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Strip equation in pagels approximation and composite nucleon
8 . ~ A . 8 Nuctear Physics B l l (1969) 439-448. North-Holland Publ. C o m p . , A m s t e r d a m
STRIP EQUATION IN PAGELS APPROXIMATION A N D COMPOSITE NUCLEON Salil ROY Indian As." ocialion f o r the C u l t i c a t i o n Of Scienc('. J a d a v p u r
and
P a d m a n a b h a DASGUPTA Saha I n s t i t u t e o f N u c l e a r I~ltysics. Calcutta 9
R e c e i v c d 24 May 1968
Abstract: Pagels, method is adopted with some modification to solving the new s t r i p approximate N / D equations. The method is applied to generate the nucleon t r a jectory in the rrN system from N, N*, and p e x c h a n g e s in the c r o s s e d channels. Enormous simplification of the computational task could be achieved by replacing the Pagels pole fitting by Gaussian interpolation. Reasonablevalue of the nuc'leon mass and the slope of the trajectory has been obtained for a small range of values of the strip parameter.
1. INTRODUCTION
In t h i s p a p e r , we p r e s e n t a d y n a m i c a l c a l c u l a t i o n of t h e n u c l e o n R e g g e t r a j e c t o r y b a s e d on a n e w s t r i p a p p r o x i m a t i o n . In t h e new s t r i p a p p r o x i m a t i o n [1], u n i t a r i t y i s i n t r o d u c e d t h r o u g h t h e N / D m e t h o d , not t h r o u g h t h e M a n d e l s t a m i t e r a t i o n p r o c e d u r e of t h e o l d e r s t r i p a p p r o x i m a t i o n . But e x a c t e l a s t i c u n i t a r i t y i s r e q u i r e d to hold only w i t h i n a l o w - e n e r g y s t r i p , the s t r i p of d o m i n a n c e of t h e d i r e c t c h a n n e l R e g ge p o l e s , t h e h i g h - e n e r g y p a r t b e i n g w e l l t a k e n into a c c o u n t by t h e c r o s s e d channel Regge poles. The resulting N/D equation are non-Fredholm and special methods, such as the Wiener-Hopf technique, are required to solve them. Some times back, Pagels [2] has developed a convenient method for solving the usual N/D equations. The method is to replace the dispersion relations by a set of linear algebraic equations which can be solved by matrix inversion. We shall adopt this method to the strip equation for the pion-nucleon system. Since we shall be interested in calculating Regge trajectory, which implies we have to deal with non-integer values of angular momentum, and also because of the rather complicated structure of the pion-nucleon partial wave singularities, we find it convenient to employ the trick of Gaussian
440
S. ROY and P. DASGUPTA
interpolation instead of the direct pole-fitting procedure of Pagels to obtain the linear algebraic equations. The previous attempts of generating the nucleon as a pion-nucleon bound state have shown convincingly that just as the nucleon exchange alone can account for the (3,3) resonance, the exchange of N* in its turn provides the major part of the force which binds the nucleon. The most important tchannel contribution comes from the p-meson, which is the most conspicuous Regge trajectory in the 7T~ system. Therefore in our dynamics we include only the forces arising from the exchange of the nucleon trajectory, the N* trajectory in the u-channel and the p trajectory in the t-channel. One may use either the Khuri representation or the Chew-Jones representation for the contributions of these Regge exchanges to obtain the generalized potential. However, we shall simplify the present calculation by taking the fixed-pole approximation for these contributions. Or in other words, simple Born contributions will constitute the input in the strip equations for our calculation.
2. STRIP EQUATIONS In the new strip approximation [ I] elastic unitarity is not assumed over the entire physical cut, but over a finite strip starting from the threshold. The pion-nucleon partial wave amplitude has the following singularity structure. The nucleon exchange give r i s e to the branch cuts: - ( M 2+2/j.2)½ ~ W 5 - ( M 2- /~2)/M,
(i)
( M 2- ll2)/M ~ W "-~(M2+2/j2) ½ ,
(ii)
(iii) -i~o to0 ,
and
0toi~o .
T h e crossed pion-nucleon cuts are (i)
-(M-
(ii) The cuts associated
-i~ to0
,
and
with/-channel (i)
Circular
(ii) The dispersion written as
/~) < W -~ M + / I
,
0 toi~
exchanges
are
c u t a l o n g !W[ = M 2 - /~2 ,
- i ~ t o i~o .
relation for the /th partial-wave
1
n (w) = ~ ( W )
.
-Wt
+ -~, J" _ W1
Im
dW'
amplitude
BI(W' )
W'
W -
1 +-
Bl(W )
W1 .f
7r W t
can be
Im BI(W') dW'
W'- W
(I)
STRIP EQUATION
441
H e r e , B~I(W) i s t h e g e n e r a l i z e d p o t e n t i a l o b t a i n e d f r o m t h e c r o s s e d c h a n n e l e x c h a n g e s , Bl(W ) i s a s s u m e d to s a t i s f y e l a s t i c u n i t a r i t y in t h e s t r i p : Wt2 -'~ W2 ~- W12. T h a t i s Im (Bl(W))- 1 = _p/(W) , O/(W) : (E- M)(k/W) 2l- 1 ,
(2)
if
2 2 2 Wt ~ W < W 1 , where k denotes the centre-of-mass
momentum.
We w r i t e
B l( W) : N l( W) / Dl( W) ,
(3)
w h e r e Nl(W ) a n d Dl(W) s h a r e t h e c u t s of B/(W), a n d D/(W) h a s c u t s f o r - W 1 < W < - W t a n d W t g W < W 1. We c a l l t h e s e c u t s t h e R - c u t s . F u r t h e r , Nl(W) h a s t h e r e s t of t h e s i n g u l a r i t i e s (cuts) of Bl(W ). W e c a l l t h e s e c u t s t h e L - c u t s . B e c a u s e of t h e s e cut s t r u c t u r e s of DI(W ) a n d NI(W) we c a n write
gl(W ) : 1 .fdW, Im N/(W') ~' L
Dl(W ) : 1 +
W'- W
'
(4)
w - Wo f dW' - - Im DI(W' ) 7r R (W' - W ) ( W ' - - ~ o i '
w h e r e Wo i s the s u b t r a c t i o n p o i n t w h e r e we a s s u m e Dl(W) to b e e q u a l to u n i t y . W i t h t h i s c h o i c e , N / D s e p a r a t i o n of Bl(W ) i s u n i q u e . E q s . (4) a n d (5) c a n b e t r a n s f o r m e d to t h e U r e t s k y t y p e of e q u a t i o n s a n d t h e r e s u l t i n g i n t e g r a l e q u a t i o n d o e s not p o s s e s s F r e d h o l m k e r n e l a n d t h e r e f o r e h a s to b e t r e a t e d in t h e W i e n e r - H o p f [3] m a n n e r . But we s h a l l a v o i d a l l t h e s e t r o u b l e s by u s i n g t h e P a g e l s a p p r o x i m a t i o n w h i c h w i l l b e d i s c u s s e d in s e c t . 3. While continuing to complex angular momenta, the partial-wave amplit u d e i s d e f i n e d by the G r i b o v - F r o i s s a r t p r o j e c t i o n f o r m u l a . T h e p r e s e n c e of t h e u - c h a n n e l l e a d s to t w o d i f f e r e n t c o n t i n u a t i o n s , n a m e l y t h e odd a n d e v e n c o n t i n u a t i o n s . T h e odd (even) c o n t i n u a t i o n c o i n c i d e s w i t h t h e p h y s i c a l a m p l i t u d e a t odd (even) i n t e g e r v a l u e s of l. T h e s e a r e t h e r i g h t a m p l i t u d e s f o r w h i c h N / D s e p a r a t i o n c a n b e d o n e f o r c o m p l e x l. S i n c e we a r e i n t e r e s t e d in g e n e r a t i n g t h e n u c l e o n in t h e p - w a v e rrN a m p l i t u d e , we s h a l l c a r r y t h r o u g h t h e c a l c u l a t i o n f o r t h e odd a m p l i t u d e only. 3. PAGELS APPROXIMATION
F r o m e q s . (2) a n d (3) one c a n e a s i l y g e t I m Dl(W ) = -Pl(W)Nl(W) ,
(6)
442
S. ROY and P. DASGUPTA
so that the expression for
DI(W)
becomes
NI(W')pl(W' )
W-W o
Dl(W )
= 1-
fdW' (W'-W)(W'-
Using integral representation
Dl(W)
IV- Wo : 1+
NI(W),
(4) f o r
~-
,
(7a)
IVoi
DI(W) t a k e s t h e f o r m
/KI(IV o, !V, x ) I m Nl(X)CkV
(7)
L where
~r R Using the relation 1
1
~t v - Tv " i v ' -
1
. {
w o = ~t'- ! %
v.~i.
W
Wo - w)
Iv'(w--%)/'
we e a n w r i t e KI(Wo, W,x) in a m o r e c o n v e n i e n t f o r m , v i z .
W FI(TV) x Fl(x ) W o Fl(W o) KI(Wo, W,x) - ( W - x ) ( W - Wo) + ( x - TV)(x- Woi + ( W o - X ) ( W o - W) '
(9)
with
Fl(X) = XGl(X)
x
: ~,
J'dTt ....lt.,2(lT" pl(W_')_ :~) .
(10)
P u t t i n g e x p r e s s i o n (9) f o r K / i W o , IV, x) in eq. (7) and u s i n g t h e i n t e g r a l r e p r e s e n t a t i o n (4) f o r N/(W), one g e t s an e x p r e s s i o n f o r DI(W) a s f o l l o w s D/(W) = 1 -
W FI(W)NI(W ) + WoFI(Wo)NI(Wo)
l
50 7 "
+-1 f dx Fl(x) imNl(x) ,~V V X-WoJ ~r L
(11)
A l l e q u a t i o n s up to eq. (11) a r e e x a c t . Now we i n t r o d u c e an a p p r o p r i a t e a p p r o x i m a t i o n a t t h i s s t a g e . If we c a n a p p r o x i m a t e the f u n c t i o n Gl(X ) in t h e L - c u t s by a s e t of p o l e t e r m s ( p r e f e r a b l y s i m p l e p o l e s ) t h e n t h e i n t e g r a t i o n on t h e r i g h t - h a n d s i d e of eq. (11) c a n b e p e r f o r m e d u s i n g eq. (4). A s P a g e l s s h o w e d , Gl(x) c a n i n d e e d b e r e p r o d u c e d on t h e L - c u t s b y a n u m b e r of p o l e t e r m s . If the L - c u t s a r e s i m p l e t h e n one c a n e x p e c t to f i t Gl(X) w i t h a f e w p o l e t e r m s . On t h e o t h e r h a n d if t h e L - c u t s a r e not s o s i m p l e a n d e x t e n d s o v e r a l a r g e p o r t i o n of t h e c o m p l e x W - p l a n e , one c a n n o t c l a i m t h a t Gl(X ) c a n be a p p r o x i m a t e d w i t h o~fly a few p o l e t e r m s . In t h a t c a s e one i s f o r c e d to u s e s u f f i c i e n t l y l a r g e n u m b e r of p o l e t e r m s . T o c h o o s e t h e p a r a m e t e r s of t h e p o i e t e r m s (i.e. p o l e p o s i t i o n a n d r e s i d u e s ) one w i l l h a v e to s o l v e c e r t a i n m a t c h i n g e q u a t i o n s s o t h a t t h e p o l e t e r m s r e p r o d u c e Gl(x) e x a c t l y a t c e r t a i n s e l e c t e d p o i n t s of t h e L - c u t s . T h e s e m a t c h i n g e q u a t i o n s a r e v e r y c o m p l i c a t e d if t h e n u m b e r of p o l e t e r m s i n c r e a s e .
STRI P I':QUATION
'[43
We h a v e a v o i d e d t h i s c o m p l i c a t e d p r o c e d u r e a n d s o l v e d t h e p r o b l e m ( i . e . p o l e f i t t i n g of Gl(X)) by an a l t e r n a t i v e m e t h o d . If we e x a m i n e t h e e x p r e s s i o n (10) f o r Gl(X) w h i c h i s a n i n t e g r a l , we s e e t h a t the i n t e g r a l d e p e n d s on t h e x only t h r o u g h t h e d o m i n a t o r ( W ' - x ) . So if we p e r f o r m i n t e g r a t i o n u s i n g s o m e s u i t a b l e n u m e r i c a l q u a d r a t u r e f o r m u l a , t h e n i m m e d i a t e l y we g e t Gl(X) a s a s u m of s i m p l e p o l e t e r m s w h o s e p a r a m e t e r s a r e e a s i l y c a l c u l a t e d . If we u s e n - p o i n t G a u s s i a n q u a d r a t u r e f o r m u l a t h e n Gl(X ) b e c o m e s
n Apl(Wi)lli ~ Ci _. Gl(X) = ~ ,2 : x_ai , i : 1 ~ w i ( ~ - x) i: 1
(12)
where
Ci=-
A Ol(Wi)Hi .... , Tr W'. 2
a i = Wi , a n d t h e W i a r e t h e v a l u e s of W' a t t h e ith G a u s s i a n p o i n t , H i is t h e ith w e i g h t f a c t o r . F u r t h e r , A is a f a c t o r due to t h e t r a n s f o r m a t i o n i n v o l v e d in numerical integration. With these,
Fl(X) = ~ x C i i=1 x - a i
(12a)
T h i s e x p r e s s i o n (12a) f o r Fl(X) i s v a l i d o n l y on t h e L - c u t s . We u s e t h i s e x p r e s s i o n f o r Fl(X ) in t h e i n t e g r a l of t h e r i g h t - h a n d s i d e of eq. (11) a n d g e t
D l ( W ) = 1 - W F l ( W ) N l ( W ) + W o ~.'I(Wo)NI(Wo)
n
+ ~ ci i=l
f dx Im
L
W° 1
gl(x ) ] - W _ _ kx-W o xUWo
x x-a i
T h e i n t e g r a t i o n c a n b e d o n e u s i n g eq. (4), w h i c h g i v e s
D l ( W ) : 1 + W o F l ( W o ) N l ( W o ) - IV F l ( W ) N l ( W ) n + ~ Ci i=1
{WNl(W)-aiNl(ai)
)-
wo Wo_ai{WoNl(Wo)-aiNl(ai)}
.
(13)
Eq. (13) i s v a l i d f o r a l l v a l u e s of W. One can e a s i l y v e r i f y t h a t eq. (13) s a t isfies unitarity, as Im Fl(X) = - P l ( X ) / X ,
(14)
on t h e R - c u t s . On t h e L - c u t s , D l ( W ) t a k e s a v e r y s i m p l e f o r m , s i n c e we u s e eq. (12a) f o r F l ( W ) a n d F l ( W o ) . U s i n g t h i s we get on the L - c u t s
444
S. ROY ;in(I P. I)ASGUI)TA n
Dl(W) = 1 - ~
i=1
r
CiaiNi(ai)
W
Wo7
iW-ai- W o - a i J
"
T h e d i s c o n t i n u i t y of the amplitude,, Br(W) i s t h e input of t h e p r o b l e m , e q u a l to t h e d i s c o n t i n u i t y of B/v(W). ~So we h a v e on t h e L - c u t s
(15) and is
Im B/(W) = I m N I ( W ) / D I ( W ) = I m B ~ ( W ) , so that Im NI(W) = D I ( W ) I m B~r(ll ') .
(16)
P u t t i n g t h e e x p r e s s i o n f o r Im NI(W) in eq. (4) a n d u s i n g eq. (15) we get an e x p r e s s i o n f o r N/(W): NI(W)
n BI~W , _ ~
i=1
CiaiNl(ai)
~-WB~(W) - aiB~(ai) ii;-a i
L
WoB~/(W) ] W o - a i j " (17)
In eq. (17), Nl(ai) a p p e a r a s u n k n o w n q u a n t i t i e s f o r w h i c h eq. (17) is to b e s o l v e d . P u t t i n g W = a l , a 2 , . . . e t c . we g e t a s e t of n s i m u l t a n e o u s e q u a t i o n s in Nl(ai) w h i c h c a n b e s o l v e d by t h e m a t r i x i n v e r s i o n m e t h o d . O n c e we g e t the Nl(ai) , we k n o w D/(W) a n d N l ( W ) f o r a l l W a n d h e n c e t h e a m p l i tude B/(W). A s we h a v e a l r e a d y m e n t i o n e d e a r l i e r , t h e e q u a t i o n f o r N in t h e n e w s t r i p a p p r o x i m a t i o n i s s i n g u l a r . T h e u s u a l m e t h o d i s to s e p a r a t e t h e s i n g u l a r p a r t of t h e k e r n e l , s o t h a t one o b t a i n s a W i e n e r - H o p f t y p e i n t e g r a l equation whose inhomogeneous term satisfies a Fredholm equation. The n u m e r i c a l w o r k i n v o l v e d in s u c h a m e t h o d is v e r y l e n g t h y . On t h e c o n t r a r y the method we have adopted is very neat and the computational time is much l e s s . It c a n b e e a s i l y shown t h a t o u r m e t h o d c o r r e s p o n d s to t h e s o l u t i o n of i n t e g r a l e q u a t i o n by m a t r i x i n v e r s i o n m e t h o d . T h i s m e a n s t h a t t h e s i n g u l a r i n t e g r a l e q u a t i o n i n v o l v e d in n e w s t r i p a p p r o x i m a t i o n c a n r e a l l y b e s o l v e d by m a t r i x i n v e r s i o n m e t h o d . So o u r a n a l y s i s m a y b e s a i d to e x e m p l i f y t h e m a t r i x i n v e r s i o n m e t h o d s u g g e s t e d by J o n e s a n d T i k t o p o u l u s [4].
4. INPUT DYNAMICS
A c c o r d i n g to t h e h y p o t h e s i s of C h e w a n d F r a u t s c h i [5], a l l s t r o n g l y int e r a c t i n g p a r t i c l e s a r e to b e t r e a t e d on t h e s a m e f o o t i n g in S - m a t r i x t h e o r y , t h a t i s , t h e r e i s no r e a s o n why a p a r t i c l e l i k e n u c l e o n s h o u l d b e t r e a t e d a s more elementary than the (3,3)-resonance. The masses and coupling cons t a n t s of a l l s t r o n g l y i n t e r a c t i n g p a r t i c l e s s h o u l d b e d y n a m i c a l l y d e t e r m i n e d from the self-consistency requirements. The stable particles should be generated as bound states while unstable particles should be determined as r e s o n a n c e s . Such a c o n j e c t u r e a b o u t t h e c o m p o s i t e n e s s of t h e n u c l e o n s e e m s to b e a l l t h e m o r e j u s t i f i e d when we c o n s i d e r t h e e x i s t e n c e of t h e T = ½, J = ~ rrN r e s o n a n c e (1680 MeV). T h e f o r c e s t h a t a r e r e s p o n s i b l e f o r t h i s r e s o n a n c e s h o u l d a l s o b i n d s o m e p - w a v e o b j e c t with T = ½ a n d J = ½ a n d t h e r e d u c t i o n in t h e c e n t r i f u g a l r e p u l s i o n w o u l d m a k e t h e net f o r c e e v e n
S'l'l~ll ) I':QUATION
4,15
m o r e a t t r a c t i v e . The p a r t i c l e that we indeed o b s e r v e with these quantum n u m b e r s is the nucleon. Investigations about the dynamical origin of the nucleon indicate strongly that the exchanges of N, N* and p in the crossed channels a r e capable of l I producing adequate attraction in the T = 2, J = l - ~, 7TN amplitude. We shall employ these exchanges to produce the nucleon Regge trajectory which contains, beside the nucleon (l = i), the N* (1680 MeV) resonance also. In the spirit of the new strip approximation, we should construct the generalised potential from the strip-approximate contributions of the N, N* and p t r a jectories in the crossed channels. However, we carry out the present calculation with much simpler potential, that corresponding to unreggeised exchanges, in order to avoid the complicated numerical calculations involved in a fully reggeised dynamics. At the same time, we demonstrate that although one may assume no reggeisation in the crossed channels, the direct channel gets reggeised none the less. The Born contribution of the said exchanges can easily be calculated in the x~arrow resonance approximation in the manner of Frautschi and WaI lecka [8]. The nucleon exchange potential in the /th partial wave with T = is given by
f½
l-,N
- g A [ - ( E + M)(W- 4k 2
M)Ql(aN)+(E- M)(W+M)QI_I(aN)]
(18)
where k2 = [(W+ M)2- /~2][(W-M)2- p2]
aN -
W2 - M 2 - 2 tz2 2k 2
1,
M and p being the m a s s e s of the nucleon and the pion r e s p e c t i v e l y , and the nucleon energy. The s u b s c r i p t l- signifies that we a r e considering the total angular momentum j = l - ~. ~ Finally, g 2 is the pion-nucleon coupling constant. The N* exchange gives a contribution f½ M .2 2 [_{(W+M)2_ /-,N* = ~ ~33
+[(W-M)2-/j2
p2}t3x~,I(M*+2M- W) ( 51"+ ~)~iJ 2
M * - 2 M + W~
+ (M ~ - :~i-2 - i ~
13X~q(M* + 2M+W) M'-2M-W (~::~//)2:)j-2 +(~-,~-~2
where * W2 + M*2- 2M2- 2p2 aN = 2k 2
1
I
.
Ol-l(aN)]'
*
Q/(aN)
(19)
S. I{()Y and P. I ) A S ( ; I ' I } ' I ' A
t4t~
, 2M*2(2M2 + 2/j. 2 - W 2 - M*2) x N = 1 + (:11"2-.112- /.,2)2-4M2/j.2 '
,'11" b e i n g the m a s s of N*, ) 3 3 a c o u p l i n g p a r a m e t e r d e t e r m i n e d f r o m the e x p e r i m e n t a l width of N*. T h e c o n t r i b u t i o n of the p - e x c h a n g e is g i v e n by f/_' ,P _ 16~W 21
[-[(W ~ ..1I)2 - IJ2][271(W - M) + ) , 2 ( 4 M ( W - M) - 2s
-
",<:[2)
I(W + M)
111
2
P
+ 2 M 2 - 2/,2)] ~
1
Q l ( a p ) - [ ( W - M) 2 - /.,.2] 9
+ "Y2(4.1,l(W + M) + 2s -+ m p - 2M 2 - 2 ~,2) i
1 2a,:~ Ql- l(ap)J
,
(20)
where Iil 2
a 0 = 1 + 2/" 2 . H e r e , mp is the m a s s of the P - m e s o n , Yl and ) 2 a r e d e t e r m i n e d f r o m the e l e c t r o m a g n e t i c f o r m f a c t o r s of n u c l e o n a n d s a t i s f y the r e l a t i o n )..1,/)2 v ..11./1.83 .
5. R E S U L T S We w r i t e N I D e q u a t i o n s f o r the a m p l i t u d e BI(W) , which is r e l a t e d to the a c t u a l a m p l i t u d e f l - in the f o l l o w i n g m a , m e r
le BI(W) - Ol(W ) f l - ,
(21)
where
OI(W) = (E- ,1]) " /" "~/)2l-1
(22)
With t h i s c h o i c e of the p h a s e f a c t o r OI(W), the input p o t e n t i a l B y ( W ) t a k e s the f o r m
.
(W)= p ~
[ / ~ - , N +'fi"-,N* +f;-,PJ "
(23)
We c h o o s e the s u b t r a c t i o n point at W = Wo = 0. In eq. (12) we have c h o s e n n = 32 which i m p l i e s the u s e of s i x t e e n - p o i n t G a u s s i a n q u a d r a t u r e f o r m u l a for the i n t e g r a l s in eq. (10). T h u s the f u n c t i o n Cl(X) has b e e n a p p r o x i m a t e d by t h i r t y - t w o - p o l e t e r m s . We take [6]
STIll P I,:QUATI()N
g r2 = 477~ N ,
where
-t47
2 ~N
Y33 = 0 . 0 6 ,
= 15 ,
~'1 = - 1 ,
M = 6.72 ,
M* = 8.8 ,
mp = 5.4 ,
a l l m a s s e s a r e e x p r e s s e d in pion m a s s units. Tahle 1 |:~estnlts o f Otll" c a l u t l l a t i o i ' L . q . IV Wl 0.5 0.6 0.7 0 .S
10.0
W1
7.04 7.46
10.5
I4'1
(i .66 7.16 7.57 -
-
11.0
6.26 6.~1 7.30 7.70
We h a v e c a l c u l a t e d the n u c l e o n t r a j e c t o r y f o r t h r e e d i f f e r e n t v a l u e s of the s t r i p p a r a m e t e r and the r e s u l t s a r e g i v e n in t a b l e 1 and fig. 1. H e r e , ce(W) s t a n d s f o r the p o s i t i o n of the R e g g e p o l e for g i v e n W. T h e '-' s i g n i f i e s that the t r a j e c t o r y g o e s a b o v e the t h r e s h o l d . It can be s e e n f r o m t a b l e 1 that the , m c l e o n p o l e c o m e s n e a r e s t to the p h y s i c a l m a s s f o r W 1 = 10.5.
0,7
06 !
05
6
70
75
W F i g . 1. T h e l t e g g e - I ~ o h _ " p o s i t i o n C~ (W) f o r the n u c l e o n p l o t t e d a g a i n s t W. t h e e n e r g y . T h e c u r v e (a) c o r r e s p o n d s t o the v a l u e 1().J3 ~)f W 1 . t h e s t r i p p a r a m e t e r , and the c u ~ ' v e (h) e c n ' r e s p o n d s t o t h e v a l u e 10. () o f W 1 .
448
S. [,I(.)Y and P. DASGUPTA
T h e a u t h o r s e x p r e s s t h e i r s i n c e r e t h a n k s t o D r . P . K. R o y a n d M r . S. Mallik for many helpful discussions. We are grateful to Professor S. N. B o s e f o r t a k i n g k e e n i n t e r e s t in t h e problem.
RE FE R E N C E S [1] G , F . C h e w and C . E . J o n e s . P h y s . Rev. 134 (1964) B208. [2] t I . P a g e l s . P h y s . Rex,. 140 (1965) B1599. [3] V . l , . T e p l i t z . P h y s . Rev. 137 (1965) B138; C . E . J o n e s , Nuovo C i m e n t o 40 (1965) 761. [41 C . F . J o n e s and G . T i k t o p o u l u s . J . Math. P h y s . 7 (1966) 311. [5] G . F . C h e w and S . C . F r a u t s c h i . P h y s . Rev. L e t t e r s 7 (1961) 394. [6] J . B a l l and D. Wong. P h y s . Rev. 133 (1964) B179. [7] E . A b e r s and C . Z e m a c h . P h y s . l t e v . 131 (1963) 2305: L . K . Pande and P. N a r a y a n s w a m i . P h y s . l~ev. 136 (1964) B1760: ( ; . D . Doolen. T . K a n k i and A . T u b i s . P h y s . Rex:. 142 (1!}66) B1072. [8] S . C . F r a u t s c h i and 3 . D . W a l e c k a . P h y s . Rev. 120 (1960) 1486.
STRIP EQUATION IN PAGELS APPROXIMATION A N D COMPOSITE NUCLEON Salil ROY Indian As." ocialion f o r the C u l t i c a t i o n Of Scienc('. J a d a v p u r
and
P a d m a n a b h a DASGUPTA Saha I n s t i t u t e o f N u c l e a r I~ltysics. Calcutta 9
R e c e i v c d 24 May 1968
Abstract: Pagels, method is adopted with some modification to solving the new s t r i p approximate N / D equations. The method is applied to generate the nucleon t r a jectory in the rrN system from N, N*, and p e x c h a n g e s in the c r o s s e d channels. Enormous simplification of the computational task could be achieved by replacing the Pagels pole fitting by Gaussian interpolation. Reasonablevalue of the nuc'leon mass and the slope of the trajectory has been obtained for a small range of values of the strip parameter.
1. INTRODUCTION
In t h i s p a p e r , we p r e s e n t a d y n a m i c a l c a l c u l a t i o n of t h e n u c l e o n R e g g e t r a j e c t o r y b a s e d on a n e w s t r i p a p p r o x i m a t i o n . In t h e new s t r i p a p p r o x i m a t i o n [1], u n i t a r i t y i s i n t r o d u c e d t h r o u g h t h e N / D m e t h o d , not t h r o u g h t h e M a n d e l s t a m i t e r a t i o n p r o c e d u r e of t h e o l d e r s t r i p a p p r o x i m a t i o n . But e x a c t e l a s t i c u n i t a r i t y i s r e q u i r e d to hold only w i t h i n a l o w - e n e r g y s t r i p , the s t r i p of d o m i n a n c e of t h e d i r e c t c h a n n e l R e g ge p o l e s , t h e h i g h - e n e r g y p a r t b e i n g w e l l t a k e n into a c c o u n t by t h e c r o s s e d channel Regge poles. The resulting N/D equation are non-Fredholm and special methods, such as the Wiener-Hopf technique, are required to solve them. Some times back, Pagels [2] has developed a convenient method for solving the usual N/D equations. The method is to replace the dispersion relations by a set of linear algebraic equations which can be solved by matrix inversion. We shall adopt this method to the strip equation for the pion-nucleon system. Since we shall be interested in calculating Regge trajectory, which implies we have to deal with non-integer values of angular momentum, and also because of the rather complicated structure of the pion-nucleon partial wave singularities, we find it convenient to employ the trick of Gaussian
440
S. ROY and P. DASGUPTA
interpolation instead of the direct pole-fitting procedure of Pagels to obtain the linear algebraic equations. The previous attempts of generating the nucleon as a pion-nucleon bound state have shown convincingly that just as the nucleon exchange alone can account for the (3,3) resonance, the exchange of N* in its turn provides the major part of the force which binds the nucleon. The most important tchannel contribution comes from the p-meson, which is the most conspicuous Regge trajectory in the 7T~ system. Therefore in our dynamics we include only the forces arising from the exchange of the nucleon trajectory, the N* trajectory in the u-channel and the p trajectory in the t-channel. One may use either the Khuri representation or the Chew-Jones representation for the contributions of these Regge exchanges to obtain the generalized potential. However, we shall simplify the present calculation by taking the fixed-pole approximation for these contributions. Or in other words, simple Born contributions will constitute the input in the strip equations for our calculation.
2. STRIP EQUATIONS In the new strip approximation [ I] elastic unitarity is not assumed over the entire physical cut, but over a finite strip starting from the threshold. The pion-nucleon partial wave amplitude has the following singularity structure. The nucleon exchange give r i s e to the branch cuts: - ( M 2+2/j.2)½ ~ W 5 - ( M 2- /~2)/M,
(i)
( M 2- ll2)/M ~ W "-~(M2+2/j2) ½ ,
(ii)
(iii) -i~o to0 ,
and
0toi~o .
T h e crossed pion-nucleon cuts are (i)
-(M-
(ii) The cuts associated
-i~ to0
,
and
with/-channel (i)
Circular
(ii) The dispersion written as
/~) < W -~ M + / I
,
0 toi~
exchanges
are
c u t a l o n g !W[ = M 2 - /~2 ,
- i ~ t o i~o .
relation for the /th partial-wave
1
n (w) = ~ ( W )
.
-Wt
+ -~, J" _ W1
Im
dW'
amplitude
BI(W' )
W'
W -
1 +-
Bl(W )
W1 .f
7r W t
can be
Im BI(W') dW'
W'- W
(I)
STRIP EQUATION
441
H e r e , B~I(W) i s t h e g e n e r a l i z e d p o t e n t i a l o b t a i n e d f r o m t h e c r o s s e d c h a n n e l e x c h a n g e s , Bl(W ) i s a s s u m e d to s a t i s f y e l a s t i c u n i t a r i t y in t h e s t r i p : Wt2 -'~ W2 ~- W12. T h a t i s Im (Bl(W))- 1 = _p/(W) , O/(W) : (E- M)(k/W) 2l- 1 ,
(2)
if
2 2 2 Wt ~ W < W 1 , where k denotes the centre-of-mass
momentum.
We w r i t e
B l( W) : N l( W) / Dl( W) ,
(3)
w h e r e Nl(W ) a n d Dl(W) s h a r e t h e c u t s of B/(W), a n d D/(W) h a s c u t s f o r - W 1 < W < - W t a n d W t g W < W 1. We c a l l t h e s e c u t s t h e R - c u t s . F u r t h e r , Nl(W) h a s t h e r e s t of t h e s i n g u l a r i t i e s (cuts) of Bl(W ). W e c a l l t h e s e c u t s t h e L - c u t s . B e c a u s e of t h e s e cut s t r u c t u r e s of DI(W ) a n d NI(W) we c a n write
gl(W ) : 1 .fdW, Im N/(W') ~' L
Dl(W ) : 1 +
W'- W
'
(4)
w - Wo f dW' - - Im DI(W' ) 7r R (W' - W ) ( W ' - - ~ o i '
w h e r e Wo i s the s u b t r a c t i o n p o i n t w h e r e we a s s u m e Dl(W) to b e e q u a l to u n i t y . W i t h t h i s c h o i c e , N / D s e p a r a t i o n of Bl(W ) i s u n i q u e . E q s . (4) a n d (5) c a n b e t r a n s f o r m e d to t h e U r e t s k y t y p e of e q u a t i o n s a n d t h e r e s u l t i n g i n t e g r a l e q u a t i o n d o e s not p o s s e s s F r e d h o l m k e r n e l a n d t h e r e f o r e h a s to b e t r e a t e d in t h e W i e n e r - H o p f [3] m a n n e r . But we s h a l l a v o i d a l l t h e s e t r o u b l e s by u s i n g t h e P a g e l s a p p r o x i m a t i o n w h i c h w i l l b e d i s c u s s e d in s e c t . 3. While continuing to complex angular momenta, the partial-wave amplit u d e i s d e f i n e d by the G r i b o v - F r o i s s a r t p r o j e c t i o n f o r m u l a . T h e p r e s e n c e of t h e u - c h a n n e l l e a d s to t w o d i f f e r e n t c o n t i n u a t i o n s , n a m e l y t h e odd a n d e v e n c o n t i n u a t i o n s . T h e odd (even) c o n t i n u a t i o n c o i n c i d e s w i t h t h e p h y s i c a l a m p l i t u d e a t odd (even) i n t e g e r v a l u e s of l. T h e s e a r e t h e r i g h t a m p l i t u d e s f o r w h i c h N / D s e p a r a t i o n c a n b e d o n e f o r c o m p l e x l. S i n c e we a r e i n t e r e s t e d in g e n e r a t i n g t h e n u c l e o n in t h e p - w a v e rrN a m p l i t u d e , we s h a l l c a r r y t h r o u g h t h e c a l c u l a t i o n f o r t h e odd a m p l i t u d e only. 3. PAGELS APPROXIMATION
F r o m e q s . (2) a n d (3) one c a n e a s i l y g e t I m Dl(W ) = -Pl(W)Nl(W) ,
(6)
442
S. ROY and P. DASGUPTA
so that the expression for
DI(W)
becomes
NI(W')pl(W' )
W-W o
Dl(W )
= 1-
fdW' (W'-W)(W'-
Using integral representation
Dl(W)
IV- Wo : 1+
NI(W),
(4) f o r
~-
,
(7a)
IVoi
DI(W) t a k e s t h e f o r m
/KI(IV o, !V, x ) I m Nl(X)CkV
(7)
L where
~r R Using the relation 1
1
~t v - Tv " i v ' -
1
. {
w o = ~t'- ! %
v.~i.
W
Wo - w)
Iv'(w--%)/'
we e a n w r i t e KI(Wo, W,x) in a m o r e c o n v e n i e n t f o r m , v i z .
W FI(TV) x Fl(x ) W o Fl(W o) KI(Wo, W,x) - ( W - x ) ( W - Wo) + ( x - TV)(x- Woi + ( W o - X ) ( W o - W) '
(9)
with
Fl(X) = XGl(X)
x
: ~,
J'dTt ....lt.,2(lT" pl(W_')_ :~) .
(10)
P u t t i n g e x p r e s s i o n (9) f o r K / i W o , IV, x) in eq. (7) and u s i n g t h e i n t e g r a l r e p r e s e n t a t i o n (4) f o r N/(W), one g e t s an e x p r e s s i o n f o r DI(W) a s f o l l o w s D/(W) = 1 -
W FI(W)NI(W ) + WoFI(Wo)NI(Wo)
l
50 7 "
+-1 f dx Fl(x) imNl(x) ,~V V X-WoJ ~r L
(11)
A l l e q u a t i o n s up to eq. (11) a r e e x a c t . Now we i n t r o d u c e an a p p r o p r i a t e a p p r o x i m a t i o n a t t h i s s t a g e . If we c a n a p p r o x i m a t e the f u n c t i o n Gl(X ) in t h e L - c u t s by a s e t of p o l e t e r m s ( p r e f e r a b l y s i m p l e p o l e s ) t h e n t h e i n t e g r a t i o n on t h e r i g h t - h a n d s i d e of eq. (11) c a n b e p e r f o r m e d u s i n g eq. (4). A s P a g e l s s h o w e d , Gl(x) c a n i n d e e d b e r e p r o d u c e d on t h e L - c u t s b y a n u m b e r of p o l e t e r m s . If the L - c u t s a r e s i m p l e t h e n one c a n e x p e c t to f i t Gl(X) w i t h a f e w p o l e t e r m s . On t h e o t h e r h a n d if t h e L - c u t s a r e not s o s i m p l e a n d e x t e n d s o v e r a l a r g e p o r t i o n of t h e c o m p l e x W - p l a n e , one c a n n o t c l a i m t h a t Gl(X ) c a n be a p p r o x i m a t e d w i t h o~fly a few p o l e t e r m s . In t h a t c a s e one i s f o r c e d to u s e s u f f i c i e n t l y l a r g e n u m b e r of p o l e t e r m s . T o c h o o s e t h e p a r a m e t e r s of t h e p o i e t e r m s (i.e. p o l e p o s i t i o n a n d r e s i d u e s ) one w i l l h a v e to s o l v e c e r t a i n m a t c h i n g e q u a t i o n s s o t h a t t h e p o l e t e r m s r e p r o d u c e Gl(x) e x a c t l y a t c e r t a i n s e l e c t e d p o i n t s of t h e L - c u t s . T h e s e m a t c h i n g e q u a t i o n s a r e v e r y c o m p l i c a t e d if t h e n u m b e r of p o l e t e r m s i n c r e a s e .
STRI P I':QUATION
'[43
We h a v e a v o i d e d t h i s c o m p l i c a t e d p r o c e d u r e a n d s o l v e d t h e p r o b l e m ( i . e . p o l e f i t t i n g of Gl(X)) by an a l t e r n a t i v e m e t h o d . If we e x a m i n e t h e e x p r e s s i o n (10) f o r Gl(X) w h i c h i s a n i n t e g r a l , we s e e t h a t the i n t e g r a l d e p e n d s on t h e x only t h r o u g h t h e d o m i n a t o r ( W ' - x ) . So if we p e r f o r m i n t e g r a t i o n u s i n g s o m e s u i t a b l e n u m e r i c a l q u a d r a t u r e f o r m u l a , t h e n i m m e d i a t e l y we g e t Gl(X) a s a s u m of s i m p l e p o l e t e r m s w h o s e p a r a m e t e r s a r e e a s i l y c a l c u l a t e d . If we u s e n - p o i n t G a u s s i a n q u a d r a t u r e f o r m u l a t h e n Gl(X ) b e c o m e s
n Apl(Wi)lli ~ Ci _. Gl(X) = ~ ,2 : x_ai , i : 1 ~ w i ( ~ - x) i: 1
(12)
where
Ci=-
A Ol(Wi)Hi .... , Tr W'. 2
a i = Wi , a n d t h e W i a r e t h e v a l u e s of W' a t t h e ith G a u s s i a n p o i n t , H i is t h e ith w e i g h t f a c t o r . F u r t h e r , A is a f a c t o r due to t h e t r a n s f o r m a t i o n i n v o l v e d in numerical integration. With these,
Fl(X) = ~ x C i i=1 x - a i
(12a)
T h i s e x p r e s s i o n (12a) f o r Fl(X) i s v a l i d o n l y on t h e L - c u t s . We u s e t h i s e x p r e s s i o n f o r Fl(X ) in t h e i n t e g r a l of t h e r i g h t - h a n d s i d e of eq. (11) a n d g e t
D l ( W ) = 1 - W F l ( W ) N l ( W ) + W o ~.'I(Wo)NI(Wo)
n
+ ~ ci i=l
f dx Im
L
W° 1
gl(x ) ] - W _ _ kx-W o xUWo
x x-a i
T h e i n t e g r a t i o n c a n b e d o n e u s i n g eq. (4), w h i c h g i v e s
D l ( W ) : 1 + W o F l ( W o ) N l ( W o ) - IV F l ( W ) N l ( W ) n + ~ Ci i=1
{WNl(W)-aiNl(ai)
)-
wo Wo_ai{WoNl(Wo)-aiNl(ai)}
.
(13)
Eq. (13) i s v a l i d f o r a l l v a l u e s of W. One can e a s i l y v e r i f y t h a t eq. (13) s a t isfies unitarity, as Im Fl(X) = - P l ( X ) / X ,
(14)
on t h e R - c u t s . On t h e L - c u t s , D l ( W ) t a k e s a v e r y s i m p l e f o r m , s i n c e we u s e eq. (12a) f o r F l ( W ) a n d F l ( W o ) . U s i n g t h i s we get on the L - c u t s
444
S. ROY ;in(I P. I)ASGUI)TA n
Dl(W) = 1 - ~
i=1
r
CiaiNi(ai)
W
Wo7
iW-ai- W o - a i J
"
T h e d i s c o n t i n u i t y of the amplitude,, Br(W) i s t h e input of t h e p r o b l e m , e q u a l to t h e d i s c o n t i n u i t y of B/v(W). ~So we h a v e on t h e L - c u t s
(15) and is
Im B/(W) = I m N I ( W ) / D I ( W ) = I m B ~ ( W ) , so that Im NI(W) = D I ( W ) I m B~r(ll ') .
(16)
P u t t i n g t h e e x p r e s s i o n f o r Im NI(W) in eq. (4) a n d u s i n g eq. (15) we get an e x p r e s s i o n f o r N/(W): NI(W)
n BI~W , _ ~
i=1
CiaiNl(ai)
~-WB~(W) - aiB~(ai) ii;-a i
L
WoB~/(W) ] W o - a i j " (17)
In eq. (17), Nl(ai) a p p e a r a s u n k n o w n q u a n t i t i e s f o r w h i c h eq. (17) is to b e s o l v e d . P u t t i n g W = a l , a 2 , . . . e t c . we g e t a s e t of n s i m u l t a n e o u s e q u a t i o n s in Nl(ai) w h i c h c a n b e s o l v e d by t h e m a t r i x i n v e r s i o n m e t h o d . O n c e we g e t the Nl(ai) , we k n o w D/(W) a n d N l ( W ) f o r a l l W a n d h e n c e t h e a m p l i tude B/(W). A s we h a v e a l r e a d y m e n t i o n e d e a r l i e r , t h e e q u a t i o n f o r N in t h e n e w s t r i p a p p r o x i m a t i o n i s s i n g u l a r . T h e u s u a l m e t h o d i s to s e p a r a t e t h e s i n g u l a r p a r t of t h e k e r n e l , s o t h a t one o b t a i n s a W i e n e r - H o p f t y p e i n t e g r a l equation whose inhomogeneous term satisfies a Fredholm equation. The n u m e r i c a l w o r k i n v o l v e d in s u c h a m e t h o d is v e r y l e n g t h y . On t h e c o n t r a r y the method we have adopted is very neat and the computational time is much l e s s . It c a n b e e a s i l y shown t h a t o u r m e t h o d c o r r e s p o n d s to t h e s o l u t i o n of i n t e g r a l e q u a t i o n by m a t r i x i n v e r s i o n m e t h o d . T h i s m e a n s t h a t t h e s i n g u l a r i n t e g r a l e q u a t i o n i n v o l v e d in n e w s t r i p a p p r o x i m a t i o n c a n r e a l l y b e s o l v e d by m a t r i x i n v e r s i o n m e t h o d . So o u r a n a l y s i s m a y b e s a i d to e x e m p l i f y t h e m a t r i x i n v e r s i o n m e t h o d s u g g e s t e d by J o n e s a n d T i k t o p o u l u s [4].
4. INPUT DYNAMICS
A c c o r d i n g to t h e h y p o t h e s i s of C h e w a n d F r a u t s c h i [5], a l l s t r o n g l y int e r a c t i n g p a r t i c l e s a r e to b e t r e a t e d on t h e s a m e f o o t i n g in S - m a t r i x t h e o r y , t h a t i s , t h e r e i s no r e a s o n why a p a r t i c l e l i k e n u c l e o n s h o u l d b e t r e a t e d a s more elementary than the (3,3)-resonance. The masses and coupling cons t a n t s of a l l s t r o n g l y i n t e r a c t i n g p a r t i c l e s s h o u l d b e d y n a m i c a l l y d e t e r m i n e d from the self-consistency requirements. The stable particles should be generated as bound states while unstable particles should be determined as r e s o n a n c e s . Such a c o n j e c t u r e a b o u t t h e c o m p o s i t e n e s s of t h e n u c l e o n s e e m s to b e a l l t h e m o r e j u s t i f i e d when we c o n s i d e r t h e e x i s t e n c e of t h e T = ½, J = ~ rrN r e s o n a n c e (1680 MeV). T h e f o r c e s t h a t a r e r e s p o n s i b l e f o r t h i s r e s o n a n c e s h o u l d a l s o b i n d s o m e p - w a v e o b j e c t with T = ½ a n d J = ½ a n d t h e r e d u c t i o n in t h e c e n t r i f u g a l r e p u l s i o n w o u l d m a k e t h e net f o r c e e v e n
S'l'l~ll ) I':QUATION
4,15
m o r e a t t r a c t i v e . The p a r t i c l e that we indeed o b s e r v e with these quantum n u m b e r s is the nucleon. Investigations about the dynamical origin of the nucleon indicate strongly that the exchanges of N, N* and p in the crossed channels a r e capable of l I producing adequate attraction in the T = 2, J = l - ~, 7TN amplitude. We shall employ these exchanges to produce the nucleon Regge trajectory which contains, beside the nucleon (l = i), the N* (1680 MeV) resonance also. In the spirit of the new strip approximation, we should construct the generalised potential from the strip-approximate contributions of the N, N* and p t r a jectories in the crossed channels. However, we carry out the present calculation with much simpler potential, that corresponding to unreggeised exchanges, in order to avoid the complicated numerical calculations involved in a fully reggeised dynamics. At the same time, we demonstrate that although one may assume no reggeisation in the crossed channels, the direct channel gets reggeised none the less. The Born contribution of the said exchanges can easily be calculated in the x~arrow resonance approximation in the manner of Frautschi and WaI lecka [8]. The nucleon exchange potential in the /th partial wave with T = is given by
f½
l-,N
- g A [ - ( E + M)(W- 4k 2
M)Ql(aN)+(E- M)(W+M)QI_I(aN)]
(18)
where k2 = [(W+ M)2- /~2][(W-M)2- p2]
aN -
W2 - M 2 - 2 tz2 2k 2
1,
M and p being the m a s s e s of the nucleon and the pion r e s p e c t i v e l y , and the nucleon energy. The s u b s c r i p t l- signifies that we a r e considering the total angular momentum j = l - ~. ~ Finally, g 2 is the pion-nucleon coupling constant. The N* exchange gives a contribution f½ M .2 2 [_{(W+M)2_ /-,N* = ~ ~33
+[(W-M)2-/j2
p2}t3x~,I(M*+2M- W) ( 51"+ ~)~iJ 2
M * - 2 M + W~
+ (M ~ - :~i-2 - i ~
13X~q(M* + 2M+W) M'-2M-W (~::~//)2:)j-2 +(~-,~-~2
where * W2 + M*2- 2M2- 2p2 aN = 2k 2
1
I
.
Ol-l(aN)]'
*
Q/(aN)
(19)
S. I{()Y and P. I ) A S ( ; I ' I } ' I ' A
t4t~
, 2M*2(2M2 + 2/j. 2 - W 2 - M*2) x N = 1 + (:11"2-.112- /.,2)2-4M2/j.2 '
,'11" b e i n g the m a s s of N*, ) 3 3 a c o u p l i n g p a r a m e t e r d e t e r m i n e d f r o m the e x p e r i m e n t a l width of N*. T h e c o n t r i b u t i o n of the p - e x c h a n g e is g i v e n by f/_' ,P _ 16~W 21
[-[(W ~ ..1I)2 - IJ2][271(W - M) + ) , 2 ( 4 M ( W - M) - 2s
-
",<:[2)
I(W + M)
111
2
P
+ 2 M 2 - 2/,2)] ~
1
Q l ( a p ) - [ ( W - M) 2 - /.,.2] 9
+ "Y2(4.1,l(W + M) + 2s -+ m p - 2M 2 - 2 ~,2) i
1 2a,:~ Ql- l(ap)J
,
(20)
where Iil 2
a 0 = 1 + 2/" 2 . H e r e , mp is the m a s s of the P - m e s o n , Yl and ) 2 a r e d e t e r m i n e d f r o m the e l e c t r o m a g n e t i c f o r m f a c t o r s of n u c l e o n a n d s a t i s f y the r e l a t i o n )..1,/)2 v ..11./1.83 .
5. R E S U L T S We w r i t e N I D e q u a t i o n s f o r the a m p l i t u d e BI(W) , which is r e l a t e d to the a c t u a l a m p l i t u d e f l - in the f o l l o w i n g m a , m e r
le BI(W) - Ol(W ) f l - ,
(21)
where
OI(W) = (E- ,1]) " /" "~/)2l-1
(22)
With t h i s c h o i c e of the p h a s e f a c t o r OI(W), the input p o t e n t i a l B y ( W ) t a k e s the f o r m
.
(W)= p ~
[ / ~ - , N +'fi"-,N* +f;-,PJ "
(23)
We c h o o s e the s u b t r a c t i o n point at W = Wo = 0. In eq. (12) we have c h o s e n n = 32 which i m p l i e s the u s e of s i x t e e n - p o i n t G a u s s i a n q u a d r a t u r e f o r m u l a for the i n t e g r a l s in eq. (10). T h u s the f u n c t i o n Cl(X) has b e e n a p p r o x i m a t e d by t h i r t y - t w o - p o l e t e r m s . We take [6]
STIll P I,:QUATI()N
g r2 = 477~ N ,
where
-t47
2 ~N
Y33 = 0 . 0 6 ,
= 15 ,
~'1 = - 1 ,
M = 6.72 ,
M* = 8.8 ,
mp = 5.4 ,
a l l m a s s e s a r e e x p r e s s e d in pion m a s s units. Tahle 1 |:~estnlts o f Otll" c a l u t l l a t i o i ' L . q . IV Wl 0.5 0.6 0.7 0 .S
10.0
W1
7.04 7.46
10.5
I4'1
(i .66 7.16 7.57 -
-
11.0
6.26 6.~1 7.30 7.70
We h a v e c a l c u l a t e d the n u c l e o n t r a j e c t o r y f o r t h r e e d i f f e r e n t v a l u e s of the s t r i p p a r a m e t e r and the r e s u l t s a r e g i v e n in t a b l e 1 and fig. 1. H e r e , ce(W) s t a n d s f o r the p o s i t i o n of the R e g g e p o l e for g i v e n W. T h e '-' s i g n i f i e s that the t r a j e c t o r y g o e s a b o v e the t h r e s h o l d . It can be s e e n f r o m t a b l e 1 that the , m c l e o n p o l e c o m e s n e a r e s t to the p h y s i c a l m a s s f o r W 1 = 10.5.
0,7
06 !
05
6
70
75
W F i g . 1. T h e l t e g g e - I ~ o h _ " p o s i t i o n C~ (W) f o r the n u c l e o n p l o t t e d a g a i n s t W. t h e e n e r g y . T h e c u r v e (a) c o r r e s p o n d s t o the v a l u e 1().J3 ~)f W 1 . t h e s t r i p p a r a m e t e r , and the c u ~ ' v e (h) e c n ' r e s p o n d s t o t h e v a l u e 10. () o f W 1 .
448
S. [,I(.)Y and P. DASGUPTA
T h e a u t h o r s e x p r e s s t h e i r s i n c e r e t h a n k s t o D r . P . K. R o y a n d M r . S. Mallik for many helpful discussions. We are grateful to Professor S. N. B o s e f o r t a k i n g k e e n i n t e r e s t in t h e problem.
RE FE R E N C E S [1] G , F . C h e w and C . E . J o n e s . P h y s . Rev. 134 (1964) B208. [2] t I . P a g e l s . P h y s . Rex,. 140 (1965) B1599. [3] V . l , . T e p l i t z . P h y s . Rev. 137 (1965) B138; C . E . J o n e s , Nuovo C i m e n t o 40 (1965) 761. [41 C . F . J o n e s and G . T i k t o p o u l u s . J . Math. P h y s . 7 (1966) 311. [5] G . F . C h e w and S . C . F r a u t s c h i . P h y s . Rev. L e t t e r s 7 (1961) 394. [6] J . B a l l and D. Wong. P h y s . Rev. 133 (1964) B179. [7] E . A b e r s and C . Z e m a c h . P h y s . l t e v . 131 (1963) 2305: L . K . Pande and P. N a r a y a n s w a m i . P h y s . l~ev. 136 (1964) B1760: ( ; . D . Doolen. T . K a n k i and A . T u b i s . P h y s . Rex:. 142 (1!}66) B1072. [8] S . C . F r a u t s c h i and 3 . D . W a l e c k a . P h y s . Rev. 120 (1960) 1486.