L~TTF_,RE AL NUOV0 CIM'ENTO VOL. 21, N. 17 29 Aprflo 1978

Comments on the Fermion Regge iza t ion w i t h o u t Parity Doubl ing .

H. F u J I s ~ i

Department o/ Physics, Rikkyo University - Tokyo 171

(ricovuto i l 3 Gennaio 1978)

The fermion Reggeization without par i ty doubling has been investigated by several authors (~-3). In par t icular i t has recently been clarified by ourselves (3) tha t the pa r i ty doubling pathology associated with the conventional Reggoization of fermions is suc- cessfully remedied in terms of the hiding cut mechanism (HCM) of Oehme type as a universal consequence of the Pomeron-Roggeon interaction. The HCI~I is inherent in the clothed physical Pomoron, i.e. the geometrical Pomeron (GP), except exact ly for the l imit ing case of the logari thmical ly increasing radius, R ( s ) = ro ln (s/so), of the opaci ty distr ibution (a.4). F rom the s-channel point of view, various interesting mo- dels (5-1~) yield the logari thmical ly growing GP disc of grey or black type. I t will there- fore be of physical importance to examine whether or not the formion Reggoization is possible without par i ty doubling even in the absence of the HCM of 0ehmo type. This is touched upon in this short communication within the general framework of Rog- geizod absorption.

Suppose the GP disc of :g(s) -~ r o In (s/so). The GP part ial-wave ampli tude ]p(u, J) is then wri t ten in the form (4)

(1) ]p(u, J) = ( J - - 1)~-~(0) ,

whore q -~ - - r ~ u ( J - - 1) -2 and 0 ~ ~ < 2. The scaling function ~ obeys the st ipulations

(2) ~(q = 0) = 2~r~8o, 0 < ~ < 1

(~) R. GARLITZ and M. KISLINGER: Phys. R6~). L ~ . , 24, 186 (1970). (s) R. S)~VIT and J. BARTFA~: Phys. l~v. D, U , 2300 (1975). (s) H. FUJISAKI" L~t. N ~ 0 Cirt~eTdo, to be published. (4) H. FUJISkKI: Riv. ~Vuovo Gimen~o, to be published. (s) J. FINKELSTEIN and F. Z~CHAR~SEN: Phys. Left., 34 B, 631 (1971). (') J. S. BALL and F. ZAOHARIASEN" Phys. Le[t., 40 •, 411 (1972); Nucl. Phys., 72 B, 149 (1974); 78B, 77 (1974); 85B, 317 (1975). (~) L. CANESCHI and A. SOHWnVIMER: NUr Phys., 44B, 31 (1972); 48B, 519 (1972). (a) H. HXR~RI: Phys. Left., 51 B, 479 (1074). (*) E. R~BI~OVICI: Phys. Rev. D, 11, 1969 (1975). (~,) H. FUXISAKI, E. AKIBA and S. 12[UTOH: Let/. Nuovo Cimento, 15, 495 (1976); H. FUJISAKI: Left. Nuovo Cimento, 19, 481 (1977). (**) ]). A.MATI, 1~. LE BELLAO and G. YlARCHESINI: .7~/'q.~CZ. Phys., 1.12 B, 107 (1976).

~92

CO~MI~NTS 01q TH]~ FER~fION RI~GGI~IZATION WITHOUT PARITY DOUBLING ~

and

(3) ~(q - . ~ o ) ~ d~-a)/~, 0 ~ < 2 .

The GP trajectory function %(~t) evidently reads

(4) ~,t.~(~) = z • ,'oVa'.

Let us turn our attention to the fermionic trajectory functions ~ l (~ r~ ) of the same signa- tare, opposite pari ty MacDowell partners. The Gribev conspiracy or equivalently the MacDowell symmetry then requires

r @ ~ ( r = ~ b ~ ( - ~ ) , ~ > o .

We are naturally led to the naive parametrization of the fermionie partial-wave ampli- tude ]~l(~v/~, J ) of definite pari ty

where

(7) ~ ( V ~ ) = ~r -~ ~ ~ / ~ + ~ ' ~ , ~ # 0 .

Since baryon states appear to be on linear trajectories, we can legitimately assume r~/4~' << 1. The ~t-dependenee of ~ 1 is then dominantly controlled by the linear term ~ 'u except for the immediate neighbourhood of ~t = 0. We are therefore confronted with the pari ty doublets which are not exactly degenerate in mass. Our next task is to examine leading absorptive Regge singularities arising from the simultaneous exchange of ~ J and ~ ] (12).

The celebrated Mandelstam pinch condition requires ~ ' # 0. The standard ~fan- delstam mechanism then yields two different pairs of absorptive Regge branch points of opposite pari ty

(s) ~ J ( ~ ) = ~ - (~o- ~'~)V~' + ~ o V ~

and

(9) t~J ~o[~(v~) = ~ - (r0 + ~ ) ~ / 4 ~ ' - ~ro~/~,

where r = • refer to the parity. In addition, we obtain asymptotically a pair of ab- sorptive Regge singularities of opposite parity, the trajectory functions of which are surely described in terms of ~• This is a natural consequence (3) of the fact that the GP disc of R(s) = re In (s/so) carries the trajectory functions a~ *J of Schwarz-type while the input fermion trajectories ~ J are legitimately approximated by

(7')

in the region I~]~O(rF/4~'2), where r~/4~'<< 1. Let us now remember elaborate arguments by BRONZAN and SUG~ (12) and by ourselves (4,18) in the case of Reggeized

(") J. B. BROIqZAI~ and R. L. SUGAR: Phys. Rev. D, 8, 3049 (1973). (1,) H. FUJIS~LKI: Prog. Theor. Phys., 51, 1503 (1974) (Erratum, ~1, 292 (1975)); 53, 593 (1975); 55, 185 (1976); Le/t. Nuovo Cimento, 19, 220 (1977).

5 9 4 m FUJISAKI

absorption of the bosonic Reggeon. I t is then possible to verify that if 0 < fl ~ 1 and/or 0 < ~ < 2, the input fermion Regge poles of both parities are unavoidably pre- sent on the physical sheet of the J-plane. Suppose both fl = 1 and ~ = 0, i.e. the ease in which the magnitude of the GP opacity is saturated at b = 0 with the uni tar i ty upper bound. Moreover let us assume r F > 0 [r F < 0] and confine ourselves to the region u>~ 0. We are then led to the following results. The input fermion pole of the positive [negative] pari ty exists on the physical sheet of the J-plane for ~/~ > (r o ~ r~ ) /2s [~/~ > (r o + rF)/2a' ] in the case r o > r F [r o > - - rF], while it is always present on the physical sheet in the case r o < r ~ [ro<--r~] . The input fermion pole of the negative [positive] pari ty is on the physical sheet for

~/~ > (to + r~)/2~' [V~ > (to-- ~)/2~'] .

In the l imiting case of both fl = 1 and ~ = 0, therefore, the input fermion trajec- tories :IF ~] Of both parities are always on the physical sheet of the J-plane at least for

( lo) V ~ > max ((r0 + rF)/2~'; (re-- rF)/2~') �9

I t is almost needless to say that the case r F ~ 0 corresponds to the completely degenerate doublets, ~[F+] = a[F -1, of opposite pari ty which inevitably exist on the physical sheet of the J-plane at least for ~ > ro/2s Consequently the pari ty doubl- ing phenomenon either exactly or effectively coincident in mass cannot be remedied in the fermion Reggeization through the absorptive mechanism of the logarithmically expanding GP disc. This fact is irrespective of both the magnitude and the scaled shape of the GP opacity distribution so long as the consistency conditions (2) and (3) are satisfied. We are therefore led to the important conclusion that the HCM of 0chine type is crucial in order to guarantee the fermion Reggeization without pari ty doubling as a universal consequence of the GP-fermion interaction. This subject undoubtedly deserves more than passing consideration.

The author aknowledges the financial support of Matsunaga Science Foundation.