, if it should tu rn out that the re a r e only t h r ee in te rmedia te s t a tes x(3410), x(3500), and x(3550); and that x(3550) does not decay into iru or KK, it is in te res t ing to note the following: It is well known [R. H. Dal i tz , in Meson Spectroscopy, edited by C. Ba l -tay and A. H. Rosenfeld (Benjamin, New York, 1968)] that in the ud s e c t o r , the spli t t ing between the P s t a t es 3 P 0 - 6 ( 9 7 0 ) , 3 P t = ^ i ( H 0 0 ) , 1 P 1 -J5(1235) , and 3 P 2

= yl2(1310) is mos t ly due to the sp in -orb i t t e r m ( i . e . , t he r e is approximate equal spacing) . In the cc" sec to r with the identification x(,3410) = 3 P 0 , X(3500) = 3 P 1 , and x(3550) = 2 i S 0 , equa l -spac ing ru le will lead to a m a s s value for the 3 P 2 s ta te of approximate ly 3680 MeV, which is c l o s e r to the value given in Eq. (5) and even c l o s e r to the m a s s of x(3684). See l a t e r d iscuss ion in the text .

1 3Recent exper iments by G. Goldhaber et al* [Phys. Rev. Let t . 37, 255 (1976)] and I. P e r u z z i et al. [Phys. Rev . Let t . 37 ,̂ 569 (1976)] suggest a m a s s value mD* ^ 2 . 0 1 GeV.

14See especia l ly Y. Nambu, Phys . Rev. D 1C>, 4262 (1974). One of the major flaws of the dual model i s i ts predict ion that form fac tors and sca t te r ing ampli tudes at l a rge momenta d e c r e a s e exponentially, in se r ious d i sag reemen t with exper iment . As suggested by Nam­bu, th is could pe rhaps be overcome by adding, to the s t r ing potential between the q u a r k s , t e r m s like a Yuka­wa fo rce . We can expect such modifications of the dual model to affect the low-lying t r a j e c t o r i e s like the daugh­t e r t r a j e c t o r i e s on which the exci ted vec tor meson s t a tes lie and unnatural par i ty t r a j ec to r i e s on which the pseudosca la r l i e , m o r e than the leading t r a j ec to ry s t a t e s . These l a t t e r s t a t es have the highest possible angular momentum for the i r m a s s . F o r a d iscuss ion of s t r ing s t a t e s within the context of the Massachuse t t s In­st i tute of Technology bag model , see K. Johnson and C. B . Thorn , P h y s . Rev. D L3, 1934 (1976); see also T . Eguchi , P h y s . Let t . 59B, 475 (1975).

1 5J . S. M. Vergees t et al., Phys . Let t . 62B, 471 (1976). 1 6C. Lovelace , P h y s . Let t . 28B, 265 (1968); J . Shapiro ,

Phys . Rev. 179, 1345 (1969). 17I have ignored isospin and s ignature f ac to r s ; these

a r e t r iv i a l to incorpora te . 18M. Gel l -Mann, Phys . Rev. Let t . 8, 263 (1962); V. N.

Gribov and I. Ya. Pomeranchuk, P h y s . Rev. Let t . %_, 343 (1962).

19Since (a i<7 ')" * is the m e a s u r e of spacing of r e sonances in the qilqj channel , the asymptotic region is c lea r ly the energy region where s » (a {/)"

i or (a^'s) » 1. The asymptot ic charge-exchange amplitude is then e s sen t i a l ­ly T ~ (a^'s)0^^. This feature of the asymptot ic sca le being different hi different qi ° 'qi channels plays a key ro le in unders tanding the o rder -o f -magni tude difference between the amplitude ty' -—p7r and ipf -+ipri; cf., J . P a s u ­pathy, Phys . Let t . 58B, 71 (1975), and Tata Insti tute for Fundamental Resea rch Report No. 75/41 (unpub­l i shed) .

20See Nambu, Ref. 15; Johnson and Thorn , Ref. 15; Eguchi , Ref. 15.

1338