Volume 67B, number 1 PHYSICS LETTERS 14 March 1977

C A N U N I T A R I T Y C O N S T R A I N I N T E R N A L S Y M M E T R Y ?

M. BISHARI l Commissariat ~ l'Energie Atomique, Division de la Physique, Service de Physique Thdorique,

CEN-Saclay, Bofte Postale No. 2, 91190 Gif-S/- Yvette, France

Received 24 January 1977

In the context of the "I/N dual unitarization" scheme, with a t-dependent factorizable model, we study the phy- sical content of the full unitarity equation for every impact parameter (or, partial wave). A non-trivial lower bound is obtained for the (effective) quark flavor number N, which has direct consequences for Oel/O T and also for the absolute strength of meson-meson cross-sections at high energies.

The " I / N dual unitarization" approach has had a considerable success [1 ] in describing hadronic phe- nomena. In particular, this approach appears to imply an intimate connection between Zweig's rule violating mechanisms (operating in the time-like region) and ex- changes in scattering amplitudes (relevant in the space- like domain).

Most of the studies in the "1 /N dual unitarization" approach have been carried out [1 ] in the framework of one dimensional models. The one dimensional ap- proximation has been rather useful in abstracting some general features of total cross-sections, but clearly is inadequate to investigate e.g. O'el in which a non-forward scattering is involved. Stated differently, one dimen- sional models are inappropriate for studying the uni- rarity equation for individual partial waves, since in such models angular momentum is effectively lost.

In any case, to the author knowledge, in all previous "1/iV dual unitarization" studies, the elastic amplitude has been determined iteratively where in the first step the elastic and the inelastic diffractive contributions were neglected in the total overlap function relative to the non-diffractive component.

The dominance of the non-diffractive component has a phenomenological [2] support, as far as total cross:sections are concerned.

However, in the study of total cross-sections one sums over all partial waves (or, equivalently, integrate over all impact parameters) which participate in the scattering process. The dominance of the non-diffrac- tive component may not be guaranteed for every

i On leave from the Weizmann Institute, Rehovot, Israel.

partial wave; for some set of partial waves, the elastic and the non-diffractive contributions may be compar- able (although, when summed over all partial waves, O'el is a small fraction of OND ). In such circumstances it may be essential to treat diffractive scattering self- consistently and not iteratively.

It may, therefore, be anticipated that the unitarity relation, when written for every partial wave, will con- tain more information which is lost upon summing over all partial waves.

Here we would like to direct an attention to the pos- sible role of the full unitarity relation when implemented in the " I / N dual unitarization" scheme, with t-dependent dynamics. The t-dependent model enables one to study the scattering also away from the forward direction as well as investigating the unitarity equation for every partial wave (or, equivalently, impact parameter).

In this work the input is provided by the "cylinder" which [3,41 represents the non-diffractive component. The planar bootstrap, being the building stone of the " I / N dual unitarization" approach, is crucial for the construction of the cylinder; without it the results below could not be derived.

With the t-dependent "cylinder" input one is able to write the unitarity relation for each impact param- eter (or, partial wave). It is pointed out that the elastic contribution must be included in the unitarity equation for small impact parameters, since it is comparable there to the non-diffractive term. Only for larger impact parameters it is justified to assume the dominance of the non-diffractive mechanism.

Physically what happens is, that the elastic compo- nent is much more central, in impact parameter space,

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Volume 67B, number 1 PHYSICS LETTERS 14 March 1977

than the non diffractive component. Thus they may be comparable for central collisions, although when inte- grated over all impact parameters one obtains Oel which is substantially smaller than aND.

Therefore the term quadratic in the elastic ampli- tude will be essential in the unitarity relation for small impact parameters. Indeed it leads to a non-trivial lower bound for &e effective quark flavor number N. Namely there exists a "critical" quark flavor number, Ncr(S ), in general energy dependent, which is not far from the "actual" [5] N and such that N>~ Ncr(S ).

It thus appears that full unitarity may provide a useful restriction on internal symmetry in terms of space-time properties.

The fact that N cannot be smaller than Nc7(s) has direct implications for hadronic cross-sections; Oel/a T must remain small and the absolute magnitude [6] of cross-sections is, in a sense, restricted not to be large.

It is remarked, that the treatment here, of the full unitarity equation, is similar to the study [10] due to Van Hove. Here, however, thanks to the planar boot- strap, we are able to study not only Oel/O T but also the absolute magnitude and the energy dependence of the various cross-sections, taking into account the im- portant restriction N ~> Ncr(S ) mentioned above; the obtained results are in accordance with expectations in the NAL-ISR regime.

The details of the derivation are given elsewhere [7] and only the necessary formalism is presented, emphasizing the central ingredients.

From the formalism in refs. [ 7 -9 ] , one may deduce that the "cylinder" (or, the bare pomeron) corresponds to the following non-diffractive component, in meson a -meson a collisions

1 ~2a(t) ( ,s)aP(t) . (1) Moo(S, t) = N 1+~

Here/3Ra(t ) is the 21anar reggeon R-meson a-meson a coupling, and ap(t) is the bare pomeron trajectory. Also 5 arises from the J-dependence of the denomina- tor of the bare pomeron J-plane solution (see refs. [7, 9]) and is approximately ~ ~ 0.45. It is reminded here that N is not necessarily an integer, due to symmetry breaking (e.g. [5] in SU(3) ,N~, 2.5).

The bootstrap is conveniently formulated in terms of the planar triple reggeon coupling g(t; t l , t ' 1) (the t's are the masses of the involved reggeons), parametrized as [81

gl =-g(t; t l , t]) = g(a(t) - o%,1) e(a/2) t e b/2(tl+t~) , (2)

where 0%, 1 = a ( t l ) + a( t~) - 1 and the meaning of the other parameters has been discussed in ref. [8].

The planar coupling t3Ra(t ) is related to the triple reggeon coupling through a linear planar equation, namely a missing mass sum-rule, with no fixed pole residue, for the forward R (t) + a ~ R(t) + a process. As discussed in refs. [7,9] , one obtains,

~2 (t) f lRa(0)g(0;t , t) Ra ~ a ~ 2 ~ ) + 1 ) . (3)

Or, with eq. (2),

~2a(t ) ~/3Ra(0 ) g/a' e bt = (g/a') 2 e bt . (4)

The crucial role of the non-linear planar bootstrap, in the determination of ap(t) and the slope parameter b, has already been emphasized in refs. [7, 8]. From ref. [7] (eqs. (12'), (13)) we have;

Otp(t) = O~p(O) + Ot~t"

ap(0) ~ 1.045 ; ~ ~ 0 .2a ' ; b/a"~ 2.33. (5)

Using eqs. (1), (4) and (5), the non diffractive input is given by

1 1 ( g , ~ 2 - ' ' Maa(S, t) = ~ X 1 ~ \ ~ ] (°t's)aP(O)e(b+~plna s)t. (6)

At high energies it is more suitable to employ the im- pact parameter, rather than the partial wave represen- tation, that is,

o o

Ms(s, t) = 8.s f p dpJo(pvCL-[) M(s, p) , 0

with the inverse relation,

(7)

1 1 f.d(x/~Z-~)X/Z--] j o ( p x F ~ ) Maa(S, t ) , M(s, p) = ~ s 0 (8)

leading to

1 1 g2 , _(o~'s)aP(0)~l e_p2/4 (b+-~'p In ~ 's) M(s, p) = ~ X 1 + g 16no b + ~t~ In o's (9)

As it stands, the "cylinder" (i.e. the input) in eq. (9) is not sufficiently constrained, since it is expressed still by both the coupling g and the slope parameter b. How- ever, we have at our disposal the non linear planar rela-

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tions, which have been vital for the study of ap(t) and b. Explicitly, the relevant relation is [8].

(g2N/2b)/16na' = 1, (10)

leading, together with eq. (9), to

2 (ot,s)aP(0)- 1 e-P2/4(b+~'P In ds) M(s,p) = 1 X - - m 2 1+~ l+(~/b)lna's ( l I )

From the above considerations, it should be recog- nized that it is the planar model, with both its linear and non-linear relations, which endows a predictive power to the present approach.

The physical non-diffractive and inelastic diffractive cross-sections are given, respectively, by,

OND(s)=l Maa(S,t=O)=8rr ? pdpM(s,p), (12) 0

OD(S ) = 8n ; p dpD(s, p) , (13) 0

where D (s, p) is the corresponding profile in the p-space. Denoting by A(s, p) the imaginary part of the elastic

amplitude, one has

aT(S ) = 8rr ? p dp A(s, p) , (14) 0

and if, as usual, the real part is neglected we also obtain

o o

%t(s) = 87r f p dp A2(s, p), (15) 0

The decomposition of the full unitarity relation, in the impact parameter space, then reads,

A(s,p) = a2(s,p) + D(s,p) + M(s,p) , (16)

with the "cylinder" (i.e. the input) in the p space given by eq. (11).

It will not be surprising if more physics is contained in eq. (16) than in the integrated unitarity relation

OT(S ) = Oel(S ) + OD(S ) + OND(S ) . (17)

For instructive purposes, the internal symmetry de- pendence of M(s, p) is extracted out, def'ming

(18) M(s,p) = (1/N2)MI(s, p) ,

where ~I(s,p) is directly obtained from eq. (11) and

may be viewed as representing the space-time proper- ties of the model. Indeed one may convince himself, that in the evaluation of~t(s, p ) N appears only through the combination g2N which is constrained by the planar model. Even the very small ap(0) - 1 ~ 0.045 is a diffe- rence of two terms each of which is of order g2N.

To simplify the discussion we begin with only the elastic contribution in the r.h.s, of eq. (16), deffering to the end the inclusion of the inelastic diffractive component. Then, one obtains the solution,

a(s,p) = ½ [1 - x/l - (4/N2)f¢(s, p)] , (19)

provided the quantity under the square root is non- negative for every impact parameter, which is guaran- teed to be so if,

N 2 >~ N2(s) = 434 (s, 0) , (20)

with

2 _ (a'S) oT(0)-I M(s, 0) = i+ ~ I + (~'p/b) in ds " (21)

The aforementioned restriction on N, from space- time properties, is displayed in eq. (20). It evidently arises from the p ~ 0 unitarity relation, in which the elastic and the non-diffractive components are com- parable.

This space-time constraint, on the underlying internal symmetry, is non-trivial, since in the NAL-ISR regime N2cr(S) ~ 5 (see ref. [7]), whereas the "actual" [5] N2is ~ 6 .

With the present bootstrap approach one can deter- mine also the magnitude of the various hadronic cross- sections. Using eqs. (12), (11), (14), (18), (19) and (20) one arrives at

OND(S ) = 4nb(l+(~/b)lna's)(N~r(S)/N2), (22)

OT(S ) = 4nb (m + (~/b)In a's) × 4f(m~r(s)/N2), (23)

with

f(x) = l - x / 1 - x + ln ( l + ~ .) . (24)

Also we have

Oel(S) U2(s)/U 2 OT(S ) - 1 4f(N2r(S)[N2) (25)

One may readily obtain from eqs. (22)-(25) that the cross-sections as well a s Oel/OT, are proportional

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[6], in leading order, to 1IN 2. Thus, since N is not allowed to be less than Ncr(S), the corresponding cross- sections cannot be too big, and the ratio Oel/O T must remain small.

For concreteness let us fix N at

N=Ncr(Sl), ¢x's 1 = 200 (NAL energy). (26)

Then, at the energy squared Sl, Oel/OT is maximal but nevertheless rather small, i.e. 0.185. Moreover, the cross sections will have their maximally allowed strength, obtaining from eqs. (5) and (22)- (24) , the following satisfactory results

OND(Sl) --~ 16.5 rob, OT(Sl) -~ 20.3 m b . (27)

The energy dependence of the various cross-sections, in the NAL-ISR regime, has been studied in ref. [7], and will not be repeated here.

Let us now briefly review the possible modifications due to the inelastic diffractive component,/~(s, #). For the sake of illustration take

D(s, p) = XlA2(s, p) + X2M(s,p) , (28)

with X1, X 2 being p independent. Then one obtains from the unitarity relation, eq. (16), a modified critical .... quark flavor number, namely

N 2/> N2,cr(S ) = (1 + Xl)(1 + X2)N2cr(S) , (29)

where Ncr(S ) is given by eq. (20). For the cross-sections, one readily obtains that Oel]O T and o T are suppressed by (1 +X1), whereas OND is suppressed by (1 +X1)(I+X2). Now, if the reasonable requirement, o D ~ Oel , is adopted, one observes that the case X 2 ~ 0 is physically unfavor- able. Indeed, in such a case X l ~ 1, thus generating a small o T (~ 10.1 mb) and a very small Oel/U T (~0.092). Obviously this substantial suppression is basically because, with X 1 ~ 1, N 2 is compelled to increase signi- ficantly, as seen from eq. (29).

An acceptable physical situation is achieved for X 1

which is rather small. For instance, with X 1 ~ 0.1 Oel/O T would be decreased by only 10%. The "actual" N2 ~ 6 is then obtained for X 2 ~ 0.14. Consequently one has, Oel/O T ~ 0.17, o T ~ 18.4 mb, Oel ~ 3.1 rob, OND ~ 13.2 mb and OD ~ 2.1 mb, and also that D(s, O) is expected to be less central than A 2(s, p).

The present work may indicate once more, as does e.g. the "Asymptotic planarity" concept [8, 11,12], that it may be important to explicitly consider the transverse motion in the framework of the " I / N dual unitarization" scheme; essential physics may be lost when the transverse momenta are averaged out.

The author acknowledges his colleagues in Saclay for interesting remarks.

References

[1] For a recent list of references, see lecture notes by: Chan Hong-Mo and Tsou Sheung Tsun, Rutherford Lab, Preprint RL-76-080 (September 1976).

[2] H. Harari, Scottish University Summer School (1973), and references therein.

[3] G; Veneziano, Phys. Letters 52B (1974) 220; Nucl. Phys. B74 (1974) 365.

[4] Chart Hong-Mo, J.E. Paton and Tsou Sheung Tsun, Nucl. Phys. B86 (1975) 479; Chan Hong-Mo, J.E. Paton, Tsou Sheung Tsun and Ng Sing Wai, Nucl. Phys. B92 (1975) 13.

[5] N. Papadopoulos, C. Schmid, C. Sorensen and D.M. Weber, Nucl. Phys. B101 (1975) 189.

[6] H.D.I. Abarbanel, G.F. Chew, M.L. Goldberger and L. Saunders, Phys. Rev. Letters 25 (1970) 1735.

[7] M. Bishari, Saclay preprint, DPh.T/77/1 (January 1977). [8] M. Bishari, Phys, Lett. 59B (1975) 461. [9] M. Bishari, Saclay preprint, DPh.T/76/97 (October 1976);

Phys. Lett. B, in press. [10] L. Van Hove, Rev. Mod. Phys. 36 (1964) 655. [11] G.F. Chew and C. Rosenzweig, Nucl. Phys. B104 (1976)

290. [12] M. Bishari, Phys. Lett. 64B (1976) 203.

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