Physical states in dual-resonance models

LET'rEI(E AL NUOVO CIMENTO VOL. 1, N. 20 15 Maggio 1971

Physical States in Dual -Resonance Models (*).

P. DI V]~CCmA (**) and S. FUBINI (***)

Laboratory ]or Nuclear Science and Department o] t)hysics Massachusetts Institute o] Technology - Cambridge, Mass.

(ricevuto il 23 Marzo 1971)

The existence of linear relations O) between the states which occur in the faetoriza- t ion of the iV-point amplitude causes the decoupling of some of these states from the external ground particles. As a consequence the space of the states can be divided in two subspaces orthogonal to each other, one spanned by the coupled states (physical states) and another formed by the spurious states which do not couple to any number of external particles. The cancellation of the negative-norm states ((, ghosts ~) is then achieved once that the subspace of the coupled states has positive definite norm (2).

A physical state Ir on the mass shell must satisfy the following conditions (3):

(1) (Lo+ 1) Iv>= O, ~ l ~ > = 0 for any n > O .

These equations are only necessary conditions because they do not allow the elimina- tion of the zero-norm states which belong to the subspace of the spurious states. The solution of equations (1) for each level a(s) = iV is a very difficult technical problem; it has been solved (4) only for one certain vahm of n at a time.

A different approach based on the construction of vertex operators associated with the physical states has recently allowed the evaluation of iV-point amplitudes for excited particles (5). These amplitudes have exactly the same form as the iV-point amplitude for ground-state particles, the only difference being that the form of

(*) This work was suppor ted in p a r t t h rough founds p rov ided by the Atomic E n e r g y Commission under Cont rac t AT(30-1)-2098. (**) On leave of absence f rom Labora to r i Nazional i del CNEN, Frasca t i (Rome). (***) On leave of absence f rom Is t i tu to di Fisica dcl l 'Universi th , Torino. (') ~. FUBINI and G. V I ~ E Z L ~ O : Nuo~,o CimeMo, 6 4 A , 811 (1969): K. BARDAKCI and S. ~IANDELSTAM: Pbps. Rer. , 184, 1640 (1969).

(~) For ~ general review of the dual- resonance mo(lels see, e.q., G. VENEZIAX0: Lecilo~c Noles at the 1970 Ericc S u m m e r School, M-I,2'. p repr in t 151 (1970).

(3) E. ]7)EL G1UD[CE and P. DT VEC~CmA: NUOCO Cimento, 7 0 A , 579 (1970). In this le t te r we restr ic t our analysis to the case ~(0)= 1.

(~) E. GALZENATI, F. GLIOZZL l~. MUSTO aI~d F. NICODEM[: Vrdvers i ty of Naples p repr in t (1970). (~) l ). CAI~IPAGNA. S. FUBINI, n . NAPOLITANO and S. SCIUTO: N~(Ot~O CimeMo, 2 A, 911 (1971); L. CLX- V]~LIA ttud P. Rkl~IOX: NAL prcpr in t (1970), to be published in Pl*~Js. Rer.

823

~2~ 1 ~. DI V]~CCHIA and S. r U B ~

the vertex operator depends on the kind of the external particle. However, the require- men~ tha t the external particles are physical states forces these ver tex operators to be (~ covariant )~ under the group of ~ , . These covariant ver tex operators Y/'(~)(x, k) are related to the correspondent physical states through the asymptot ic relations

(2) lir~o <0[ ~ -- <~I exp [ i V'2 kqo],

lira xY/'(~)(x, k) = exp [i ~/2 kqo] IV). X---~0

Therefore, the solution of the eqs. (1) has been reduced to the construction of a set of operators ~(~)(x, k) which give a finite nonzero contribution to the limits (2) and t ransform under generalized projective transformations in the following way:

(3) [Ln, $/'(~)(x' k)] d - = ,. x ~x [x-'*~(~)(x' k)] for any integer n .

Using these rules i t has been possible to evaluate hr-point ampli tudes for external particles lying on the parent t ra jec tory (~).

In this le t ter we give a rule for the construction of the operators $/~(~)(x, k) associated with the physical states. Such a procedure shows manifestly the dependence of the degeneracy of the spectrum on the number of external legs (7). As a consequence of this there is a large set of physical states which cannot decay into less than a certain number of ground-state particles (v).

For the sake of clarity, we discuss in some detail the construction of the ver tex operators associated with the physical states coupled to two ground particles. The product of two vertex operators associated with the ground particle can be wri t ten in the following form:

(4) ~(x~, x~) = ~ ( x , , k,)C~(x~, k,) =

= =,x~ exp [-- ~/~po [k, log ~ + ~ log x,]] exp [i ~/~(k, + ~,)qo][xl-- x~] -a~''-' �9

�9 exp [i v'2[k,Q(-)(x~) -[- k3~(-~(x~)]] exp [i %/2[k~Q(+)(x~) ~- k2Q(+)(x2)]],

where (5)

(5) { r ~) = exp [i ~/~kQ(-'(x)] exp [i V~kQ(O'(z)] exp [i ~/~kQ(+)(x)],

~(s ) = 1 - (kl + k~) ~ , k~ = k~ = 1 .

Due to the factor [Xl--X2] -~(s)-l, •(Xl, X2) , as a function of x~, has a branch point singulari ty for x~ = x 1. /V(xl, x2) becomes a single-value function for x 2 sufficiently close to x 1 when a(s) is equal to an integer h r. Therefore for a(s)~ hr we can define the following expression:

(6) ~V(~r)(Xl) = ~ ~ ' ( X l ' ~1) ~(~2'X2 k 2 ) dx2,

xl

(8) C. ROSENZWEIG and V. P. SUKHATME: Nr Cimento, to be published. (7) F. GLiozzl: University of Turin preprint (1970); A. RABL: ~Teizmann Institute preprint (1970).

1)H'~'SICAL STATES IN D U A L - R E S O N A N C E 5IODELS 825

where the integral is evaluated along ~ small circle of the plane x 2 with the center in the point x2 = x~. I t is easy to see that WCZC)(Xl) gives a finite contribution to the limits (2) and satisfies the commutation relations (3). The finiteness of the limits (2) is a direct consequence of the definition (4) of F(Xl, x2). On the other hand, the cova- fiance of W(~V~(x~) under the generalized projective group follows from the fact that r kl) is covariant by itself and the other term of the commutator does not contrib- ute, being an integral of a total derivative along a closed circuit. In conclusion, W(N)(x~) is the vertex operator associated with the physical states coupled to two ground particles. We stress the important role of the condition ~(s)= h r in the previous proof.

The generalization of eq. (6) for states coupled to more than two ground particles is now straightforward.

The vertex operator Y/('P)(x, P) associated with a physical state of momentum P on the mass shell coupled to i ground particles can be writ ten as

(7) 3V(~)(x,p)= ~dx2...~dx~Y/.(x~,k~)YP(x2, k2) Y/'(x,,k~) ~2 26i

2, X l x ~ - 1

The integral on the variable %+x is evaluated along a small circle of the plane x~+~ with the center in the point %+1= x~. The momenta k~ must satisfy the relations

(8)

k~= 1

kh 1 - - ~J-1

~ k a = P , h=l

for any j ,

for 2 < j ~ < i ,

where 2Vj are positive integers and 2V, = l - - p 2 . Because of these constraints the integrand function in (7) is meromorphic in the integration variables %; therefore, the path of integration is unambiguously a closed line. I t is easy to show that the operator Yf(~)(x, P) satisfies the commutation relations (3). This is a consequence of the fact that Y/'(xl, kl) is (~ covariant ,) under the generalized projective group and the other terms of the commutator do not give any contribution since they are integrals of total derivatives along a closed circuit.

The physical states can then be obtained using expression (2):

(9) (% P)--~ lira (0[ dx~ dx s . . . . . . xl-->O X 1 X 2

Wl ~ff2 J ~ - I

~(X~, ki)

X~

Such a form for the physical states shows clearly the root of the degeneracy of the spec- t rum in the dual-resonance model and its dependence on the number of external legs. In fact the expression (9) defines a physical state for any choice of the momenta k s provided that the constraints (8) are satisfied. However, these constraints leave a lot of arbitrariness to the momenta k~. Therefore the space of the physical states at a certain level ~(--p2) = h r and for a fixed number of legs is defined by the independent vectors <y,, P1 which can be constructed from (9) using the arbitrariness left in the monlcnta k 3. The degeneracy at a certain level ~(- p2) = N, increases with the

~2~ P, DI VECCHIA a n d s. FUBINI

number of external legs unt i l the number of external momenta becomes large enough to reproduce the full degeneracy of the spectrum for that given level. This fact provides the existence of a large set of physical states which are not coupled to less than a certain number of ground-state particles. We remark that the states (9) contain also zero- norm vectors which are actually decoupled from the physical states.

The expression (9) can be generalized using vertex operators associated with excited resonances. In particular, if Y/~(~)(x~, k~) is a vertex operator associated with the physical state on the mass shell <r k~l, it is easy to see that also the state

(lo) (% P] = lim dx2 dxa... ~) ax, ~"1 ~ , xv-~O d 1 X2

"//'(xi, k,) X~

will be a physical state on the mass shell. I t is worth while the notice that the norm of the state (9) gives the residue of the

2i-point amplitude with k~_ h = kt~+1 (h = 0, 1, 2 . . . . . i) at the poles of the multiperiph- eral configuration shown in Fig. 1.

Fig. 1. 1

In order to illustrate how to extract the physical states from (9) let us discuss in some detail the states coupled to two ground particles. Using the expression (6) these states are given by

(11) (~01, P [ = ~ . <0, ~o[~x 2g e x p ,

or equivalently by

(12)

where

(13)

x [iV~k~Q(+'(x)] ~-o <~-, Pl = ~., <o, v I ~e~ ~xp

. P = k a + k ~ , A = k l - - k ~ , 1 - - P ~ = .N'.

In order to separate the even spins from the odd ones it is more convenient to work with the following linear combinations:

(l~) <w+.~'l=~-[<w~ Pl§176 e~p ,v,~Q+(x) cos v'~]l,=o'

1 1 8 "v P + .

PHYSICAL STATES IN DUAL-RESONANCE MODELS 827

At the level N = 1 the previous expressions give the physica l states

(16) <~+, PI = <0, PI(P.a l ) . <~_, v 1 = <0, P](z . .1) ,

which are the coupled states a l ready obta ined in ref. (3). The same result as in ref. (a) has t hen been obta ined for the physical states a t the level N = 2. B u t at the level ~V = 3 the vectors (14) and (15) do not give all the physical states. In fact using only two legs i t is impossible to generate a state 1 + and to dis t inguish between the two states of spin 1- which are called IB,> and IC,> in ref. (a). However, the s tate 1- ob ta ined from <~_, P] is a l inear combina t ion of <C,I and <B,[. The two s tates 1- have been finally separated cons t ruct ing the states of the level N = 3 which are coupled to three ground particles.

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Physical states in dual-resonance models