Parity doublets and the Mandelstam-Sommerfeld-Watson transformation in backward K+p elastic scattering

Nuclear Physics B69 (1974) 157-184. North-Holland Publishing Company

PARITY D O U B L E T S AND THE M A N D E L S T A M - S O M M E R F E L D - W A T S O N T R A N S F O R M A T I O N IN BACKWARD K+p ELASTIC S C A T T E R I N G E.J. SACHARIDISt Rutherford High Energy Laboratory, Chilton, Didcot, Berkshire, England

Received 7 May 1973 (Revised 28 September 1973) Abstract: K+p backward elastic scattering and polarization are described in terms of u-channel exchanges. In the Reggeization of the scattering amplitudes, use is made of the MandelstamSommerfeld-Watson transformation and rotation functions of the second kind. The residues are assumed for convenience to depend meromorphically on u rather than on ~/u. A description is obtained in terms of only two adjustable parameters of all existing measurements on elastic K+p scattering and polarization for incident laboratory momenta above 1.5 GeV/c and centreof-mass scattering angles larger than 120° (cos 0< -0.50).

1. Introduction

Ever since the original work of Regge [1] demonstrating the existence of poles in the complex angular momentum plane for potential scattering, and the subsequent extension of the ideas therein [2] to relativistic scattering, Regge poles have dominated hadron phenomenology with considerable success. Lack of knowledge of Regge trajectories and residue functions, however, has limited their application to high energies and small momentum transfers. Analyticity and Regge asymptotic behaviour of the scattering amplitude has led then to the so called finite energy sum rules [3] in which effectively the low energy part of the amplitude imposes constraints on the high energy Regge parametrization and vice versa. In applying finite energy sum rules in various reactions [4], it was realized that high energy Regge amplitudes when extrapolated in the low energy region, represent there an average behaviour of the amplitude rather than only a non-resonating background. Such studies together with assumptions of resonance dominance, exchange degeneracy, and SU(3) symmetry (absence of exotics) led to the various aspects of duality [5] which culminated in the construction of closed crossing-symmetric expressions of the amplitude of the Veneziano type [6]. In particular duality suggests that if a reaction in the s-channel has exotic quantum numbers and hence no resonances, then its Regge amplitude derived from high energy data should describe the process adequately at quite low energies. Such a case was noticed in the forward K ÷ p elastic scattering before duality was invented [7, 33]. t

Now at CERN, Division NP, 1211 Geneva 23, Switzerland.

E. J. Sacharidis, Backward K + p elastic scattering

158

In what follows we describe backward K + p scattering in terms of u-channel exchanges. We avoid the usual high energy approximations in the parametrization and use the full rotation functions of the second kind as derived in applying the Sommerfeld-Watson transformation ~t la Mandelstam. Kinematic singularities are evaded by plausible assumptions and with very simple residue functions, which allow equivalent status to parity doublets, we have been able to describe K + p elastic scattering and polarization over a wide range of energies and momentum transfers.

2. The model

Kaon nucleon elastic scattering, as is well known, can be described in terms of two relativistically invariant amplitudes A and B related to spin non-flip and spin-flip respectively [8]. They are related to the usual scattering amplitudes f l and fa by

1_~A = w + m f~ 4n E+M

W - M f2 ' E-M

(la)

1 B= l___f + 1 4---~ E + M 1 E--~MMf 2 ,

(lb)

where the centre-of-mass variables W and E correspond to the total energy of the kaon nucleon system and the total energy of the nucleon respectively and M is the rest mass of the nucleon. Furthermore, f~ and f2 can readily be expressed in terms of the helicity amplitudes [9]f+ + (0, 40 and f+ _ (0, ~b) and their corresponding partial wave expansions. Thus:

fl

_

1 2 cos ½0f + - +

exp(iq~) 2 sin ½0 f+

dJ~,.~( O) = a=~z,;i~.... ( J + ½ ) ( f a - ~ + f s + O 2cos½0 d s_+,~(0) - s=~,l~.... ( J + ½ ) ( f s - ~ - f s + ~ ) 2 sin ½0 ' 1 2 cos

+ exp(i4) e 2 sin ½ 0 ~+ -

Y~ (J+½)(A-,+/J+0 g~,do)

J=¢, k....

2 cos ½0 J

+ s=~r, ~ ....

0

z sin ~tt

E.J. Sacharidis, Backward

K +p e l a s t i c s c a t t e r i n g

159

with 0 the centre-of-mass scattering angle and ¢ the azimuthal angle of the scattering plane, dS, u(O) are the usual rotation functions [10] and fs-,, fs+, are the partial wave amplitudes of total angular m o m e n t u m J and orbital angular m o m e n t u m 1 = J - k , l = J + ½ respectively. Incidentally, in all formulae in this p a p e r the helicity indices 2, p will invariably have the value 2 = p = ½ so that certain relations might differ for other values of 2 and/~. For the unique continuation of the amplitudes f~ and f2 in the complex J-plane, as we know, it is necessary to split them up into an even signature partf~, 2 and an odd signature partlY,2 according to whether J - ½ is even or odd and write in an obvious notation: e

o

f~,z = f~,z+f~,z,

(2)

with

+(0) Z . . . . (J +½)(f~-++ f~+O d~, 2 cos ½0

2--

J=$,,~

T

(J +½)(f~-~- f,+ O d~-~'~(o)

£ J=~,~

2 sin ½0

.... 1 e e ½(J+~,)(fj_~+fj+~)

~,

3+

½,~(z, J)

J=~., ?r.... 1 e e --+ z(J +½)(fj-~-fs+.Od-~,~(z, J) ,

E J=½, k ....

~ ,2

d~, ~(0) Z J = ~z, k, ...

(J+½)(fs-~:+fs+÷)

2 cos 30

-v- J=~g,E~ ....

(J + ½)(fa- ¢ - - f s + ~ )

d~.~,+(0)

~

=

2 sin ½0

½(J +½)(fy_~ + f;+ ~)a~, ~(z, J)

E

J=:t, ~ , . . .

-v J = ½ ,E~

½(J+½)(f?-~- f L Oa;-+,~(z, J), ....

where the upper and lower signs in -T- correspond to the subscripts 1 and 2 respectively and the signatured rotation functions d x,u e are given by a~. ~ (z, J) = ½ [dJ, ~ (z) + o ( - l) ~- ~ aJ, _. ( - z)], with the reduced rotation functions, d Ja,¢(z), given by

dS, u(z)

r\ - - DJ-a[lJ t__z~1, ) ~., - # \

and z = cos 0, a = + or a = e, o as appropriate.

off;~,'~

160

E.J. Sacharidis, Backward K +p elastic scattering

In the conventional reggeization of the amplitudes one makes use of the S o m m e r f e l d - W a t s o n transformation to replace the sum over J by an integral over a suitable contour in the complex J-plane, e.g. C' in fig. 1. H o w e v e r such a representation is valid only for Re ~i > ½ where ~ is the complex angular m o m e n t u m . ImJ

-M ReJ

- N -1

C'

C*

Fig. 1. Complex angular momentum plane.

For large m o m e n t u m transfers in the crossed channel and linear Regge trajectories, however, it is necessary to have an expansion valid for Re e~ ~ ½. The possibility of such an expansion for the spinless case was first shown by M a n d e l s t a m [ll]. F o r particles with spin, one substitutes [13] the rotation functions daS,, by their expressions in terms of the rotation functions of the second kind [12] e J2, g and pushing the contour to the left, e.g. C" in fig. 1, one obtains the following MandelstamSommerfeld-Watson representation of the signatured amplitudes*: . . . . .

f~ = ~. (~i+½) K e s t j ~ , - O

+ ~ ( ~ + ½ ) Res(f~,+÷) i

a + i~

exp(-i~pz~i) 4 sin rc~

__~,_1 [ez., (z) -- ~ . ~

a + i q e x p ( - - ir/~zcq) i- _~,_ 1 ( z ) + ~]_,~1 (z)] C z.i, , 4 sin ~ i

. a + i t / e x p ( - iq=cq) [ez_~,'-* (z) + ~ E ; ' f ~ = ~ (a~+½) Res(f~,_ 0 4 sin Tccq ~,

+ '}-"(a,+½) Res(f~,+÷) i

l (z)]

a+i*l

(3)

(z)]

e x p ( - # p z c q ) [ea.~, l(z) - - = ' - * --='- e a . - . (z)], 4 sin ~a~

* For details on the derivation of formulae (3, 4) and the assumptions therein see Sacharidis [43].

(4)

E. J. Sacharidis, Backward K + p elastic scattering

161

where

~s,u(z)=(12____f)-~l~-ul(~)-½1~+Ules,u(z) is the reduced rotation function of the second kind, ~/= sign I m z and a = + or a = e, o as appropriate. Background terms in the above formulae have been omitted since they behave like z - ~ for large z where - M is the position at which the contour intersects the negative real axis (see fig. 1) and this position is supposed to be far to the left.

3. u-channel exchanges In our description of the amplitudes so far, no mention has been made of a particular channel. In this section we specify our notation according to the diagram of fig. 2 where K N scattering corresponds to the s-channel and K N scattering to the u-channel. We shall express the amplitudes in the u-channel in terms of Regge poles and analytically continue them to the s-channel to obtain the amplitudes for K N backward scattering. We shall be concerned mainly with K + p scattering. In the following a subscript u or s will be used when necessary to indicate the channel in which the corresponding quantities refer. K

. . . .

~

. . . .

•

N

A.I $" N

l° Fig. 2. Diagram for KN scattering.

The internal quantum numbers of the K N system are: Isospin [ = 0 or 1, baryon number B = 1 and strangeness S = -10 In addition to A and 2 there is a host of Y~ and Y* resonances which share these quantum numbers. A m o n g the best known to fit the SU(3)-Chew-Frautschi scheme [14] are the A ° and As for I = 0 and the 27° and zD for I = 1, where the superscripts S, O, D, denote SU(3) singlet, octet and decuplet respectively and the subscripts ~,/~, ~, 6 stand for even parity-even signature ( P - - + , J - ½ = even), odd parity-even signature, odd parity-odd signature (P = - , J - ½ = odd) and even parity-odd signature respectively. Fig. 3 shows the Chew-Frautschi plot of the well established A and 27 states [19] as well as the above mentioned trajectories.

162

E. J. Sacharidis, Backward K+p elastic scattering

9/2

I

{¢:

7/2

J:

I

[

I

Parity Signature +

*

S : Singtet O : Octet D '. D¢cuptet 5/2

3/2

1

1/2

1

~

~. (1750 1/z )

,

A s (Y =0, I = 0 )

I

'7/2

/

~-t

~h-)

5/2

/

~-t

jf 3/2

(

I/Z

1

I 0

0

1

1

1

5

~

5

//

,

VZ-)

) ( ( I 670, ~/z-)

I

I

2

3

M2 [GleV z ]

Fig. 3. Chew-Frautschi plot of established A a n d s states [19].

4

=

E. J. Sacharidis, Backward K+p elastic scattering

163

Since the K Z N coupling, as derived f r o m forward dispersion relations [15], is small c o m p a r e d with the K A N coupling we shall ignore the contributions of the 27° and 2;o trajectories and restrict ourselves to the A ° and AS trajectories only. We reserve for a subsequent publication the inclusion of the Z ° and S~ trajectories as well as the as yet not so well established A ° and A~ trajectories for the case when, in addition to the K + p the rest of the reactions of the K N system will be considered. Clearly, both A ° and A s contribute only to the I = 0 amplitudes f~,2 in the u-channel. Their corresponding contributions to the partial wave amplitudes are as follows: A ° has even parity and even signature i.e. P = + and J - ½ = even. Since the K N system has odd intrinsic parity, its orbital angular m o m e n t u m l must be odd, i.e. 1 = J + ½ . A ° therefore contributes t o f f + ~ ( + W.) and through McDowell's reflection principle [16] to the parity doublet ff_½(+ W.) = - f f + ~ ( - W.). Thus: 1+ f , .c = (C~A. + :1) Res[ --f~a. + ~ ( -- W.)]

+ (~A,, + ½)

Res

iq e x p ( - ir#rc~a. ) [g%.~ _, (z,) -o~,-. z- SAlt--1 (~.)] 4 sin recta.

[f~-A. + k ( "b Wu)] 1 + it/exp ( - i~]7~O~A") [ex.- IA'. --1 4 sin

(Zig) .~- ~-,'~ - IA._. --1 (Zlg)]"

~(~a~

(5a) Similarly 1 Res [ - f 2 a . + -~( - W.)] 1 + iq exp( - irlT~O~A~.)[~,~A,, --1 (Zu) + ea.f~.e = (c~a. + z) - .a:_.- . (z.)] 4 sin 7~0CA~

+(CA.+½) Res[f2~.+~r(+ 14/,)3 l + i q e x p ( - it/roe a - ) rr.-=A.-, t, .~ . ,~- o ~7,--etA. -1 (z.)]. :~,. _. 4 sin 'K0~A: (5b) A s has odd parity ( P = - ) as well as odd signature ( J - ½ = odd); it couples, therefore, to even orbital angular m o m e n t a o f the K N system ( l - - - J + ½ ) and hence contributes to ff+~(+W,,) and through McDowell's reflection principle to fs°_~(+ W . ) = --fs°+½( - IV.). Hence:

f

o

= (ca, +½) R e s [ - f °

+ ~ ( _ W.)]

+ (C~A, + ½) Res If°A, + ~ ( + I4/,)]

- 1 + it/exp( - i~lrrOta~,)[O2,'uA~'- ~ (z,3 - e,~,--'~'~'-i_u (z,,)] 4 sin rosa,

-- 1 + i t / e x p ( - - i~lnaa,) [ ~ , ~ a , - , (z.) - z - ~ , ~ - a -~-~,-~ (zu)], 4 sin nc~A. (6a)

E. J. Sacharidis,BackwardK + p elasticscattering

164

and

f~o = (0% + 5) R e s [ - f ° y

+ ~ ( _ !~L)]

--1 +iqexp(--ulnaay)[O].~.y_

- 1 + iq exp(

+ (0~ay+ ½) Res [flay +-I ( + W.)]

1

( z . ) ~A_a7.J-- ul A y -

irlTrOtAv) r~-.~.- 1

---~ay--

(z.)]

1

4 sin 7C0~ay

-

4 sin 7~0CA~'

L'a.,-

(z.)--ea,-.

(z.)]. (6b)

N o w if we assume for simplicity that the residues depend meromorphically on u rather than on IV. = ~/u i.e. if Res [f~A~ +~ (-- 141.)] = Res [f~^. + ~ ( + '~,)] = Res [f2A.+ ~(U)]'

(7a)

Res [flAy + ÷ ( - 14I.)] = Res [ f o y + ~ ( + I41.)] = Res [ f o y + ~(u)],

(7b)

and if in addition the trajectories are even in x/u then eqs. (5) and (6) yield:

f~,, = - f ~ . = (aa~+½) Res[f2~.+~(u)]

1 + it/exp( - itln~a) - - . A . - 1 (z.), e2, - p 2 sin g~A~

(8a)

f10 = _f~o = (aay+½) Res[f~°y+~(u)]

- 1 + itl exp ( - itlnO~av) e~., __ .Aya (z.)]. _N

(8b)

2

s i n ~OCAy

With the help of eq. (1) we obtain the invariant amplitudes in the u-channel

A ° = 4n W ' + M f l . - 4 ~ E.+ M B° --

4n --fl.

E,,+M

+

W"-Mfz., E.-M

(9a)

4n

(9b)

E . - - M f2""

Here and subsequently the superscripts 0. and 1 in the A, B amplitudes imply isospin I = 0 and I = 1 respectively. Finally using the s<->u isospin crossing matrices [17] we obtain the amplitudes A, B in the s-channel:

(BO~=( A'/

~ \AU

kB~/

1 --~(B°~'l 1 _1

_:/\~.1

In particular for the K + p system which is a pure I = 1 state we have: Asl=l

1 ~(A,,o +Au),

1 o +Bu). t B~1 = --~(B,,

E. J. Sacharidis, Backward K+p elastic scattering

165

Since A~ and B.a are assumed to be negligibly small due to small K S N couplings, and taking into account eq. (9) we find: A~ = 2n +------~M W" f , . - 2n W , , - M fz,, = 2 n u - m 2 - - M z f l u , E.+M Bs1 --

2n -

-

E. + M

E.-M f

2n lu

- -

E.- M

f 2u

(10a)

k~

=

2 M f~, 2n Z'5"k.

(10b)

In the last step of the above equations use has been made of the equality f l . = - f 2 , , of eq. (8) and the kinematical relations: w. = ~/u,

(~la)

u - rnZ + M 2 E,, =

k2

2 x/u

'

(1 lb)

[u - (M + m y ] [~ - (M - m) :]

(l l c)

4u

where m is the kaon mass and k. the incident centre-of-mass momentum in the u-channel. Let us now define the reduced residues: Re(u) = ~(aA.+½) k ~ s i n n ~ A R e s [ f 2 A . + ~ (u)]'

(12a)

RO(u) = n(eA.+½) Res[f~Av+l ( o U)].

(12b)

k~z sin lr~a~ With the help of eqs. (2), (8), (10) and (12) we can write the final form of the invariant amplitudes for K + p scattering: A~ (s, t) = R ~ ( u ) ( u - m 2 - M2) [1 + iq e x p ( - iqnaA.)] ~a,z-~*-'-u

(z.)

+ R ° ( u ) ( u - m 2 -- M 2) [ - - 1 + iq exp( - - itlTraA~)] ~ Z--~A~'-, _ . 1 (z,,), • ---- ~[A~-B2 (s, t) = R~(u) 2 M [1 + it1 e x p ( - ltlTCO~A.)] ea,-u

1

(13a)

(z.)

+ R°(u) 2 M [ - - l + iq e x p ( - ir/nea,)] e~. --,A~,_. 1 (z,,).

(13b)

4. Kinematic singularities and their evasion As can be seen from eqs. (12) and (I3) the amplitudes A~ and B~ will have simple poles at integer values of ~a~ and aa~ respectively due to the sine terms• We shall

166

E. d. Sacharidis, Backward K+p elastic scattering

assume t h a t such poles are cancelled b y c o r r e s p o n d i n g zeros in the residues o f f £ ~ + ~ a n d f £ ~ +÷ so t h a t Re(u) a n d R°(u) are regular in the physical region o f the s-channel. F r o m the k i n e m a t i c relations: zu=cOs0~=

= I +

2ut

1+

t 2k~

= i +

2ut , [ u - ( m + M ) Z ] [ u - ( m - M ) 2]

(14a)

( M 2 -- m2) 2 - - - 2k~(1 + c o s 0~), S

u =

(14b)

i [s-(M+m) 2][s-(M-m)2],

k~--

(14c)

where ks(k.) a n d 0s(0.) are the centre-of-mass incident m o m e n t u m a n d scattering angle for the s-channel (u-channel) respectively, we deduce the following b e h a v i o u r o f these quantities in the physical region o f the s-channel: (M 2 _/912)2 COS 0 s = -- 1 :

= - -

U =

s

/ ~ XA / ~ XB / z +. l ~

0 ~

1 Is -- ( M + m) 23 [s--(M--m) 2]S-oo + oo /'~

"~--s+2(M2+m

/ 7 _ Is -- ( M -- m) 2] Is - ( M + m) 2]

4s

z~ = - 1

/+

2)

[ s - 2 ( M 2 + m2)]

1 Z z . . . . " ~ + 1,

where 2(M 2 X A = --1 +

rr/2) 2

[S -- ( M + m) ~] Is -- ( M -- m)2] ' 2 ( M 2 - m 2)

x B = -1

+

[s - ( M + m)23 ~ [s - ( M - m)23 -t-'

and Zumax ~

+

--] S2

[ ( M 2 + m 2) s - ( M 2 -- m2) 2] + ( M 2 -- m 2) [s -- ( M + m)2] ~ [-s - ( M - m)2]"

E.J. Sacharidis, Backward K÷p elastic scattering

is t h e m a x i m u m XAXX~

v a l u e o f z , . I t will b e n o t i c e d t h a t

for

XA, X a ~ < I

167

s~2(M

for

2+m z)=2.25GeV

2

i.e.

p ,~ 0.339 G e V / c ,

s~>2(MZ+m2).

Also for sufficiently large s s __P -- 2 M 2 -- M '

z ....

x A "~ x a ----- - - 1 .

T h e b e h a v i o u r o f z= as a f u n c t i o n o f c o s 0~ f o r v a r i o u s i n c i d e n t l a b o r a t o r y m o m e n t a p c a n b e s e e n i n fig. 4. z, s t a r t s f r o m t h e v a l u e - 1 a t c o s 0s = - l, rises s t e a d i l y t o t h e v a l u e + 1 a t c o s 0, = XA w h e n u = 0, r e a c h e s its m a x i m u m

value

z ....

a t c o s 0s = xB

a n d t h e r e a f t e r falls s t e a d i l y t o t h e v a l u e 1 a t c o s 0 s = + 1. I

i

i

I

[

I

]

I

I

l

i

I

i

!

[

i

[

I

l

15

=15 G eV/c

10

11

~ 5

\

\

\

3 \ N

-1

-0.5

0

0,5

coses

Fig. 4. z= = cos 0~ as a function of cos 0s for various incident laboratory momenta.

t

168

E. J. Sacharidis, Backward K+p elastic scattering

os,_u(z) is regular in this range of values of z. except at z. = _+ 1. It has a logarithmic singularity at both these points and in addition a simple pole at z. = + 1 for 2 = ~ = 1. As noted above z. is in the n e i g h b o u r h o o d of unity only in the very backward and forward angular regions. T o circumvent the singular behaviour of the amplitudes in these regions due to the ~s~, _.(z) term, the latter is replaced by its linear extrapolation in zu, the extrapolation taking place from a point z 0 > 1. In other words for z. < Zo we replace in eq. (13) 0~, _~(z) by:

O~,-.(Zo)+(z.-zo)

d _j _ .

(15)

The point z o will be decided below when fitting the data.

5. Fit to the data and discussion

In this section we describe existing measurements on b a c k w a r d elastic K + p scattering and polarization in terms of the invariant amplitudes A and B as given by eq. (13) with the isospin and s-channel indices suppressed. We shall restrict ourselves to centre-of-mass angles 0 >~ 120 ° i.e. cos 0 ~< - 0 . 5 and incident laboratory m o m e n t a p ~> 1.5 GeV/c. Using GeV and h = c = 1 units throughout we can write [18] for the elastic differential cross section:

dt=zrs\4k/

~

IA'I=+~-M- 7 s

I_(t/4M2)AIB[2

[GeV-4],

(16a)

and for the polarization P dg 1 P -- = - sin 0 I m ( A ' *B), dt 16n x/s

(16b)

where k is the centre-of-mass incident m o m e n t u m , o) is the total energy of the incident kaon in the laboratory system and the amplitude A' is expressed in terms of A and B by

A' = A + co+(t/4M) B. 1 --(t/4M 2)

(16c)

F r o m eq. (14a) we see that if we are to continue to s+ ie, i.e. to the upper lip of the cut s-plane, we must set in eq. (13) t/ = sign I m z. = + .

(17)

We assume real linear Regge trajectories for A ° and ASs with c o m m o n slope as derived from the Chew-Frautschi plot of fig. 3 by a least squares fit under the following conditions:

E.d. Sacharidis, Backward K+p elastic scattering

169

(a) Common slope but non-degenerate intercepts of the trajectories. Thus: cG= = - 0 . 7 1 + 0 . 9 7 u ,

(18a)

aa~ = --0.74+0.97U.

(18b)

(b) Degenerate trajectories i.e. ~A, = ~A~ = ~ = --0.68+0.95U.

(19)

For the non-degenerate trajectories above (eq. (18)) we consider the following five cases of reduced residue functions Re(u) and R°(u) of eq. (13): 1) The simplest case, namely that they are constant with Re(u) = C1,

R°(u) = C2.

(20)

1I) The more conventional case of exponentials in u, i.e. Re(u) = C t exp(b~u),

R°(u) = C2 exp(b2u).

(21)

li D Explicit expression of the dependence on the angular momentum factor

(c~+½) in eq. (13) i.e. Re(u)

=

CI(O{A -}-½) ,

R°(u) = Cz(c~a+½).

(22)

IV) Attach an exponential dependence in u to the case 01D: Re(u) = C1(C~A +½)exp(blU),

R°(u) = C2(~A +½)exp(bzu).

(23)

V) Attach a gamma function to the case (llI) to cancel the wrong signature nonsense zeros, i.e. Re(u) = CI(O~A -{-1)I"[½(O~A q-½)],

R°(u) = C2(O~Av'-[-½)IF'[I(O~A~-b~2) 3.

(24)

For the degenerate trajectories (eq. (19)) we consider the following two cases of degenerate reduced residue functions: VI) Explicit dependence on the angular momentum factor (c~+½), i.e. R°(u) = R°(u) = C , ( ~ + ½ ) .

(25)

VII) In addition exponential dependence in u: Re(u) = R°(u) = C,(a+½) exp(blU),

(26)

where CA, Ce, bl, b2 are constants to be determined by the data. Finally the reduced rotation function of the second kind, ~,_~(z), in eq. (13) is computed using its expression [12] in terms of the hypergeometric series F: -s (z) = ea, -~

(j + ½) [F (j + ½)32 1 z -2 F ( 2 j + 2)

F j+3,j+};2j+2;

2 1+z

,

(27)

E.J. Sacharidis, Backward K +p elastic scattering

170

where /arg(1 +z)[ 2, hence in the case of real positive z for z > 1. For rapid convergence near z = 1, however, we use the analytic continuation of the right-hand side of eq. (27) [20] into the domain [I - z [ < 2 and l a r g ( z - I)[ < n:

0~_,(z)-2j+l

1--

,=~

n!(n+l)[

"\

2 J_JJ'

where Vo = l o g ( - ~ 2 1 ) + 2 0 ( 1 ) + l - 2 ~ b ( j + ½ ) -

V n + l = l)n -{-

1-~,

j+½

j--½

j+½

(n+2)(j+)+n)

(n+l)(-j+½+n)

In the above expressions F denotes the usual g a m m a function and ~ the logarithmic derivative of F, i.e. ~ ( x ) = F ' ( x ) / F ( x ) , while (a). is a shorthand script for a(a+ 1) ... ( a + n - 1). In the range of values of zu considered in the present fit, i.e. z, real and greater than one, usually the first twenty terms of the series gave a precision better than 10-5. In the actual fit, however, to be on the safe side, the series was truncated at the 50th term. The regularization of ~, _u(zu) in the b a c k w a r d region near z, = 1 was obtained by extrapolating linearly in zu for z, < z o (see eq. (15)). F o r each incident m o m e n t u m p, z o was taken to be

z o = 1 + r(z . . . . - 1),

(29)

with z ..... the m a x i m u m value of z, corresponding to the incident m o m e n t u m p under consideration. In the final fit we set r = 0.90.

(30)

Trials with different values o f r indicated that the fit to the data was insensitive to values of r in the range 0.60 ~< r ~< 1.0, but it deteriorated rapidly as r decreased below 0.60 feeling strongly the singularity of ~ , _u(z,) at z, = ]. The o p t i m u m values of the parameters C1, C2, bl, b2 were obtained by minimizing the chi-square, X z, which was c o m p u t e d using existing data on K + p elastic scattering and polarization for incident K + laboratory m o m e n t a greater than 1.50 GeV/c and centre-of-mass scattering angles 0 greater than 120 ° ( c o s 0 < - 0 . 5 0 ) and the predictions of this model as calculated using eqs. (13)-(30)• The references of the experimental data used are indicated in tables 2 and 3. Because of their large 2~2

E. J. Sacharidis, B a c k w a r d K +p elastic scattering

171

contribution, the differential cross sections of ref. [24] were not included in the minimization procedure. Also the last point ( c o s 0 = - 0 . 9 4 ) o f ref. [26] being abnormally low was omitted. The minimization procedure was based on the variable metric method as incorporated in the C E R N program " M I N U I T " and its adaptation at R H E L [21]. In order to reduce large contributions to the Z a due to the uncertainties in the normalization o f the experimental data, we assigned a normalization parameter, a, to each distribution. This normalization parameter was assumed to have a Gaussian distribution about the value one with standard deviation, a, equal to the normalization error as stated in the corresponding experiment or approximated by us. F o r each trial set of the parameters C1, C2, bl, bz of the model, the normalization parameters, a, were analytically set to the values which rendered m i n i m u m the Z 2 o f the corresponding distributions (see appendix).

Table 1 Optimum values of the adjustable parameters C1, bl, C2, b2, of the model and overall chi-square per point, ZZ/point, for different cases Case

I 1I II1 IV V VI VII

C1

bl

(52

bz

[GeV- 3]

[GeV- z]

[GeV- a]

[GeV- 2]

6.757 6.347 11.88 15.26 2.859 10.08 10.38

-- 0.499 0.708 0.206

0.082 0.514 9.608 9.323 1.056

Z2

I. 100 - 0.207

point 2.87 2.55 2.73 2.42 2.54 2.77 2.67

Proceeding along these lines we have obtained the o p t i m u m values of the parameters shown in table 1. Also in table 1 we indicate the chi-square per point we obtained for each case. This X2/point is the ratio of the overall Z 2 over the number o f degrees of freedom, the latter being the difference between the total n u m b e r o f measured points and the n u m b e r o f adjustable parameters (including the normalization parameters). In tables 2 and 3 we show how the Z 2 is distributed a m o n g various experiments. Here X2/point is simply the Z 2 of the angular distribution divided by the n u m b e r o f measured points in the distribution. The column marked by ___a indicates the normalization error while the o p t i m u m value o f the normalization parameter a is contained in column a for each case. I n figs. 5a and 5b we plot all K + p differential cross sections referred to in table 2 together with the predictions & t h e model for the cases I, III, V and VI. A comparison of all cases o f the model for some m o m e n t a is shown in figs. 6a-6d. Finally only

172

E. J. Sacharidis, Backward K +p elastic scattering

some of the polarization data of table 3 are plotted in fig. 7 since the polarization is consistent with zero in the m o m e n t u m and angular range considered here. From an inspection of tables 1-3 and figs. 5-7 we can infer the following: (i) Backward K + p elastic differential cross sections are adequately described by this model for centre-of-mass angles greater than 120 ° (cos0 < - 0 . 5 0 ) and incident K + laboratory momenta greater than 1.50 GeV/c. The fit improves as the incident m o m e n t u m increases and is in general very good above 1.7 GeV/c. An exception to the latter trend are the data of ref. [25] which yield higher chi-squares at the upper end of their m o m e n t u m range. It is hard to understand the abnormally high chi-square at the two lowest momenta of ref. [24] except that the differential cross sections of this experiment were obtained as a by-product of polarization I

[-

I p=1-501

R¢f.[221

I

q4

I

GcV/¢~ 3

I

I

_

Rcf. [23]

t~..-J- I

i

3

p=1.546 GcVIc ~,~

~ ;

- =~ _,

T ~

.

0.6

T

12 T

2

1.59~

±

•

~ .

• i-

1.744J 2 ~

T1 , 6 2 7 ~

l

'~ ~""

li

2

,~ ,,{

-~t

0,8

;

0.6

,;2 q ;....Z"; ~tt '~6 L_T i j ........~- ~-~o.,.

":r2

~

'"

-

1.97o 1

Ir i

~

.~
'"

0.8

0.3

0.2 i

__I_

0.4

0.2 cosO

-o~ _~6

-o'.~ _o'6 _&

0-2 -,

-0.5

-0-6

-0-?

-0-8

-0-9

Fig. 5a

-1

-0.5

-0.6

-0.7

-0.8

-0.9

-1

0.1

E. J. Sacharidis, Backward K +p elastic scattering

173 0.5 0.4

Ref.[25]

~.~.~:~- .....

=xtro

~polat~on _/z

Ij : ~ ._.#:

06

"

.

Ref. {24]

l

//"~--

{ 0.2

"

I ..T ~ I ~ 0.6

0.4 0.3

..................,~./"4

0-2 0.1 0,08

0.06 0.1

~J-- I~'_~"~-~.~

4

.~ -~-----1.69o r

./..-'-~

-

0.05

0.6

Oo:; 0.01

F

Rcf,[271

I /,','f'i ~0.3

5LtLt:-,I:L. . . . . .

0.05

.--J

E 0.01 _.[ ..... 7_ c:~,~-- ~

~ 0.2 0.005

? +/i o,

~L- -

T2.530 ]

.... -~/

T I 0.01

0 m001 0.0005

0.0001 -0,6

-0.7

-0.8

-0-9

-I

-0-5

-0.6

-03

-0.6

-0.9

-I

-0-5

-0.6

-0.7

-0-8

-0-9

-I

Fig. 5b Fig. 5a, b. Backward K+p elastic differential cross sections versus cos 0. The solid, dashed, dotted and dot-dashed lines through the data indicate the predictions of the model in cases l, II[, V and VI respectively. The vertical arrows indicate the position of u = 0 and u = -0.5 respectively while the angle indicates the place from which the extrapolation of the rotation functions was made at each incident laboratory momentum. Also indicated on the graphs are the corresponding incident laboratory momenta p.

m e a s u r e m e n t s an d hence, a l t h o u g h they have high statistics, they m i g h t hide systematic uncertainties. Suppression o f the dependence o f the reduced residues Re(u) an d R°(u) on the a n g u la r m o m e n t u m factor (~+½) i.e. in the cases I and II leads to d o m i n a n c e o f the A= trajectory and hence p r o n o u n c e d dips at w r o n g signature zeros o f this trajectory. Such p r o n o u n c e d dips are not exhibited by the data as can easily be seen at all incident m o m e n t a where m e a s u r e m e n t s extend to

E.J. Sacharidis, Backward K + p elastic scattering

174 Table 2

Chi-square per point, XZ/point, and o p t i m u m normalization factor, a, for various experiments on elastic K + p differential cross sections, p: Incident m o m e n t u m in the laboratory system [GeV/c] Ref.: Reference. ±or: Normalization error p

REF

-+(~

OASE I a

1.501

22 0.10

1.530 1.563

CASE II !~OllT T

CASE I£I a

CAS5 IV a

CASE V a

CASE VI

CASE Vll

~

a

a

POINT

0.91 2.37

0.90

1.24

0.87

1.45

0.91 1.06

0.89 1.23

0.90

1.24

0 . 9 1 1.66

a

POINT

1.04 3 . 7 4 0.98 3 . 2 5

l.O0 2.38 0.96 2.02

0.98 0.94

2.41 2.23

1.02 2.24 0.96 1 . 8 7

1 . 0 0 2.33 0.95 1.98

1.00 2.32 0.95 1 . 8 9

1 . 0 2 2.91 0 . 9 7 2.41

1. 594

1.20 7 . 3 2

1 . 1 4 5.38

1.11 5.59

1.15 5 . 3 7

1.13 5 . 4 2

1 . 1 3 5.30

1 . 1 6 6.14

1.627

0.93

3.69

0.92 2.29

0.89

2.27

0.92 2.17

0.9]

0.92

0.93 2.87

1.663

0.98

3.32

0.96 2.17

0.95

2.20

0.97

0 . 9 6 2.12

0 . 9 6 2.10

0.97

2.58

1.696

0.94 2.34

0.94

1.39

0.92

1.36

0.95 1.36

0.94

0.94

0.94

1.74

l. 728 l . 765

0.92 4.19 0.85 2.70

0 . 9 1 2.69 0.87 2 . 5 0

0.89 0.87

2.58 2.93

0 . 9 2 2.63 0.87 2.35

0 . 9 1 2.62 0.87 2.58

0 . 9 1 2.67 0.87 2.48

0 . 9 2 3.28 0 . 8 6 2.47

1. 799

0.89

1.86

0.90" 1.53

0.90

1.67

0.90

1.48

0.90

0,90

0.90

1.834

0.98

1.28

1.02 1.56

1.01 1.64

1.03

1.63

1.02 1.58

1.02 1.56

1 . 0 0 1.39

1.873

0.93 2.17

0.93

1.27

0.93

1.23

0.94

1.26

0.93

1.22

0.94 1.23

0.93

1.57

1,909

0.95

2.00

0.97

1.52

0.97

1.49

0.98

1.55

0.98

1.50

0.98

1.52

0.97

1.66

1,946

0.85

1.65

0.87

1.18

0.87

1.34

0.87

1.16

0.87

1.18

0.88

1.14

0.86

1.25

2,006

0.94

1.75

0.96 1.17

0.96 1.20

0.96 1.15

0.96

1.16

0.96 1.17

0 . 9 5 1.36

2,087

0.91 2 . 4 6

0.93

1.63

0.94

1.42

0.93

1.77

0.94

1.54

0.94

1.60

0.93

1.88

2.167

0.93

0.97

1.45

0.98

1.55

0.97 1.46

0.97

1.45

0.97

1.42

0.95

1.46

2.259

0.06 1 . 7 4

0.97 0.99

0.97 0.75

0.97

0.97

0.86

0.97 0.88

0.97

1.16

1 . 0 4 3.71

1.05 3.28

1 . 0 3 3.05

1.01 2.68

1 . 0 4 3.66 1 . 0 9 3.16

1.70

2.08

1.14

2.25 1.35

1.53

2.29 1.37

1.50

1.59

1.546

23 0.05

1.10 5.57

l.O5

3.36

1.595 1.642

"

1.14 4.47

1.08

2.91

1.07 3 . 1 1

1 . 0 9 2.84

1 . 0 7 2.71

1.06 2.44

l.O0

3.30

0.97

1.64

0.96 1.66

0.97

1.58

0.96

1.55

0.96

1.57

0 . 9 7 2.16

l . 744

"

1.01

2.58

0.98

1.65

0.97

0.99

1.62

O.9B

1.61

0.98

1.67

0.99

l . 798

"

1.22 4.07 0.99 0.90

0.99

0.51

24 0.15

0.87

18.1

0.86

I0.8

1.64

0.79

21.3

0.80

10.9

0.76

1.74 1.79

0,80

8.45

0.83

2.84

0.91 6.97

1.89

0.80

1.907 1.54

3.25

1.17 2.69

1.61

2,00

1.18 2.71

1.16 2 . 5 2

1 . 1 6 2.32

1 . 1 8 2.97

1 . 0 0 0.77

0.99

0.45

0.99

0.55

0.99

0.50

0.99

0.55

0 . 8 1 9.54

0.88

9.98

0.86 10.7

0.87

11.8

0.88

14.8

7.87

0.82

II.I

0.80

10.3

0.81

11.8

0.81

16.0

0.80

2.14

0,85

2.63

0,83

2.68

0.84

3,30

0,83

5.42

0 . 9 7 2.55

0.94

0.98

0.99

3.13

0.97 2.09

0.98

2.70

0.95 4.38

0.86

0.84

1.93

0.87 0.79

0.86

0.87

1.14

0.84

1.06

1 . 1 6 2.63

1.18

1.73

1.50 5 . 5 4

1 . 3 6 4.30

1 . 4 0 6.10

1 . 3 8 4.15

1.30 3.89

1.21 3.41

1.30 3.92

1.67

1.56 3.28

1.42 2.71

1 . 4 6 3.91

1.44 2.67

1 . 3 6 2.34

1.26 1.93

1 . 3 5 2.15

1.71

1.74 1 . 4 5

1.60 1.37

1 . 6 4 1.80

1.63

1.40

1.54 1.23

1.44 1.06

1 . 5 4 1.05

l . 79

1.27 2 . 3 5

1.18 1.39

1.18 2.74

1.17 1.32

1.12 1.19

1.06 1.39

1.13

1.88

1.26 1 . 0 5

1 . 1 6 0.70

1 . 1 6 0.94

1.17 0.65

1 . 1 2 0.78

1.07

1.28

1 . 1 4 1.14

2. O0 2.20

1.24 l . S l 1.00 2 . 8 0

1.16 2.78 0.94 4.89

1.14 1.96 0.91 3.48

1.17 2.94 0.95 5 . 1 6

1 . 1 3 3.32 0 . 9 2 5.48

1.11 5.50 0.92 7 . 6 3

1 . 1 6 4.03 0.95 5.84

2.33 2.45

0.82 4.36

0.77

7.89

0.75

5.60

0.78

8.31

0 . 7 6 8.83

0.75

0.93

0.92 4.93

0.87

4.13

0.94

5.14

0 . 9 1 5.18

0.93 6.14

0.94

0.87 1.33

0.90

0.90 0.44

0.90

0.39

0 . 9 1 0.36

0.91 0.37

0.89 0.66

1.04 3 . 4 5

1.07 4.37

1 . 0 9 6.47

1.06 3.62

1.08 5.04

1 . 0 7 4.67

1 . 0 6 3.83

0.98 0.98

1.00 0.99

1.03 1.01

1.00 0.99

1.01 1.00

1.01 1.00

1.00 0.99

1.61"

25 0.50

1.97

26 0.15

2.11

27

0,05

2.31 2.53 2.72 2.53

28

0.05

2.76 3.20

3.37

0.79 0.97

0.39

0.66 1.07

1.80 2.11

0.55 0.93

1.01 1.39

12.3

0.93 1,27

1.44

0.78 9.28 5.11

0.48 0.88

1.00 1 . 0 4

1.02 2.09

1 . 0 5 4.02

1.01 1 . 7 9

1 . 0 3 2.83

1 . 0 3 2.78

1.01 1.70

1.00 1.28

l.Ol

1.88

1 . 0 1 1.96

l.Ol

1.01

1.92

1 . 0 2 2.02

l.Ol

0.99 0 . 7 9

0.99

0.79

0 . 9 9 0.72

0 . 9 9 0.92

0.99 0.75

1.00 0.84

0 . 9 9 0.75 1 . 0 2 0.80

2.02

1.61

1.02 1 . 0 2

1.02 0.55

1 . 0 2 0.97

1 . 0 2 0.47

1 . 0 2 0.73

1.02 0.76

3.55

29 0.15

1.18

1.00

1.20

0.66

1.21

0.50

1.19

0.69

1.21

0.53

1.23

0.61

1.21

0.66

5.00

35

0.20

0.72

7.47

0.95

7.57

1,24

II.0

0.91

1.87

1.09

6,27

1.16

5.72

0,95

3.65

5.20 7. O0

30

0.15

0.98 1.16 0.99 0.66

1.02 0.98

1.19 0.31

0,97 0.96

1.14 0.26

1.02 0.98

1.26 0.32

1.01 1.17 0.98 0.29

1.06 1.23 1.01 0.28

1.04 1,17 1.01 G.41

* The data of r e f . [25] were m u l t i p l i e d by 1.50 before the f i t t i n g procedure and were assigned a noril;alizat i o n e r r o r -+0.50 instead of _+0.('7 stated i n the corresponding paper. For di']crepancies in the ncrmalization betwcen this e~perin~l I ~::J ~L~LII/ c i , ~ e r data see R.W.B1and et a1., Phjs. Letters 29~ (1969) 610.

175

E.J. Sacharidis, Backward K + p elastic scattering

1~_ b)

0.1

o.1

,

-0.5

°°~

-O-fi

;

-0,7

-0.8

,

-1.0

-0.5

~°

-0.6

-0-7

~

-0.8

I

0.1

I

J

GeV/<

I

~ , ,

• o,o1~

-0-9

2.53 GeWc iRef.[28] I

[

-0.9 I

-1.0 /.

0.01

t

,ef.{29]

~

~

d)

5.0

OeV/c

Ref.{3<]

=-0.99 (GeV/c) 2

Z/~/~

l

0.01 ,

.

..

.

.

\

I

0.001 -05

.

-0.6

'x

cos8

t -0.7

;" '

~

f -0.8

P -0.9

--

.0001 -1.0 -0.5

-0.6

/ / I -0.7

cosO ~ -0.8

. J -0.9

-1.0

Fig. 6a-d. Comparison between the predictions of all cases I-VH of the model at various incident laboratory momenta.

176

E . J . Sacharidis, Backward K + p elastic scattering

1.0

I

I

I

I

p = 1.54 GeVIc

I

I

I

p= 2.10 GeVlc

0.5

e Ref. [321 -0.5

[]

,,

[2z,] [31]

1.0

.--. _~ 0.5

1.73, 1,74

2.29

1.89

2.58

no

-0.5

1.0'

0.5

/ I

-0.5

-1'00-5

I -0.6

-0-7

-0.8

-0-9

"-05

•

t -0.6

I -0.7

I -0.8

I -0.9

-1.0

cose

Fig, 7. Polarization P versus cos 0 in b a c k w a r d elastic K + p scattering. T h e solid lines t h r o u g h the data represent the predictions of the m o d e l in all cases while the n u m b e r s indicate the corresponding incident K + laboratory m o m e n t a p.

large m o m e n t u m transfers approaching the first wrong signature value ~a= = --~ in the physical region of the s-channel. In particular the 5.0 GeV/c differential cross sections of ref. [34] which extend to large m o m e n t u m transfers in the backward region, indicate structure suggesting the presence of wrong signature dips of both A ° and A s trajectories. To account for this structure, an appropriate balance of contributions from the A ° and A~s trajectories is needed and is afforded by case IV. All other cases yield a high chi-square for these 5.0 GeV/c data.

E. J. Sacharidis, Backward K+p elastic scattering

177

(ii) The m o d e l yields zero p o l a r i z a t i o n in all cases which is consistent with existing d a t a as i n d i c a t e d b y the Z2/point in table 3 a n d the graphs o f fig. 7. It should be e m p h a s i z e d , however, t h a t the vanishing o f the p o l a r i z a t i o n is a general feature o f o u r m o d e l r a t h e r t h a n the result o f a specific case o f o u r residues Re(u) a n d R°(u). It stems f r o m o u r a s s u m p t i o n s o f linear trajectories in u a n d the m e r o m o r p h i c d e p e n d e n c e o f the residues on u (eq. (7)) which led to eq. (10) a n d hence to a real r a t i o B/A. S h o u l d p o l a r i z a t i o n other than zero be detected in the b a c k w a r d region the a b o v e a s s u m p t i o n s m u s t be modified. Table 3 Chi-square per point, X2/point, for various experiments on K+p polarization, p: incident momentum in the laboratory system [GeV/c]. Ref: Reference. The normalization parameters are irrelevant here since the model predicts zero polarization p

Ref

1.71 1.89 1.89

31 31 31 24 24 24 24 24 32 32 32 32 32 32 32 32 32 32 32 32

1.54 1.64

1.74 1.79 1.89 1.54

1.64 1.73 1.82 1.91 2.01 2.10 2.19 2.29 2.39 2.48 2.58

~2 Point 1.14 0.58 0.58 1.93 1.56

0.67 0.24 0.97 1.39

2.06 1.11 2.28 1.59 0.38 0.21 0.50 1.68 0.20 0.62 0.24

(iii) A t t a c h i n g a n exponential d e p e n d e n c e in u in the a b o v e cases I, I I I a n d VI to o b t a i n cases II, IV a n d VII led to a very small i m p r o v e m e n t o f the overall fit to the d a t a h a r d l y justifying the d o u b l i n g o f the n u m b e r o f p a r a m e t e r s o f the model. One exception, however, as m e n t i o n e d above, are the 5.0 G e V / c d a t a which are a d e q u a t e l y described only b y case IV. I n fig. 8 we indicate m o r e clearly the range o f cross sections we are covering by d e p i c t i n g m e a s u r e m e n t s at conveniently spaced incident m o m e n t a . I t is r e m a r k a b l e t h a t cross sections r a n g i n g over f o u r orders o f m a g n i t u d e are faithfully r e p r o d u c e d

178

E.J. Sacharidis, B a c k w a r d K +p elastic scattering

by this model with at most two adjustable parameters. This is more so since we arrived at the relevant amplitudes by a coherent derivation using the simplest possible assumptions such as reality and linearity of trajectories as derived from the SU(3)Chew-Frautschi plot, no cuts, meromorphic dependence of residues on u and evasion of kinematic singularities on grounds of physical continuity. Furthermore, to the best of our knowledge, this is the first comprehensive application of the Mandelstam-Sommerfeld-Watson transformation in the Reggeization of the K N amplitudes.

1 0.5

I 0.1 "G" 0,05

E 0.01

0.005

0.001 0.0005

0.0001 -0.5

-0.6

-0-7

-0.8

-0.9

-1

Fig. 8. Backward K + p elastic differential cross sections versus cos 0 at widely spaced K + laboratory momenta.

At this point it might be proper to compare our assumptions to similar ones postulated earlier by other authors. As early as 1962 Gribov [35] demonstrated, on account of analyticity of the A and B amplitudes at u = 0, that fermion Regge poles

E.J. Sacharidis, Backward K +p elastic scattering

179

should occur in complex conjugate pairs corresponding to states of the same total angular m o m e n t u m but opposite parity. Moreover the residues at a pair of such poles should be related by complex conjugation and both residues and trajectories should be real analytic functions of energy i.e. of 14/, = x/u in the present case. Thus if we designate by c~-+ and r + the trajectories and residues of the members of the parity doublet, Gribov's statement is:

~+ (4~) = ~- (-,Y~),

(31 a)

r + (x/~)

(31b)

=

- ~- (-

X/~).

Real analyticity for u < 0 would then imply: + (x/~)

=

[~- (~/~)]*,

r + (x/~)

=

- [r - ( + X/~)]*

(32a) = Jr+ ( - x/~)]*,

(32b)

where* denotes complex conjugation (for more details see ref. [36]). If the trajectory ~+(x/u) is linear in u as is strongly suggested by the ChewFrautschi plot, then e+ (x/u) = e - (x/u) and it is easy to prove using eq. (32b) together with eqs. (5), (6) and (9) that the polarization due to the exchange of a parity doublet is zero. Non-zero polarization for the exchange of one parity doublet would, therefore, indicate the presence of odd x/u terms in e+ (x/u). Our assumption of meromorphic dependence of the residues on u rather than on x/u, i.e. eq. (7) is consistent with eq. (32b) as long as we choose real residues, as we have already done. However, it should be emphasized that as long as eq. (7) is satisfied and the trajectories are even functions of x/u, eqs. (8)-(10) follow and hence the polarization is zero for the simultaneous exchange of any number of parity doublets. To produce polarization with residues meromorphic in u, therefore, one needs the presence of odd x/u terms in c~± (x/u) for some of the exchanged parity doublets or alternatively the presence of cuts. Although there are some indications for the existence of parity doublets particularly at higher masses, there is difficulty in classifying all well established fermion resonances into doublets due to large mass shifts in trajectories with the same signature but opposite parity [36]. This lack of confidence in the existence of parity doublets is reflected in the attempts to suppress the unobserved twins by introducing a fixed cut in the angular m o m e n t u m plane forcing thus the undesirable member into the unphysical sheet [37] or by introducing highly asymmetric residues which vanish when - x / u equals the mass of the undesirable member [38, 39]. Clearly our approach is at variance with the above schemes since it gives equivalent status to both members of the parity doublet, but there seems no reason to prevent such a case for the negative values of u considered here. Furthermore our method of regularization of the reduced rotation function

180

E. J. Sacharidis, Backward K+p elastic scattering

~,_u(z) (eq. (15)) and thereby of the invariant amplitudes in the very backward region, although mathematically not rigorous carries none the less, a good deal of common sense and practical weight. For if a representation of the amplitude is good enough for z > z o say, where presumably the singular terms have died off, then a linear extrapolation to z < z 0 should be a reasonable approximation as long as it extends to a small range of the physical region as in our case. The singular behaviour of our representation of the amplitudes, as has already been mentioned above, is due to the usual partial wave expansion which involves rotation functions which are singular at the points z = + 1. At these points the background term is also singular since it involves the same rotation functions and is therefore expected to cancel the corresponding singular behaviour of the Regge amplitude so that the total amplitude is regular near the points z = ± 1. In fact it is known [13, 40] that a SommerfeldWatson transform of the amplitudes exists with regular behaviour at the points z = ± 1. A m o n g the ways proposed to regularize the conventional Regge-pole expressions at z = ± 1, that of Drechsler [13, 40] consists in subtracting effectively the singular part of the rotation functions and showing that for the remaining part a Sommerfeld-Watson transform exists. This method, from the practical point of view, does not differ from ours since the singular terms die off effectively at the point Zo > 1 from which we perform the extrapolation in the backward region. Similarly the daughter-like solution [41] to the regularization at z = ± 1 which establishes the usual asymptotic behaviour ~ s " in the backward direction makes no definite statements for the non-leading terms which are very important for nonasymptotic energies and large m o m e n t u m transfers. To obtain non-leading terms one has to assume a definite behaviour of the daughter trajectories and their singular residues which apart from complicating the Regge picture would inevitably carry a certain degree of arbitrariness. We feel that the use of the full rotation function of the second kind (eq. (27)) with its linear extrapolation (eq. (15)) in the very backward direction is an interesting practical approach in linking asymptotic and nonasymptotic regions. However, it is clear that this approach also involves an assumption about the daughter poles; namely that they are unimportant away from the singular points z = _ 1. Adequate fits to backward K + p scattering at quite low energies and large m o m e n t u m transfers have been obtained by Barger [39] using only the leading term, s ~, and asymmetric residues to suppress the parity twins. However, apart from the fact that a few adjustable parameters (at least four) are needed to fit the data, the use of a residue function consisting of a linear superposition of two exponentials in u and the presence of only leading terms in s obscure the Regge picture and give the impression of a mere successful parametrization of the data. We believe that we have captured both energy and m o m e n t u m transfer dependence in one single expression: the full rotation function of the second kind with its linear extrapolation, and this allows us to describe adequately existing data on K + p elastic scattering and polarization with at most two adjustable parameters.

E. J. Sacharidis, Backward K +p elastic scattering

181

It would be interesting to try to fit backward data on K+n~nK

+,

K+n~pK ° ,

K°p~pK °,

since the amplitudes for these reactions can be derived from the same u-channel amplitudes, eqs. (5)-(9), through s-u crossing. Already backward K ° p ~ P K s ° scattering measurements [42] indicate that the cross sections for this process in which only 2; trajectories can be exchanged, are much smaller than corresponding K + p ~ p K + cross sections in which, as we have seen, both A and I7 trajectories can be exchanged. Our neglect of the 2; contributions in the present work is thus justified. In concluding we may add that more measurements and better precision on backward K + p elastic scattering and polarization at medium and higher energies are required to decide whether this model has more physical content than an accidentally successful economic parametrization of the scattering amplitudes consistent with the accuracy of present measurements. This work was initiated while the author was collaborating with a University College London group in an experiment to measure K-+p elastic scattering at intermediate energies [22]. The encouragement of and discussions with my colleagues are gratefully acknowledged and in particular the forbearance of Professor F.F. Heymann and Drs. B.G. Duff, R.C. Hanna and D.C. Imrie is appreciated. I have profited from discussions with Drs. R.J.N. Phillips and R.P. Worden whose constructive criticism is greatly appreciated. My thanks are due to Dr. J. Barlow for his help in using the R H E L version of the " M I N U I T " minimizing routines. The excellent service at the R H E L computer has been indispensable and last but not least I am indebted to Drs. G . H . Stafford and G. Manning for an extension of my stay at R H E L which made possible the conclusion of this paper.

Appendix Due to its practical importance, we give here details of the way in which the normalization parameters, a, were treated. Let a distribution of n measured points Yi-+ ~ be fitted by a curve Yi and let the normalization error be + ~ so that the true points are a y i .-I- a~rl,

where a, the normalization parameter, is assumed to follow a Gaussian distribution about unity with standard deviation a. Now the probability for a particular set of measurements y~_ a~, i = 1, n would be: P

=

Co F ~

1/ 1[ Y i - a Y l

exp(--:(--} L--i=1 \ \ aai

\z\-]

/

/a--l\2\

)|exp(-½(--} J, //A \ \ a / /

E. J. Sacharidis, Backward K+p elastic scattering

182

w h e r e Co is a constant. C l e a r l y , i n o r d e r t o m a x i m i z e t h e p r o b a b i l i t y P we n e e d t o minimize the chi-square*

i= ~ \ a0-i / _-- ~ (Y/a,)_-Y(12\ 0-,

] +\

~((1/a)-l)a ' ' - ~ ((Y~/a-)-Y')2+((1/a: - 1 )

i=1

i=l

where in the last step we have assumed Substituting b

1

A=

a/a "~ a

~,,

2'

O"i

s i n c e a is i n t h e v i c i n i t y o f u n i t y .

I+~y~a

a

i = 1 0-2

1 ~ B = ~-~+ i= 1

1

yiY,.

y2 --,

C = ~-~+

0-2

i=l

0 -2

we obtain

Z2 = Ab2-2Bb+C, S i n c e A > 0 we see t h a t

Zz

has a minimum

at

B

b = A

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183

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