duality diagrams and the pomeron

IL NUOVO CIMENTO VOL. 2A, N. 1 1 Marzo 1971

Duality Diagrams and the Pomeron.

1~. DELBOURGO and P. I~OTELLI

Physics Department, Imperial College, London

(ricevuto il 10 Luglio 1970)

S u m m a r y . - - A model for the pomeron is suggested, based upon an interpretation of all planar quark diagrams. The model involves as a subclass those <( legal ~> diagrams used to generalize the Veneziano model to supermultiplet scattering. In consequence the definition of duality is enlarged and more specifically the Harari-Freund identifica- tion of resonances with Regge poles and pomeron with background is modified.

I . - I n t r o d u c t i o n .

To define dua l i ty in a precise wa y has p r o v e d difficult. P a r t l y because

of this, the t e r m s tands for a n u m b e r of concepts (1), to which we shall be

gu i l ty in this pape r of add ing another . H o w e v e r in the H a r a r i - F r e u n d identi-

f icat ion (5) of resonances wi th Regge poles on the one hand, and the p o m e r o n

wi th b a c k g r o u n d on the other , we have a verifiable if somewha t vague con-

c e p t - t h e vagueness or ig ina t ing in p a r t f rom the prac t ica l difficulty of sepa-

r a t ing resonances f r o m b a c k g r o u n d in expe r imen ta l da ta . The Veneziano

model (3) expl ic i t ly exemplifies the resonance-Regge p a r t of this idea. I t is

however subject to a n u m b e r of faults, a m o n g which is the absence of any

P o m e r o n and the inappl icab i l i ty of the model to b a r y o n - b a r y o n scat ter ing.

(1) See for instance the review article by M. JACOB. (2) P . G . O . ]~REUND: Phys. Rev. Lett., 20, 235 (1968); It. HARARI: Phys. Rev. Lett., 20, 1385 (1968). {3) G. VENEZlANO: NUOVO Cimento, 57A, 190 (1968).

67

68 R. DELBOURGO 3,nd P. ROTELLI

One of the most impor tan t steps in the s tudy of the Veneziano model was its in terpreta t ion via legal duali ty diagrams (4), which are simply those planar quark-scattering diagrams consistent with only resonance exchanges in both channels. Not only did they provide an automatic way of incorporating exact SU3 invariance into the theory (5), but they have lately been used to incor-

porate spin into the theory automatically (¢.7) in the form of U12 invariant

amplitudes. In this paper we shall present a model for the pomeron, suggested by a

simple phenomenological version, which is akin to the generalization of the Veneziano model by the present authors, and is also interpretable in terms of planar quark diagrams. Since these diagrams are necessarily of the previ- ously called ((illegal ~ class, we shall extend our model to all planar quark diagrams. As a result we correctly reproduce the leading asymptot ic Regge or pomeron behaviours in all channels even for baryon-baryon scattering. We shall also provide an ansatz for the high-energy behuviour due to the

exchange of so-called exotics (s). In Sect. 2 we shall b r i e fy describe our in terpreta t ion of duali ty diagrams

in the context of U12- In Sect. 3 the properties of the pomeron are reproduced by a simple phenomenologieal model portraying many of the features of our

fin~l version, which is presented in Sect. 4 and applied to quark-quark scattering. Section 5 generalizes our approach to all planar quark dual i ty diagrams and its s t ructure for meson-meson, meson-baryon and baryon-baryon scattering given. Fits to some of the latest Serpukhov data involving only the pomeron are given in Section 6, and we conclude in Sect. 7 with a restrospec- t i r e definition of our form of dual i ty and a discussion of its limitations.

2. - Duality diagrams and supermuhiplet traces.

In recent papers (6) we have introduced a method for incorporating spin as well as isospin into the Veneziano model by a generalized interpretat ion

of the Imachi-I tarar i -Rosner dual i ty diagrams. These diagrams for meson-

(4) 1~¢[. IMACHI, T. MATSUOKA, K. NINOMIYA and S. SAWADA: Progr. Theor. Phys., 40, 353 (1968); H. HARARI: Phys. Rev. Lett., 22, 562 (1969); J. ROSNER: Phys. Rev. Lett., 22, 689 (1969). (5) H.M. CHA• and J. PATO~: Nucl. Phys., 10B, 516 (1969); J. ROSNER: Phys. _Rev. Lett., 22, 689 (1969). (8) R. DELBOURGO and P. ROTELLI: Phys. Lett., 30B, 192 (1969); and ICTP/69/1. (7) S. MANDELSTAM: Phys. Rev., 184, 1625 (1968), and to be published; K. BARDACKI and M. HALPER~: Phys. Rev., 183, 1456 (1969). (s) Exotics are by definition those resonances outside the naive quark model, e.g. (q~)~ with N ~ 1.

DUALITY DIAGRAMS A:ND THE I*OMERON 6 9

meson scattering (or at least all the independent ones) are shown in Fig. 1 a). By associating e~eh incoming particle with the corresponding S U3 matrix, and the quark lines linking these particles with S U3 contractions, each diagram

t t u

2 4 4 2 2 3

v(1-~(s), 1-%(0) v(1-%(u), ]-%(t)) v(1-%~8), ]_-~(u)) a)

t

S ~

V ( z 4 - - c4s, - - 1 - - ct)

U

U S •

V ( z 4 - e 4 u , - 1 - - ct) V ( h - - ~ s , - - 1 - - c u )

Fig. 1.

t t u

V(-- 1 -- cu, z 4 -- c a t) V ( - - 1 - - cs, z 4 - . c4 t)

b)

V ( - - 1 - - cs, z 4 - CaU )

will represent an S U3 trace. For scalar-scalar scattering we need only multiply each trace by the appropriate Veneziano form factor exhibiting dual poles in the Mandelstam variables shown, (in this case the C ( X , Y ) function) add the three terms to give crossing symmet ry and we have the conventional

7 0 R. DELBOURGO ~ d P. R O T ~ L L I

Veneziano model for this process (9). External spin is a complicating factor in this ~ppro~ch, but it may in certain cases such us ¢o ~-~ 3u be put in by hand by a suitable choice of kinematic fuctor.

Two ways have been suggested for standardizing the procedure for deuling with external spins. The first is based on a bootstrap philosophy (~0) and consists in projecting out the required scattering amplitude from ~ multi- Veneziano formalism. This relies on the general faetorization of the multi- Veneziano and is restricted to bosons unless an explicit formalism for the introduction of half-integer external fermions is given (~). The second approach, which may complement the first, wus advanced by the present authors and independently by MANDELSTAM ~nd co-workers. I t involves the introduction of U~2 supermultiplet w~ve functions for the externul particles (~). This permits a generalization of the Harari-Rosner rules by interpreting the duality diagrams not merely us S U~ traces but as U~ traces. Thus spin is treated relativistically and in as automatic a way as S U~. :Not only does this procedure incorporate many of the successful predictions of SUw~ but, with the same Veneziano form factors, it correctly reproduces the leading-signatured high- energy l~egge behaviours in each channel for all the processes incorporated in the scattering of supermultiplets.

The basic building blocks of this procedure are the U~ wave functions for the boosted representations of U~ ® U~, The lowest 36-dimensionul meson representation consists of ~ mass-degenerate nonet of vector mesons (J~ = 1-) ~nd the pseudoscalar mesons (0-),

(1) ¢~(p) = (;,.p + m) ( ;~ i + ~,~)~ (~')~o,

with A , B = I , . . . , 12; a, f i~--l , . . . , 4, and a , b = l , 2,3. While the lowest representation for the baryons is the fully symmetric 364-dimensional representation, containing an octet of 1+ nucleons ~nd u decuplet of ~+ b~ryons, again mass degenerate in the symmetry limit,

(2) ÷ 3[(~.p ÷ M)~l , C]~D~.~b~,

where C is the usual charge-conjugation operator.

(9) C. LOVELACE: Phys. Lett., 28B, 264 (1968); K. KAWARABAYASHI, S. KITAKADO and H. YABUKI: Phys. Lett., 28B, 432 (1969). (10) See for example, K. ]~ARDACKI and H. RUEGG: Phys. Lett., 28B, 242 (1968); D. I. OLIV]~ and W. J. ZAK~.WSKI: Phys. Lett., 30B, 650 (1969); S. MANDELSTAlVI: Comm. -~Yucl. Part. Phys., 3, 147 (1969). (11) M. B. G~EEN and :R. L. H~IMANN: Phys. Lett., 30B, 642 (1969); I. MONTVA~r: Phys. Lett., 30B, 652 (1969); D. FAIRLIE and K. JONES: Durham preprint. (12) R. DELBOURGO, M. A. •ASHID, A. SALAM and J. STRATHDEE: The U:~ Symmetry (Vienna, 1965).

D U A L I T Y D I A G R A M S AND TH E P O M E R O N 71

Thus the ampli tude corresponding to the diagrams in Fig. 1 a) for meson- meson scattering reads,

(3) V~(s , t) Tr (¢(p~)q)(p2)qS(p3)q)(p~))÷ (2~-~ 4, se-+u) -Jr- (3~-+ 4, te-+u) ,

where s = (p~ ~- p~)~, t = (p~ ÷ p~)~ and u ~ (p~ + p~)~. V~(s , t) stands for the Veneziano form factor suitably normalized and which to leading order has been shown elsewhere to be given by

(4) v~(s , t) ~ ~ r (~ - ~(8)) r O - %(0) _r(2- %(s ) - %(0) '

where fl~ is a constant. In general it can be assumed tha t a number (possible infinite) of daughter terms reside in V~(s, t) as well as the simple beta-funct ion of eq. (4). Consistent with present data, the resonances are assumed to lie on straight-line Regge trajectories, e.g.

(5) %(s) = ~o + s a ' ,

where ~' for all known l~egge trajectories ~ 1 (GeV/c) -~. Since the poles of the beta-function occur at integer positive values of %, exchange degeneracy is implied. I t is of course the al ternative expansions of the V-function in terms of an infinite number of poles in one or other of the dual variables which

characterizes duality. The above expression reproduces the original Vene- ziano model for o - ~ 3~ and unifies this ampli tude with ~ - 0 ~ : ~ , ~p--~Hp coo--> p~ etc. To leading order, the beta-funct ion form for V serves admirably at high energies bu t gives poor results at low energies for such things as seat- tering lengths, which suggests the need of additional daughter terms in V at the very least (1~).

The amplitude for meson-baryon scattering described by the diagTams of Fig. 2 a) is

(o)

where

(7)

and

(s)

T ~ o ( _ p,) LP~(p~)[¢~(p~)¢~(p~) V~¢(s, t) ÷ (1+-+3, s~-+ u)] ÷

(la) Low-energy results will also be modified by the inclusion of ~< illegal ,) diagrams and by unitarization procedures.

72 R . D Z L B O U R G O ~ n d P . R O T E L L I

t t u

a)

Fig. 2.

t t

S • U

2 ~ & 2 ~ & V(zs - -%s , - - 1 - -c t ) V ( z s - -c ~ u , - - 1 - -c t )

b)

the various Ws differing only in their intercepts. Again the low-energy results for ~2~"--> ~A ~ are poor, and a set of ghosts degenerate with the physical particles confronts us with the need of breaking the symmet ry and eliminating the a t tendent par i ty partners (19 before the model can even begin to represent the physical resonance structure. We may temporari ly by-pass this

problem by directing our a t ten t ion to the high and possible intermediate energy regions. There the model with its few parameters is very compact, and i t has been (15) applied to the decays at rest of ~n--> 3~ and ~ p - + 3~.

Restricting our a t tent ion to high energies, the major failing of this model is the absence of the pomeron and of any legal duali ty diagrams for baryon- baryon scattering. The problems are not of course unrelated since the pomeron

will contr ibute to elastic baryon-baryon scattering.

(14) First noted and studied in the ease of ~-9 scattering by P. G. O. F~UND and E. SC~O~B~G." Phys. Left., 28B, 600 (1969). (as) K. J. BARN]~S and S. C. SACKER: private communication.

D U A L I T Y D I A G R A M S AND T t t E POMERON 73

3. - A p h e n o m e n o l o g i c a l p o m e r o n .

The Pomeron, like the quarks and the intermediate vector boson, is one of the most intriguing objects in elementary-particle-physics. By definition it dominates all elastic high-energy scattering data. For large s and fixed t i t corresponds to an exchange in the t-channel of a particle with the quantum numbers of the vacuum. This ampli tude AP(s, t) is assumed to become pure imaginary in the forward direction as s - ÷ c~, and if the total cross-section is asymptotical ly flat then A~(s, O) ~ is for large s. To the extent tha t elastic scattering at high energies is the shadow effect of all the inelastic modes, then the pomeron's properties might be uniquely determined by the uni tar i ty equation, bu t in practice it may prove more fruitful to use the phenomenological properties of the pomeron to assess the behaviour of the multiparticle-produc- tion processes, about which we know so little experimentally. A fur ther feature of the pomeron is the exponential f~ll-off in t of d(rJdt for fixed s.

Summarizing these properties for the ease of scalar-scalar meson scattering, and doing so in an s-u crossing symmetric Regge formalism gives for large s

(9) A~(s, t).-~ flF-~(~(t)) exp [at] (1 + exp [iz~(t)]) (s_/~'") s i ~ ( ~ ( t ) ) ~.So! '

where a(t) ---- 1 + a'~t and fl is a real constant. The 1/sin (na(t)) not only appears natural ly from a Sommerfeld-Watson t ransformation but plays the essential role of cancelling the zero in the signature factor at t : 0. We now also know from the latest Serpukhov data tha t a'~ ~ 0.5 (GeV/e) -2. Although no longer considered zero a ' is still much smaller than the slope of conventional Regge-poles.

That i t is not a Regge pole is par t of the ga ra r i -F reund concept of duality. We may interpret this to mean tha t it is devoid of the poles in a(t) say for real values of t, poles which would characterize the Veneziano amplitudes in the narrow-resonance approximation. We can easily modify eq. (9) to incorporate this feature, without da/dt vanishing in the forward direction, by writing

(10) AP,-~fi (1 + exp[i:~e(t)]) (s__] ~'('' sinh (~bt) \So/ '

where b is real. More generally b may be complex but when chosen as real it is the antithesis of the Regge form (eq. (9)). The 1/sinh (zbt) incorporates

the exponential fall in t for large t and has the added advantage of being bounded for both positive and negative real t, i.e.

(11) d a (pomeron),-~ exp [2(=b + o:'~ln(s/So))itl] for It[ ~ 0. d~

74 R. D E L B O U R G O a n d I ~. R O T E L L I

I0 o

,o -, ~.

\

\ \

\ \

\ \

\ \ \

\ \

~ \ \ \ \ \

lO.-4

I I I I

0 0.3

G r a p h I . - - - - p h e n o m e n o l o g i c a l p o m e r o n ,

- - e x p [ l l . 3 t ] f i t t o d a t a f o r It] < 0 .12 .

\ \

\ \

L \ \ \

\ \ \ \ \

\ \

\ \

I I I "

0.6 - t 0.9

- - - - - - p o m e r o n a F ( - - 1 - - et)(1 --- 0 .2 t ) "~,

DUALITY DIAGRAMS AND TIIE POMERON 7 5

To be exact we must note tha t the pomeron considered is an S U3 singlet in the t-channel and allow for pomeron exchanges in both the s and u channels. This gives in total

[ sinh (zbt) \so/ I~ -~ eycl. (s, t, u) ,

where I~ represents isotopic spin zero in the t-channel etc. Observe the desir- able feature tha t for say fixed t and large s, in the case with no shinkage the second and third terms fall exponential ly with s and are soon negligible. However in general eq. (12) represents the pomeron contr ibution as an energy- dependent mixture of isotopic spins.

The most interest ing proper ty of this simple phenomenological form for the Pomeron is the appearance of poles for imaginary values of s, t and u. This novel p roper ty will characterize the pomeron throughout this paper. As i t stands this model is not entirely satisfactory even phenomenologically due to a discrepancy between a pure exponential fall-off in t for all t < 0 and tha t given by 1/sinh (7&t). To illustrate, we plot the d(r/dt given by eq. (10) with zb = 6.5 (GeV/c) -2 in Graph 1.

4. - The pomeron diagrams and quark-quark scattering.

Let us examine the dual i ty diagrams which could possibly represent the

exchange of a pomeron in one of the channels. I t is easy to identify them because the pomeron carries the quantum numbers of the vacuum. They are those diagrams in which no quark or ant iquark lines are exchanged.

Fig. 3.

Examples of such diagrams for various processes are shown in Fig. 3. To determine what form factor we have to associate with each diagram let us start by considering the specific, if hypothetical , case of quark-ant iquark scattering. All four diagrams involve the pomeron and are shown in Fig. 4. Figure 4 a) obviously corresponds to meson resonance (gegge) exchanges in the s-channel and pomeron in the t-channel. If we recall the form-factor s tructure for the

7 ~ R. ])ELBOUGGO a n d e . ROTELLI

t

S

a)

V(1 -- a¢(s), -- 1 - - c t )

Fig. 4.

t

2 4

b)

V ( - - 1 - - cs, 1 - - % ( 0 )

t (/

g_~ s >

4 2 2 3

c) d)

V ( z ~ - - c 2 u , - - 1 - - ct) V ( - - 1 - - cs, z 2 - - c.zu)

legal dual i ty diagrams given in Section 2, they consists of a product of two gamma-functions, one for each of the dual variables, divided by a gamma- function which eliminates all the douple poles, lqow we know tha t the gamma- function for the exchange of a Regge pole in the s-channel is F ( I - %(s)). We might ask what is the equivalent form for the pomeron exchange. Our analysis in Sect. 3 suggests we look for a function with poles along the imaginary axis. The simplest choice is F ( i t ) and indeed,

(13) I F ( i t ) 12 = u~-'/sinh (~t).

More generally we shall consider T ' ( - - n - - c t ) where n is a positive integer and e predominant ly imaginary. The exact value of n can be determined from the high-energy behuviours in b o t h s and t. Figure 4 a) has the Ulz trace ~ 3 u l ~ d u ~ , - ~ ( t - ~ - c o n s t ) and multiplied by its form factor (with no double poles) is to leading order

F(1 - - a ~ ( s ) ) F ( - - n - - e t )

(14) u 3 u l u , u ~ F ( - - m + 1 - - n - - a ~ ( s ) - - c t ) '

where m is ~ positive integer. For large t and fixed s this behaves as t-~-~(s)

and we conclude tha t m ---- 0 is needed to reproduce the desired Regge behaviour. For large s and fixed t eq. (14) goes like s n+ct. Since this corresponds

to a pomeron exchange we must set n = 1. Thus for Fig. 4 a) the form factor to leading order is also a beta-funct ion

(15) v ( s , t) ,.~ 1"(1 - ~ ( s ) ) r ( - 1 - ct) r ( - - a)

So far our pomeron is wi thout signature. As with Regge poles it is the addi-

t ion of two diagrams which produces the signature factor. In this case where

D U A L I T Y D I A G R A M S A N D T H E P O M E R O N 77

the pomeron is exchanged in the t-channel we mus t consider Fig. 4 a) and 4 c) together. However the u-ehalmel in Fig. 4 c) corresponds to a double quark or exotic s ta te! We know almost nothing about exotics in general, bu t to reproduce the conventional pomeron signgture 4 c) mus t correspond to the form (1/c..)F(z2- c~u) where z: and c~ are complex numbers . I t will be sug-

gested in Sect. 5 t ha t the Re z~ depends only upon the to ta l nmnber 1 of quark

and an t iquark lines exchanged, and tha t in this case Re z2 = ½. Thus 4 a) and 4 c) together give (for the p t ra jectory)

(16) ~ F ( 1 - - s ) F ( - - 1 - - c t ) 1 F(½ ÷ i y 2 - - c ~ u ) F ( - - 1 - - c t ) }

tip ~3 u ~ 4 u2 | - - F ( ~ ~ - -~s - - c t i - - ÷ c.~ F(iy2 - - ~ - - c~ u - - ct) '

where we have used the definition Yt = Inlzz.

The equivalen~ expression for 4 b) and 4 d) is obta ined by crossing sym- m e t r y f rom eq. (16), i.e. 3 ~ 2, s ~-+ t.

However our formal ism as it stands is only good to leading order in s(u), since the pole at t = 0 in F ( - - 1 - ct) is only cancelled by the signature fac-

tor of the leading and even-daughter trajectories in s. This suggests tha t we

eliminate all odd-daughters f rom eq. (16) both for the resonance and the exotic

channels. Since aesthet ic a rguments have already been made by MANDEL-

STA~ (,6) for el iminating the odd daughter trajectories of the beta function (~;) in the case of legal dual i ty diagrams it is t empt ing to require this of all channels in all diagrams. For example, to el iminate the first daughter we m a y add to (.16) the ex t ra form factors

(17) fl~u3u~u'u~ { ( ~ m~ 1) p(½ - s ) F ( - 1 - ct) ÷

P ( ÷ 1 __ s - - ct)

1 - z2~ r ( z ~ - e 2 ~ ) r ( - 1 - ct)~ +

2 is the sum of the square of the external masses. But the exact where ~ mq

structure for V mus t awai t fur ther analysis and we shall henceforth only concern ourselves wi th leading-order effects.

5. - The pomeron and physical scattering amplitudes.

The diagrams for meson-meson scat tering are shown in Fig. 1. We see tha t

those in Fig. 1 b) correspond to a pomeron exchanged in one channel and to

a (2q2~) exotic in the other. To determine the value of l~e z4 for these exotics,

(1~) S. ~[ANDELSTAM: Phys. Rev. Lett., 21, 1724 (1968). (17) Due to the kinematic factor m tile UI~. trace this does not mean the elimination of odd daughters in general.

7~ R. DELBOURGO and P. ROTELLI

and hence their high-energy behaviour in the pomeron variable, we note the following:

F ( ½ - s) corresponds to a p exchange in s-channel,

F(1 ~ - s) corresponds to a 5e exchange in s-ehann@

17"(--1- cs) corresponds to a Pomeron exchange in s-channel.

Of course these give only the parent - t ra jec tory intercepts correctly (18) (the pion t ra jec tory must be artificially raised from its first daughter role in I " ( 1 - %(s))). However they satisfy the following mnemonic

(18) V(--1 + ~l--e~s) ,

corresponding to total quark number 1 exchanged in the s-channel. For non- exotic resonances ct is of course ~' ~ 1 (GeV/c) -2. Now this rule has encom- passed the pomeron structure (l = 0) as well as the l~egge poles and we shall assume it applies equally to the Re zz for the exotics. In all physical scattering processes exotics occur for l > 3 and this rule will mean tha t these objects never dominate the high-energy behaviour. The imaginary par t of z~ does not affect the high-energy behaviour bu t will have bearing on whether any of the exotics manifests itself as a recognizable physical particle or is too broad to be identified. If we te rm all cases with 1 > 3 as exotic, then the deuteron is an outstanding example of an exotic resonance lying on the real axis below the nucleon-nucleon threshold. Thus for the deuteron t ra jec tory we must have F(%(M]--s)) where M~ is the mass of the deuteron. Since here 1 = 6 , it is interesting to note tha t our rule gives M ] R e c 6 = 3 1 and

hence l~e c e _~ 1.0 (GeV/c) -2. For meson-meson scattering our formalism gives for the diagrams in Fig. 1 b),

(19) fi~v ~ | [ -F(1-~i~yd ~cdsZ-c~ -~- /"(l + iyd--c,u--ct) "

- - t/4m~) ~ ~- cycl (s, t, u)}. I~(1 I

The 1/c~ is included so that fl~ be real. The additional terms for meson-baryon scattering are shown in Fig. 2 b)

and are related by crossing symmetry . For meson-nucleon scattering they

(is) We have taken the liberty of raising the he intercept from its physical value of ~ 0.18. The physical value must, of course, be used in curve fitting and the difference may be attributed to our simplifications of degenerate multiplet mass and universal slopes for the Regge trajectories.

DUALITY DIAGRAMS AND THE I~OMERON 79

contr ibute

( 1 ) / F ( 2 ~ - , i y s - - c s s ) F ( - - 1 - - c t ) (2o) ~ ; / ~ ~ ~ P(2 i + i y~ - - c s u ) l ' ( - - 1 - - col + F(1} -~- i y5 - - c s u - - c t ) I *

, I ~ , u , ( 1 - - t/4M~)(1 - - t /4m2),

where M is the mass of the nucleon and m tha t of the meson. Thus our ansatz

for the in tercept of the exotic channels means t ha t for large t these te rms go like t -t+c'', while those in Fig. 2 a) will go like t i+* and dominate at least in the

forward direction s = 0.

t s t t

(2 (J

2 ~ 4 ~ 2 V ( e d 3 ~ - s ) , - - 1 - e t ) V ( - ~ - c u , c~(31-s)) V(z~-cou,- -1 ~t) V ( - 1 - - e u , z~ -~ t )

Fig. 5.

t

V ( z , - c~u, 1 - %(t))

t 1

V( 1 -- %(u), z~-- c~ t)

The only diagrams for ba ryon-ba ryon scat ter ing are shown in Fig. 5. Applying the constraints of crossing s y m m e t r y and Fermi statistics to the case of nucleon-nucleon scat ter ing leads to the ampl i tude

[ F ( c ~ ( 3 } - - s ) - - l - - c t ) -~ I ' ( z 6 - - c 6 u - - l - - c t ) J

[F(~o(31- s))F(- ~ - c~) F(~o--c.t)F(--~--cu)].

_ _ R J

• I t ( 2 + iy, - e~u)r(1 - ~ ( t ) ) r ( 2 + iy~ - - c~t) F ( 1 - - ~ ( u ) ) K;(u, t) [

80 R, DELBOURGO and P. ROTELLI

where the k inemat ic fac tor K~(t, u ) ~ ~ 3 u ~ u ~ represents the exchange of a meson nonet in the t-channel and has the high-energy behaviour of K~(t, u)--~

t2u + O(t). The R and P superscripts on fl~v distinguish between those te rms

with Regge and pomeron exchanges respectively. I t is worth re-emphasizing t ha t a l though none of the diagrams in Fig. 5

of the legal type they do incorporate the Regge and pomeron exchanges in the allowed channels~ which any feasible model mus t provide.

6 . - P a r a m e t r i z a t i o n .

The latest Serpukhov da ta on high-energy p-p scat tering (19) has yielded

the best determinat ion to date of the pomeron parameters . Specifically if the

diffraction slope is given by

d ~ / d ~ ,-o (22) d t / d ¢ = exp [(bo + 2bl ha (S/So))t],

then a fit is found with

(23) bo = 6.8 ~= 0.3, bl = 0.47 ± 0.09, So = 1 (GeV/c) ~ .

I n our formal ism since the pomeron has been assumed to be universal these values determine c, and a suitable choice is

(24) c ---- 0.4 -4- i2 .0 .

This provides for a stLrinkage effect in the ampl i tude of 0.4, consistent with eq. (23). The expression for da/dt (normalized to 1.0 at t----0) for p-p scattering is to leading order given in our model (so) by

d a / d ~ = { 1 + exp[4~t]--exp[2~t]cos(O.4~tl}[F(--1--(O.4+i2.0)t) l~* (25) d t / d t t=o

, (s/s°)°'st (1 - - 0.2 t) 6,

(19) G. G. BEZNOGIKtI, A. BUYAK, K. J. JOVCHEV, L. F. KIRILLOVA, P. K. MARKOV, B. A. MOROZOV, V. A. •IKITIN, P. V. •OMOKONOV, M. G. SHAFRANOVA, V. A. SVI]~IDOV, T. BIEN, V. J. ZAYACHKI, N. K. ZHIDKOV, L. S. ZOLIN, S. B. NURUSHEV and V. L. SOLOVIANOV: Phys. Lett., 30B, 274 (1969). (20) We have assumed % = 1 in deriving equation (25). I t is a first approximation and does imply that the deuteron recurrences would be on the real axis, i.e. appear as physical resonances. There is evidence for such a dibaryon trajectory from the analysis of L. M. LIBBY and E. PREDAZZI: Lett. Nuovo Cimento, 2, 881 (1969).

DUALITY DIAGRAMS AND THE POMERON 81

where we have chosen for M in the ~12 trace a mean mult ip le t mass M ~ = = 1 ~ (GeV/c) ~, and all Mandels tam variables are in (GeV/c) 2. A plot of this funct ion againt t for Pz = 58.1 GeV/c is shown in Graph 1. The s t raight line is a pure exponent ia l in t, exp [11.3t] and is a very good approx imat ion to the curve given by eq. (25). The exper imenta l points lie in the very-small- t range 0 .008< Itl < 0.12 and are not shown on our Graph, bu t are in excellent agree- m e a t wi th the exponent ia l curve shown and hence with our model. For com-

parison we exhibi t upon the same graph a sample plot based on eq. (10). I t is

p redominant ly the k inemat ic factor which causes eq. (25) to be such a

be t te r approx imat ion to a pure exponential diffraction peak. All evidence

suggests tha t addit ional da ta at larger values of It[ than the present Serpukhov points (up to at least It I = 1 (GeV/c) z) will lie on an exponential extrapola- t ion through those points.

The determinat ion of c2 and Yz for the exotic channels is more difficult and will not be given here. The quanti t ies will in general depend upon the

to ta l numbe r exchange l, and m a y also depend upon the ba ryon number (-= number of quarks less number of antiquarks). When evaluat ing U12 traces, the

problem of what mass to use always arises, mean mul t ip le t or physical. One

way of resolving this is to consider ~ -3 f scat ter ing (at say P~b = 19 GeV/c) ; i t

is obvious t ha t use of the physical pion mass in ( 1 - t/dm 2) makes the kine-

mat ic factor dominate ]F(--1--ct)] 2, and da/dt actual ly rises with ]t] up to

It I --~ 0.5 (GeV/c) 2 ; this is completely unacceptable. Consequently we conclude tha t mean mult ip le t masses should be used (21) in the Ux2 trace. This result has already been used in our ten ta t ive fit to the highest p-p elast ic-scattering

da ta discussed above, where the choice has only a small effect for [t]< 1 (GeV/c) 2. Although our formalism for the pomeron terms predicts tha t eventual ly

(for large [tl) the differential cross-section will rise above t ha t give~ by a pure

exponential fall off, the tailing behaviour seen in high-energy da ta (~) is a

consequence of the contr ibut ions f rom the Regge exchange terms. This fol-

lows f rom the (fixed-t) energy dependence of these tails, and can be used to es t imate tha t p ,.~hfi~ox ~ •

7 . - C o n c l u s i o n s .

As we have seen in the previous Sections our model is designed to produce qual i ta t ive agreement wi th high-energy elastic or quasi-elastic scat tering data,

(21) The same feature appears in other supermultiplet fits, Reggeized or not. See for example: R. DELBOURGO and A. SALAM: Phys. Lett., 28 B, 497 (1968); K. MORIARTY, B. HARTLEY, R. )/[00RE and P. COLLINS: Phys. Rev., 187, 339 (1969). (22) j . V. ALLABY, F. BINON, A. 1~. DIDDENS, 1 :). DUTEIL, A. KLOVNING, n. MEUNIER, J. P. PEIGNEUX, E. J. SACHARIDIS, K. SCHLUPMANN, M. SPIGHEL, J. P. STROOT, A. M. THORNDIKE and G. M. WETHERELL: Phys. Lett., 28B, 229 (1968).

6 - l l N u o v o C i m e n t o A .

8 2 R. DELBOURGO a n d P. ROTELLI

and some preliminary fits such as tha t given for the pomeron in Sect. 4 are most encouraging. Of course many parameters are yet to be determined, and i t is well to remember tha t our estimates of c for the pomeron and of Re z for the exotics are open to future modification. Even for the Regge terms we have talked of universal slopes for Regge trajectories and intercepts in the degenerate mass supermultiplet limit. All of these simplifying assumptions

can be broken by hand in specific applications of the model and are not too troublesome. A more difficult feature is S U3 breaking at vertices, which manifests itself, for example, in the difference between ~2~(' and K~;' total cross- sections. Thus one needs a consistent way of introducing S Us-breaking effects. However the latest Serpukhov data on total cross-sections (53) not only show this S U3 violation, bu t also a baffling energy-independent difference of ~ ( ~ + p ) - --a~(=-p) and a~(K+p)--a~(K-p) etc. For this breakdown of the pomeron theorem we have no explanation. Finally we note the uncer ta in ty in extra- polating our model to low energies due to the unphysical internal resonance s tructure described in Sect. 2 and our lack of a complete form for V(s, t).

Undoubtedly the most interesting feature of our model is the description of the pomeron as a string of poles lying almost at r ight angles to the physical resonances. Wi th the exotics also as poles in the complex s, t and u planes the whole / ' -mat r ix can be expanded in terms of complex poles. Leaving the narrow-resonance approximation the physical particles must also lie in the complex s, t and u planes, whence the distinction between resonances and exotics becomes only one of degree. The physical scattering ampli tude must of course have cuts. Our model manifestly does not, a fault it shares with the more conventional ununitar ized Yeneziano models. Thus all our poles formally lie upon the physical Riemann sheet. This noucausal state of affairs can be understood if our pomeron, and possibly exotic poles are approxima- tions to the physical scattering cuts. Alternat ively all poles might be made to lie on lower Riemann sheets when uni tar i ty or any smoothing procedure (24) is taken properly into account. In Fig. 6 we see the sort of pole pa t tern tha t emerges for resonances, pomeron and exotics. Exotics may manifest them- selves as resonances but (with the exception of the deuteron) their elusiveness to date implies tha t they are too massive and too broad to be easily identified.

The spirit of the Harar i -Freund concept of dual i ty was a segregation of

background (pomeron) and resonance (Regge) effects. We have of course

(23) J'. V° .A_LLABY, YU. B. BUSHNrN, S. P. DENISOV, A. N. DIDDENS, l:{,. W. DOBINSON, S. V. DONSKOV, G. GIACOMELLI, YU. P. GORIN, A. KLOVNING, 2~. J. PETRUKHIN, YU. D. PROKOSHKIN, t~. S. SttUVALOV, C. A. STAHLBRA.ND and D. H. STOY&NOBA: Phys. Left., 3OB, 500 (1969). (34) A. MARTIN: Phys. ~Sett., 2 9 B , 431 (1969); K. HUANG: Phys. Rev. ~Sett., 23, 903 (1969).

D U A L I T Y D I A G R A M S AND THE POMER ON 8 3

generalized what we mean by background to include exotics, but~ in addi- tion, we have considered diagrams in which resonances ---- pomeron (Fig. 4 a)), and resonances ---- exotics (Fig. 5). Obviously we have depar ted from the dif- fractive dual i ty principle.

Fig. 6.

- - 4

E

- s

is

PesoDcLDce C .". n ~: l y . X "w. 1 y .

×

x

x

X

>(

X

X

X

"4

X

x

x

\

x

x

x

x p o m e r o n

x i

3

x exot / 'CS

×

-120 6 ; 12

Let us therefore conclude with a resume of our in terpreta t ion of dual i ty:

1) Scattering amplitudes are described by the legal and illegal planar quark diagrams.

2) These diagrams are in terpre ted as U12 traces.

3) Each diagram has as associated form factor with the proper ty of being expandible as a sum of poles in either of the dual variables.

4) The pomeron appears as a string of poles, running close to the imaginary axis in the variable in which it is exchanged.

5) The segregation of resonances and Regge poles on the one hand, and

background and pomeron on the other no longer exists.

We would like to thank Dr. I. HALLIDAY for some interesting discussions upon the pomeron.

84 R. DELBOURGO and P. ROTELLI

ADDENDU~

Since in our model we can no longer pretend that all trajectories have the same slope, the asymptot ic behaviour of the beta-terms in which both Mandelstam variables tend to infinity may grow exponentially, i.e.

p ( Z l _ C _ z _ e ~ s _ c 2 u ) , - , o x p s c~in c ~ _ c ~ + c , i n \ ~ / _ l j

• F(Zl, z2, e~, e2, t ) { l + O(1/s)), as s -+ o% t fixed.

Since the coefficient of s in the exponent is not a function of z~ or z2, it is in principle possible to add an infinite number of daughter terms to caJaeel the effect. I n practice we may mult iply these offending terms by entire functions of s, u, and t such as (exp[--s~s2]q - exp[--e2u~]) for the example shown above.

We are grateful to Prof. R. A~NOWIT~ for bringing this difficulty to our attention.

• R I A S S U N T O (*)

Si suggerisee un modello del pomerone basato su un'interpret~zione di tutti i diagrammi piani dei quark. I1 modello eoinvolge come sottoclasse quei diagrammi (( legali )) usati per generalizzare il modello di Veneziano allo scattering di supermultipletto. In con- seguenza si estende la definizione di dualith e pifi speeificamente si modifica l'identi- ficazione di Harari-Freund dells risonanze con i poll di Regge e del pomerone con il rondo.

(*) Traduzione a cura della Redazione.

~na rpaMMbl )IBOfiCTBeHIIOeTH H noMepoH.

Pe3IoMe (*). - - H p e ~ n a r a e T c a MO~eJIb ~J/fl r ioMepoHa, OCHOBaHHaSt Ha nnTepnpeTaUHri

a c e x rtnocKrix ~ H a r p a M M KBapKoB. ~ T a MO~eJIb BKYth3~taeT KaK r to~i(nacc Te << 3aKOHHble >>

~rtarpaMM~,t, KOTOpble 6bUIH I4CYIOYlb3OBaHbI npn 0 6 0 6 m e n n n MO~enH B e n e u a a n o Ha

c y r ~ a ~ p a c c e n n n ~ cyIIepMyJllbTHIUIeTOB. B pe3yJ/bTaTe 3 T o r e p a c m n p n e T c f l o n p e ~ e n e n n e

~BO~CTBeHHOCTI4 14, B qaCTItOCTH, BH~Ott3Melt~teTC~I H~eHTI, I~IdKaI~Hfl Xapapr~-~pyn~a pe3oHaHCOB C n o n m c a M n P e ~ e n n o M e p o n a c (~OHOM.

(*) Hepe6eOeno peOamlue&

duality diagrams and the pomeron

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