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The generalized Veneziano model for charged pion photoproduction
~
Nuc[ear Physics B29 (1971) 189-203. North-Hol|and Pubhshlng Company
THE GENERALIZED VENEZIANO MODEL FOR CHARGED PION P H O T O P R O D U C T I O N E. REYA
Daresbury Nuclear Pfiyszes Laboratory
Received 14 December 1970
Abstract: Charged pion photoproduetion is investigated in the framework of Venezlano B 5 amplitudes using Sugawara's "spurion" technique in order to introduce fixed J-plane pores. The Re~ge exchanges are taken to be evasive. The agreement w~th expemment for the differential cross section, the ?r-/~ + ratio and the polarisatlon asymmetries for photon energies between 3.4 and 18 GeV ~s quite reasonable.
1. INTRODUCTION In explaining the e x p e r i m e n t a l data on c h a r g e d pion photoproduction t h e r e a r e e s s e n t i a l l y t h r e e different m e c h a n i s m s p r e s e n t l y known: (i) a p a i r of c o n s p i r i n g pion Regge t r a j e c t o r i e s [1], (ii) a s e l f - c o n s p i r i n g Regge cut or a b s o r p t i o n c o r r e c t i o n s to evasive Regge poles [2], (iii) right signature fixed poles in the angular m o m e n t u m plane [3, 4]. All t h r e e models give a s a t i s f a c t o r y d e s c r i p t i o n of experiment. In this p a p e r we study c h a r g e d pion photoproduction in the f r a m e w o r k of the g e n e r a l i s e d Veneziano model, n a m e l y the five point function B5, and using S u g a w a r a ' s " s p u r i o n " technique (see below) in o r d e r to introduce fixed J - p l a n e poles. T h e r e f o r e we a r e using the third m e c h a n i s m mentioned above, i.e. explaining the s h a r p f o r w a r d peak in da/dt by a right s i g n a t u r e fixed pole w h e r e a s the t - c h a n n e l Regge exchanges a r e taken to be evasive. That such fixed p o l e s (and also K r o n e c k e r deltas [5]) can o c c u r in p r o c e s s e s initiated by photons, i.e. in amplitudes satisfying linear unitarity r e l a t i o n s to lowest o r d e r in e, has been shown by Bronzan et al. [6] and Singh [7]. They p r o v e d this for Compton s c a t t e r i n g by c o m p a r i n g the D a s h e n G e l l - M a n n - F u b i n i s u m rule with the F r o i s s a r t - G r i b o v continuation into the c o m p l e x J - p l a n e . Using a simple field t h e o r e t i c a l model, Dosch [8] has shown that a fixed power behaviour, and hence a fixed J - p l a n e pole, may o c c u r also in m e s o n photoproduction whether the p r o d u c e d meson is " e l e m e n t a r y " or " c o m p o s i t e " . Some speculations upon the p r e s e n c e * of fixed * If, however, one performs an explicit separation of the elementary particle pole terms, superconvergence relations for certain invariant amplitudes may occur and therefore fixed poles in the J-plane are not required.
190
E.Reya, Pion photoproduction
poles in photoproduction amplitudes have also been made by Ball and J a c o b [9]. Dashen and Lee [10] have given a r g u m e n t s against right signature fixed J - p l a n e poles in b a c k w a r d uo photoproduction c o m p a r i n g with e x p e r i m e n t 1 the s-dependence of da/du of a fixed J = y and a moving Regge pole r e s p e c tively. However, Dombey [11] pointed out that these a r g u m e n t s a r e not c o r r e c t , since the Coulomb t e r m s give a contribution p r o p o r t i o n a l to sin 20 and so vanish in the backward direction. To apply these a r g u m e n t s also to the f o r w a r d direction is much m o r e difficult, since in this c a s e t h e r e is no such difference between the energy dependence of a fixed J = 0 pole and of the a p p r o p r i a t e Regge exchange. Recently, on the g e n e r a l b a s i s of the F r o i s s a r t - G r i b o v projection, F i n k l e r [12] has shown that unitarity allows fixed poles at c e r t a i n nonsense points of either right or wrong signature. Moreover, such fixed poles which survive at wrong signature nonsense points need not be r e l a t e d to the GribovP o m e r a n c h u k singularities, nor to the third double s p e c t r a l function. Obviously, t h e s e r e s u l t s apply also to amplitudes d e s c r i b i n g p r o c e s s e s initiated by photons. In this c a s e the fixed poles still r e m a i n even when we include exact unitarity. This c l e a r l y r e f u t e s the c o m m o n belief (see above) that fixed poles can exist only as long as linear unitarity r e l a t i o n s a r e considered, w h e r e a s exact bilinear unitarity would forbid such a fixed power behaviour. F o r a f u r t h e r d i s c u s s i o n on this point and also on the existence of such fixed s i n g u l a r i t i e s strongly favouring the evasive m e c h a n i s m for Regge exchanges we r e f e r the r e a d e r to ref. [12]. F r o m a m o r e phenomenological point of view the fixed poles p r o p o s e d by Amati et al. [3] * t o g e t h e r with the usual Regge exchanges yield a v e r y good explanation of the e x p e r i m e n t a l data. The s a m e quite convincing r e s u l t s , even for the u - / ~ - ratio, a r e obtained by Bender et al. [4], using a right signature fixed Khuri pole at ~ = -1 yielding an a s y m p t o t i c 1/v behaviour. F u r t h e r m o r e t h e r e is c e r t a i n l y no contradiction to fixed power behaviour in the invariant amplitudes since s2(da/d t) s e e m s to be n e a r l y independent of s and no shrinkage can be found. To the best of our knowledge v e r y little work has been done up to now u s ing the Veneziano model [13] to d e s c r i b e pion photoproduction. In p a r t i c u l a r no extensive fits to the e x p e r i m e n t a l data have been made. Within the f r a m e work of the original f o u r - p a r t i c l e Veneziano amplitude different models for the invariant amplitudes and helicity amplitudes have been suggested, using a p a r i t y - d o u b l e t c o n s p i r a c y of the pion [14, 15]. In these p a p e r s only the basic f e a t u r e s of such models in the v e r y f o r w a r d direction a r e discussed. F o r instance the r a p i d v a r i a t i o n of the p a r i t y c o n s e r v i n g helicity amplitudes 2 in the i n t e r v a l - m 2 --< t --< m a r i s e s quite naturally within the f r a m e w o r k of the Veneziano m o d e l A r a p i d variation of this kind is known to be n e c e s s a r y in o r d e r to fit both the magnitude of the differential c r o s s section and the p o l a r i s e d photon a s y m m e t r i e s . F u r t h e r m o r e it has been e m p h a s i z e d by * It should be mentioned that the origin of such a smooth background (represented by those right signature fixed poles at J = 0, -1) can also be due to Regge cuts or absorption corrections to an evasive Regge pole.
E.Reya, Pion photoproductzon
191
Ader et al. [16] that the e v a s i v e m e c h a n i s m is a l s o c o n s i s t e n t with the V e n eziano model, n a m e l y that the a m p l i t u d e s a r e c o m p a t i b l e with the B o r n t e r m c o n t r i b u t i o n s at low energy. R e c e n t l y a v e r y i n t e r e s t i n g a t t e m p t to d e s c r i b e the photoproduction data h a s been m a d e by Bender, Rothe, Dosch and MUller [17]. Using Veneziano B 5 a m p l i t u d e s , p r o p e r l y modified to get fixed poles, they obtained r e m a r k a b l e fits to the d i f f e r e n t i a l c r o s s s e c t i o n at photon e n e r g i e s f r o m 8 to 16 GeV in a t - r a n g e up to -1 (GeV/c) 2. It is worthwhile e m p h a s i z i n g that this model d o e s not contain any f r e e p a r a m e t e r , b e c a u s e the Veneziano n o r m a l i z a t i o n c o n s t a n t s w e r e fixed by the e l e c t r i c B o r n - t e r m s , since only N~ e x c h a n g e s a r e taken into account. F o r the s a k e of c l a r i t y we b r i e f l y s u m m a r i z e in sect. 2 all the r e l e v a n t notations and f o r m u l a e we use. In sect. 3 we d e s c r i b e the model and give the e x p r e s s i o n s f o r the different i n v a r i a n t amplitudes. Finally sect. 4 c o n tains the fits to the e x p e r i m e n t a l data and s o m e d i s c u s s i o n of t h e m , and our c o n c l u s i o n s a r e s u m m a r i z e d in sect. 5.
2. KINEMATICS AND BASIC FORMULAE A c c o r d i n g to CGLN [18] the m o s t g e n e r a l f o r m of the p h o t o - m e s o n T m a t r i x e l e m e n t m a y be written a s 4 T = ~ Ai(s , t)Mi, i=1
(1)
w h e r e the g a u g e - i n v a r i a n t spin m a t r i c e s take the f o r m M1 = ~'5 ¢]¢ , M 2 = 2~,5(P.k q. E - P . E q . k ) , M 3 = ~,5(4 q . k -]~ q.E) , M 4 = 275(¢P.k
-~P.
~ - mN~(~) ,
(2)
E and k a r e the photon p o l a r i z a t i o n v e c t o r and i t s f o u r - m o m e n t u m , r e s p e c tively, q i s the f o u r - m o m e n t u m of the outgoing pion with m a s s mTr. The f o u r - m o m e n t a of the initial and final nucleons with m a s s m N a r e denoted by P l a n d P 2 , and the total m o m e n t u m P = ½(Pl + P2)- T h e i n v a r i a n t s s and t a r e given by s = (k + p l ) 2 , t = (q - k) 2. In i s o s p i n space, the i n v a r i a n t a m p l i t u d e s A i ( s , t) h a v e the following d e c o m p o s i t i o n : A i = 5 ~3 A i(+) + : [1T O l ' T3 ]A j (-) i + ~"~ A!Z°) '
(3)
being the i s o s p i n index of the outgoing pion and A (i° ) ("It " = 1 , G = +1) i n v o l v e s i s o s c a l a r photons w h e r e a s both Ai+)(/t = ' 0, G = - 1 ) a n d A i - ) ( I t = " 1, G = -1) involve the i s o v e c t o r photon. So one h a s f o r c h a r g e d pion p r o d u c t i o n
E. Reya, Pion photoproduction
192
Ai(Tp-~ ~r+n) = ,/'~(AI °) + A~-)) , Ai(Tn --" 7r-p) = "/'g(al °) - AI-)) .
(4)
F u r t h e r m o r e the parity-conserving t-channel helicity amplitudes [ 1 9 ] f ~ (where X = X~ - XTr, ~ = XN - ~N) can be e x p r e s s e d in t e r m s of the invariant amplitudesA'i (s, t), namely
t_m 2 f;0 : ~ -
~r
(A1
_
2tuNA4) ,
t-m 2~/ t-4rn2 ½
_
~/~ \---~----) (A1 + tA2)'
fl-0-
t_m 2 f l l = _ - + ~r ('t-1(2reNAl _ tA4) , t_ m 2
(5)
f-l l = - 7 u (t - 4m2)½A 3 With this normalization the differential c r o s s section is given by d~
1 _ _
I
dt - 647r (s_m2)2
{(cos2ot_ 1)(if1+012+ [f~o12)
+ (c°s20t +
1)(1~1112 + tfll 12) + 4c°sOtRe(f~1111[)},
(6)
with 2
2st+t2-t(24+mr) cos
0 t
-
( t - m 2) ~/ t( t-4m 2)
The polarized photon a s y m m e t r y ~ is defined by d (Tl/d t - d ~ " / d t
= daZ/dt+do~,/dt,
(7)
where d~±/dt is the differential c r o s s section for photons polarized perpendicular to the production plane, having contributions only f r o m the natural parity amplitudes ~ 0 and f ~ l ' while the unnatural parity amplitudes f i 0 and f l l contribute only to the differential c r o s s section for parallel polarized photons, dd'/dt (ref. [20]).
E.Reya, Pion photoproduction
193
Finally we i n t r o d u c e the a s y m m e t r y of the c r o s s section f o r Vp -. lr+n f r o m a p o l a r i z e d t a r g e t with incident u n p o l a r i z e d photons [21]
A(t) =
- 2 , / ~ I m [A~ A4+A~(AI+tA2)]
]All 2+1 tl IA4] 2+1 t I IA3)2+IAI+tA212
(8)
w h e r e the t a r g e t is p o l a r i z e d e i t h e r p a r a l l e l or a n t i p a r a l l e l to the n o r m a l to the production p l a n e defined by k × q.
3. THE MODEL Since we a r e going to study photoproduction u s i n g the e v a s i v e m e c h a n i s m we need, in o r d e r to explain the s h a r p f o r w a r d p e a k in the d i f f e r e n t i a l c r o s s section, e i t h e r fixed pole o r cut c o n t r i b u t i o n s to the i n v a r i a n t a m p l i t u d e s A~s,t). As we have a l r e a d y e m p h a s i z e d in sect. 1, we confine o u r s e l v e s to the f i r s t p o s s i b i l i t y , n a m e l y fixed poles. A p o s s i b l e way of introducing such fixed p o w e r b e h a v i o u r in the Veneziano a m p l i t u d e s was s u g g e s t e d by Sugaw a r n [221 He p r o p o s e d a method of f o r m u l a t i n g the o f f - m a s s - s h e l l a m p l i tudes ( e s p e c i a l l y f o r Compton s c a t t e r i n g ) with the help of a " s p u r i o n " having v a c u u m quantum n u m b e r s t. G e n e r a l l y s p e a k i n g this m e a n s that one c o u p l e s a " s p u r i o n " with f o u r - m o m e n t u m P 5 , say, to a c u r r e n t , and then goes to the l i m i t p 5 -~ 0 in the c o r r e s p o n d i n g (N + 1) point function. Ohba [23] showed that it is indeed p o s s i b l e to d e r i v e also N - b o d y o f f - m a s s - s h e l l a m p l i t u d e s s e l f - c o n s i s t e n t l y f r o m ( N + 1) body a m p l i t u d e s in the g e n e r a l i z e d Veneziano model. T h e r e , also, the a m b i g u i t y in S u g a w a r a ' s original method of e l i m i nating unwanted s i n g u l a r i t i e s such a s double a n d / o r t r i p l e p o l e s (which a r e a c o n s e q u e n c e of the l i m i t p 5 --. 0) was r e m o v e d and a unique way of e l i m i nating such p o l e s w a s given. In o u r c a s e we s t a r t f r o m the f i v e - p o i n t a m p l i t u d e and c o n s i d e r the l i m i t P5 --" 0, a s shown in fig. 1. In o r d e r to get a c a n c e l l a t i o n of the above m e n tioned u n p h y s i c a l double p o l e s we have to put all v a r i a b l e s c o r r e s p o n d i n g to c h a n n e l s dual to channel (1, 5) equal to a constant (>i 1), which can e a s i l y be s e e n f r o m the definition of the f i v e - p o i n t function. T h i s y i e l d s S
1
4
2
Pff'O
2
5
5
Fig. 1. J' Obviously this "spurion" has nothing to do with that commonly used, e.g. in ~ - and K-decays, where it carries finite isospin in order to balance isospin conservation.
194
E . R e y a , P z o n photop~,oduction
lim B5(-c~ 12' -~23' -a34' -c~45' -~ 51)
pS-~0
: B5(-~23(s23) , -~34(s34),a, -~51(P~),b) ,
(9)
with s i j = ( P i + P j ) 2 and a, b >/ 1. According to ref. [23] we take a = b = 1 getting an exact (i. e. no f u r t h e r non-singular contributions of the appropriate channels) cancellation of the unwanted double poles. F r o m now on we keep this convention whenever possible. Since the photon couples essentially to vector mesons we take for ~51 the p - t r a j e c t o r y . Consequently a typical exp r e s s i o n for an s - t o f f - m a s s - s h e l l amplitude, say, using the kinematic variables defined in sect. 2 is given by
(10)
Bs(-ot(s), -~(t), 1, 1 - C~p(k2), 15.
Such amplitudes have not only the c o r r e c t Regge behaviour but also the r e quired fixed power behaviour in s and t. F u r t h e r m o r e , if k 2 = m 2 , for instance, we encounter the p-pole and the residue is exactly the Veneziano B 4 amplitude for the purely hadronic p-production. Ademollo and Del Giudice [24 ] studied f u r t h e r mathematical p r o p e r t i e s of those amplitudes and also applied such ideas to electroproduction mad Compton scattering. Before we write down the final expressions for the invariant CGLN a m plitudes, we briefly s u m m a r i z e some of the basic r e q u i r e m e n t s they have to
(+,o) , A^(+,o5,
fulfil. F r o m the s - u c r o s s i n g properties, it follows that A .
A~") and A~+' o) must be even while A~-) , A~-) , A~+' o ) a n d A ~ -) areZ odd functions under the substitution s - . u . F u r t h e r m o r e the nucleon and pion pole contributions to the A i ' s should have the c o r r e c t r e s i d u e s (electric and magnetic Born t e r m s ) , namely l i m (s
2 A(.~, -raN) ~ o) = r(.% ~ o)
2
s-~m N
,
tim ( t - 2 )
A(?, z o) : R(~, o)
t-.m 2 7r
where
(2, o)
r1
1
= --seg,
r~. o) '
-
eg
2
'
t-m
e v ~o)__q )__½g~(,p+Un ),
R~-) 2 e g =
2' s-m N
R~, o)__R~+.o)_ R~~ o5= R~ o)=0 "
(11)
E.Reya, Pzon photoproduct~on
195
H e r e ~ and Pn a r e the anomalous magnetic m o m e n t s of the proton and n e u tron r e h p e c t i v e l y and g is the ~r - N coupling constant *. Other well known p r o p e r t i e s of the i n v a r i a n t amplitudes, such as the asymptotic behaviour e t c . , can be found for instance in ref. [15] or [18]. We know [16] that the e v a s i v e m e c h a n i s m only r e q u i r e s that natural p a r i t y contributions go to z e r o f a s t e r than s a~r(°)-I as s ~ m The a m p l i t u d e A (-)" 1 r e c e i v e s contributions only f r o m n a t u r a l p a r i t y t-channel exchanges. All such m e s o n t r a j e c t o r i e s known at p r e s e n t have an i n t e r c e p t g r e a t e r than a~r(o). Now a c c o r d i n g to eq. (11) we see that the nucleon contribution does not vanish at t = 0. T h e r e f o r e we take f o r this t-channel exchange the f i r s t daughter of the exchange d e g e n e r a t e (p - f - co - A2) t r a j e c t o r y in o r d e r to be c o n s i s t e n t with the evasive mechanism. Within the f r a m e w o r k of the Veneziano model one is led quite naturally to a s s u m e this t r a j e c t o r y in such a c a s e and its e x i s t e n c e is c e r t a i n l y not c o n t r a d i c t e d by e x p e r i m e n t . T h e r e f o r e the generation of the e l e c t r i c Born t e r m s in the v e r y f o r w a r d d i r e c t i o n using this method s e e m s to us to be l e s s dubious than a s s u m i n g a natural parity__conspirator of the pion. Thus using for the N a contributions__ in A~-)" and A~-) the s a m e ansatz as in ref. [17], we obtain for the A~/-) amplitudes the following e x p r e s s i o n s : l
-a
T
A -)(s, t) = - ~ e g B(~-~,~o ) ×{ B 5(~1 - ~Na(S), 1 - aA~(t), 1, ao, 1) - B5(~1 - aNa(U), 1 - aA~(t), 1, ao, 1)} +/31 t{B5( ~ -aNT(S), 1 - a A 2 ( t ) , 1, ao, 1) -B5(~ -aNT(U) , 1 - a A 2 ( t ) , 1, ao, 1)}, (12a)
A
~-)(s, t) = e g -a' t_m2 B(c, ao) ~Y
× {B 5(Y1 - aNa(S), 1 - an(t), c, ao, d ) - B5(½ - aNa(U), 1 - air(t), c, ao, d) } - i l l {B5(~" aNT(S), 1 - aA2(t), 1, ao, 1) - BS( ~ - aNT(U), 1 - aA2(t), 1, ao, 1)}, (12b)
. e2/4ff = 1 -1~ ,
~2/4ff = 14.6, gp'
=
1.78, ~tn = - 1 . 9 1 .
E, Reya, Pzon photoproduct ion
196
A(-)(s t) = $eg 3 ’ x @5(+ - c.uN,(s), 1 - @A&t ), 1, @o, 1) + B5(+ - @N,(u), +& (B5(# - aN$s),
1 - @A#),
1, Qo, 1) + B5(+! -aN$U),
1 -a Al@ ), 1, ffo, I)> 1 -a A#),
1, ~I-J,I)), WC)
A4C-J (s, t) = aeg
&N($3- %I)Be)
+@3 {B5(+ -@NY(S), 1 - @Aa(
1, oo, 1) - B5(+-aNY(4,
1 -@Aa(
l,oo,
1))) (12d)
2 withu=2m +m2-st and B( X, y ) being the Euler B-function; cy’ is the N T common slope of all Regge trajectories and a0 = 1 - ao(k2 = 0). The gauge invariant pion pole structure (eq. (11)) requires c and d in eq. (12b) to satisfy the following relation l+c=
d.
(13)
Taking as usual c = 1, eq. (13) fixes d. Concerning the isoscalar amplitudes Ai(0) it is not possible required variance,
pole structure
consistently
using only B5’s.
A(z+$‘) also contains a factor
l/( t - mz),
must not exhibit a pole at t = m,“. Therefore
to fulfil the
Because of gauge in-
but at the same time it
the function multiplied by this
singular term must go to zero as t -+ rnfo Solving for the zeros of a fivepoint function (still not a solved problem in one variable), one obtains as solutions different functions of the remaining four arguments. Therefore one has different constraint equations for different energies s, say, which turns out to be inconsistent in our case. To get the right pole structure in A1(0) and A’z”) we use for the nucleon contribution function which exhibits the required
an s - u Veneziano four-point
zero at t = rnf . This is certainly
not
unreasonable as a first approximation, since the essential structure of the various measured quantities in the forward direction is given by t-channel exchanges. Furthermore, the requirements of crossing symmetry, signature and the appropriate asymptotic behaviour of the amplitudes A3(O) and A4(0) do not allow the nucleon to contribute, as already pointed out in ref.
E. Reya, P~on photoproductzon
197
[15]. The absence of this pole in the Veneziano f o r m u l a is c o n s i s t e n t with the a p p r o x i m a t e equality of ~p and - g n . Thus we a r e finally led to the f o l lowing e x p r e s s i o n s f o r the i s o s c a l a r amplitudes:
A(°)(S'l
t) = -yeg(-o~ 1 , ) B(~~ - a Sc~(s), 5 - aNc~(U))
+f14 t{B5(~ - aNy(S), 1 - a p ( t ) , 1, ao, 1)+ B5( ~ - aNy(U), 1 - ap(t), 1, ao, 1)}, (14a)
eg , d~°)(s, t) : t - - ~ ( - a
~,~ ) ~ ( y - aNa(S), ½- a j a ( u ) )
7f -f14 {Bs(J - aNy(S), 1 - ap(t), 1, ao, 1) +B5( ~ - aNy(U), 1 - ap(t), 1, ao, 1)} +f15 {Bs(~ - aNy(S), 1 - aB(t), 1, ao, 1) +B5( ~ - aNT(U), 1 - aB(t), 1, ao, 1)} (14b)
A~°)(s, t) = O,
(14c)
A~°)(s, t) = f16 {B5(~ - a N y ( S ) , 1 -ap(t), 1, ao, 1)
+Bs( -aNy(U), 1
1, ao, 1)}.
(14d)
Thus we a r e left with six f r e e n o r m a l i z a t i o n constants fli. However, it t u r n s out that the contributions of the low-lying B and A 1 m e s o n s a r e p r a c t i c a l l y negligible, e s p e c i a l l y at higher e n e r g i e s , and t h e r e f o r e only the four r e maining p a r a m e t e r s a r e important. In o r d e r to avoid f u r t h e r f r e e p a r a m e t e r s we also neglected possible A 5 contributions. This s e e m s to be r e a s o n able b e c a u s e it is well known that the A 5 c o n t r i b u t e s p r e f e r a b l y in the b a c k ward direction, and t h e r e only about 25% w h e r e the main contribution is due to the Ny. The t r a j e c t o r i e s a r e taken a s aNa(S ) = -0.27+a's,
aNy(S) = - 0 . 8 + a ' s ,
ap
(t) = a A 2 ( t ) = 0 . 4 8 + a ' t ,
a~
it) = - 0 . 0 1 8 + a ' t ,
aA~(t) = - 0 . 5 2 + a ' t ,
aAl(t ) = -0.1+a't,
aB(t ) = -0.3+a't ,
with ot' = 0.89 (GeV/c)-2. To see the basic f e a t u r e s of our amplitudes, we c o n s i d e r the h i g h - e n e r g y beh~viour of the B 5 functions. E x p r e s s i n g the B 5 function in t e r m s of a gene r a l i z e d h y p e r g e o m e t r i c function [25] one easily finds for the h i g h - e n e r g y limit, keeping only f i r s t o r d e r s in s,
198
E.Reya, P~on photoproductzon
B5(~ - ~ ( s ) , 1 - ~ ( t ) , a, % , b) ~ {(-c~ 's)(~(t) -1 F(1 - ~ ( t ) ) B ( ~ ( t ) + b - 1, ~o)[1 - ( - ~ ' s ) - l ( 1 - ~(t)) ] + ( - ~ , s ) -b r(b) B(1 - b - a ( t ) , a)[1 - ( - ~ ' s ) - l b ] } { 1 +0(1) + . . . } .
(15)
U s i n g t h i s e x p r e s s i o n the a s y m p t o t i c a l l y l e a d i n g t e r m of A~-)- g o e s like i / s , i.e. it r e p r e s e n t s the c o n t r i b u t i o n of a r i g h t s i g n a t u r e fixed pole at the n o n s e n s e v a l u e J = 0. T h u s dcr/dt at t = 0 is a p p r o x i m a t e l y given by the n u c l e o n p o l e t e r m only, in a g r e e m e n t with e x p e r i m e n t . The r e m a i n i n g two a n t i s y m m e t r i c a m p l i t u d e s A~-)- and A~-)" r e c e i v e a l s o c o n t r i b u t i o n s f r o m a J = 0 f i x e d pole, but h e r e the l e a d i n g t e r m s h a v e the u s u a l R e g g e b e h a v i o u r . Only in the f i r s t t e r m of eq. (12b) d o e s no 1 / s fixed p o l e a p p e a r ( b e c a u s e of c o n dition (13)), which is due to the R e g g e i z e d pion c o n t a i n e d in A(2-)--. B e c a u s e of the c r o s s i n g b e h a v i o u r of the C G L N a m p l i t u d e s no r i g h t s i g n a t u r e fixed J ~O~
p o l e at J = 0 c a n a p p e a r in A -) and A i
(i = 1, 2, 4). I n s e r t i n g eq. (15) into
the a p p r o p r i a t e e x p r e s s i o n f o r t h e s e a m p l i t u d e s s h o w s t h a t the c o n t r i b u t i o n s of such a pole do i n d e e d c a n c e l , and a p a r t f r o m the asymptoticall]¢ l e a d i n g R e g g e t e r m the f i r s t c o n t r i b u t i n g fixed p o l e h a s J = -1, i.e. a 1 / s z f i x e d p o w e r b e h a v i o u r *. F i n a l l y we s h o u l d m e n t i o n that in the c a l c u l a h o n s we u s e d the e x a c t e x p r e s s i o n s f o r the A(.$ - ' o) a m p l i t u d e s ,
a s given in eqs. (12) and (14), only in
t h o s e c a s e s w h e r e the a c c u r a c y of the e x p a n s i o n (157 w a s not sufficient. T h i s w a s done in o r d e r to s i m p l i f y the n u m e r i c a l c a l c u l a t i o n s .
4. R E S U L T S AND DISCUSSION In t h i s s e c t i o n we p r e s e n t the r e s u l t s f o r the v a r i o u s q u a n t i t i e s d e f i n e d in sect. 2. T h e n o r m a l i z a t i o n c o n s t a n t s / 3 i w e r e fixed so a s to get the b e s t fit to the d i f f e r e n t i a l c r o s s s e c t i o n f o r 7r+ p r o d u c t i o n f o r i n c i d e n t photon e n e r g i e s E 7 = 8 and 16 GeV. The r e s t being p r e d i c t e d by the model. T h e lr + d i f f e r e n t i a l c r o s s s e c t i o n and the 7r-/~ + p r o d u c t i o n r a t i o R at e n e r g i e s b e tween E y = 3.4 and 18 GeV a r e shown in figs. 2, 3 and 4. Although the s h a p e and m a g n i t u d e of d ~ / d t is r e p r o d u c e d v e r y well, the e n e r g y d e p e n d e n c e s e e m s a little bit too s t r o n g . This, and a l s o the f a c t that we a r e not able to r e p r o d u c e the r i g h t s t r u c t u r e of R at E v = 16 GeV, n a m e l y a dip at ~ ~ 0.6 * The same applies to yo photoproductlon. Furthermore we obtain in this case (using eq. (15)) that for instance the asymptotically leading terms of the dominant A i(+) amplitudes (cO-exchange) are multiplied by a factor (1-exp (i ~'~ cO(t)) which also explains the dip in d(7/dt at t ~ -0.55 (GeV/c) 2 very well. Such a h t is under investigation and will be pubhshed elsewhere.
E . R e y a , Pion photoproduction
10°I! ! !
101
t
J
1¢
199
~ 4
i
GeV
100
,o ,o
t
o
0.5
/~
(aev/c)
0
1.o
F i g . 2. D i f f e r e n t i a l c r o s s s e c t i o n f o r y p ~ ?r+n b e t w e e n 3.4 and 8 GeV. The data a r e f r o m r e f . [26].
"~.le GeV
0.5
(oev/c~
1.0
F i g . 3. D i f f e r e n t i a l c r o s s s e c t i o n f o r ~p ~ ~'+n b e t w e e n 11 and 18 GeV. The data a r e f r o m r e f . [26].
1.0 R
0.5
o
o'.5
1:o GeV/c
F i g . 4, The r a t i o R = (d(r/dt)(yn--* ~r-p)/(dcr/dt)(yp -~ 7r+n) f o r i n c i d e n t photon e n e r g i e s of 3.4 GeV (solid curve) and 16 GeV (dashed c u r v e ) . The e x p e r i m e n t a l data a r e f r o m r e f . [27]: e E y = 3.4 GeV, o E y = 16 GeV.
E.Reya, Pion phoLoproduction
200
GeV/c, strongly suggest that the p r e s e n t f o r m of the i s o s c a l a r amplitudes
(o) is not s a t i s f a c t o r y . If, however, one finds another way of ful-
~o ) andA 2
filling the Born t e r m pole s t r u c t u r e of these amplitudes without using the somewhat a r t i f i c i a l four point functions within the B 5 f r a m e w o r k , it is quite probable that these deficiencies disappear. The p o l a r i z e d photon a s y m m e t r i e s ~+ and ~ - , for ~r+ and ~- photoproduction r e s p e c t i v e l y , a r e shown in fig, 5 for E~ = 3.4 GeV and in fig. 6 we give a prediction of these quantities for E~ = 16 GeV. The a g r e e m e n t with the e x p e r i m e n t a l data is v e r y good for ~ t --< 0.9 GeV/c. F o r higher values of ~-t~the a s y m m e t r i e s d e c r e a s e r a p idly w h e r e a s the not v e r y a c c u r a t e e x p e r i m e n t a l data in this region indicate a m o r e i n c r e a s i n g behaviour. As in the p r e v i o u s case, the question r e m a i n s open whether, at h i g h e r - m o m e n t u m t r a n s f e r s , possible cut contributions b e c o m e i m p o r t a n t or whether the d e s c r i p t i o n using fixed poles is sufficient but only the p r e s e n t f o r m of the i s o s c a l a r amplitudes is r e s p o n s i b l e for such difficulties. Fig. 7 p r e s e n t s the a s y m m e t r y of the c r o s s section for Vp--. ~r+n f r o m a p o l a r i z e d t a r g e t with incident u n p o l a r i z e d photons. The qualitative a g r e e m e n t with e x p e r i m e n t is f a i r l y good. In fig. 8 a prediction for this quantity for E~ = 8 GeV is shown.
0.8 mm ÷ 04
0
-0,4
O8 O4 0 -04
0.5
/:~ (Gev/c)
1.0
Fig. 5. The polarised photon asymmetries ~+ and ~- for zr+ and if- photoproduction respectively. The solid curves are the results for F_~ = 3.4 GeV. The data are from ref. [28]: oE~= 3 GeV, eE)/= 3.4 GeV.
E.Reya, Pion photoproduction
201
OB ! ÷ 0,4 O
-0.4
O.4 0
-0.4 J
0
0.5
~/~(GeV/c)
1'0
Fig. 6. P r e d i c t i o n s of the a s y m m e t r i e s N+ and N - at E,,j = 16 GeV.
O.tl X(t)
E~= 5 6eV 081
Eli=Boer
0,4
-0.4 -04
"I
-0.8
0 0.8 A(t) 0,4
0'5
,/~-(Qev/c)
110
E~t6 6or Fig. 8. Prechction o f A (t) f o r E~--- 8 G e V
0 -0.4.
-0|
o'.,
J~(e.v/c)
1'.o
Fig. 7. The Left-right a s y m m e t r y A (t) i n ~ + photoproductlon f o r m a p o i a r i s e d t a r g e t at E ~ = 5 and 16 GeV c o m p a r e d with the data f r o m ref. [29].
202
E.Reya, Pion photoproduction
F i n a l l y we w o u l d l i k e to r e - e m p h a s i z e t h a t , a s one w o u l d e x p e c t , t h e l o w l y i n g B a n d A 1 m e s o n s t u r n out to b e r a t h e r u n i m p o r t a n t , e s p e c i a l l y at h i g h p h o t o n e n e r g i e s . T h e B - t r a j e c t o r y c o n t r i b u t e s o n l y a f e w p e r c e n t to t h e c r o s s s e c t i o n , a n d by n e g l e c t i n g , f o r i n s t a n c e , t h e A 1 c o n t r i b u t i o n , i . e . p u t t i n g f [ 1 = 0, o n l y t h e r e s u l t s f o r t h e a s y m m e t r y p a r a m e t e r s at low e n e r g i e s a l t e r s l i g h t l y . T h e r e f o r e , t h o u g h we h a v e n o m i n a l l y s i x f r e e p a r a m e t e r s o n l y f o u r of t h e m a r e r e s p o n s i b l e f o r the r e s u l t s of t h i s m o d e l .
5. CONCLUSIONS U s i n g t h e V e n e z i a n o f i v e - p o i n t a m p l i t u d e s with a p p r o p r i a t e l y i n t r o d u c e d f i x e d J - p l a n e p o l e s we a r e a b l e to o b t a i n a f a i r l y c o n s i s t e n t a n d r e a s o n a b l e d e s c r i p t i o n of c h a r g e d p i o n p h o t o p r o d u c t i o n . A p a r t f r o m t h e u n s a t i s f a c t o r y r e s u l t f o r t h e n-/Tr + p r o d u c t i o n r a t i o a t a p h o t o n e n e r g y of 16 GeV, t h e s u c c e s s of t h i s m o d e l in e x p l a i n i n g t h e d a t a with m o v i n g R e g g e p o l e s a n d f i x e d J - p l a n e p o l e s only s u g g e s t s t h a t f i x e d p o l e s a r e i n d e e d a b l e to a c c o u n t f o r p h e n o m e n a w h i c h c u r r e n t l y a r e t h o u g h t to n e c e s s i t a t e c u t s . T h e r e f o r e t h e f i x e d p o l e s r e v i v e t h e h o p e t h a t R e g g e c u t s a r e u n i m p o r t a n t at l e a s t a t h i g h e n e r g i e s not only in p h o t o p r o d u c t i o n but a l s o in p u r e h a d r o n i c p r o c e s s e s like n-p charge exchange or n-N scattering. The relatively simple propert i e s of s u c h p o l e s a n d t h e i r e c o n o m i c a l p a r a m e t r i z a t i o n a r e s u f f i c i e n t r e a sons for investigating these possibilities. I w o u l d l i k e to e x p r e s s m y t h a n k s to P r o f e s s o r A. D o n n a c h i e a n d D r s . R. P. B a j p a l , B. C a r r e r a s , J. F. L. H o p k i n s o n a n d J. N. J. W h i t e f o r m a n y discussions and useful remarks.
REFERENCES [1] J. S. BaH, W.R. F r a z e r and M. Jacob, Phys. Rev. L e t t e r s 20 (1968) 518; F. S. Henyey, Phys. Rev. 170 (1968) 1619. [2] J. Froy[and and D. Gordon, Phys. Rev. 177 (1969) 2500; M. L. B[aekmon, G. ICramer and K. Schil[ing, NucL Phys. B12 {1969) 495; J . D . Jackson and C.Quigg, Phys. L e t t e r s 29B (1969) 236, see a[so r e f e r e n c e s therein. [3] D. Amati, G. Cohen-Tannoudji, R. Jengo and Ph. Sahn, Phys. L e t t e r s 26B (1968) 510. [4] I. Bender, H.G. Dosch and B. J. Rothe, Nuovo Clmento 62A (1969) 1026. [5] D. J. Gross and H. Page[s, Phys. Rev. L e t t e r s 20 (1968) 961. [6] J. B. Bronzan, I.S. Gerstem, B.W. Lee and F. E. Low, Phys. Rev. 157 (1967) 1448, [7] V. Singh, Phys. Rev. L e t t e r s 18 (1967) 36. [8] H.G. Dosch, Phys. Rev. 173 (1968) 1595. [9] J. S. BaH and M. Jaeob, Nuovo Cimento 54A (1968) 620. [10] R. Dashen and S. Y. Lee, Phys. Rev. L e t t e r s 22 (1969) 366. [11] N. Dombey, Phys. L e t t e r s 30B (1969) 646. [12] P. F i n k i e r , Phys. Rev. DI (1970) 1172.
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[13] Chan Hong-Mo, Review paper p r e s e n t e d at the Royal Society Meeting on Duahtyreggeons and resonances in e l e m e n t a r y particle p r o c e s s e s , 1969, and CERN p r e p r i n t TH. 1057. [14] M. Ahmad, Fayyazuddm and Riazuddm, Phys. Rev. L e t t e r s 23 (1969) 504; F . D r a g o , Nuovo Cimento L e t t e r s 2 (1969) 712. [15] I. Bender and H. J. Rothe, Nuovo Clmento L et t er s 2 {1969) 477. [16] J . P . Ader, M. Capdeville and Ph. Salin, Nucl. Phys. B17 (1970) 221. [17] I. Bender, H . J . Rothe, H.G. Dosch and V. F. Miiller, Nuovo Cimento Let t er s 3 (1970) 625. [18] G. F. Chew, M. L. Goldberger, F . E . Low and Y. Nambu, Phys. Rev. 106 (1957) 1345. [19] M. Gell-Mann, M.L. Goldberger, F . E . Low, E. Marx and F. Zachariasen, Phys. Rev. B133 {1964) 145. [20] P.Stlchel, Z. Phys. 180 (1964) 170. [21] J. D. Jackson and C.Quigg, U m v e r s i t y of California Radiation Laboratory p r e print UCRL-19410, 1970. [22] H. Sugawara, Tokyo University of Education p r e p r m t T U E P - 6 9 - 4 , unpubhshed. [23] I. Ohba, Prog. Theor. Phys. 42 (1969) 432. [24] M. Ademollo and E . D e ! Giudice, Nuovo Cimento 63A (1969) 639; Nuovo Clmento L e t t e r s 2 (1969) 345. [25] A. Bialas and S. Pokorski, Nucl. Phys. B10 0969) 399. [26] A. M. Boyarski, F. Bulos, W. Busza, R. Diebold, S.D. Ecklund, G.E. F i s c h e r , J. R. Rees and B. Richter, Phys. Rev. L e t t e r s 20 (1968) 300; Contribution to the 14th Int. Conf. on high-energy physics, Vienna, 1968 paper 430, unpublished; P. Heide, U. KDtz, R . A . L e w i s , P. Schmfiser, H . J . Skronn and H. Wahl, Phys. Rev. L e t t e r s 21 (1968) 248. [27] A. M. Boyarksi, R. Diebold, S.D. Ecklund, G.E. F i s c h e r , Y. Murata, B. Richter and W. S. C . W il li a m s , Phys. Rev. L e t t e r s 21 (1968) 1767; P. H e i d e e t a l . , ref. [26]; Z. B a r - Y a m , J. dePagter, M.M. Hoenig, W. Kern, D. Luckey and L. S. Osborne, Phys. Rev. L e t t e r s 19 (1967) 40. [28] C. Geweniger, P. Heide, U. K~tz, R.A. Lewis, P. SchmUser, H . J . Skronn, H. Wahl and K. Wegener, Phys. L e t t e r s 29B (1969) 41; F u r t h e r res u l ts from both the DESY and CEA experiments have been reported by K. LUbelsmeyer, P r o c . 4th Int. Symp. on electron and photon intereactions at high en erg i es , Liverpool, 1969, ed. D.W. Braben (Daresbury, 1969) p.45. [29] C. C. Morehouse, M. Borghini, O. Chamberlain, R. Fuzesy, W. Gorn, T. PoweIl, P. Robrish, S. Rock, S. Shannon, G. Shapiro, H. Weisberg, A. Boyarski, S. Ecklund, Y. Murata, B. Richter and R. Siemann, Phys. Rev. Let t er s 25 (1970) 835.
Nuc[ear Physics B29 (1971) 189-203. North-Hol|and Pubhshlng Company
THE GENERALIZED VENEZIANO MODEL FOR CHARGED PION P H O T O P R O D U C T I O N E. REYA
Daresbury Nuclear Pfiyszes Laboratory
Received 14 December 1970
Abstract: Charged pion photoproduetion is investigated in the framework of Venezlano B 5 amplitudes using Sugawara's "spurion" technique in order to introduce fixed J-plane pores. The Re~ge exchanges are taken to be evasive. The agreement w~th expemment for the differential cross section, the ?r-/~ + ratio and the polarisatlon asymmetries for photon energies between 3.4 and 18 GeV ~s quite reasonable.
1. INTRODUCTION In explaining the e x p e r i m e n t a l data on c h a r g e d pion photoproduction t h e r e a r e e s s e n t i a l l y t h r e e different m e c h a n i s m s p r e s e n t l y known: (i) a p a i r of c o n s p i r i n g pion Regge t r a j e c t o r i e s [1], (ii) a s e l f - c o n s p i r i n g Regge cut or a b s o r p t i o n c o r r e c t i o n s to evasive Regge poles [2], (iii) right signature fixed poles in the angular m o m e n t u m plane [3, 4]. All t h r e e models give a s a t i s f a c t o r y d e s c r i p t i o n of experiment. In this p a p e r we study c h a r g e d pion photoproduction in the f r a m e w o r k of the g e n e r a l i s e d Veneziano model, n a m e l y the five point function B5, and using S u g a w a r a ' s " s p u r i o n " technique (see below) in o r d e r to introduce fixed J - p l a n e poles. T h e r e f o r e we a r e using the third m e c h a n i s m mentioned above, i.e. explaining the s h a r p f o r w a r d peak in da/dt by a right s i g n a t u r e fixed pole w h e r e a s the t - c h a n n e l Regge exchanges a r e taken to be evasive. That such fixed p o l e s (and also K r o n e c k e r deltas [5]) can o c c u r in p r o c e s s e s initiated by photons, i.e. in amplitudes satisfying linear unitarity r e l a t i o n s to lowest o r d e r in e, has been shown by Bronzan et al. [6] and Singh [7]. They p r o v e d this for Compton s c a t t e r i n g by c o m p a r i n g the D a s h e n G e l l - M a n n - F u b i n i s u m rule with the F r o i s s a r t - G r i b o v continuation into the c o m p l e x J - p l a n e . Using a simple field t h e o r e t i c a l model, Dosch [8] has shown that a fixed power behaviour, and hence a fixed J - p l a n e pole, may o c c u r also in m e s o n photoproduction whether the p r o d u c e d meson is " e l e m e n t a r y " or " c o m p o s i t e " . Some speculations upon the p r e s e n c e * of fixed * If, however, one performs an explicit separation of the elementary particle pole terms, superconvergence relations for certain invariant amplitudes may occur and therefore fixed poles in the J-plane are not required.
190
E.Reya, Pion photoproduction
poles in photoproduction amplitudes have also been made by Ball and J a c o b [9]. Dashen and Lee [10] have given a r g u m e n t s against right signature fixed J - p l a n e poles in b a c k w a r d uo photoproduction c o m p a r i n g with e x p e r i m e n t 1 the s-dependence of da/du of a fixed J = y and a moving Regge pole r e s p e c tively. However, Dombey [11] pointed out that these a r g u m e n t s a r e not c o r r e c t , since the Coulomb t e r m s give a contribution p r o p o r t i o n a l to sin 20 and so vanish in the backward direction. To apply these a r g u m e n t s also to the f o r w a r d direction is much m o r e difficult, since in this c a s e t h e r e is no such difference between the energy dependence of a fixed J = 0 pole and of the a p p r o p r i a t e Regge exchange. Recently, on the g e n e r a l b a s i s of the F r o i s s a r t - G r i b o v projection, F i n k l e r [12] has shown that unitarity allows fixed poles at c e r t a i n nonsense points of either right or wrong signature. Moreover, such fixed poles which survive at wrong signature nonsense points need not be r e l a t e d to the GribovP o m e r a n c h u k singularities, nor to the third double s p e c t r a l function. Obviously, t h e s e r e s u l t s apply also to amplitudes d e s c r i b i n g p r o c e s s e s initiated by photons. In this c a s e the fixed poles still r e m a i n even when we include exact unitarity. This c l e a r l y r e f u t e s the c o m m o n belief (see above) that fixed poles can exist only as long as linear unitarity r e l a t i o n s a r e considered, w h e r e a s exact bilinear unitarity would forbid such a fixed power behaviour. F o r a f u r t h e r d i s c u s s i o n on this point and also on the existence of such fixed s i n g u l a r i t i e s strongly favouring the evasive m e c h a n i s m for Regge exchanges we r e f e r the r e a d e r to ref. [12]. F r o m a m o r e phenomenological point of view the fixed poles p r o p o s e d by Amati et al. [3] * t o g e t h e r with the usual Regge exchanges yield a v e r y good explanation of the e x p e r i m e n t a l data. The s a m e quite convincing r e s u l t s , even for the u - / ~ - ratio, a r e obtained by Bender et al. [4], using a right signature fixed Khuri pole at ~ = -1 yielding an a s y m p t o t i c 1/v behaviour. F u r t h e r m o r e t h e r e is c e r t a i n l y no contradiction to fixed power behaviour in the invariant amplitudes since s2(da/d t) s e e m s to be n e a r l y independent of s and no shrinkage can be found. To the best of our knowledge v e r y little work has been done up to now u s ing the Veneziano model [13] to d e s c r i b e pion photoproduction. In p a r t i c u l a r no extensive fits to the e x p e r i m e n t a l data have been made. Within the f r a m e work of the original f o u r - p a r t i c l e Veneziano amplitude different models for the invariant amplitudes and helicity amplitudes have been suggested, using a p a r i t y - d o u b l e t c o n s p i r a c y of the pion [14, 15]. In these p a p e r s only the basic f e a t u r e s of such models in the v e r y f o r w a r d direction a r e discussed. F o r instance the r a p i d v a r i a t i o n of the p a r i t y c o n s e r v i n g helicity amplitudes 2 in the i n t e r v a l - m 2 --< t --< m a r i s e s quite naturally within the f r a m e w o r k of the Veneziano m o d e l A r a p i d variation of this kind is known to be n e c e s s a r y in o r d e r to fit both the magnitude of the differential c r o s s section and the p o l a r i s e d photon a s y m m e t r i e s . F u r t h e r m o r e it has been e m p h a s i z e d by * It should be mentioned that the origin of such a smooth background (represented by those right signature fixed poles at J = 0, -1) can also be due to Regge cuts or absorption corrections to an evasive Regge pole.
E.Reya, Pion photoproductzon
191
Ader et al. [16] that the e v a s i v e m e c h a n i s m is a l s o c o n s i s t e n t with the V e n eziano model, n a m e l y that the a m p l i t u d e s a r e c o m p a t i b l e with the B o r n t e r m c o n t r i b u t i o n s at low energy. R e c e n t l y a v e r y i n t e r e s t i n g a t t e m p t to d e s c r i b e the photoproduction data h a s been m a d e by Bender, Rothe, Dosch and MUller [17]. Using Veneziano B 5 a m p l i t u d e s , p r o p e r l y modified to get fixed poles, they obtained r e m a r k a b l e fits to the d i f f e r e n t i a l c r o s s s e c t i o n at photon e n e r g i e s f r o m 8 to 16 GeV in a t - r a n g e up to -1 (GeV/c) 2. It is worthwhile e m p h a s i z i n g that this model d o e s not contain any f r e e p a r a m e t e r , b e c a u s e the Veneziano n o r m a l i z a t i o n c o n s t a n t s w e r e fixed by the e l e c t r i c B o r n - t e r m s , since only N~ e x c h a n g e s a r e taken into account. F o r the s a k e of c l a r i t y we b r i e f l y s u m m a r i z e in sect. 2 all the r e l e v a n t notations and f o r m u l a e we use. In sect. 3 we d e s c r i b e the model and give the e x p r e s s i o n s f o r the different i n v a r i a n t amplitudes. Finally sect. 4 c o n tains the fits to the e x p e r i m e n t a l data and s o m e d i s c u s s i o n of t h e m , and our c o n c l u s i o n s a r e s u m m a r i z e d in sect. 5.
2. KINEMATICS AND BASIC FORMULAE A c c o r d i n g to CGLN [18] the m o s t g e n e r a l f o r m of the p h o t o - m e s o n T m a t r i x e l e m e n t m a y be written a s 4 T = ~ Ai(s , t)Mi, i=1
(1)
w h e r e the g a u g e - i n v a r i a n t spin m a t r i c e s take the f o r m M1 = ~'5 ¢]¢ , M 2 = 2~,5(P.k q. E - P . E q . k ) , M 3 = ~,5(4 q . k -]~ q.E) , M 4 = 275(¢P.k
-~P.
~ - mN~(~) ,
(2)
E and k a r e the photon p o l a r i z a t i o n v e c t o r and i t s f o u r - m o m e n t u m , r e s p e c tively, q i s the f o u r - m o m e n t u m of the outgoing pion with m a s s mTr. The f o u r - m o m e n t a of the initial and final nucleons with m a s s m N a r e denoted by P l a n d P 2 , and the total m o m e n t u m P = ½(Pl + P2)- T h e i n v a r i a n t s s and t a r e given by s = (k + p l ) 2 , t = (q - k) 2. In i s o s p i n space, the i n v a r i a n t a m p l i t u d e s A i ( s , t) h a v e the following d e c o m p o s i t i o n : A i = 5 ~3 A i(+) + : [1T O l ' T3 ]A j (-) i + ~"~ A!Z°) '
(3)
being the i s o s p i n index of the outgoing pion and A (i° ) ("It " = 1 , G = +1) i n v o l v e s i s o s c a l a r photons w h e r e a s both Ai+)(/t = ' 0, G = - 1 ) a n d A i - ) ( I t = " 1, G = -1) involve the i s o v e c t o r photon. So one h a s f o r c h a r g e d pion p r o d u c t i o n
E. Reya, Pion photoproduction
192
Ai(Tp-~ ~r+n) = ,/'~(AI °) + A~-)) , Ai(Tn --" 7r-p) = "/'g(al °) - AI-)) .
(4)
F u r t h e r m o r e the parity-conserving t-channel helicity amplitudes [ 1 9 ] f ~ (where X = X~ - XTr, ~ = XN - ~N) can be e x p r e s s e d in t e r m s of the invariant amplitudesA'i (s, t), namely
t_m 2 f;0 : ~ -
~r
(A1
_
2tuNA4) ,
t-m 2~/ t-4rn2 ½
_
~/~ \---~----) (A1 + tA2)'
fl-0-
t_m 2 f l l = _ - + ~r ('t-1(2reNAl _ tA4) , t_ m 2
(5)
f-l l = - 7 u (t - 4m2)½A 3 With this normalization the differential c r o s s section is given by d~
1 _ _
I
dt - 647r (s_m2)2
{(cos2ot_ 1)(if1+012+ [f~o12)
+ (c°s20t +
1)(1~1112 + tfll 12) + 4c°sOtRe(f~1111[)},
(6)
with 2
2st+t2-t(24+mr) cos
0 t
-
( t - m 2) ~/ t( t-4m 2)
The polarized photon a s y m m e t r y ~ is defined by d (Tl/d t - d ~ " / d t
= daZ/dt+do~,/dt,
(7)
where d~±/dt is the differential c r o s s section for photons polarized perpendicular to the production plane, having contributions only f r o m the natural parity amplitudes ~ 0 and f ~ l ' while the unnatural parity amplitudes f i 0 and f l l contribute only to the differential c r o s s section for parallel polarized photons, dd'/dt (ref. [20]).
E.Reya, Pion photoproduction
193
Finally we i n t r o d u c e the a s y m m e t r y of the c r o s s section f o r Vp -. lr+n f r o m a p o l a r i z e d t a r g e t with incident u n p o l a r i z e d photons [21]
A(t) =
- 2 , / ~ I m [A~ A4+A~(AI+tA2)]
]All 2+1 tl IA4] 2+1 t I IA3)2+IAI+tA212
(8)
w h e r e the t a r g e t is p o l a r i z e d e i t h e r p a r a l l e l or a n t i p a r a l l e l to the n o r m a l to the production p l a n e defined by k × q.
3. THE MODEL Since we a r e going to study photoproduction u s i n g the e v a s i v e m e c h a n i s m we need, in o r d e r to explain the s h a r p f o r w a r d p e a k in the d i f f e r e n t i a l c r o s s section, e i t h e r fixed pole o r cut c o n t r i b u t i o n s to the i n v a r i a n t a m p l i t u d e s A~s,t). As we have a l r e a d y e m p h a s i z e d in sect. 1, we confine o u r s e l v e s to the f i r s t p o s s i b i l i t y , n a m e l y fixed poles. A p o s s i b l e way of introducing such fixed p o w e r b e h a v i o u r in the Veneziano a m p l i t u d e s was s u g g e s t e d by Sugaw a r n [221 He p r o p o s e d a method of f o r m u l a t i n g the o f f - m a s s - s h e l l a m p l i tudes ( e s p e c i a l l y f o r Compton s c a t t e r i n g ) with the help of a " s p u r i o n " having v a c u u m quantum n u m b e r s t. G e n e r a l l y s p e a k i n g this m e a n s that one c o u p l e s a " s p u r i o n " with f o u r - m o m e n t u m P 5 , say, to a c u r r e n t , and then goes to the l i m i t p 5 -~ 0 in the c o r r e s p o n d i n g (N + 1) point function. Ohba [23] showed that it is indeed p o s s i b l e to d e r i v e also N - b o d y o f f - m a s s - s h e l l a m p l i t u d e s s e l f - c o n s i s t e n t l y f r o m ( N + 1) body a m p l i t u d e s in the g e n e r a l i z e d Veneziano model. T h e r e , also, the a m b i g u i t y in S u g a w a r a ' s original method of e l i m i nating unwanted s i n g u l a r i t i e s such a s double a n d / o r t r i p l e p o l e s (which a r e a c o n s e q u e n c e of the l i m i t p 5 --. 0) was r e m o v e d and a unique way of e l i m i nating such p o l e s w a s given. In o u r c a s e we s t a r t f r o m the f i v e - p o i n t a m p l i t u d e and c o n s i d e r the l i m i t P5 --" 0, a s shown in fig. 1. In o r d e r to get a c a n c e l l a t i o n of the above m e n tioned u n p h y s i c a l double p o l e s we have to put all v a r i a b l e s c o r r e s p o n d i n g to c h a n n e l s dual to channel (1, 5) equal to a constant (>i 1), which can e a s i l y be s e e n f r o m the definition of the f i v e - p o i n t function. T h i s y i e l d s S
1
4
2
Pff'O
2
5
5
Fig. 1. J' Obviously this "spurion" has nothing to do with that commonly used, e.g. in ~ - and K-decays, where it carries finite isospin in order to balance isospin conservation.
194
E . R e y a , P z o n photop~,oduction
lim B5(-c~ 12' -~23' -a34' -c~45' -~ 51)
pS-~0
: B5(-~23(s23) , -~34(s34),a, -~51(P~),b) ,
(9)
with s i j = ( P i + P j ) 2 and a, b >/ 1. According to ref. [23] we take a = b = 1 getting an exact (i. e. no f u r t h e r non-singular contributions of the appropriate channels) cancellation of the unwanted double poles. F r o m now on we keep this convention whenever possible. Since the photon couples essentially to vector mesons we take for ~51 the p - t r a j e c t o r y . Consequently a typical exp r e s s i o n for an s - t o f f - m a s s - s h e l l amplitude, say, using the kinematic variables defined in sect. 2 is given by
(10)
Bs(-ot(s), -~(t), 1, 1 - C~p(k2), 15.
Such amplitudes have not only the c o r r e c t Regge behaviour but also the r e quired fixed power behaviour in s and t. F u r t h e r m o r e , if k 2 = m 2 , for instance, we encounter the p-pole and the residue is exactly the Veneziano B 4 amplitude for the purely hadronic p-production. Ademollo and Del Giudice [24 ] studied f u r t h e r mathematical p r o p e r t i e s of those amplitudes and also applied such ideas to electroproduction mad Compton scattering. Before we write down the final expressions for the invariant CGLN a m plitudes, we briefly s u m m a r i z e some of the basic r e q u i r e m e n t s they have to
(+,o) , A^(+,o5,
fulfil. F r o m the s - u c r o s s i n g properties, it follows that A .
A~") and A~+' o) must be even while A~-) , A~-) , A~+' o ) a n d A ~ -) areZ odd functions under the substitution s - . u . F u r t h e r m o r e the nucleon and pion pole contributions to the A i ' s should have the c o r r e c t r e s i d u e s (electric and magnetic Born t e r m s ) , namely l i m (s
2 A(.~, -raN) ~ o) = r(.% ~ o)
2
s-~m N
,
tim ( t - 2 )
A(?, z o) : R(~, o)
t-.m 2 7r
where
(2, o)
r1
1
= --seg,
r~. o) '
-
eg
2
'
t-m
e v ~o)__q )__½g~(,p+Un ),
R~-) 2 e g =
2' s-m N
R~, o)__R~+.o)_ R~~ o5= R~ o)=0 "
(11)
E.Reya, Pzon photoproduct~on
195
H e r e ~ and Pn a r e the anomalous magnetic m o m e n t s of the proton and n e u tron r e h p e c t i v e l y and g is the ~r - N coupling constant *. Other well known p r o p e r t i e s of the i n v a r i a n t amplitudes, such as the asymptotic behaviour e t c . , can be found for instance in ref. [15] or [18]. We know [16] that the e v a s i v e m e c h a n i s m only r e q u i r e s that natural p a r i t y contributions go to z e r o f a s t e r than s a~r(°)-I as s ~ m The a m p l i t u d e A (-)" 1 r e c e i v e s contributions only f r o m n a t u r a l p a r i t y t-channel exchanges. All such m e s o n t r a j e c t o r i e s known at p r e s e n t have an i n t e r c e p t g r e a t e r than a~r(o). Now a c c o r d i n g to eq. (11) we see that the nucleon contribution does not vanish at t = 0. T h e r e f o r e we take f o r this t-channel exchange the f i r s t daughter of the exchange d e g e n e r a t e (p - f - co - A2) t r a j e c t o r y in o r d e r to be c o n s i s t e n t with the evasive mechanism. Within the f r a m e w o r k of the Veneziano model one is led quite naturally to a s s u m e this t r a j e c t o r y in such a c a s e and its e x i s t e n c e is c e r t a i n l y not c o n t r a d i c t e d by e x p e r i m e n t . T h e r e f o r e the generation of the e l e c t r i c Born t e r m s in the v e r y f o r w a r d d i r e c t i o n using this method s e e m s to us to be l e s s dubious than a s s u m i n g a natural parity__conspirator of the pion. Thus using for the N a contributions__ in A~-)" and A~-) the s a m e ansatz as in ref. [17], we obtain for the A~/-) amplitudes the following e x p r e s s i o n s : l
-a
T
A -)(s, t) = - ~ e g B(~-~,~o ) ×{ B 5(~1 - ~Na(S), 1 - aA~(t), 1, ao, 1) - B5(~1 - aNa(U), 1 - aA~(t), 1, ao, 1)} +/31 t{B5( ~ -aNT(S), 1 - a A 2 ( t ) , 1, ao, 1) -B5(~ -aNT(U) , 1 - a A 2 ( t ) , 1, ao, 1)}, (12a)
A
~-)(s, t) = e g -a' t_m2 B(c, ao) ~Y
× {B 5(Y1 - aNa(S), 1 - an(t), c, ao, d ) - B5(½ - aNa(U), 1 - air(t), c, ao, d) } - i l l {B5(~" aNT(S), 1 - aA2(t), 1, ao, 1) - BS( ~ - aNT(U), 1 - aA2(t), 1, ao, 1)}, (12b)
. e2/4ff = 1 -1~ ,
~2/4ff = 14.6, gp'
=
1.78, ~tn = - 1 . 9 1 .
E, Reya, Pzon photoproduct ion
196
A(-)(s t) = $eg 3 ’ x @5(+ - c.uN,(s), 1 - @A&t ), 1, @o, 1) + B5(+ - @N,(u), +& (B5(# - aN$s),
1 - @A#),
1, Qo, 1) + B5(+! -aN$U),
1 -a Al@ ), 1, ffo, I)> 1 -a A#),
1, ~I-J,I)), WC)
A4C-J (s, t) = aeg
&N($3- %I)Be)
+@3 {B5(+ -@NY(S), 1 - @Aa(
1, oo, 1) - B5(+-aNY(4,
1 -@Aa(
l,oo,
1))) (12d)
2 withu=2m +m2-st and B( X, y ) being the Euler B-function; cy’ is the N T common slope of all Regge trajectories and a0 = 1 - ao(k2 = 0). The gauge invariant pion pole structure (eq. (11)) requires c and d in eq. (12b) to satisfy the following relation l+c=
d.
(13)
Taking as usual c = 1, eq. (13) fixes d. Concerning the isoscalar amplitudes Ai(0) it is not possible required variance,
pole structure
consistently
using only B5’s.
A(z+$‘) also contains a factor
l/( t - mz),
must not exhibit a pole at t = m,“. Therefore
to fulfil the
Because of gauge in-
but at the same time it
the function multiplied by this
singular term must go to zero as t -+ rnfo Solving for the zeros of a fivepoint function (still not a solved problem in one variable), one obtains as solutions different functions of the remaining four arguments. Therefore one has different constraint equations for different energies s, say, which turns out to be inconsistent in our case. To get the right pole structure in A1(0) and A’z”) we use for the nucleon contribution function which exhibits the required
an s - u Veneziano four-point
zero at t = rnf . This is certainly
not
unreasonable as a first approximation, since the essential structure of the various measured quantities in the forward direction is given by t-channel exchanges. Furthermore, the requirements of crossing symmetry, signature and the appropriate asymptotic behaviour of the amplitudes A3(O) and A4(0) do not allow the nucleon to contribute, as already pointed out in ref.
E. Reya, P~on photoproductzon
197
[15]. The absence of this pole in the Veneziano f o r m u l a is c o n s i s t e n t with the a p p r o x i m a t e equality of ~p and - g n . Thus we a r e finally led to the f o l lowing e x p r e s s i o n s f o r the i s o s c a l a r amplitudes:
A(°)(S'l
t) = -yeg(-o~ 1 , ) B(~~ - a Sc~(s), 5 - aNc~(U))
+f14 t{B5(~ - aNy(S), 1 - a p ( t ) , 1, ao, 1)+ B5( ~ - aNy(U), 1 - ap(t), 1, ao, 1)}, (14a)
eg , d~°)(s, t) : t - - ~ ( - a
~,~ ) ~ ( y - aNa(S), ½- a j a ( u ) )
7f -f14 {Bs(J - aNy(S), 1 - ap(t), 1, ao, 1) +B5( ~ - aNy(U), 1 - ap(t), 1, ao, 1)} +f15 {Bs(~ - aNy(S), 1 - aB(t), 1, ao, 1) +B5( ~ - aNT(U), 1 - aB(t), 1, ao, 1)} (14b)
A~°)(s, t) = O,
(14c)
A~°)(s, t) = f16 {B5(~ - a N y ( S ) , 1 -ap(t), 1, ao, 1)
+Bs( -aNy(U), 1
1, ao, 1)}.
(14d)
Thus we a r e left with six f r e e n o r m a l i z a t i o n constants fli. However, it t u r n s out that the contributions of the low-lying B and A 1 m e s o n s a r e p r a c t i c a l l y negligible, e s p e c i a l l y at higher e n e r g i e s , and t h e r e f o r e only the four r e maining p a r a m e t e r s a r e important. In o r d e r to avoid f u r t h e r f r e e p a r a m e t e r s we also neglected possible A 5 contributions. This s e e m s to be r e a s o n able b e c a u s e it is well known that the A 5 c o n t r i b u t e s p r e f e r a b l y in the b a c k ward direction, and t h e r e only about 25% w h e r e the main contribution is due to the Ny. The t r a j e c t o r i e s a r e taken a s aNa(S ) = -0.27+a's,
aNy(S) = - 0 . 8 + a ' s ,
ap
(t) = a A 2 ( t ) = 0 . 4 8 + a ' t ,
a~
it) = - 0 . 0 1 8 + a ' t ,
aA~(t) = - 0 . 5 2 + a ' t ,
aAl(t ) = -0.1+a't,
aB(t ) = -0.3+a't ,
with ot' = 0.89 (GeV/c)-2. To see the basic f e a t u r e s of our amplitudes, we c o n s i d e r the h i g h - e n e r g y beh~viour of the B 5 functions. E x p r e s s i n g the B 5 function in t e r m s of a gene r a l i z e d h y p e r g e o m e t r i c function [25] one easily finds for the h i g h - e n e r g y limit, keeping only f i r s t o r d e r s in s,
198
E.Reya, P~on photoproductzon
B5(~ - ~ ( s ) , 1 - ~ ( t ) , a, % , b) ~ {(-c~ 's)(~(t) -1 F(1 - ~ ( t ) ) B ( ~ ( t ) + b - 1, ~o)[1 - ( - ~ ' s ) - l ( 1 - ~(t)) ] + ( - ~ , s ) -b r(b) B(1 - b - a ( t ) , a)[1 - ( - ~ ' s ) - l b ] } { 1 +0(1) + . . . } .
(15)
U s i n g t h i s e x p r e s s i o n the a s y m p t o t i c a l l y l e a d i n g t e r m of A~-)- g o e s like i / s , i.e. it r e p r e s e n t s the c o n t r i b u t i o n of a r i g h t s i g n a t u r e fixed pole at the n o n s e n s e v a l u e J = 0. T h u s dcr/dt at t = 0 is a p p r o x i m a t e l y given by the n u c l e o n p o l e t e r m only, in a g r e e m e n t with e x p e r i m e n t . The r e m a i n i n g two a n t i s y m m e t r i c a m p l i t u d e s A~-)- and A~-)" r e c e i v e a l s o c o n t r i b u t i o n s f r o m a J = 0 f i x e d pole, but h e r e the l e a d i n g t e r m s h a v e the u s u a l R e g g e b e h a v i o u r . Only in the f i r s t t e r m of eq. (12b) d o e s no 1 / s fixed p o l e a p p e a r ( b e c a u s e of c o n dition (13)), which is due to the R e g g e i z e d pion c o n t a i n e d in A(2-)--. B e c a u s e of the c r o s s i n g b e h a v i o u r of the C G L N a m p l i t u d e s no r i g h t s i g n a t u r e fixed J ~O~
p o l e at J = 0 c a n a p p e a r in A -) and A i
(i = 1, 2, 4). I n s e r t i n g eq. (15) into
the a p p r o p r i a t e e x p r e s s i o n f o r t h e s e a m p l i t u d e s s h o w s t h a t the c o n t r i b u t i o n s of such a pole do i n d e e d c a n c e l , and a p a r t f r o m the asymptoticall]¢ l e a d i n g R e g g e t e r m the f i r s t c o n t r i b u t i n g fixed p o l e h a s J = -1, i.e. a 1 / s z f i x e d p o w e r b e h a v i o u r *. F i n a l l y we s h o u l d m e n t i o n that in the c a l c u l a h o n s we u s e d the e x a c t e x p r e s s i o n s f o r the A(.$ - ' o) a m p l i t u d e s ,
a s given in eqs. (12) and (14), only in
t h o s e c a s e s w h e r e the a c c u r a c y of the e x p a n s i o n (157 w a s not sufficient. T h i s w a s done in o r d e r to s i m p l i f y the n u m e r i c a l c a l c u l a t i o n s .
4. R E S U L T S AND DISCUSSION In t h i s s e c t i o n we p r e s e n t the r e s u l t s f o r the v a r i o u s q u a n t i t i e s d e f i n e d in sect. 2. T h e n o r m a l i z a t i o n c o n s t a n t s / 3 i w e r e fixed so a s to get the b e s t fit to the d i f f e r e n t i a l c r o s s s e c t i o n f o r 7r+ p r o d u c t i o n f o r i n c i d e n t photon e n e r g i e s E 7 = 8 and 16 GeV. The r e s t being p r e d i c t e d by the model. T h e lr + d i f f e r e n t i a l c r o s s s e c t i o n and the 7r-/~ + p r o d u c t i o n r a t i o R at e n e r g i e s b e tween E y = 3.4 and 18 GeV a r e shown in figs. 2, 3 and 4. Although the s h a p e and m a g n i t u d e of d ~ / d t is r e p r o d u c e d v e r y well, the e n e r g y d e p e n d e n c e s e e m s a little bit too s t r o n g . This, and a l s o the f a c t that we a r e not able to r e p r o d u c e the r i g h t s t r u c t u r e of R at E v = 16 GeV, n a m e l y a dip at ~ ~ 0.6 * The same applies to yo photoproductlon. Furthermore we obtain in this case (using eq. (15)) that for instance the asymptotically leading terms of the dominant A i(+) amplitudes (cO-exchange) are multiplied by a factor (1-exp (i ~'~ cO(t)) which also explains the dip in d(7/dt at t ~ -0.55 (GeV/c) 2 very well. Such a h t is under investigation and will be pubhshed elsewhere.
E . R e y a , Pion photoproduction
10°I! ! !
101
t
J
1¢
199
~ 4
i
GeV
100
,o ,o
t
o
0.5
/~
(aev/c)
0
1.o
F i g . 2. D i f f e r e n t i a l c r o s s s e c t i o n f o r y p ~ ?r+n b e t w e e n 3.4 and 8 GeV. The data a r e f r o m r e f . [26].
"~.le GeV
0.5
(oev/c~
1.0
F i g . 3. D i f f e r e n t i a l c r o s s s e c t i o n f o r ~p ~ ~'+n b e t w e e n 11 and 18 GeV. The data a r e f r o m r e f . [26].
1.0 R
0.5
o
o'.5
1:o GeV/c
F i g . 4, The r a t i o R = (d(r/dt)(yn--* ~r-p)/(dcr/dt)(yp -~ 7r+n) f o r i n c i d e n t photon e n e r g i e s of 3.4 GeV (solid curve) and 16 GeV (dashed c u r v e ) . The e x p e r i m e n t a l data a r e f r o m r e f . [27]: e E y = 3.4 GeV, o E y = 16 GeV.
E.Reya, Pion phoLoproduction
200
GeV/c, strongly suggest that the p r e s e n t f o r m of the i s o s c a l a r amplitudes
(o) is not s a t i s f a c t o r y . If, however, one finds another way of ful-
~o ) andA 2
filling the Born t e r m pole s t r u c t u r e of these amplitudes without using the somewhat a r t i f i c i a l four point functions within the B 5 f r a m e w o r k , it is quite probable that these deficiencies disappear. The p o l a r i z e d photon a s y m m e t r i e s ~+ and ~ - , for ~r+ and ~- photoproduction r e s p e c t i v e l y , a r e shown in fig, 5 for E~ = 3.4 GeV and in fig. 6 we give a prediction of these quantities for E~ = 16 GeV. The a g r e e m e n t with the e x p e r i m e n t a l data is v e r y good for ~ t --< 0.9 GeV/c. F o r higher values of ~-t~the a s y m m e t r i e s d e c r e a s e r a p idly w h e r e a s the not v e r y a c c u r a t e e x p e r i m e n t a l data in this region indicate a m o r e i n c r e a s i n g behaviour. As in the p r e v i o u s case, the question r e m a i n s open whether, at h i g h e r - m o m e n t u m t r a n s f e r s , possible cut contributions b e c o m e i m p o r t a n t or whether the d e s c r i p t i o n using fixed poles is sufficient but only the p r e s e n t f o r m of the i s o s c a l a r amplitudes is r e s p o n s i b l e for such difficulties. Fig. 7 p r e s e n t s the a s y m m e t r y of the c r o s s section for Vp--. ~r+n f r o m a p o l a r i z e d t a r g e t with incident u n p o l a r i z e d photons. The qualitative a g r e e m e n t with e x p e r i m e n t is f a i r l y good. In fig. 8 a prediction for this quantity for E~ = 8 GeV is shown.
0.8 mm ÷ 04
0
-0,4
O8 O4 0 -04
0.5
/:~ (Gev/c)
1.0
Fig. 5. The polarised photon asymmetries ~+ and ~- for zr+ and if- photoproduction respectively. The solid curves are the results for F_~ = 3.4 GeV. The data are from ref. [28]: oE~= 3 GeV, eE)/= 3.4 GeV.
E.Reya, Pion photoproduction
201
OB ! ÷ 0,4 O
-0.4
O.4 0
-0.4 J
0
0.5
~/~(GeV/c)
1'0
Fig. 6. P r e d i c t i o n s of the a s y m m e t r i e s N+ and N - at E,,j = 16 GeV.
O.tl X(t)
E~= 5 6eV 081
Eli=Boer
0,4
-0.4 -04
"I
-0.8
0 0.8 A(t) 0,4
0'5
,/~-(Qev/c)
110
E~t6 6or Fig. 8. Prechction o f A (t) f o r E~--- 8 G e V
0 -0.4.
-0|
o'.,
J~(e.v/c)
1'.o
Fig. 7. The Left-right a s y m m e t r y A (t) i n ~ + photoproductlon f o r m a p o i a r i s e d t a r g e t at E ~ = 5 and 16 GeV c o m p a r e d with the data f r o m ref. [29].
202
E.Reya, Pion photoproduction
F i n a l l y we w o u l d l i k e to r e - e m p h a s i z e t h a t , a s one w o u l d e x p e c t , t h e l o w l y i n g B a n d A 1 m e s o n s t u r n out to b e r a t h e r u n i m p o r t a n t , e s p e c i a l l y at h i g h p h o t o n e n e r g i e s . T h e B - t r a j e c t o r y c o n t r i b u t e s o n l y a f e w p e r c e n t to t h e c r o s s s e c t i o n , a n d by n e g l e c t i n g , f o r i n s t a n c e , t h e A 1 c o n t r i b u t i o n , i . e . p u t t i n g f [ 1 = 0, o n l y t h e r e s u l t s f o r t h e a s y m m e t r y p a r a m e t e r s at low e n e r g i e s a l t e r s l i g h t l y . T h e r e f o r e , t h o u g h we h a v e n o m i n a l l y s i x f r e e p a r a m e t e r s o n l y f o u r of t h e m a r e r e s p o n s i b l e f o r the r e s u l t s of t h i s m o d e l .
5. CONCLUSIONS U s i n g t h e V e n e z i a n o f i v e - p o i n t a m p l i t u d e s with a p p r o p r i a t e l y i n t r o d u c e d f i x e d J - p l a n e p o l e s we a r e a b l e to o b t a i n a f a i r l y c o n s i s t e n t a n d r e a s o n a b l e d e s c r i p t i o n of c h a r g e d p i o n p h o t o p r o d u c t i o n . A p a r t f r o m t h e u n s a t i s f a c t o r y r e s u l t f o r t h e n-/Tr + p r o d u c t i o n r a t i o a t a p h o t o n e n e r g y of 16 GeV, t h e s u c c e s s of t h i s m o d e l in e x p l a i n i n g t h e d a t a with m o v i n g R e g g e p o l e s a n d f i x e d J - p l a n e p o l e s only s u g g e s t s t h a t f i x e d p o l e s a r e i n d e e d a b l e to a c c o u n t f o r p h e n o m e n a w h i c h c u r r e n t l y a r e t h o u g h t to n e c e s s i t a t e c u t s . T h e r e f o r e t h e f i x e d p o l e s r e v i v e t h e h o p e t h a t R e g g e c u t s a r e u n i m p o r t a n t at l e a s t a t h i g h e n e r g i e s not only in p h o t o p r o d u c t i o n but a l s o in p u r e h a d r o n i c p r o c e s s e s like n-p charge exchange or n-N scattering. The relatively simple propert i e s of s u c h p o l e s a n d t h e i r e c o n o m i c a l p a r a m e t r i z a t i o n a r e s u f f i c i e n t r e a sons for investigating these possibilities. I w o u l d l i k e to e x p r e s s m y t h a n k s to P r o f e s s o r A. D o n n a c h i e a n d D r s . R. P. B a j p a l , B. C a r r e r a s , J. F. L. H o p k i n s o n a n d J. N. J. W h i t e f o r m a n y discussions and useful remarks.
REFERENCES [1] J. S. BaH, W.R. F r a z e r and M. Jacob, Phys. Rev. L e t t e r s 20 (1968) 518; F. S. Henyey, Phys. Rev. 170 (1968) 1619. [2] J. Froy[and and D. Gordon, Phys. Rev. 177 (1969) 2500; M. L. B[aekmon, G. ICramer and K. Schil[ing, NucL Phys. B12 {1969) 495; J . D . Jackson and C.Quigg, Phys. L e t t e r s 29B (1969) 236, see a[so r e f e r e n c e s therein. [3] D. Amati, G. Cohen-Tannoudji, R. Jengo and Ph. Sahn, Phys. L e t t e r s 26B (1968) 510. [4] I. Bender, H.G. Dosch and B. J. Rothe, Nuovo Clmento 62A (1969) 1026. [5] D. J. Gross and H. Page[s, Phys. Rev. L e t t e r s 20 (1968) 961. [6] J. B. Bronzan, I.S. Gerstem, B.W. Lee and F. E. Low, Phys. Rev. 157 (1967) 1448, [7] V. Singh, Phys. Rev. L e t t e r s 18 (1967) 36. [8] H.G. Dosch, Phys. Rev. 173 (1968) 1595. [9] J. S. BaH and M. Jaeob, Nuovo Cimento 54A (1968) 620. [10] R. Dashen and S. Y. Lee, Phys. Rev. L e t t e r s 22 (1969) 366. [11] N. Dombey, Phys. L e t t e r s 30B (1969) 646. [12] P. F i n k i e r , Phys. Rev. DI (1970) 1172.
E.Reya, Pion pholoproduction
203
[13] Chan Hong-Mo, Review paper p r e s e n t e d at the Royal Society Meeting on Duahtyreggeons and resonances in e l e m e n t a r y particle p r o c e s s e s , 1969, and CERN p r e p r i n t TH. 1057. [14] M. Ahmad, Fayyazuddm and Riazuddm, Phys. Rev. L e t t e r s 23 (1969) 504; F . D r a g o , Nuovo Cimento L e t t e r s 2 (1969) 712. [15] I. Bender and H. J. Rothe, Nuovo Clmento L et t er s 2 {1969) 477. [16] J . P . Ader, M. Capdeville and Ph. Salin, Nucl. Phys. B17 (1970) 221. [17] I. Bender, H . J . Rothe, H.G. Dosch and V. F. Miiller, Nuovo Cimento Let t er s 3 (1970) 625. [18] G. F. Chew, M. L. Goldberger, F . E . Low and Y. Nambu, Phys. Rev. 106 (1957) 1345. [19] M. Gell-Mann, M.L. Goldberger, F . E . Low, E. Marx and F. Zachariasen, Phys. Rev. B133 {1964) 145. [20] P.Stlchel, Z. Phys. 180 (1964) 170. [21] J. D. Jackson and C.Quigg, U m v e r s i t y of California Radiation Laboratory p r e print UCRL-19410, 1970. [22] H. Sugawara, Tokyo University of Education p r e p r m t T U E P - 6 9 - 4 , unpubhshed. [23] I. Ohba, Prog. Theor. Phys. 42 (1969) 432. [24] M. Ademollo and E . D e ! Giudice, Nuovo Cimento 63A (1969) 639; Nuovo Clmento L e t t e r s 2 (1969) 345. [25] A. Bialas and S. Pokorski, Nucl. Phys. B10 0969) 399. [26] A. M. Boyarski, F. Bulos, W. Busza, R. Diebold, S.D. Ecklund, G.E. F i s c h e r , J. R. Rees and B. Richter, Phys. Rev. L e t t e r s 20 (1968) 300; Contribution to the 14th Int. Conf. on high-energy physics, Vienna, 1968 paper 430, unpublished; P. Heide, U. KDtz, R . A . L e w i s , P. Schmfiser, H . J . Skronn and H. Wahl, Phys. Rev. L e t t e r s 21 (1968) 248. [27] A. M. Boyarksi, R. Diebold, S.D. Ecklund, G.E. F i s c h e r , Y. Murata, B. Richter and W. S. C . W il li a m s , Phys. Rev. L e t t e r s 21 (1968) 1767; P. H e i d e e t a l . , ref. [26]; Z. B a r - Y a m , J. dePagter, M.M. Hoenig, W. Kern, D. Luckey and L. S. Osborne, Phys. Rev. L e t t e r s 19 (1967) 40. [28] C. Geweniger, P. Heide, U. K~tz, R.A. Lewis, P. SchmUser, H . J . Skronn, H. Wahl and K. Wegener, Phys. L e t t e r s 29B (1969) 41; F u r t h e r res u l ts from both the DESY and CEA experiments have been reported by K. LUbelsmeyer, P r o c . 4th Int. Symp. on electron and photon intereactions at high en erg i es , Liverpool, 1969, ed. D.W. Braben (Daresbury, 1969) p.45. [29] C. C. Morehouse, M. Borghini, O. Chamberlain, R. Fuzesy, W. Gorn, T. PoweIl, P. Robrish, S. Rock, S. Shannon, G. Shapiro, H. Weisberg, A. Boyarski, S. Ecklund, Y. Murata, B. Richter and R. Siemann, Phys. Rev. Let t er s 25 (1970) 835.