A one-dimensional approximation for the pomeron in the dual unitarity scheme

Nuclear Physics BOO (1975) 167 188 © North-Holland Publishing Colnpany

A ONE-DIMENSIONAL APPROXIMATION FOR THE POMERON IN THE DUAL UNITARITY SCHEME Norisuke SAKAI * Theory Division. Rutherford Laboratory, Chillon, Didcot, Oxon O X I 1 0 0 X , UK Received 9 June 1975 (Revised 11 August 1975)

Using a one-dimensional approximation, tile pomcron in the dual unitarity scheme is examined analytically in a nmltiperipheral cluster model. Clusters have negative short range correlations due tothe tmi n effect. In dual models, the pomeron has a crossing odd part, especially in BB and BB scattering. By taking all reggeon loops into account. we show explicitly in our approximation that the crossing-odd part has only lower-lying singularities in the j-plane (typically complex poles with Re a "~- 0.2) whereas the crossing-even part has the leading singularity at ,/= Cep(we imposed ap = I ). Tile dual unitarity scheme leads to a modified version of the f-dominance of the pomeron. We calculate the SU(3) breaking of the pomeron due to the SU(3) breaking among Regge trajectories without additional assumptions on the SUt3) property of the pomeron.

1. Introduction R e c e n t l y a m e t h o d was suggested [ 1 - 4 ] for calculating the elastic scattering as the shadow of multiparticle processes which possess Regge behaviour, semi-local duality and exchange degeneracy. In this dual unitarity scheme, the p o m e r o n is built up by the sum o f ( c r o s s e d ) reggeon loops as shown in fig. 1. The crossed loop corresponds to the quark diagram in fig. 2 and the wavy lines are discontinuities of reggeons which replace p r o d u c e d resonances (clusters) via the semi-local duality (fig. 3). In ref. [21 , these loops are integrated numerically. The purpose of this paper is to use longitudinal rapidity variables neglecting transverse m o t i o n (one-dimensional a p p r o x i m a t i o n ) [5] in order to solve the problem analytically. The hope is that we can obtain more insight into the structure of the model, e.g. singularities in angular m o m e n t u m plane, dependence on various parameters such as s y m m e t r y breaking. In terms of quark diagrams [6] which represent duality constraints, the p o m e r o n is conventionally given [7] by the single-loop diagram with crossed mesonic reggeons as shown in fig. 4 in the case of meson-meson (MM) scattering, t t o w e v e r in scattering with baryons, we can draw the single-loop diagram only for baryon* Address after September 1975: Department of Physics, Tohoku University, Sendai 980, Japan.

168

N. Sakai / Approximation for the pomeron

I z

~

• • 0

x

II - I . - . . l l

0

0

\

Fig. 1. The pomeron in the dual unitarity scheme. Crossed reggeon loops are connected by reggeons which is dual to averages of resonances.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 2. Quark diagram for the crossed reggeon loop. Fig. 3. Semi-local duality which connects resonances and reggeons, Fig. 4. Single loop diagram with crossed mesons for MM scattering. t:ig. 5. Single loop diagram with crossed mesons for BB scattering.

baryon (BB) scattering (rig. 5), but n o t for b a r y o n - a n t i b a r y o n (BB) scattering. This apparent lack of crossing s y m m e t r y was considered [8, 9] sometimes as a serious defect of dual models. In this paper, we would like to suggest that this difficulty can be overcome if we take all higher-order crossed loops into account to build the pomeron. In fact the dual unitarity scheme in ref. [2] can be regarded as a specific example of the p o m e r o n model based on qu~lrk diagrams. Our one-dimensional approx. imation makes the model sufficiently simple to extract the singularity structure in the angular m o m e n t u m plane (/'-plane). We shall show that if we put the leading pole at j = ~p = 1 in the crossing-even part of the pomeron, we obtain in the crossing-odd part only lower-lying singularities, typically complex poles with Re a ~ 0.2 which"

N. Sakai / Approximation for the pomeron

169

Fig. 6. f-dominance of the pomeron. The central blob B is connected to external particles through rcggeon f or f'.

represent damped oscillations. Therefore BB and BB total cross sections become almost equal except at low energies *. In the dual unitarity scheme, the pomeron couples to particles through reggeons such as f a n d f' (the f-dominance [10] of the pomeron) as shown in fig. 6. The conventional approach of the f-dominance assumes that the central Nob B in fig. 6 is an SU(3) (or SU(N) in general) singlet, and the SU(3) breaking comes only from the symmetry breaking among reggeon trajectories which connect B to particles. However the SU(3) symmetry property of the pomeron in the dual unitarity scheme is completely determined by reggeon trajectories without additional assumptions on the SU(3) property of B. Our one-dimensional treatment enables us to evaluate explicitly the SU(3) property of the pomeron. The model for the pomeron can be regarded as a multiperipheral production of resonant clusters which are described by reggeons on the average. They may not be isotropically decaying objects that appeared in conventional cluster models [ 11]. We can explicitly evaluate the cross section for producing the n clusters and hence obtain the nmltiplicity distribution of clusters (not the multiplicity distribution of particles decaying from a cluster) in terms of pomeron and reggeon intercepts. We find a negative correlation between clusters. This point has been noticed in similar models [12, 131. The negative correlation arises from tmi n effects and is of short range in rapidity space. We do not discuss the t-dependence of the elastic scattering, because we suppressed the transverse degree of freedom. The decay of the clusters and the bootstrap condition [2,4], which can determine the reggeon parameters, are not treated. The one-dimensional approximation to the multiperipheral cluster model is introduced in sect. 2. The crossing symmetry property of the pomeron in the dual unitarity scheme is analyzed in sect. 3. The f-dominance and SU(3) symmetry breaking of the pomeron are treated in sect. 4. Multiplicity distribution and distributions and correlations of clusters in rapidity space are calculated in sect. 5. In

* The crossing~odd part of the pomeron occurs most dramatically in BB and BB scattering, but a small crossing-odd part can appear also in MB or MM scattering if SU(N) symmetry breaking trajectories is taken into account. This point has been noticed by the authors of ref. [ 1] (private communication).

N. Sakai / Approximation for the pomeron

170

sect. 6, our results are compared with the numerical results in ref. [21 and with a recent work 114] on a similar model. Some of the details of the calculations are found in the appendix.

2. One-dimensional multiperipheral cluster model In the dual unitarity scheme of ref. [2], the pomeron is constructed via unitarity as the shadow of multiperipheral production of resonances (fig. 1). Semi-local duality (fig. 3) is used to describe the average over resonances by the discontinuity through a reggeon line (wavy line) in fig. 1 which we shall call a cluster. The cross on loop reggeon lines indicates that quark lines are twisted (fig. 2) so that no quantum number can be exchanged. Loops with uncrossed reggeon lines are taken into account by assuming that the reggeon (wavy line) is already a sum of uncrossed loops as shown in fig. 7. Let us consider meson-meson (MM) or meson-baryon (MB) scattering with SU{N) symmetry. We define the invariant mass squared of the ith cluster as si, that of the neighbouring i and i + lth clusters as si, i+l, and that for the incident channel as s. If we denote the loop reggeon by a and the cluster reggeon by o%1, we obtain the cross section for producing n clusters (n - 1 loops): Un

1

(si, - - i+l]2~ddPn

s~cl

Oc--

S i=1

i=1 ~SiSi+17

ds i

(l)

=

where d~n represents the phase space factor

dOn = ~ 4 ( i ~ = l k i - P a - P b

d3ki(k2+si)-l/2

(2)

The momenta of incident particles and of the ith produced cluster are denoted by Pa, Pb and k i respectively. Rapidity variables are defined for incident particles a and b with masses rn a and rn b Pa = (ma, O, O, 0),

Pb = (mb cosh Y, O, O, m a sinh Y) ,

and for produced clusters

Fig. 7. A reggeon (wavy line) is a sum of uncrossed loop diagrams.

(3)

N. Sakai / Approximation for the pomeron

Y2

171

Yn

I

°l

Fig. 8. Rapidly variables for the multiperipheral cluster model. Center and interval for the ith tluster are Yi and x i. Gap between the ith and i + 1th clusters is z i. ki=(wicoshYi,

kij_,wisinhYi),

w2=si+k2

l .

(4)

We use the strong-ordering a p p r o x i m a t i o n [5] for the cluster production Y l < Y 2 < '" < Y n - I

(5)


with large rapidity difference. We assume that k± m a y be neglected. We can define a variable x i which corresponds to a rapidity interval occupied by the ith cluster when s i is large (6)

s i = m 2 e xi ,

where m is a parameter which may be interpreted as a certain averaged transverse mass of produced particles. The rapidity gap z i between the ith and i + I th clusters is defined by Zi = (Yi+I

(7)

1

l X i + I ) - - ( Y i + 2Xi)

which is illustrated in fig. 8. If we take mare b = m 2 for simplicity, the n cluster cross section becomes in the one-dimensional a p p r o x i m a t i o n * ll

On = Va,Tb(Ng2)n- 1 I~ dx. e (c%l-1)xi i=l

(8)

t

X I-[ dz/.e2(a-1)zJ8 ]=1 \i=1

Y j=l z]--

,

where g is the triple-regge coupling c o n s t a n t smeared over transverse m o m e n t a and Ta and 7b are the reggeon coupling to external particles. Because of N different quark lines in the loop, g is multiplied by N. We should stress that the integration region is l i m i t e d * * to * Similar approximation has been considered in ref. [ 13]. The restriction on z/comes from the tmi n effect and can be generalised to z] >1b using a parameter b which is estimated in sect. 6. The case with b ~ 0 is treated in the appendix.

17 2

N. Sakai / Approximation for the pomeron

x;/> o,

(9)

§.>o.

(lO)

The strong-ordering approximation for the multiperipheral cluster production may be justified by the following reasons: (i) As the rapidity gap decreases, the tmi n increases. Therefore the cluster overlap is suppressed [ 12]. (ii) The strong-ordering approximation for the multiperipheral production of clusters is expected to work better than for the multiperipheral production of particles [ 15], because of the relatively low density of clusters. (iii) Reggeons of the loop are crossed so that no resonance is possible across the gap which could provide attraction in rapidity space. In evaluating the cross section, it is convenient to introduce the/. representation (angular momentum) ~/.) = ~ d Y e - q 1)Yo(y) "

(11)

o

0 Integrating over rapidity variables, we obtain the n cluster cross section in the/" presentation ~n(/') = TaT b (/" - O~cl)-n {Ng2/(/" - 2c~+ 1) } n- 1

(12)

Since the pomeron consists of at least one loop (two clusters), the total cross section is given by

~'(/') = n~__2~'n (J) Ng 2

= ")'aTb {(/"- O~cl) ( / - 2ol + 1) - N g 2 } (] - Olcl)'"

(13)

We obtain two new poles as roots of the equation ( / - ~cl)(/'- 2~+ 1) - N g 2 = 0.

(14)

By requiring the leading pole for the pomeron at/" = C~p,we can determine the tripleRegge coupling constant g (15)

g2 = A B / N

A =%

%1'

B=c~p-2~+

1.

(16)

N. Sakai / Approximation for the pomeron

17 3

The other new pole lies rather low a t / = C~Q aQ~-acl+2C~-ape.g.,

(17)

1,

C~Q = -- 0.5

if we take *

O~p = 1,

~cl = 0~= 0.5 .

(18)

The one-dimensional a p p r o x i m a t i o n may not be realiable to discuss/, plane structure below 2a - 1. Therefore the numerical value of C~Q should not be taken too seriously. It is interesting to note that the numerical value of the coupling strength g2N is compatible with that c o m p u t e d previously from the planar bootstrap by Rosenzweig and Veneziano [4] (in their n o t a t i o n g2N/1670. The energy dependence of the total cross section (Y representation) is illustrated in fig. 9. The constribution f r o m the single Regge exchange and the sum of the reggeon and the p o m e r o n c o n t r i b u t i o n s are also shown.

3. Crossing p r o p e r t y of the p o m e r o n in the dual unitarity model In duality schemes which are nicely described in terms of quark diagrams [6], the p o m e r o n is conventionally identified [7] with a crossed meson l o o p shown in fig. 4 in the case of meson-meson (MM) scattering. The diagram ensures vacuum

t.,: OtQ[ Logo

\\ 0 0 " ' 0 " 0 0 . . . . . \

Pomcron

con

Y=Iog (s/m2~ Fig. 9. The total cross section as a function of the total rapidity Y. Pomeron contribution (solid line), reggeon (dashed line) and their sum (dotted line).

* In this choice of trajectories, the model becomes almost the same as that of Huan Lee [3]. His model for the pomcron would be obtained if we modified the semi-local duality by adding the direct-channel single-particle state to the reggeon exchange in the right-hand side of fig. 3, because his reggeon is constructed by the sum of two or more particle productions with uncrossed reggeons.

174

N. Sakai / Approximation for the pomeron

Fig. 10. Pomeron for qq scattering consists of even number of clusters.

Fig. 1 1. Pomeron for q~ scattering consists of odd number of clusters.

quantum number exchange in the t-channel and gives a non-resonant background in the s-channel [16]. In the scattering of baryons, however, we can draw the single loop diagram with crossed mesons only for baryon-baryon (BB) scattering (fig. 5), but not for baryon-antibaryon (BB) scattering. Thus the pomeron is not crossing symmetric. This was considered as a serious defect [8, 9] in the dual scheme. In order to obtain a leading singularity in the vacuum quantum number exchange amplitude n e a r / = 1, we ought to consider all diagrams with any number of crossed loops. In general, diagrams with an odd number of crossed meson loops contribute to BB scattering, and those with an even number of loops to BB scattering. Asymptotically even and odd numbers of loops give equal contributions to the total cross section which is determined by a large number of loops. In this section we shall show explicitly in our model that BB and BB total cross sections have exactly the same leading singularity and residue.. Hence the pomeron is asymptotically crossing even. Let us first consider quark-quark (qc0 and quark-antiquark ( q ~ scattering in the SU(N) symmetric case. Diagrams with an even number of clusters contribute to the pomeron in qq scattering (fig. 1O) Oqq = o 2 + o 4 + o 6 + . . . ,

(19)

and those with odd number of clusters to q~ scattering (fig. I l) Oq~ = 0 3 + 0 5 + o 7 + .... The crossing-even part of the pomeron is given in the / representation by

(20)

175

N. Sakai / Approximation for the pomeron ' 2.0

I

+

T

-'

•

-+

t

~

T

'

Y

!

(:lp = 1 , !'

C{

= etcl = 0 . 5

10~-

0 . . .......................................................................... .,., ' "

-'"

0.5~

Oqq

~-i

o 0.2

i o.1

0 05

~

o8

2

4

6 8 Y = log (sire 2)

I0

12

Fig. 1 2. Pomeron contribution to the total cross section for qq (dash-dotted line) and q~ (dashdouble-dotted line) scattering. Crossing-even part o+ (dotted line), odd part o (dashed line) and contribution a n from n cluster production (solid line}.

~o =+~ -- Oqq + ~o q~ -- £ ~

2

Ng 2

(2t)

= Tq {(]. 0Q1)(/_ 2 a + 1) - X g 2 } ( / ' - a c l ) ' which is exactly the same as eq. (13) for MM scattering except the external quarkreggeon coupling 7q- The crossing-odd part of the p o m e r o n is given by

2

Ng2

-"

= Vq { ( / . 0~cl)(]'-- 2 a + 1 ) + N g 2 } ( / ' - a c l ) Given the leading singularity/" = Otp in ~'+, we can determine the triple-Regge coupling c o n s t a n t g and consequently the complete j plane structure. We obtain two new poles in the crossing-odd part ~ ( / ) from the e q u a t i o n ( / ' - o%1)(/" - 2 a + I) + (ffp - o%1)(o~ P - 2oe + 1) = O .

(23)

176

N. Sakai /Approximation fi>r the pomeron

These are lower-lying and usually complex poles (e.g./ = 0.25 + 0.66i in the parametrization (18) which represent a rapid oscillation damping. In fig. 12, we illustrate the energy dependence of the crossing-even part o+ and odd part o_, together with Oqq, Oqq and contributions from diagrams with a definite number of clusters (using tire parametrization (18)). The qq and q~ cross sections are very close to each other except at very low energies where the concept of the pomeron may not be well-defined. In our specific model, the main part of the low-energy difference o_ is cancelled if the single-Regge contribution (n = 1) is added to q~ cross section *. Experimental data [ 18] on multiparticle production from pp collision shows that baryons in the final state carry about half** of the incident momenta (the leading particle effect). The non-annihilation part of BB collision appears to 0e similar to BB. This suggests that the pomeron in BB and BB scattering is as a first approximation dominated by diagrams with B (or B) type clusters at both ends of the multiperipheral chain, i.e. diagrams consisting of crossed meson loops like those in figs. 10 and 11. In this case, the formulas for BB and BB scattering are exactly the same as those for qq and q~ scattering except for the external particle-reggeon couplings. Therefore we obtain eBB ~ OB~ asymptotically. In BB scattering, we can exchange not only an ordinary meson (fig. 13) but also an exotic meson [19] (fig. 14), which is expected to contribute less because it is a lower-lying singularity. For the vacuum quantum number exchange part of BB scattering, we can draw many more diagrams involving exotic exchange in addition to those consisting of crossed meson loops, in particular, the diagram in fig. 15 was proposed in ref. [9] to recover the crossing symmetry of the pomeron and to break the factorization OMM OBB = (OMB)2 at the same time :~. In our model, however, we have shown that the pomeron becomes asymptotically crossing-even, without introducting additional diagrams such as exotic exchanges. If exotic mesons are taken into account, our model becomes a two-channel (ordinary and exotic mesons) problem. The pomeron now consists of three types of diagrams shown in fig. 16, (a), (b) and (c), where all the quark lines flow back after forming meson or baryon loops in the central blob. There is still a crossing-odd part of the pomeron in general, because BB and BB scattering cannot have exactly identical diagrams. Similarly to the diagrams with ordinary mesons, one can show that the crossing-odd part has only lower-lying singularities compared to the leading one (/" = e~p = 1) ap-

* Phenomenology [17] of the total cross section indicates that the vacuum quantum number exchange part hasj = 1 andj = 0.5 singularities with positive residues. In our model, however, the reggeon pole (/= e~clm 0.5) in the reggeon contribution (n = l) is cancelled, if it is added to the pomeron contribution (n ~>2) because of the negative residue of the pole atj = C~clin eq. (21). This may be due to the inadequacy of our approximation to describe low-energy behaviour (Huan-Lee's model [3] possesses the same defect). ** This seems to hold even after subtracting the so-called diffractive component. $ The authors of ref. [9] are now considering that the diagram is important only at low energies.

N. Sakai / Approximation for the pomeron

f

Fig. 13.

1

Fig. 14.

177

Fig. 15.

Fig. 13. The qutuck diagram for an ordinary meson exchange in BB scattering. Fig. 14. The quark diagram for an exotic meson exchange in BB scattering. Fig. 15. The diagram proposed in ref. [9] with a baryon loop connected to an ordinary and an exotic meson exchange.

(b)

(a)

\

(-

.

/

(c) Fig. 16. Pomeron diagrams for two-channels (ordinary and exotic mesons) in BB and BB scattering ta) ordinary-ordinary transition (b) ordinary-exotic (c) exotic-exotic. pearing in the crossing-even part. We obtain poles in the j plane in our approximation, and hence the factorization should hold asymptotically * even if we incorporate these extra diagrams. Eylon and Harari [20] e x a m i n e d BB annihilations into mesons using a dualitydiagram model of Huan-Lee type. T h e y showed that BB annihilations contribute do d o m i n a n t l y to the p o m e r o n . An example o f relevant diagrams is shown in fig. 17. If BB annihilations are not taken into account, their model of the p o m e r o n has a crossing-odd part which vanishes asymptotically. They argued that the contributions from BB annihilations can cancel the crossing-odd part even at low energies, provided the trajectory and coupling constant of exotic exchanges are suitably ar* Since the pomeron in dual models couples to particles always through factorizable Reggc poles, it seems difficult to avoid the factorization irrespective of the nature of the leading / plane singularity, provided it is non-degenerate.

178

N. Sakai / Approximation for the pomeron

Fig. 17. A qm,rk diagram for BB annihilations which contributes to the pomeron.

ranged. A similar argument seems to apply in our case, although we have very little phenomenological information on exotic exchanges. Therefore we conclude that BB annihilations and other diagrams may help to fill the difference between the crossed meson contributions to OBB and OBE. However, we should stress that these conventional diagrams (crossed meson loops) are dominant and ensure equal asymptotic cross sections for BB and BB scattering. For MM and MB scattering, both an odd and an even number of loops can contribute to the pomeron, which is therefore crossing even if SU(N) symmetry is exact. If the symmetry breaking among Regge trajectories are taken into account, however, the pomeron can have a crossing-odd part which vanishes asymptotically. This crossing-odd part is very similar to that of BB and BB scattering, but is small (proportional to the symmetry breaking).

4. f-dominance of the pomeron

In the dual unitarity scheme, the pomeron couples to particles always through reggeons (f, f' and so on) as shown in fig. 6. This kind of pomeron coupling is called f-dominance of the pomeron [21 ]. The central blob B in fig. 6 is conventionally assumed to be a SU(3) (or SU(N) in general) singlet, and the symmetry breaking of the pomeron coupling comes from the breaking among Regge trajectories which connect the pomeron to particles. The reggeon couplings are always assumed to satisfy the symmetry. Since the pomeron is constructed from reggeon loops (fig. 1) in the dual unitarity scheme, however, the symmetry property of the central blob B is completely determined by the symmetry breaking among Regge trajectories without additional assumptions. The one-dimensional approximation enables us to evaluate explicitly the SU(3) (or SU(N)) property of the pomeron. Let us take the broken SU(3) case where k quark is distinguished from p and n quarks (normal quarks). In order to calculate the crossed meson loops, we define two possible states according to the quark content jl } = X/-~(p~ + n g ) ,

i2}= XX,

(24)

corresponding to f (or co) and f' (or ~b) respectively. The pomeron can be described by a two-by-two matrix whose row and column represent couplings to the left and

N. Sakai / Approximation for the pomeron

179

right external particles respectively. The cross section for producing n clusters is given by ~-n(j) = TaR(J) (g2L(I.)R(/.) ) n-1 7b ,

Rq,=[O.-%)-' o 0 k(j)

0 - ~2)-~

]

(25) '

[ 2(/'- 2% + 1) I X/2(/'- 2ee3 + 1 ) - 1 ] = L,V'~_(/'- 2ee3+ 1) -1 (/'- 2ee2 + 1) -1

d

where R(j) represents the reggeon propagator and L(j) the crossed meson loop whose first (second) row and column correspond to the state I1) (12)). Reggeons are classified according to the quark content eel " normal quarks only, &2 ;k quarks only , ee3 " normal and X quarks,

(f, co) ; (ix, ~)

(26)

(K**, K*).

The triple-Regge coupling is assumed to be SU(3) symmetric and is denoted as g. The particle-reggeon couplings are denoted as Ta and Yb. The crossing-even and odd parts of qq ( q ~ scalte.ring are given by ~ + 0 " ~ = - ~Oqq + ~Oq~ = n ~=2 ~0 n = T ~ g 2 R L R ( l - g 2 L R )

!

~'_ (/') ~ O'qq -- O'q~ --- n~=2(-- 1)n O'n = T~g2RLR(I +g2LR)-I '

(27)

(28)

whose ]" plane singularities are determined by det(1 -g2L(j)R(j)) = 0

for

~+ ,

(29)

det{l +g2L(j)R(j)} = 0

for

~" ,

(30)

respectively. The triple-Regge coupling constant g is determined by requiring the leading singularity in ~'+ at/--- C~p as in the previous sections. If we take a parametrization of reggeons ee1 = 0.5 (f, co),

ee2 = 0.2 (f',0),

ee3 = 0.35 (K**, K * ) ,

and eel? = 1, we obtain in ~" (/) only complex poles (those with largest real part are

180

N. Sakai / Approximation for the pomeron

~,

p.n/p.n

P,

Fig. 18. Diagramatic definition of the asymptotic ratio r of h and normal quark scattering to normal and normal quark scattering.

Table 1 Ratio r of X -normal quark scattering to normal quark scattering for several values of c~f, and C~K*; other intercepts are fixed at C~p= 1 and c~f = 0.5; our results (dual unitarity scheme) are compared to the prediction using ref. [10] Dual unitarity scheme cef,

C~K. =0.25

0.3

0.35

Carlitz et al. ref. [10]

0

0.34

0.36

0.38

0.5

0.1

0.38

0.40

0.42

0.56

0.2

0.44

0.46

0.49

0.63

at R e j ~ 0. 1). T h e r e f o r e the crossing-odd p a r t dies o u t quickly. In ~ + ( J ) the n e w singularity n e x t to O~p = 1 lies r a t h e r low a t j --~ 0.28. The ratio r o f the a s y m p t o t i c total cross section for X- a n d n o r m a l q u a r k scattering to t h a t for n o r m a l a n d n o r m a l q u a r k s c a t t e r i n g ( f i g . 18) is given by the ratio o f the residue m a t r i x e l e m e n t s at the j = ~p pole r = N/~2l

(/" = ~ p ) / ~11 (j- = O~p)

=2{(C~P-2°~3+l)(aP g2

~2)

°~P-2~3+1}-1

,

(31)

~p - 2c~2 + 1

where g2 is given by the q u a d r a t i c e q u a t i o n ¢ ( 2 9 ) with j = exp. T h e ratio r is pres e n t e d :~+ in table 1 for several values o f Regge i n t e r c e p t s c~2 = ~ , , if3 = °tK* fix-

* One has to choose the solution with the smaller value o f g 2 in order to makej = C~pthe leading pole and to give the correct solution in the previous sections in the symmetry limit. ** Our value is numerically not so much different from the ratio of single loop diagrams [22] obtained by replacing the blob B in fig. 6 by a crossed meson loop.

N. Sakai / Approximation for the pomeron

181

ing O~p = 1 and o~f = 0.5. It is compared with that obtained by assuming the central blob B is a SU(3) singlet [21]. Using the ratio r, we can obtain the ratio of asymptotic total cross sections such as

OKN/°TrN = L( 1 2 + r) ,

(32)

O~N/OpN = r .

(33)

A phenomenological analysis of OrrN and OKN by Barger and Phillips [ 17] gives r--~ 0.61. Experimental data on pseudoscalar photoproduction [23] suggest a somewhat smaller value r --~ 0.4. Considering also the uncertainty in the theoretical wdue r due to the uncertainty of reggeon intercepts, we cannot favor or disfavor our dual pomeron model over the assumption of B being a SU(3) singlet. Similarly we can consider the broken SU(4) symmetry to incorporate charmed quarks (c quarks), hr table 2, the corresponding ratio r c (c-normal quark scattering to normal-normal quark scattering) is presented for several values of Regge intercepts C~D* (charmed reggeon) and c~fc (c~- reggeon) fixing ap = l, c~f ~-- aN* ~ c~f, 0.5. We obtain much smaller r c compared to the value [24] obtained by assuming that the central blob B is a SU(4) singlet. Since the one-dimensional approximation may involve errors in treating very low-lying singularities the numerical wdue in table 2 should be taken as a rough estimate.

5. Multiplicity and correlations of clusters Let us take for sinrplicity MM scattering with SU(N) symmetry. The distribution of the multiplicity n of the clusters (not the multiplicity of particles decaying from the cluster) are completely determined once pomeron C~p and reggeons c%1 and ct are given. Using generating functions [25] given in the appendix, we obtain the average number of clusters in terms of parameters A and B defiued in eq. (16) (.4 = 0.5, B = 1 in the parametrization O~p = 1, acl = oe = 0.5)

AB

2AB

7~=A--g-/~Y- (A +B) ~

B A +~+ 2,

(34)

and the second Mueller moment f2 - n(n - l) - ~2 (AB)2 Y+ 8(AB)2 + - B-2 + . -2AB - 2 TZ=-2(A+B)~ (A+B)~ ( A + B ) 2 ( A + B ) 3

2

(35) •

It is interesting to note t h a t f 2 is negative asymptotically, i.e. the distribution of the number of clusters is narrower than the Poisson. This result has been obtained in a similar m o d e l [13]. We find that all the higher multiplicity momentsj')c are a lin-

182

N. Sakai / Approximation for tlle pomeron

Table 2 Ratio r c of charmed-normal quark scattering to normal-normal quark scattering for several values of C~D* (charmed reggeon) and ~fc (c?" reggeon); pomerons and other reggeons are set at cep = 1, ~f = C~K*= c~f, = 0.5; our results (dual unitarity scheme) are compared with the prediction using ref. [10] Dual unitarity scheme ~fc

~D* =

4

- 3

- 2

1

Carlitz et al. ref. [10]

- 8

0.006

0.007

0.009

0.014

0.056

- 6

0.007

0.009

0.012

0.018

0.071

-4

0.010

0.01 3

0.017

0.025

0.10

-2

0.017

0.021

0.028

0.042

0.17

ear function of Y. This feature is a general p r o p e r t y of Regge-pole d o m i n a t e d models [25] which gives short range correlations. We can also calculate exclusive and inclusive distributions and correlations of clusters in rapidity space. (i) The distribution of cluster mass M 2 is obtained via the relation (6) by calculating the distribution of the rapidity interval x occupied by a cluster. The asymptotic behavior for Y - x - + ~ o f the n-cluster (semi-inclusive) and fully inclusive distributions is given by 1 don ondx

n_ l( 1 Y

1 d°in dx ~ e - a x

-o-

~)n-2

(36)

A 2B ,~-~

(r

-

x)

(37)

.

In terms of the cluster mass M 2, it reads 1 d°n ~n1 on d M 2 m 2

o dM 2

M 2 \ m 2,

2(lo ,r \ ~]

og

~

(38) '

'

where m 2 is the scale parameter used in eq. (6). The inclusive average of the rapidity interval x occupied by a cluster is given by

N. Sakai / Approximation ./br the pomeron

~-=

do m / jx-~-dx

fdom ~-ct~ ~ .

1

18 3

(40)

if we take the parametrization cq, = 1 and c%1 = c~ = 0.5, we obtain 2 ~ 2, which is larger than that of the conventional isotropic cluster and suggests that our cluster decay may be elongated along the longitudinal axis. (ii) The distribution of the rapidiO' gap z between clusters is given asymptotically ( Y ~ ~) by l don o n dz

(%1

1 d°in dz ~

e

2~+l)e

(41)

{e~cl 2c~+l)z,

Bz AB2y A +B

"

(42)

The inclusive average of the rapidity gap is given by ~ fz

d°in

d°in

1

(43I

/

which becomes one unit of rapidity in tire parametrization eq, = 1, c%1 = c~ = 0.5. (iii) The distribution of the (central) position oJ'a clustery, is given in the central region 0', Y - - v -+ ~)

1 d°in

~

o dydM 2

M[, -'~ A A2B M 2 ~m 2 ] A +B " 1

(44)

representing the central plateau (independent of 3'). (iv) Two-cluster correlation in the central region is given by

C(v, M2;y,,M,2) _= 1

d°m o dydM2dy'dM~2

1 dam

1 d°m

o dydM 2 o d3,'dM'2

1 ( M 2~ A 1 [ M ' 2 ) A ( A2Bt2 a 2 \m 2 ] M,2 \ m 2 \A~-B-] ~ '

(45)

~, = _ 0 ( . _ r / ) + 0 ( r / ) B e-{ap c~@~ r/-lY'--3'[-

Iog(MM"~

\m2 ] '

which represents a negative short-range correlation (the first term in C) followed by

184

N. Sakai / Approximation for the pomeron

C(y, M2; y',M '2)

-log (MM')

log ( Mm..~.)M'

Fig. 19. Correlation C(y, M 2, ?." M' 2) of two clusters with masses M and M' as function of the rapidity difference y ' - y . The negative C for small ly' - yl represents the suppression of cluster overlap due to the tmin effect and is responsible for f2 being negative.

a small positive short range correlation due to the secondary singularity o~O. The correlation function is illustrated in fig. 19, as a function o f f - y. The negative correlation for small L v ' - y l comes from the repulsion between clusters due to the tmi n effect and gives rise to the negative Mueller moment f2 in eq. (35). In our approximation, the positive short-range correlation comes only from O~Q and is insufficient to compensate for the negative short-range correlation due to the tmi n effect. However, the one-dimensional approximation may not be reliable to discuss lower-lying singularities such as ~Q(O~Q < 2o<-- 1 provided acl < O~p). In particular the Mueller moment f2 may become asymptotically positive if we have a secondary vacuum singularity (such as the f meson) with sufficient strength.

6. Comparison with other works (i) Our analytic calculation seems to agree well with tile numerical evaluation in ref. [2]. If we estimate our scale parameter m 2 by using the ratio of the reggeon (f) to the pomeron calculated in ref. [2], we obtain a reasonable value rn 2 "- 0.4 GeV 2. Our ~ appears to be slightly smaller than theirs. This indicates that the/rain effect, which suppresses the overlap of clusters, may not be well-approximated. We can bring our g into closer agreement with theirs, if we allow a small overlap of clusters z/~> b,

b < 0,

instead of tire restriction (10). The maxinmm possible overlap [26] by using the cut-off l 0 of momentum transfers b ~

We estimate pendix.

log(itol/rn2 ) .

(46) b may be estimated

(47)

0.5 ~< b ~< 0. Some details o f the b 4 : 0 case are described in the ap-

N. Sakai / Approximation for the pomeron

185

(ii) Finally we refer to a very recent analysis by Squires and Webber [14] of a similar nrultiperipheral cluster model using nmltiplicity sum rules. They included an exchange degenerate (EXD)Tr nonet besides an EXD p nonet for loop reggeons and determined triple-Regge coupling constants from inclusive data. With certain simplifying assumptions, they obtained the average multiplicity g of clusters which increases faster than g in ref. [2] (and ours). Their value of B/Y seems to be compatible with other estimates based on isotropic clusters *. However, they did not consider the constraint that Otot should have ] = Oep = I leading singularity. Taking this constraint (15) into account, we have obtained the expression (34). Therefore, once ~i/Y and ~e are given, we can determine Regge trajectory c%1 dual to the cluster. In order to obtain the large g / Y compatible with isotropic clusters, one has to take rather low-lying trajectories for % F e.g. if one uses B/Y ~ 0.7 (ref. [ i4] ) and o~= 0.25 (the average o f p and 7r), one obtains acl = 0.31. Hence we conclude that the inclusion of more trajectories (EXD a nonet) with similar coupling effectively increases the strength of triple-Regge coupling (Ng 2 in our case) and makes g / Y larger, but at the same time clusters become concentrated to small masses and dual to very low-lying trajectories. After writing up this paper, we were informed of two preprints discussing some of the points considered here. Schmid and Sorensen [27[ used the Chew-Pignotti approximation and a duality diagram approach to examine the symmetry structure. They discovered the crossing-odd part of the pomeron which has a lower-lying singularity compared to the crossing-even part. Our model is a more explicit multiperipheral cluster model in which the possibility of variable cluster mass is taken in-to account via semi-local duality. Rosenzweig and Chew [28] discussed the summation of crossed meson loops (they call cylinders) and the SU(3) breaking of the pomeron due to c~f 4: oef, (f dominance). However they neglected the structure of the crossed meson loop (L(/') in our eq. (25) was taken as an SU(3) singlet and independent of]'). Therefore the j plane structure becomes different and more importantly, their physical interpretation is quite different from ours. We wish to thank Drs. J. Dias de Deus, D.P. Roy and J. Paton for careful reading of the manuscript and many useful discussions. We are grateful to Drs. J. Kwiecinski, Chan Hong-Mo and R.G. Roberts for stimulating discussions. We are indebted to Drs. J. Paton and Chan Hong-Mo for bringing preprints by Schmid and Sorensen and by Rosenzweig and Chew to our attention.

Appendix We shall evaluate the ] and Y representation of cross sections of the multiperipheral model (8). The rapidity gap z/is restricted a~ in eq. (46) using the parameter b. * However, they calculated only the leading order in Y. It is not clear whether the linear term in Y alone is sufficient to desclibe ~- at present energies.

N. Sakai/ Approximationfor the pomeron

186

The cross section for producing n clusters is given by _ TaTb

[Ng2e q 2a+ 1)b}n--1

a'.0) (/-~cO"[

:72~+i

in the/representation and becomes asymptotically (Y--> ~,)

TaTb {Ng2 e (C%l-2C~+l)by }n %1 2a+l

°n(Y)~(n- 1)!

e(%l--l)Y

The pomeron contribution ~'(/) to the total cross section is obtained if the coupling constant Ng 2 in eq. (13) is replaced by NgXe q 2c~+l)b Eq. (14) now becomes a transcendental equation which gives complex poles [26] for o'(/). However the asymptotic total cross section is controlled by the/" = ap = 1 pole B e(C~p- 1 ) Y o(Y)~ TaTb A +B+ABb where A andB are defined in eq. (16). Multiplicity distribution is conveniently calculated by using the generating function 125] in the / representation

"Z(~',/) - ,,i=2 ~

~-"a'.f:)

TaTb~-2ABe-q ~p)b ((] apj2+(A+B)(j

~ p ) + A B ( 1 - ~ ' e q ~p)b)}

Taking the leading singularity in ~(~', ]), we can obtain the asymptotic behavior of Mueller's moments [25] (integrated inclusive correlations)fk logZ (~', Y)= ~ ( ~ - 1 ) k f k / k ! + l o g o ( Y ) . k=l To leading order in Y, we obtain

fi/Y~ AB/(A +B +ABb) , f2/Y~

(AB) 2 {2+2(A+B) b + A B b 2}/(A+B+ABb) 3.

Higher moments are more complicated and are given in the case of b -- 0

.fk/Y~(2k-3)!!(

2)k - I ( A B ) k(A+B) 2:~+t

N. Sakai / Approximation for the pomeron

187

D i s t r i b u t i o n s in rapidity space in the case o f b 4 : 0 are a s y m p t o t i c a l l y a h n o s t the same as t h a t for b = 0 (eqs. ( 3 6 ) ( 4 4 ) ) e x c e p t for the s u b s t i t u t i o n

A +B-+A +B+ABb, z-~z-b.

References [1] Chan Hong-Mo and J. Paton, Phys. Letters 46B (1973) 228; Chart Hong-Mo, J. Paton and Tsou Sheung Tsu, Nuch Phys. B86 (1975) 479. [2] Chan ttong-Mo, J. Paton, Tsou Sheung Tsun and Ng Sing Wai, Nuch Phys. B92 (1975) 13. [3] ttuan Lee, Phys. Rev. Letters 30 (1973) 719. [4] G. Veneziano, Phys. Letters 43B (1973) 413; Nuch Phys. B74 (1974) 365; Phys. Letters 52 52B (1974) 220; C. Rosenzweig and G. Veneziano, Phys. Letters 52B (1974) 335; M.M. Schaap and G. Veneziano, Nuovo Cimento Letters 12 (1975) 204; G. Marchesini and G. Veneziano, Phys. Letters 56B (1975) 271. [5] G.F. Chew and A. Pignotti, Phys. Rev. 176 (1968) 2112; C.E. DeTar, Phys. Rev. D3 (1971) 128. [6] H. Itarari, Phys. Rev. Letters 22 (1969) 562; J.L. Rosner, Phys. Rev. Letters 22 (1969) 689. [7] P.G.O. Freund and R.J. Rivers, Phys. Letters 29B (1969) 510; G. Frye and L. Susskind, Phys. Letters 31B (1970) 589; C. Lovelace, Phys. Letters 34B (1971) 500. [8] M.B. Einhorn and G.C. Fox, Nuch Phys. B89 (1975) 45. [9] M.R. Pennington and A. Gula, The meson and the baryon loop to make the pomeron, Rutherford Laboratory preprint RL-75-024, Y. 106 (1975); Nuch Phys. B96 (1975) 535. [10] R. Carlitz, M.B. Green and A. Zee, Phys. Rev. D4 (1971) 3439. [11 ] A. Bialas, Proc. 4th Int. Symposium on multiparticle hadrodynamics, Pavia 1973, p. 93. [ 12] L.I. Perlovsky, Phys. Letters 56B (1975) 45; A. Gula, ls the absence of semi-inclusive correlations compatible with the cluster model? Rutherford Laboratory preprint RL-75-030, T. 108 (1975). [13] S. Matsuda, K. Sasaki and T. Uematsu, Inclusive bootstrap for multiperipheral cluster emission model, Kyoto Univ. preprint, KUNS294 (1974). [14] E.J. Squires and D.M. Webber, A self-consistent multi-Regge cluster model, Durham preprint (April, 1975). [15] S. Jadach and J. Turnau, Phys. Letters 50B (1974) 369. [16] P.G.O. Freund, Phys. Rev. Letters 20 (1968) 235; H. Haraxi, Phys. Rev. Letters 20 (1968) 1395. [17] V. Barger and R.J.N. Phillips , Nucl. Phys. B32 (1971) 93. [18] L. Van Hove, Phys. Reports 1 (1971) 347. [19] J.L. Rosner, Phys. Rev. Letters 21 (1968) 950, 1142(E); D.P. Roy and M. Suzuki, Phys. Letters 28B (1968)558; tl.J. Lipkin, Nucl. Phys. B9 (1969) 349. [20] Y. Eylon and H. Harari, Nucl. Phys. B80 (1974) 349. [21] R. Carlitz, M.B. Green and A. Zee, Phys. Rev. Letters 26 (1971) 1515; Phys. Rev. D4 (1971) 3439. [22] J. Dias de Deus and J. Uschersohn, Rutherford Lab. preprint RL-75-042, T. 112 (1975).

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N. Sakai / Approximation for the pomeron

[23] G. Wolf, Proc. Int. Symposium on electron and photon interactions at high energies, Cornell Univ. (1971) p. 189. [24] T. Inami, Phys. Letters 56B (1975) 291; V. Barger and R.J.N. Phillips, Implication for ~ N scattering of universal features of other elastic interactions, Univ. of Wisconsin preprint COO-881-445 (1975). [25] A.H. Mueller, Phys. Rev. D4 (1971) 150. [26] G.F. Chew and D. Snider, Phys. Letters 31B (1969) 75. [27] C. Schmid and C. Sorensen, Nuch Phys. B96 (1975) 209. [28} C. Rosenzweig and G. Chew, A Systematic lifting of exchange-degeneracy that clarifies the relationship between pomeron, reggeons and SU(3) symmetry violation, Berkeley preprint LBL-3834 (1975).