DUAL RESONANCE MODELS Peter Goddard

To cite this version:

Peter Goddard. DUAL RESONANCE MODELS. Journal de Physique Colloques, 1973, 34 (C1), pp.C1-160-C1-166. 10.1051/jphyscol:1973117. jpa-00215197

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DUAL RESONANCE MODELS

Peter GODDARD

Department of Mathematics, University of Durham

1.- INTRODUCTION. - Giving a review of developments Before I can explain the solutions to problems lis- in dual models [I) presents problems, because the ted above, I shall first have to explain what the techniques that have been developed are not as wide- problems were and why they lead to significant ad- ly familiar as those of Lagrangian field theory, say. vances in building a dual theory of strong interac- This is particularly true this year when some of the tions. most important developoents have had a highly technical aspect. The field has advanced remarkably in 2.- BACKGROUND : THE VENEZIANO MODEL. - I start the last few months and these developments have by discussing gauge conditions and the elimination brought us to a very exciting point at this very of ghosts in the Veneziano N point function, the moment. There now is the prospect in the near future prototype dual model. In the operator formalism, of building consistent dual models for fermions, and which makes factorization evident, this N point perhaps, models which permit some direct comparison function is defined as with nature. The main areas in which our understanding has pro- gressed in the last year are the following : where i) the structure of physical states and the construction of physical state projection operators. (It is this technical development which initiated and facilitated much of the other recent progress) ii) proof that the Pomeron singularity, generated by the model, is a pole (under suitable circunstan- ces) and proof that the consequent model for Pomeror? (The integral is performed over a circle in the scattering is ghost free. complex plane through the fixed points Za,Zb,Zc, iii) proof of the gauge propcrties of the dual fer- the cyclic order of the Z's being maintained) and mion (Ramond) model, that it is ghost free if the the function F is defined by the operator expression femion mass is zero, and that it couples consistently to dual (Neveu-Schwarz) mesons. iv) the detailed way in which the dual-model spectrum and Born terms can be pictured in terms of a quantized model of interacting strings. I shall explain something of the first three developments in this review. Unfortunately space (and evaluated in the Fock space defined by the annihilation and creation operators a? aV and position and time)limitations have not permitted me to deal with m' n the fourth (see refs. 12-71). momentum operators qt, pi : Among the things these developments emphasize are igWV i) the importance of critical dimensions for dual [a:.a;' 1 - gV.' $1,". m,n > 0 ; [~~.P;I (4) models : each model is really only satisfactory in aL10) = ppl~) = 0 g = diag(-l,L,l, ...) a particular dimension of space-time, and ii) that the Pomeron singularity is an essential feature : it with V(Z,p) - :exp[ ip.Q(Z)j: (5) seems churlish to hope it will go away when the model goes to such efforts t~ make it consistent.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973117 DVAL RESONKiCE MODELS C1-161

Equation (1) then defines a Regge-behaved meromor- rators ait. Thus rather than three dimensions of so- phic amplitude which factorizes at each pole into a lutions to (10) we have effectively two. Any part of finite number of terms, the maximum number needed a state with a' in it can be neglected. Such a part being independent of the external legs. a. is the in- is null : it is orthogonal to all physical states. tercept of the leading Regge trajectory. Independen- This fact, that we only get two polarization states ce of which three points Za,Zb,Zc are kept fixed is for a photon (D-2 in D dimensions) isfamiliar. guarenteed by the Moebius group invariance property Rather remarkably, the dual model arranges a similar of F : transversality foraspectrum including massive particles. There is always one gauge condition in dual models, and this is produced by the Moebius invariance des- cribed above. But one gauge condition is not suffi- The presence of the Lorentz metric tensor g'V in cient to remove the ghosts produced by an infinity equation (4), (notice we have one time and an unspe- of time components. Virasoro [8] pointed out that cified, (D-1) say, number of space dimensions), means if a. = 1 there are an infinity of gauge conditions, that we are not working in a positive definite which could potentially remove all the ghosts. This Hilbert space. Each time component a: will create a means that the states that couple at a given mass negative-norm state : shall satisfy the condition

We can only avoid negative probabilities if it is and the Virasoro conditions not the whole Fock space which couples at the poles but rather some smaller space of "physical" states which contains no negative-norm states. The problem where is one of reconciling positivity with Lorentz invariance. It also occurs in a covariant treatment of quantum electrodynamics, where the Fourier components of the free electromagnetic potential A are quantized according to the condition,

[aw(k), aY = ko gpv 6(k-k1) . (9) Here the potential negative norm or ghost states The L 's close to form the Virasoro algebra arising from ao(kJt are removed by the Lorentz condition a.A = 0 which quantum-mechanically produces

gauge conditions in which the c-number term is of crucial importance. It turned out to be a highly non trivial problem to describe whether or not equations (12) and (13) have any solutions with negative norm. The step that pre- to be satisfied by the "physical" states I*). Thus cipitated the solution was taken by Del Giudice, Di in the dual model we need gauge conditions to remove Vecchia and Fubini r9) (DDF) who constructed opera- the ghosts. Before we return to dual models let us i tors A which behave like annihilation and creation remember the structure of the solutions to (10). m operators 1101 : k is a light like vector. We work in the frame in i j *jt Aj which k = (l/fi) (1,1,0,0,. ..) and use the notation IAm,Anl = m6ij 6m,-n 9 -n (18) 1 $ = (xO+X ) (1/n) . Condition (10) becomes

but generate only physical states, because they commute with the Ln3s. The definition of A: depends and, as [a-(k),a-(k)t]- - =O and la-$,a +t(5) 110, not only on the momentum of the state being construc- it follows that 14') must be built of ait (i ?2) and a-t. The a- operators do not matter. They commute ted, hut also on a choice of a lightlike vector k. Then i runs over the (D-2) values transverse to the with the two (if we work with D =4) transverse ope- momentum of the state being constructed and k. We can regard the DDF construction as giving the complete solution to (12) and (13) together with the additional conditions

We need another dimension of oscillators to get the solutions to (12) and (13) above. However when D Fokt Troce takes the critical value 26, the DDF states are complete in the same sense that transverse photon states Planor Loop are complete -all the solutions orthogonal to them are null physical states and can be ignored. Thus we have the following no ghost theorem [11,12] : if I$) In addition to the planar loops illustrated in satisfies the on-shell gauge conditions (Lo-1) = 0, I+) figure 2a there are also non planar contributions L,]$) = 0, n > 0 and D = 26, constructed as indicated in figure 2b. It is this non planar diagramthatpotjsesses additional singularities.

a) Plonor Loop where If) is in the space spanned by the DDF states and Ins} is a null spurious state (orthogonal to all solutions of the conditions). b) NonXI-# flanor loop For D<26 we do not have this transversality but we can embed the problem in the 26 dimensional one and deduce the absence of ghosts that way. For D>26 there are ghosts. Before its occurence in the ghost problem the magic dimension D=26 and the idea that two dimensions of states should decouple had already occured as a rather Delphic prediction of Lovelace [13], as a condition for the dual Pomeron to he a pole. To see how Fig. 2 this property came true I move on to consider dual It has a Regge-Regge cut in the s-channel, affecting loops. the t-channel high energy behaviour (in agreement with Mandelstam's third double spectral function cri- 3. - LOOPS AND PRO.1ECTION OPERATORS. - The amplitu- terion). In the t-channel there are singularities des I have been discussing up to now are to be re- with vacuum quantum numbers lying on trajectories garded as the Born terms in a strong-interaction with half the slope of the input Reggeons and an in- theory. These amplitudes in general are required to tercept dependent on D. In general, these singulari- bc Regge behaved, analytic with suitable crossing ties are unitarity-violating cuts. However, Lovelace symmetry, and Lorentz invariant. To obtain unitarity 1131 pointed out that if the dimension, D, of space- we add higher-order terms. Factorization has been re- time were allowed to vary, the singularity would pro- quired so that this can be done consistently. Loops bably be an intercept-2 pole at D=26 if the gauge are formed by factorizing N point functions to ob- conditions removed two dimensions of states. This, tain amplitudes for excited states, taking a trace remarkably, is exactly what happens when D = 26. As over these excited states and performing an integral yet one does not understand "&I' this miraculous over the loop momentum in addition to the z-integra- coincidence occurs, why the number D=26 (and the tions (see Fig.1). transversality of physical states) should occur both Loops constructed in this Feynman diagram-like way in the condition for the Pomeron to be a pole and in will possess the correct normal-threshold-type sin- the proof of the absence of ghosts. gularities. They may also possess other singularities That it is a pole does not follow immediately from not put there by design, thrown up by other parts the no-ghost theorem. One has to find out how to cal- of the integration region. culate loops in 26 dimensions ensuring. that only phy- DUAL RESONANCE MODELS C1-163

sical states contributed to the discontinuities.Brink and Olive [14,15,16] showed how to do this early this year by constructing a projection operator onto the To do this they expressed E in terms of gauge opera- space of physical (DDF) states, which works on shell. tors and the Knls (implicitly selected by the DDF The idea behind it is simple and it is important be- states) : cause a number of other developments have followed from it. Consider the projection operator onto a gi- 12 ven mass shell p = 1 - N ; N is an integer. The on- shell states satisfy the condition The D's are functions of the K's alone and form a clo- sed algebra with the L's :

D 2 P,.Ln1 = (m-n)L,,,+n + n(n -l)6n,-m Since (L -1) has an integral spectrum when p is on some mass shell we can construct the projection ope- Pm,Dn] =- (2m+n)Dmtn (33) rator onto mass shell W , by using Cauchy's theorem. [Dm, Dnl = 0 Then (30) follows using the gauge conditions satisfied by the (off-shell) states I+), 1 m) : (The integral is over a small wide about 2x0). Similarly we can construct a physical-state projec-

tion operator T if we can find an operator E, with As T works only on shell, care is needed in cons- an integral spectrum such that the DDF stares are tructing loops,where off shell integrations are per- just the solutions of formed. Brink and Olive [IS] circumvented this diffi- culty by using the Feynman tree theorem developed for dealing with gauge conditions in Yang-Mills theo- Then ry. Actually it is not essential to use this theorem. One may prove directly that the conjectured loop integrand has the rightsingularities. (Alternatively, It is easy to show-that a satisfactory E is : it is possible to develop a projection operator which works off shell 1171, but so far it has only been successfully used for planar loops). Using clever extensions of the techniques developed for trees, Brink and Olive 1161 were able to prove that the Pomeron is indeed a pole in 26 dimensions. If we write the N point (tree) amplitude in the ver- Having proved that the Pomeron is a pole, the next tex propagator form(which follows from (1)) : step is to discuss its factorization [18,19], and, in particular, whether there are ghost states in the Pomeron sector. Having solved one ghost problem, one (27) is faced with another. At first it appeared that we see that theno -ghost theorem is equivalent to there might be a ghost at the second excited level [18], but a careful analysis by Ejeveu showed that all was well. where This analysis also produced further of evidence of the similarity between the amplitudes for the Pomeron and thevirasoro-Shapiro (VS)mo&lr20,21]. This model, and similarly for (m 1. which is obtained, roughly speaking, by integrating Brink and Olive [14] were able to show this di- the Veneziano integrand over a disc rather than a rectly (thus constructing another proof of the absen- circle, also has leading intercept 2. Both it and ce of ghosts) by showing the Pomeron sector of the Veneziano model are factorized by two commuting sets of annihilation and crea- (al.4tyE~1") = o (30) tion operators like the a's, a and a say. In the VS from which (28) follows by integrating twice because model, the gauge conditions are very similar to two C1-164 P. GODDARD

sets of Virasoro conditions space of states. The model is initially factorized in a Fock space defined by the a operators of the Veneziano model together with anticommuting b operators :

(Lo -a>lvs) = 0 (where the Ln and En are made out of a and a operators respectively). The Pomeron gauge conditions are and the physical states satisfy gauge conditions rather different :

(L, - in) 1 P ) = o . (3 6) Another difference between the two cases is in the propagators. The VS propagator is rather like the and a mass shell condition conventional one 1 (Lo - 7) 19) = 0 . (43) The L's are defined slightly differently to equation (14) and the L's and G's close to form the algebra while the Pomeron one has an additional factor,

Olive and Scherk [22] showed that in general the dif- ferences in propagators and gauge conditions exactly This model (in 4 dimensions at least) has a spectrum cancelled. They proved that which is more reminiscent of the physical world. furthermore, one can prove exactly the same results about the absence of ghosts and the transversality of physical states [12,25,26], but this time the critical dimension is 10. Again there is a Pomeron singularity generated with the same intercept and where the IPi) are Pomeron physical states and T~~ slope as before. Thus again we have the relations is the propction operator for the VS model, constructed in the obvious way

The relation a R(0) = 1/2a P (0) has obvious attrac- This means that the Pomeron sector has the same spec- tions and people have tried to argue its model indepen- trum as the VS model. It is still an open question dence. This model has the attractive feature that the whether thc scattering amplitudes are the same, tachyon on the leading Regge trajectory is absent though the shortage of available models makes it very and it is possible to formulate the version with no likely. tachyons at all (outside the Pomeron sector). Also the model has a conserved G-parity-like quantum num- 4. - THE NEVEU - SCHIJARZ (PION) AND RAMOND (FERMION) ber which enables us to think of one particle as a a.- Up to now I have been describing develop- pion. ments in terms of the Veneziano model, partly for One may construct projection operators for this simplicity. The class of known models, which can model as well. Here the Pomeron sector is also some- achieve the degree of consistency I havedescribed what improved, with the tachyon missing on the lea- for the Veneziano mode1,isvery restricted. The other ding trajectory. Unfortunately there is a negative- model which has been most thoroughly studied is the G-parity Pomeron with intercept 1. However, there are dual pion model of Neveu and Schwarz [23,24]. This indications that theNeveu -Schwarz Pomeron may improve model maintains the ghost killing mechanisms of the further when fermions are taken into account. 1271 Veneziano model enlarged to deal with the larger It is in understanding fermions that the most DUAL RESOtiIANCE MODELS C1-165

recent progress hasoccurred.Soon after Neveu and figure 5. This m-0 condition had previously been Schwarz proposed their model for mesons they and deduced as a necessary condition for the decoupling Thornr28,29] showed how to produce a model for fer- of meson states, by Thorn1361 and also by Schwarz mions with the physical states satisfying the gauge [I], as a condition for the fermion no-ghost theo- conditions and generalized Dirac equation proposed rem to work. In retrospect it is easy to deduce it by Ramond [30]. The amplitude first written down des- as a necessary condition for the fermion states to cribed a dual fermion emitting mesons (Fig.3). By be transverse at the first excited level. dualising to the configuration of figure 4 to fac- One reason that it was important to get the fer- torize at a meson pole they showed that the meson mion emission vertex correct, particularly in rela- sector of the Ramond model was the Neveu-Schwarz tion to the meson states which couple to it, is model. that it is necessary for the construction of the Figure 4, or, more generally, figure 5, defines a amplitude for fermion scattering (Fig.6) . vertex for fermion emission 131,321. Whereas the vertex for meson emission (used in Fig.3) had simple properties with respect to the fermion gauge conditions, the fermion-emission vertex of figure 5 is a very much more complicated object[33,34]. With some effort it was shown that fermior gauges convert into sums of

Fig. 6

As Schwarz has pointed out [311, this amplitude is interesting for the light it throwson the duality properties of our fermions. Here the gauge properties of the fermion vertex have a non-negligible effect. Figuratively speaking, they are reflected backwards and forwards to produce a correction factor. This has been calculated by Olive and Sct~erk [37]. They showed that, for the absence of meson ghosts, a factor ~(x)has to be introduced into the meson propagator. Then

L -1 = (1@ & A 1). (46)

So a candidate for the amplitude of figure 6 is Figs.3-4-5 meson gauges. But unfortunately meson gauges did not convert into sums of fermions ones. Thus there was a danger of being able to generate meson ghosts by Corrigan [38] has gone some way towards evaluating fermion-antifermion pairs. In fact the fermion ver- (47), and obtained tex changed an "incident" meson gauge into a sum of "transmitted" fermion gauges and a sum of "reflec- { dxx-a(~)-l(~ - x)k2k3 C(x) , (48) ted" meson gauges; This reflection and transmission behaviour had been observed between the Pomeron and where 2 )5 D det(1-A (X ) Reggeon sectors of the conventional Veneziano model, C(x) = A (x) where it caused no problem 1341. Brink, Olive, Rebbi k=O (49) and Scherk [35] were able to show that if the ground S are tensor quantities formed from the Dirac ma- state fernion nass were zero the transverse meson k trices (normally y5, y', [y', yv], yPy5, 1 when states would be complete at the point indicated in D=4), and C1-166 P. GODDARD

2 ~(x)= dct(1- M(x) ) (50) So, soon we should known that singularities are in m+n+l the t-channel of figure 6, and consequently what the n + + - 112 - 112 Mm(x) ' (-XI 2-m+n+l ( ) ( ) duality properties of Rarnond fermions are. Schwarz 1311 has suggested that they be interpreted as quarks (51) a.>d gluons and, in a very recent paper1411 shows how and (Am) and (wTlm are the antisytmnetric and sym- the addition of N colour degrees of freedom to the metric parts of M respectively. Thus we have 3 un- T 2 -1 Neveu Schwarz model produces a critical dimension known functions, det (1 - ~~(x)),~(x) and V (1-A ) V. of Ds10-2N. Obviously N=3, D=4 is an attractive It is pointed out that there is at least one simple possibility. relation between them [39] :

2 ACKNOWLEDGEMENT. - I am grateful to Edward CORRIGAN, det - *) = 1 - IvT(l.~~-l~I(52) 2 det (1 -A ) David FAIRLIE, David OLIVE and John SCHWARZ for ex- plaining things to me and helpful conversations Nobody has a yet shown how to calculate these functions, but Schwarz and Wur40] found, using a computer that In a comment after the talk Claudio EBB1 drew 114 A(x) = (1 - x) , attention to a paper of S. MANDELSTAM ("Manifestly at least to six decimal places. dual formlation of the Ramond model" Berkeley preprint, August 1973) which calculates fermion-fernion scattering amplitudes using his string formalism 171.

REFERENCES

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