P H Y S I C 4 L R E F I L W 1) b O L I l 1 1 E 1 5 , N L I \ l B E K 3 1 F I > B I { L i R ) 1 9 7 7

Elastic widths of charmed particles from duality*

Keiji Ig i t Brookhaven National Laboratory. Upton, New York 11973

(Received 28 September 1976)

Dynamical consequences based on charm and duality are examined. Elastic widths of J ( + ) - x resonances on leading trajectories to 06 are predicted to be about an order of magnitude smaller than those of p- f to aa. Further, the D* width is predicted to be extremely small, but even the D** far from threshold still keeps a small width. These features are explained as a consequence of the FESR (finite-energy sum rules) duality and kinematical considerations.

I. INTRODUCTION

Recent discoveries of new particleslm4 strongly suggest a new degree of freedom (charm5) in part ic le physics . This new degree of f reedom leads to the prediction of a r i ch spectrum of ad- ditional hadrons. On the other hand, the concept of duality f o r hadrons in the sense of finite-en- ergy sum ru les (FESR) a l s o has convincing ex- perimental support .

Therefore , i t is of g rea t use for the new hadron dynamics t o accommodate the new degree of f ree - dom i n the duality scheme of hadrons. The pur- pose of th i s paper is t o investigate a consequence of the above dynamical scheme, specializing main- ly to part ia l widths of the new hadrons. As we shal l s e e , recent experiments appear t o indicate that the Regge s lopes of new hadrons a r e s m a l l while those of ordinary ones a r e known to be al- mos t universal . One can wri te the dual amplitude explicitly when s lopes of Regge t ra jec tor ies a r e equal both in the s and 2 channels. When the two slopes a r e different, however, the B, amplitude of the Veneziano type would lead to an increasing exponential behavior a s s approached infinity in the physical region with the scat ter ing angle fixed.' Therefore , one might wonder i f new hadrons could be related t o ordinary hadrons through duality.

Recently, however, an interesting possibility has been suggested by Callan ei n1 .7 and by Kang and Schnitzer8 in o r d e r t o explain different s lopes of ordinary and new hadrons in the resonance re - gions. They considered a quark and a n antiquark of m a s s F I I , and i ~ , , attached to a m a s s l e s s s t r ing, whose r e s t tension is T o , and considered the rigid rotation of the sys tem, which would give r i s e t o the leading Regge t rajectory if quantized. It has been shown that in the nonrelativistic limit th i s t ra jec tory sat isf ies

while in the relat ivis t ic l imit

Thus, s lopes of new and ordinary hadrons will ap- proach the universal slope at infinity while these slopes a r e different in the resonance regions. If this is the case , the new and ordinary hadrons could be related through duality without contradict- ing Mandelstam's arguments.' Since it is not known, however, how to construct an explicit dual model f o r a r b i t r a r y t ra jec tor ies , we wish t o work h e r e in a phenomenological way in o r d e r to ab- s t r a c t physical implications.

Suppose we can approximate t ra jec tor ies such a s p-f , D*-D**, and J ( 4 ) in t e r m s of s t raight l ines in the resonance regions. Then one can wri te the B, amplitudes f o r DB, sD, KR, nK, and an scat ter ing in o r d e r to evaluate resonance part ia l widths on leading t ra jec tor ies a s the FESR bootstrap solution. Here ure normalized our am- plitude assuming the /-channel p coupling tu ex- t e rna l par t ic les t o satisfy SU(4) and assumed a possible SU(4) breaking to come from the experi- mental m a s s breaking. According t o the past ex- perience in dual models,"he 13, amplitude pre - dicts e last ic widths a t resonance poles on the lead- ing t ra jec tor ies in fair ly good agreements with experimental values once it i s normalized a t one resonance, but it appears not to predict re l iable values f o r daughters because of absorption. We will show that e las t i c widths on leading t ra jec tor ies satisfy r ~ * - D * * < r ~ * - K * * I r p - f alld r ~ - x < r~-f*

<rP'f fo r the sarlle value of spin. Especially, e las t i c widths of $-I s e r i e s to DB a r e about a n o r d e r of magnitude s m a l l e r than those of p-f s e r i e s to na. F u r t h e r , the D* (vector) width i s predicted to be extremely smal l , owing partly to the smal l Q value, but ever1 the D** ( tensor) which i s not s o close to the nD threshold s t i l l has a smal l width. We will elucidate that the above feature is understandable a s a general con- sequence of the FESR duality and kinematical considerat ions.

866 K E I J I I G I

11. GENERAL APPROACH A"'-(t, s ) = - hnB4(a(t), cu(s)). (2.5)

As we have discussed in the previous sect ion, we begin by writing the B, amplitudes f o r Dii, ;iD, K x , TIC, and iiir scat ter ing.

4 DD and I I ~ scattering

Let us choose the DD sys tem a s s-channel. Then we wri te the s-channel amplitudes a s AXs(s, t , 11) (I, = 1 , O), where Is is the isospin super - sc r ip t in the s channel. A's" (A's") is even (odd) under i - u crossing, which leads t o

A'S=' = - X D [ ~ , ( a u ( t ) , o(u))+B,(a( l ) , a,(u))l,

H e r e o ( t ) denotes the p-,f t ra jectory function, a$( / ) is the J ( $ ) t ra jectory function, and

The t-channel amplitudes a r e denoted by A1t(s, t , u ) (1, = 1,O) and a r e related to the s-channel ampli- tude f rom cross ing by

Axt=' =$(AIs=l -AIS=O)

B. n D scattering

The scat ter ing anlplitude f o r this case is de- composed a s follows:

T(s, f , u ) = € J ~ ~ A ( + ) ( ~ , t , LL) f$rB, r ,]A(-)(s , t ,u ) .

(2.3)

Here

= ' ( ~ 1 ~ = 1 / 2 - ~ 1 ~ = 3 / ? ) , (2.4b)

and a D * ( s ) is the Regge t rajectory for D*-D**.

C. nn scattering

We define A"'"- a s

A s f o r the KK and K f l scattering,' ' one has only t o replace ID and a+ in E q s . (2 . l a ) and (2.lb) by hK and cu *, respectively. Similar ly, fo r the T K scat ter ingL0 one should replace hTD and in Eqs . (2.4a) and (2.4b) by hTK and a,*.

We then normalize the above amplitudes assurn- ing the t-channel p coupling t o the DD, K z , and n r channels to satisfy SU(4). The SU(4) Clebsch- Gordan coefficientsl1 immediately lead us t o the relation

Thus once the value X is determined a t t h e p pole of the nn amplitude, a l l the leading e las t i c widths in the Dii, nD, K f i , nK, and n n channels a r e easily calculable in t e r m s of h and empir ical s t raight- line Regge t ra jec tor ies in the resonance regions.

F o r the p rocess A + B -A + B of the B , type, we obtain

using (A5) and (A6) in the Appendix. Here the i ( j ) denotes the quantum number of the d i rec t (crossed) channel, and w e have

and C'(AB -AB) is the known constant. In particul- a r , when I =S, Eq. (2.7) reduces t o

Using Eqs . (2.2a), (2.2b), (2.4a), (2.4b), (2.5), and s i m i l a r amplitudes f ~ r K f l and nK scat ter ing with (A5) and (A6), we obtain the following values of ~ ' ( A B -AB), a i , a j f o r respect ive cases :

15 - E L A S T I C W I D T H S O F C H 4 R M E D P A R T I C L E S F R O M D U A L I T Y 867

f o r J ( h ) (in ~ D - ~ i 5 ) , c ' = l , a i = a , , a j = a P . 9

f o r D*-D** (in nD-nD), c i = 2 , a i = a D * , a j = f f , ;

f o r $-f' (in KK-KK), c i = l , a i = a m , a j = a P) . (2.12)

f o r f-A, (in KR-KR), c i = $ , a i = a p , a j = a 0, . f o r K*-I(** (in nK-aI(), c i = $ , ali=CYK+, LYj=Q ' P 7

f o r p ( i n l r n - n a ) , c i = l , a i = c u . = a . J P ?

f o r J (in ?in-nrr), ci=;, a i = a . = a J P.

111. PREDICTlOhS A h l ) COMPARISON TO EXPERIMENTS

The equation (2.11) together with Eq. (2.12) gives u s the pnr elast ic width to be''

Using Eq. (3.1) with the experilmental p mass13 (773 MeV) and p e las t i c width13 (152 MeV), we ad- just X to be

We approximate t ra jec tor ies such a s J($) , D*-Dt*, a-f', K*-K**, and p-f in t e r m s of s t raight l ines passing the f i r s t two resonances except f o r the J ( d ) > D*-D** c a s e s . F o r the J(4) t ra jectory we assumed hr(3684) to be the second daughter of the leading $ with a , (s) = 3. Since only one resonance is observed on the D*-D** trajectory we a rb i t ra - r i ly set" (a&) ' = a:a I . Thus we obtain the follow- ing t ra jec tor ies :

a ( s ) = a ( O ) t a's =0.415.+-0.978s . We take part ic le m a s s e s f r o m experiment a s I I I , ,

= 140 MeV, ?i7,0 = 135 MeV, 1 1 1 , = 494 MeV, and t1iDo=1865 MeV. F o r nD -710 scat ter ing, we used a s external m a s s e s Do and s ince we took Do* m a s s f o r the D* t ra jec tory .

In Tables I and 11 we give the predicted e las t i c widths and m a s s e s of resonances on the J ($ ) -x and D*-D** t ra jec tor ies , respectively. We ob- s e r v e the following charac te r i s t i c fea tures in these tables:

(i) Elast ic widths of 4)-x resonances to DfS a r e about a n o r d e r of magnitude s m a l l e r compared with those of p-1. resonances to na f o r 6 z12: 2 4.

(ii) The D* (vector) e last ic width to nD is pre- dicted t o be extremely smal l (-100 keV) in spi te of i t s allowed decay. This is partly due t o the

TABLE I. Elastic widths of J(;l')-x resonances on the leading trajectory. Bound states below the DB threshold a r e not listed.

DD resonances [ J ( ; i j ) - ~ se r ies] J P Mass (MeV) Predicted width (MeV)

s m a l l Q value but even the D** ( tensor) which is not s o close t o the nD threshold s t i l l keeps a s m a l l width (-3 MeV).

Although d i rec t comparison with experiments is s t i l l difficult, let us compare the predictions with experiments fo r ordinary resonances12 in o r d e r t o check the reliability of our approach. As is easi ly seen , the calculated values of (3-f', p-A,, and f resonances in the KI? channel (Table 111), those of K*-K** resonances in the nK channel (Table IV), and those of p - j resonances i n the s n channel (Table V) a r e in reasonable agreement with exper- iments . The following observations a r e a l so made:

(iii) As is seen in Table 11, the e las t i c width of d (vector) to KI? is s m a l l due t o the s m a l l Q val- ue , while Q-f' resonance widths with higher sp ins appear t o not decrease s o rapidly a s .'V inc reases .

(iv) As is a l so seen in Table 111, the coupling of ordinary resonances such a s p-A, and f to K z i s fair ly s m a l l and d e c r e a s e s rapidly a s N inc reases .

IV. DISCUSSIONS AND COUCLUDINC REMARKS

Let u s consider h e r e the way in which the char- ac te r i s t i c fea tures (i) and (ii) in the previous sec- tion could be understood.

We will show that these fea tures will b e under- stood a s a general consequence of the FESR duality and kinematical considerations. Let u s s t a r t f r o m (i). Using the Appendix, one can make the part ia l- wave expansion of the D'D- amplitude in t e r rns of the s-channel i-x resonances a s

TABLE 11. Elastic ~viclths of D*-D'* resonailces on the leading trxjcctory.

TD resonances (D*-D*" ser ies) J' Mass (MeV) Predicted width (MeV)

K K I J I I G I

TABLE 111. Elast ic widths of 9-f', p-A2, and f reso- nances on the leading t ra jec tor ies . Bound s ta tes below the Ki? threshold a r e not l i s ted . -

KZ resonances (6-f' s e r i e s ) Width (MeV)

JP Mass (MeV) Predicted Experimental

TABLE V. Elast ic widths of p,f resonances on the leading t ra jec tor ies . The unclerlinecl value i s used a s input.

T K rcsonanccs ( p-f s e r i e s ) Width (MeV)

J Mass (MeV) Predicted Experimental

1- 1020 r y , , -4 .54 3.26*0.34

2+ 1520 r;;, =51.2 4 0 1 10

3 - 1890 r f , 3 = 4 7 . 6

4+ 2190 l-f' 4 , 4 - - -39.3

KE resonances ( p-A2, f s e r i e s ) Width (MeV)

J P Mass (MeV) Predicted Experimental

2+ 1270 A > U ) r2,2 '3.22 4.79*0.75 UiZk2)

4 . 8 6 1 1.62 ( ~ i ? f l

3- 1630 r i , 3 - 3 . 0 1 smal l

4+ 1910 1-44':)- 1 .83

5- 2170 - 1 . 0 2

with d i = $ fo r i = $,x. Hence, in the average sense of FESR duality, we obtain around u,(s) -M a t t = 0 ,

di((w;)2ri 2 I ) - ) , O ~ ( O ) ~ V ~ ~ ) . i = $, x 24,?q(~D) ave

S imi la r discussions f o r ria in t e r m s of the s-chan- nel p- j . resonances lead t o

TABLE IV. Elastic widths of K*-K** resonances on the leading t ra jec tor ies .

TTK resonances W*-K** s e r i e s ) Width (MeV)

JP Mass (MeV) Predicted Experimental

with di = + f o r i=f, f o r i = p . Using Eqs . (4.2), (4.3) with yFD(0)/y7(0) = i, we have

F o r the s a m e values of N, one has ;> ,4,igf which reduces the right-hand s ide of Eq. (4.4) ap- preciably, e.g., fo r "r = 4 , 5 , 6 , 7 the ra t ios become 0.080, 0.119, 0.146, 0.167, respectively. This explains why (r$,xl(21 t l)),, is much smal le r than (r$:l(21 + I)),,,. Similar discussions f o r the riD scat ter ing a l so explain the property ( i i ) . Ex- t remely smal l value (-100 keV) f o r D* (vector) is a l s o due to the extremely s m a l l Q value.

The property (iii) that h-f' resonance widths with higher sp ins do not d e c r e a s e s o rapidly a s iY in- c r e a s e s could be explained by duality through p ex- change with a ( 0 ) - + . The property (iv) is a l so un- derstandable by duality arguments through 4-f' ex- change having the low intercept . We conclude with a few r e m a r k s : (1) In Sec. I11 we confined our discussions only

t o resonances on leading t ra jec tor ies , but argu- ments based on the FESR duality predict both 41-X, D*-D** resonances t o be s m a l l in the average s e n s e including daughters .

(2) Final ly, we r e m a r k that the study of the new hadrons may clarify the role of daughter s ta tes in connection with duality.

ACKNOIVLEDGMENTS

The author would like to thank D r . T . L. True- man and D r . L.-L. Wang and other colleagues of the Theory Group f o r useful discussions and con- versa t ions , and Dr . D. P. Sidhu f o r programming of numerical computations. He is a l so thankful to D r . T. L. Trueman f o r the kind hospitality ex- tended t o him a t Brookhaven.

15 - E L A S T I C W I D T H S O F C H A R M E D P A R T I C L E S F R O M D U A L I T Y 869

APPENDIX: KINEMATICS AND NOTATION

We consider the scattering amplitude for the process A + B -A + B. In te rms of the amplitude f ( s , 0) (I =the s- channel isospin) which satisfies du/dC2 = If(s, 0) l z , we have Af(s, t , u) = ($x/ 2) f ' ( s , ~ ) , where

for DF, nD, K x , and nK scattering, and

for nn scattering, where the factor of 2 is due to Bose statistics.

From Eq. (Al) , we have

and from Eq. (AZ),

Near a resonance at a ( s ) =N, the Breit-Wigner form gives ei6sin6 =lll,I?/(121,~ - s - i121NI?), where M N and r a r e the mass and width of the resonance. Thus we find the elastic width a t (A\r, 1) a s

1 lim (]\I ,' - s ) -

" N s - ~ h ; 2 fi

x l l l d ( c o s o ) ~ , ( c o s o ) ~ ~

for DiS, nD, KE, and nK scattering, and

for nn scattering

*Work supported by Energy Research and Development Administration.

t o n leave f r o m University of Tokyo, Tokyo, Japan . 'G. Goldhaber e t a l . , Phys . Rev. Lett . 37, 255 (1976);

I . Peruzz i e ta2 ., ibid. 37, 569 (1976). 'B. Knapp e t a2 ., Phys. Rev. Lett . 37, 882 (1976). 3 ~ . G. Cazzoli e t d ., Phys . Rev. Lett . 2, 1125 (1975). 4 ~ . J . Aubert e t a1 ., Phys. Rev. Lett . 2, 1404 (1974);

J.-E. Augustin e t al . , ibid. 2, 1406 (1974). 's. L. Glashow, J . Iliopoulos, and L. Maiani, Phys. Rev.

D 1, 1285 (1970); J. D . Bjorken and S. L . Glashow, Phys . Lett. 2, 255 (1964); Y. H a r a , Phys . Rev. 3, B701 (1964); Z. hlaki, Prog . Theor . Phys . 2, 331 (1964); 31, 333 (1964); V. Tepli tz and P . Tar janne , Phys . Rev. Lett . 11, 447 (1963). F o r a review s e e &I. K . Gail lard, B . W . L e e , and J . L . Rosner , Rev. Mod. Phys . 47, 277 (1975).

6 ~ . Mandelstam, Phys. Rev. Lett . 21, 1724 (1968).

7 ~ . G. Callan e t a1 ., Phys. Rev. Lett. 2, 52 (1975). *J. S. Kang and H. J . Schnitzer , Phys . Rev. D 2, 841

(1975). or a review, s e e LI. Fukugita and K. Igi, Phys. Rep.

(to b e published). 'OK. Kawarahayashi, S. Kitakado, and H . Yabuki, Phys.

Lett. z, 432 (1969). "v. Rahl, G. Campbell , J r . , and K . C. Wali, J . Math.

Phys . 16, 2494 (1975). 1 2 ~ . Eguchi and K. Igi, Phys . Lett . e, 486 (1971).

This paper compares predictions of elast ic widths for ordinary hosons with experiments f r o m SU(3) and duality.

13part icle Data Group, Rev. Mod. Phys. 4J, S 1 (1976). 14using the c lass ica l s t r ing model ( s e e Refs. 7 and 8 ) ,

one can argue 0 5 < cub < 0'. The author i s thankful to P r o f e s s o r H . J . Schnitzer for clarifying this point.