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Off shell πK amplitudes consistent with chiral symmetry breaking and the Bjorken limit
~
Nuclear Physics B27 (1971) 193-204. North-Holland Publishing Company
OFF SHELL ~rK AMPLITUDES CONSISTENT WITH CHIRAL SYMMETRY BREAKING AND THE BJORKEN LIMIT Paul RORWITZ * Department o f P h y s i c s and Institute of Theoretical Science, University of Oregon, Eugene, Oregon Received 19 October 1970
A b s t r a c t : We p r e s e n t a t w o - p a r a m e t e r family of YK --07rK amplitudes with two ~" or a ~r and a K off the m a s s shell. They d e s c r i b e scattering with a r b i t r a r y daughter states and reduce to a finite s e r i e s of Veneziano-like t e r m s (satisfying the Adlerz e r o condition) when at l e a s t one momentum is on a p a r t i c l e or daughter pole. They p o s s e s s Bjorken l i m i t behaviour in a finite region of the s t u plane. With one ~r or one K soft, they a r e consistent with u^U + c u 8 s y m m e t r y breaking. The exact Bjorken l i m i t behaviour d e t e r m i n e s one of the p a r a m e t e r s . A modified KSFR relation, and calculations of c a n d f k / f + ( O ) f ~ make possible a comparison with experiment.
1. I N T R O D U C T I O N T h e V e n e z i a n o a m p l i t u d e [1] f o r f o u r - p o i n t f u n c t i o n s i s an e x t r e m e l y e l e g a n t m o d e l e m b o d y i n g n a r r o w r e s o n a n c e s l y i n g on l i n e a r t r a j e c t o r i e s , t o g e t h e r w i t h d u a l i t y in t h e s t r i c t m a t h e m a t i c a l s e n s e t h a t t h e p o l e s c o r r e s p o n d i n g to t h e r e s o n a n c e s e x c h a n g e d in o n e c h a n n e l s u m to g i v e R e g g e a s y m p t o t i c b e h a v i o u r in t h e c r o s s e d c h a n n e l . Any a t t e m p t to c o n n e c t t h e m o d e l w i t h t h e p r e d i c t i o n s m a d e by c u r r e n t a l g e b r a t e c h n i q u e s n e c e s s a r i l y i n v o l v e s t h e e x t r a p o l a t i o n of the a m p l i t u d e to u n p h y s i c a l v a l u e s of o n e o r more external masses. The simplest possible ansatz for this extrapolation i s , of c o u r s e , t h a t t h e V e n e z i a n o a m p l i t u d e i t s e l f c o n t a i n s , t h r o u g h i t s d e p e n d e n c e on t h e M a n d e l s t a m v a r i a b l e s , a l l the n e c e s s a r y d e p e n d e n c e on ext e r n a l m a s s e s . Such an a s s u m p t i o n l e a d s , f o r t h e c a s e of p s e u d o s c a l a r m e s o n e l a s t i c a m p l i t u d e s , to r e a s o n a b l e m a s s r e l a t i o n s [2] (with the e x c e p t i o n of t h e ~7 ( r e f . [3])), w h e n t h e A d l e r z e r o c o n d i t i o n i s i m p o s e d . Such a condition, however, cannot give information concerning purely multiplicat i v e f u n c t i o n s , s u c h a s f o r m f a c t o r s . I n d e e d , f o r the c a s e of 7rK s c a t t e r i n g i t i s known [4] t h a t the a b s e n c e of s u c h f o r m f a c t o r s l e a d s to a v i o l a t i o n of c o n s i s t e n c y r e q u i r e m e n t s i m p o s e d by s y m m e t r y b r e a k i n g of the t y p e p r o * Work supported in p a r t by the US Atomic Energy Commission under contract AT(45-1) -2041.
194
P. HORWITZ
posed by Cell-Mann, Oakes and Renner [5]. F u r t h e r m o r e , i n s i s t e n c e on these consistency conditions leads to f o r m f a c t o r s which explicitly exhibit poles at points c o r r e s p o n d i n g to the m a s s e s of daughter p a r t i c l e s [6]. It is quite easy to show, however, that the m e r e introduction of a single multiplicative function of the off-shell m o m e n t a cannot r e a s o n a b l y (we shall be m o r e explicit below) give r i s e to yet another r e q u i r e m e n t f r o m field theory: the Bjorken limit [7]. In o r d e r to i n c o r p o r a t e this feature one is f o r c e d to introduce an infinite number of satellite t e r m s . This has the effect, of c o u r s e , of i n c r e a s i n g the a r b i t r a r i n e s s of the amplitude. However, the imposition of the c o n s t r a i n t s a r i s i n g f r o m assumption of the (3,3*) ~9 (3",3) nature of the s y m m e t r y breaking m e c h a n i s m , r e f e r r e d to above, r e d u c e s this a r b i t r a r i n e s s . In this p a p e r we p r e s e n t a two p a r a m e t e r family of amplitudes for the reaction 7r+K- --* ~+K- with the final p a r t i c l e s off their m a s s shells. In cons t r u c t i n g these amplitudes we have not been concerned with "deriving" them, e.g. f r o m the truncation of higher n-point functions [8]; r a t h e r we have s p e cifically chosen an amplitude for which c u r r e n t a l g e b r a i m p o s e s as many c o n s t r a i n t s as possible, and have endeavored to satisfy these c o n s t r a i n t s while maintaining such d e s i r a b l e c h a r a c t e r i s t i c s as duality for the o n - s h e l l amplitudes, absence of ghosts and a n c e s t o r s , f a c t o r i z a t i o n , Adler z e r o s for the amplitudes involving daughter p a r t i c l e s , and as far as Oossible a s y m m e t r i c t r e a t m e n t of external and internal off-shell p a r t i c l e s . Our amplitudes do not satisfy the Bjorken limit e v e r y w h e r e on the st u plane, but the b r e a k down is consistent with, though not r e q u i r e d by, the derivation of the Bjorken limit. Although we make no claim for the uniqueness of the amplitudes we have constructed, we have, w h e r e v e r possible, c o m p a r e d their p r e d i c t i o n s to experimental data. To this end we r e q u i r e a slight generalization to the case where the two off-shell p a r t i c l e s a r e pions. This generalization turns out to be r a t h e r elegant, w h e r e a s the equivalent one to two kaons off shell is not. We also d i s c u s s briefly the case of off-shell p a r t i c l e s whose combined quantum n u m b e r s a r e exotic. The amplitudes which a r i s e in this case vanish f a s t e r than any power in the Bjorken limit, a r e s u l t which is at l e a s t c o n s i s tent with the Weltanschaung e x p r e s s e d by the Veneziano model.
2. CONSTRAINTS IMPOSED BY ( 3 , 3 * ) • (3",3) SYMMETRY BREAKING We consider the p r o c e s s 7T+(#i) + K-(ki) --" ~+(pf) + K ' ( k f ) , with
? 2 Pt = mTr '
k.2 = m 2 . 1
The relevant amplitude is, a s s u m i n g PCAC for pions and kaons:
OFF SHELL ffK AMPLITUDES ~/r~rK (2~)4 54{pi+ki-Pf-kf),,%rfKfX
= i(2~)3(4p°k°)~
195
i s t.~,2 k2~
, ,vf, fJ
fd4xd4y e ipfx e ~hfy
<01T(D;(x)D~-(y))I~+(pi) K-(ki)>,
(1)
where D~ - ~ A ~ a n d the subscripts denote the particles which are off shell.2 We define the Mandelstam invariants s = (Pi+ki) 2, t = (Pi _pf)2, u = ~i -kf) . Here u is an exotic channel, so that the on-shell amplitude is described in the Veneziano model by a single term [9]
ATrK(s,t) -- -fiV(s,t) - -fl
F(1 - aK,(S)) F(1 - ap(t)) F(1 - aK,(S) - ap(t))
(2)
T h e n o r m a l i z a t i o n of M in eq. (1) is fixed by the r e q u i r e m e n t m K-k
~K
2
2
M~rfKf(s,t;p ,k ) p2=m 2 =A~g(s,t).
(3)
k2---~ T a k i n g the l i m i t Pf/x -~ 0 in eq. (1) and i n t e g r a t i n g by p a r t s , we obtain 1
~K 2 = M#fKf(s,mTr;O,s)
(2~)3(4pTkT)~<0I~+K-(0)I~+(pi)K - (ki)),
(4)
where we have defined
(5) and s i m i l a r I y , in the l i m i t k f ~ -~ 0 1
M~f~f(s, ~ m~; s, 0) : (2~)3(4~%°)~<01~-~+(0) l~+~i) K-@i)>-
(6)
If, following Gell-Mann, Oakes and Renner [5], we make the assumption that the SU(3) ® SU(3) symmetry of the world is broken by a term that transforms like uo + cu8, we can relate the matrix elements in eqs. (4) and (6), the ratio of which becomes [4] simply a constant, dependent on c:
<°]%+K-L~> ~ + c n ~ <0~K_ + I~K>=~ .
(7)
We must, then, impose on our functions M the requirement that M(p~~ O) be proportional to M(kp-~0), the constant of proportionality then leading to a determination of c. This requirement is not met by the Veneziano form, eq. (2), alone [4, 6]. We shall have occasion below, in comparing our results to experiment, to make use of the relation
196
P. HORWITZ
I
-
--/abc <8
]
(8)
c ]~ c where D V - @ V~. This relation zs easily proved, starting from the equations of the current algebra: a b [Ap(x),Ao(Y)]6(x o _yO) = i/ab c vC (x) 64(x _ y ) + s . t . ,
[Aa(x),db(y)]
5(xo _yO) = //abe
vCp.(x) 64(x -Y)
+ s.t.,
(9)
(10)
w h e r e "s.t." stands for "Schwinger t e r m s " which we take to be in the form of gradients of delta functions. Operating on eq. (9) by ~/axg and on eq. (10) by a/ayg, f r o m the left, and taking the difference of the resulting equations, the Schwinger t e r m s and all other t e r m s involving d e r i v a t i v e s of delta functions cancel out and we a r e left with
{(Aa(x),Db(y)] _ [Ab(y),D~(x)]} 5(x o _ yO) = if abcD~(x) 8 4 ( x . y ) ,
(11)
f r o m which eq. (8) follows upon taking m a t r i x elements.
3
CONSTRUCTION O r
TT~
The s i m p l e s t explicit f o r m for the off-shell extrapolation of the Veneziano amplitude is simply a multiplicative function of the two off-shell m o menta. The n u m b e r of linearly independent L o r e n t z s c a l a r s which can be f o r m e d f r o m pf~ and kf~ is three, and we choose them as p2, k2, s. We have, then
~K
2 k 2) =A~K(s,t)
M~fKf(S , t; p f ,
2
2
FufKf(S; pf, kf ).
(12)
However, if we r e q u i r e that M be m e r o m o r p h i c in all v a r i a b l e s (which is physically r e a s o n a b l e , since we a r e working in a n o n - u n i t a r y a p p r o x i m a tion) we see that F must, in fact, be independent of s. If it were not it would produce either unwanted poles,, in s n°r a n c e s t o r s (polynomials in s). T h e r e f o r e F is a function o n l y of p~ and k~. But in this c a s e it is quite im possible for M to satisfy the Bjorken limit. In this limit we have Pfo --' i°° , kfo ~ -i~o, with the r e s u l t that -t ~ _p2 ._. _kf2 ~ +x ~ + o% up to t e r m s of o r d e r x~. In this limit we r e q u i r e ~K
2
2
1
M~fKf(s,t;p f , k f ) ~ qo l = x --~l ,
(13)
w h e r e q stands for either pf o r kf, and l is s o m e positive integer, independent of s, in principle d e t e r m i n a b l e f r o m the commutation p r o p e r t i e s of the fields and their time derivatives. In the case at hand the exact value of l is not in fact known, but all we need here is that l is independent o f x . Since we have
OFF SHELL 7rK AMPLITUDES
A~K(s,t} B~
197
xaK*(S)'
(14)
eqs. (12) and (13) a r e c l e a r l y incompatible. As the next m o s t c o m p l i c a t e d a n s a t z we c o n s i d e r adding s a t e l l i t e V e n e ziano t e r m s to eq. (12). The f o r m - f a c t o r functions a r e then allowed to be finite p o l y n o m i a l s in s. C o n s i d e r a t i o n s s i m i l a r to those above indicate, h o w e v e r , that even with this amount of latitude no finite s e r i e s can a c c o m m o d a t e the r e q u i r e m e n t eq. (13). In the c a s e of an infinite s e r i e s we can a r r a n g e for it to c o n v e r g e n o n - u n i f o r m l y in x and exploit this to e l i m i n a t e the s - d e p e n d e n c e which the Regge b e h a v i o u r has bequeathed us. Such s e r i e s have been written down, e.g. by Bander [10], and o u r investigation of the a s y m p t o t i c b e h a v i o u r (see appendix) is b a s e d in p a r t on his. We c o n s i d e r the function M~'K~fKfis"
t;p2'k2f)
=
r(1-aK*(s))r(t+n-ap(t)) r(l+n-aK*(s)-ap(t))
(-1) n "Y~TKn= 0 F i n + B )
X
r(n-%(P2)) r(n-ag(k2)) r(n+A-s) r(n+A-p
)
r(n + A -
tea-s)
(15)
'
w h e r e A and B a r e a r b i t r a r y c o n s t a n t s ; a~T(X) = Up(X) - ½, aK(X) = aK*(X) - ½, a r e the t r a j e c t o r i e s which p a s s through the n and K r e s p e c t i v e l y . Eq. (3) l e a d s to
We note f i r s t of all, that r ( n + m 2) Fin + ½)
M~-K ~ f K f"t s , m 2 ", 0 ' s ) = - ~ K f ( s ) n ~= 0 (-1)n F(n+A)F(n+B) '
~ (-1)n r(n+m2)r(n+X+m v2- m K2)
~rK ( ,m2K;S,0) = -Y~K f ( s ) MTrfKf s n=0
F(n +A) F(n + B)
(17)
(18) '
where, ,f(s) - F(1 - aK*(S))/F(A -s) and we a r e using m a s s units for which 2 ( m ~ - r n ~ ) = 1. In the two z e r o - m a s s l i m i t s the s - d e p e n d e n c e f a c t o r s out of the s e r i e s and is identical, leaving the ratio independent of s , as it m u s t be in o r d e r to s a t i s f y eq. (7). This function is thus c o n s i s t e n t with a (3,3*) • (3",3) s y m m e t r y b r e a k i n•g m e c h a n i s m . yK We next point out that the reszdues of poles in MTrfKf at finite values of p2 and k 2 a r e finite s u m s , the b e h a v i o u r of which f o r l a r g e values of s o r t is s t r a i g h t f o r w a r d l y seenn to ben consistent, with the Regge hypothesis. The r e s i d u e of the pole at p~ = m~, k~ = m 2, in p a r t i c u l a r , is s i m p l y A 7TK Only when both of the " o f f - s h e l l " legs a r e on d a u g h t e r m a s s e s do we get m o r e than one t e r m in the s e r i e s , and in this c a s e the f a c t o r
198
P. HORWIT Z
F(n +A - s ) / F ( A - s ) e n s u r e s the c o r r e c t a n g u l a r m o m e n t u m f o r the t - c h a n n e l p o l e s in t h e s e s a t e l l i t e t e r m s . T h e o n - s h e l l a m p l i t u d e d e s c r i b i n g the r e a c tion ~K ~ urnKn - w h e r e u m i s the s c a l a r d a u g h t e r of the ruth r e c u r r e n c e on t h e u t r a j e c t o r y , and s i m i l a r l y f o r K n - i s a s e r i e s c o n t a i n i n g m i n ( m , n ) t e r m s . T h i s i s s u f f i c i e n t to e n s u r e t h a t t h e A d l e r c o n d i t i o n c o n t i n u e s to h o l d , e v e n when o n e of the p a r t i c l e s i s a d a u g h t e r . T h e p r e s e n c e of the f a c t o r F(1 - a K . ( S ) ) w i l l e n s u r e t h a t the t h r e e - p o i n t f u n c t i o n d e s c r i b i n g the d e c a y of K* into two o f f - s h e l l p a r t i c l e s h a s the s a m e B j o r k e n l i m i t b e h a v i o u r a s the full a m p l i t u d e . T h i s m e a n s , h o w e v e r , t h a t w e m u s t b e c a r e f u l n o t to c h o o s e a v a l u e f o r A w h i c h would c a u s e t h e f a c t o r F ( A - s ) in t h e d e n o m i n a t o r to c a n c e l the K* p o l e . In p a r t i c u l a r , the s y m m e t r i c f o r m f o r the a m p l i t u d e o b t a i n e d when A - s = 1 - aK*(S) i s to b e a v o i d e d f o r t h i s r e a s o n . (It a l s o g i v e s f ( s ) = 0 in eqs. (17) and (18)). F r o m eq. (8) w e s e e t h a t the m a t r i x e l e m e n t of D V w i l l o b e y an u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n f o r A > m 2 + ½. A s s u m i n g o n l y t h e c u r r e n t c o m m u t a t i o n r e l a t i o n s and the s u p p r e s s i o n of e x o t i c s t a t e s , l F u c h s and Kuo [11] h a v e shown t h a t the K I 3 f o r m f a c t o r s a r e b o u n d e d by s - L T h i s i m p l i e s A >/ m 2 + 1 in our model. T h e e x a m i n a t i o n of the b e h a v i o u r of M in the B j o r k e n l i m i t ( d e f i n e d a s p~J . -~ k~J . -~ t -~ - x - ~ - ~ ) i s r e l e g a t e d to an a p p e n d i x . T h e r e i t i s shown t h a t the s e r i e s of eq. (15) c o n v e r g e s in a l l c a s e s of p h y s i c a l i n t e r e s t , and ~K M~fKf ~
XY'
where ~' = m a x ( ½ + m 2 - 2 A + s , ½+ m 2 _ A _ B ) . Setting A + B = ½(1+ Z) +
(19)
w e h a v e the d e s i r e d r e s u l t , eq. (13), f o r s --< 2A - m 2 - ½ ( l + 1). F o r s l a r g e r than t h i s v a l u e the a m p l i t u d e f a l l s l e s s r a p i d l y than w o u l d be p r e d i c t e d by t h e B j o r k e n l i m i t . But in t h i s c a s e the d e r i v a t i o n of the B j o r k e n l i m i t b r e a k s down b e c a u s e t h e f a c t o r ( p o ) - l c a n n o t b e r e m o v e d f r o m the i n t e g r a l o v e r Im M, w h i c h c e a s e s to c o n v e r g e u n i f o r m l y . T h e b e h a v i o u r of o u r a m p l i t u d e i s t h u s c o n s i s t e n t w i t h t h e d e r i v a t i o n of the B j o r k e n l i m i t , though i t i s u n n a t u r a l , f r o m t h e p o i n t of v i e w of f i e l d t h e o r y , t h a t the e x i s t e n c e of t h e m a t r i x e l e m e n t of a p r o d u c t of two o p e r a t o r s s h o u l d d e p e n d on the e x t e r n a l s t a t e s 5. T h u s , " w e a k " B j o r k e n l i m i t b e h a v i o u r , s u c h a s we h a v e h e r e , i s m a t h e m a t i c a l l y c o n s i s t e n t , b u t at v a r i a n c e w i t h c o m m o n l y a c c e p t e d i d e a s of f i e l d t h e o r y . W e s h a l l b e c o n t e n t w i t h i t , h o w e v e r , in t h i s p a p e r . 4. G E N E R A L I Z A T I O N S O F T H E A M P L I T U D E W e h a v e c o n s t r u c t e d a s c a t t e r i n g a m p l i t u d e w h i c h s a t i s f i e s a l l the c o n :~ I am indebted to J. Weis and K. Wilson for conversations on this point.
OFF S H E I L ~'K AMPLITUDES
199
s t r a i n t s r e q u i r e d by (3",3) $ (3,3*) s y m m e t r y b r e a k i n g and p o s s e s s e s a c o r r e c t B j o r k e n l i m i t in the l i m i t e d s e n s e d e s c r i b e d above. U n f o r t u n a t e l y , it d e p e n d s on two p a r a m e t e r s w h i c h a r e in p r i n c i p l e u n d e t e r m i n e d , a l t h o u g h eq. (19) would p r o v i d e one r e l a t i o n b e t w e e n t h e m if we knew l. C e r t a i n g e n e r a l i z a t i o n s , h o w e v e r , a r e p o s s i b l e without the addition of new p a r a m e t e r s and t h e s e m a y be u s e d e.g. to fix the n o r m a l i z a t i o n of the a m p l i t u d e in t e r m s of the p-width. T h e m o s t u s e f u l g e n e r a l i z a t i o n is that w h i c h i n v o l v e s t a k i n g the two p i o n s off t h e i r m a s s s h e l l s . F o r this we c h o o s e , in a r a t h e r o b v i o u s f a s h i o n , the function
7rK
t;p2,p2)=
M~fgi(s '
~
(-1) n n =0 F ( n + B )
r(l+n-az*(s))V(i-%(t)) F ( l + n - aK*(S ) -
up(t))
× r ( n - %(p2)) r ( n - %(p2}} r ( n +A - t )
r(n+A_p2) r(n+A_p2)
(20)
r(A-t)
T h e n o r m a l i z a t i o n condition a n a l o g o u s to (3) g i v e s
~,~ = ~f2m4F(B)[F(A-m2)] 2 . T h e p a r a m e t e r A m u s t be i d e n t i c a l f o r M~ K and M .nK b e c a u s e of the r e q u i r e m e n t M~JK(p 2 =m 2) = M~K(k 2 =m2); B is t"hen deter"mined by the B j o r k e n l i m i t co'h'~lition (n'~te tha"t'~n this c a s e the a p p r o p r i a t e l i m i t is s -~ _oo, t finite); and a s s u m i n g that l is the s a m e f o r the nn and nK c a s e s , B is a l s o the s a m e . If we t r y to take the two k a o n s off the m a s s shell, we m u s t u s e a function of the f o r m of eq. (20) with an r e p l a c e d by a K. In this c a s e , h o w e v e r , the B j o r k e n l i m i t b e h a v i o u r is d i f f e r e n t f r o m that o b t a i n e d by u s e of eq. (19); ~K by the s a m e c o n s i s t e n c y c o n s i d e r a A m u s t be the s a m e f o r M ~ as f o r M~K tions a s a b o v e , but now we a r e f o r c e d to v a r y B in o r d e r even to a c h i e v e an i n t e g r a l p o w e r law in the B j o r k e n limit. In p a r t i c u l a r , if we c h o o s e :
M~ K ts ÷.k2 k2~ KfKiX ,~, f ,
ij
~ (-1) n r(l+n-aK*(s))F(1-ap(t)) =n=oF(n+B,) F(l+n-aK.(s)-ap(t)) r(nx C(n+A-k 2)
+A-t) r(n+A-k
2)
r(A-t)
'
(21)
the r e l a t i o n a n a l o g o u s to (19) is A + B ' -- ½(1+/') + 2m 2 - m 2
7T'
(22)
w h e r e l ' is the i n t e g e r a p p r o p r i a t e to the m a t r i x e l e m e n t of a c o m m u t a t o r of n o n - s t r a n g e o p e r a t o r s b e t w e e n kaon s t a t e s . In p r i n c i p l e , l ' could d i f f e r f r o m l, but m e r e l y r e q u i r i n g that l ' be an i n t e g e r f o r c e s B' to d i f f e r f r o m B.
200
P. HORWITZ
This is an unnatural feature of the model. In c o m p a r i s o n s with experiment, t h e r e f o r e , we shall avoid taking two kaons off the m a s s shell. This r e d u c e s the n u m b e r of quantities which we can calculate, but at l e a s t we shall be using a s e l f - c o n s i s t e n t model. The p r o b l e m encountered here is a puzzling one which may be related to a s i m i l a r difficulty which afflicts the standard Veneziano amplitude for KK s c a t t e r i n g [6, 12]. F o r amplitudes like M~.Ku~o, where the off-shell p a r t i c l e s a r e in an exotic n 1~I channel, the c o m m u t a t o r s "of the fields with their v a r i o u s time d e r i v a t i v e s all have exotic quantum n u m b e r s and if the theory has no local exotic o p e r a t o r s they must all vanish. The Bjorken p r o c e d u r e then p r e d i c t s that such amplitudes must fall off f a s t e r than any power. The Veneziano amplitude and all its satellites have p r e c i s e l y this behaviour, so p r a c t i c a l l y any f o r m we choose for the extrapolated amplitude will do. A p a r t i c u l a r l y s y m m e t r i cal choice is
~K k2,p2) MniKf(s,t;
~ (-1) n = -~nK n=O ~
F(l+n-aK*(S))F(l+n-ap(t)) F ( l + n - aK*(S) - ap(t))
×
r(n+A -p2i ) r(n+A -k2)
,
(23)
w h i c h falls off exponentially in the B j o r k e n limit (-s -~ -t -~ _p2 _~ _k 2 _, x -
+cO )o
5. COMPARISON TO E X P E R I M E N T Since the p r e s e n t model r e d u c e s to the standard Veneziano f o r m when all p a r t i c l e s a r e on their (lowest) m a s s shells, all quantities derivable f r o m the o n - s h e l l amplitude (phase shifts, s c a t t e r i n g lengths) will be identical to those p r e d i c t e d by the Veneziano amplitude. There a r e t h r e e ways, however, in which we can use the extrapolated amplitude to make contact with experiment: (i) the A d l e r - W e i s b e r g e r relation [13] fixes the n o r m a l i z a tion of the amplitude, providing a modified KSFR relation; (ii) through eq. (7), we can d e r i v e a value for c, which can also be e s t i m a t e d , e.g. through m a s s relations [5]; (iii) eq. (8) enables us to d e r i v e information concerning certain KI3 p a r a m e t e r s . If we knew l, we would have only one f r e e p a r a m e t e r in M ~ a n d / W ~ ~. Since the amplitudes we a r e c o n s i d e r i n g r e f e r only to connected d i a g r a m s , the l we a r e seeking is the s m a l l e s t i n t e g e r for which l-1 + -
-
oum
o
,
Udxo2 and we a s s u m e that the s a m e l applies to the 7rn case. F r o m strong PCAC and the canonical commutation relations it is c l e a r that l >/ 3. We shall make the c o m p a r i s o n with e x p e r i m e n t using I = 3,4, 5, 6. We fix the n o r m a l i z a t i o n of f n by
OFF SHELL yK AMPLITUDES 3
201
1
(2y)Y(2qo)~ {~ ~
1
2
2
(24)
in which case the experimentalvalue is fy = 0.67 rn~. The Adler-Weisberger relation [13] for this amplitude is
a ~(~, o;o,o)i
i
(25)
s=m 2 = 5 '
a--~-
w h i c h f i x e s the v a l u e of yv~. In t e r m s of A and l, t h e n , we d e t e r m i n e the p r o d u c t ~ f 2 , /3 i n t u r n m a y b e d e t e r m i n e d f r o m the o b s e r v e d K* width, w h i c h g i v e s /3/4~ = 2.24. F o r the p a r t i c u l a r c a s e of i n t e r e s t eq. (8) g i v e s , a f t e r an i s o s p i n r o t a t i o n :
<0 I%+g- - ~K-~+ ]K-~+> = i <0[DK+IK- o>.
(26)
We define the K13 form factors by
~[&(s)(k-p)~ + f_(s)(k+p).] --(4k°F)~, (27) setting s = 0
/+(0) =
2 rM~_K ,,, 2 M~_K ,^ 2 ^ 0)] A m 2 t ufKftU, m u ; 0, 0) ~fKfLU, i n K ; u, ,
(28)
w h e r e A m 2 - m 2 - m~. 2 The m o s t a c c e s s i b l e e x p e r i m e n t a l quantity is fK/(f+(O) fu), f o r w h i c h the l a t e s t v a l u e i s 1.22 ~ 0.06 (ref. [14]). F i n a l l y , we can c a l c u l a t e c t h r o u g h eq. (7). T h i s q u a n t i t y i s r o u g h l y e s t i m a t e d to b e a b o u t - 1 . 2 5 (ref. [5]). We p r e s e n t the r e s u l t s of the c a l c u l a t i o n s i n t a b l e 1. F o r e a c h v a l u e of l, A h a s b e e n r o u g h l y c h o s e n to m a x i m i z e a g r e e m e n t f o r a l l t h r e e q u a n t i t i e s . We i n c l u d e in the t a b l e the v a l u e of A w h i c h r e s u l t e d f r o m t h i s p r o c e d u r e ; c i s v e r y i n s e n s i t i v e to A o v e r a wide r a n g e ; i3 and f+(0) a r e q u i t e s t r o n g l y d e p e n d e n t on A. T h e c a l c u l a t i o n s w e r e p e r f o r m e d on a c o m p u t e r , s i n c e s o m e of the s e r i e s i n v o l v e d do n o t c o n v e r g e very rapidly. Table 1 Values of/3/4/r,
fK/f+(O)f~ and c corresponding to different choices of l.
l
A
fl/4y
fK/f+(O)f~
3
0.95
2.67
0.74
-1.18
4
1.4
2.32
1.00
-1.18
5
2.1
2.24
1.19
-1.18
6
2.8
2.25
1.29
-1.18
The p a r a m e t e r A has been chosen in each case to give the best overall fit, and the value so arrived at is that displayed in the table.
202
P. HORWITz
6. DISCUSSION AND CONCLUSION F r o m table 1 it is evident that values of I > 3 a r e p r e f e r r e d . The a g r e e ment with e x p e r i m e n t is p a r t i c u l a r l y good for l = 5 and l = 6. F o r l = 3 the p r e f e r r e d value for A violates the Fuchs-Kuo bound [11], and in any case the fit to e x p e r i m e n t is quite poor. The value for c is essentially independent of l and is close to that which is obtained f r o m s i m p l e r f o r m - f a c t o r models
[6,151. The model amplitude proposed in this p a p e r is not unique, and no attempt has been made here to "derive" it, o r something like it, f r o m a dual n-point function. Its a g r e e m e n t with experiment, where c o m p a r i s o n is possible, is very good, if one is willing to allow some latitude in the choice of l. The app a r e n t inability of the model to d e s c r i b e , in a s y m m e t r i c fashion, p r o c e s s e s with two off-shell kaons, is a s e r i o u s drawback, which may point the way to an i m p r o v e m e n t of the model. At p r e s e n t it r e p r e s e n t s the only attempt known to the author to combine l o w - e n e r g y chiral c o n s t r a i n t s with Bjorken limit behaviour a r i s i n g out of a sum of poles in external m a s s variables. We feel that r e q u i r i n g "good" behaviour at low and high e n e r g i e s is likely to be important in exploring the relation between chiral s y m m e t r y breaking, c u r r e n t algebra, and the d u a l - r e s o n a n c e model.
APPENDIX In this appendix we shall derive the Bjorken limit behaviour of the a m p l i tude M~n~Kf, which we r e p r o d u c e below: nK 2 kf2) M~fKf(S, t;pf,
~
(-1) n
= -~'nK n= 0 F---(-~B)
x
F(1-aK*(S))F(l+n-ap(t)) F(1 - aK.(S) - ap(t))
F(A -s)
F(n+A_pf2)r(n+A-f)k 2
(15)
We break the series into two parts: M Bj-~ 19 F(1 -aK*(S)) 2 ( ~ 1 1 (_l)n r(n+Ar(n+B)-S)+ ~ F(A - s ) n=0 n =2N-1
a =A - B - s ,
(_l)nna(n+x)fl, (A. 1)
i 2 _ 2A + s and we have a s s u m e d that fl =-~+mTr
x(--- -t ~ _p2 ~ _k 2 >> 2N) is l a r g e enough for the Sterling approximation to be a r b i t r a r i l y good for the F-functions in which it a p p e a r s , and s i m i l a r l y for n >/ 2 N - 1. The finite s e r i e s obviously behaves like xfl; the infinite one, however, does not converge uniformly for a > -1 and must be examined separately. Consider the integral oo
Ia~=- f O
(277)a(21?+l)fid~?,
OFF SHELL 7rK AMPLITUDES
203
w h i c h c o n v e r g e s of and only if, a > -1, a+/3 < -1. If this condition is s a t i s fied we have
M lim lim ( 2 rln ) a ( 2 rln + 1) fl A rln , Ia/3 = lira 77N--' 0 M--' oo ZX77n--,0 n =N
(A.2)
where lim
r/n = ~o,
A~?n --- ~?n+l - rTn.
n----~ oo
We m a y , without l o s s of g e n e r a l i t y , c h o o s e ~n ~ n/x, in w h i c h c a s e ZXrln = 1/x, and we h a v e :
S(aN)~( x) x ~ - ~ xa+/3+lIa /3
f o r a > -1, /3 < - 1 ,
(A.3)
w h e r e we have u s e d the notation
S(~fl(x) - ~
n=N
(2n)a(2n+x)/3.
(A.4)
S~nce I,~a is i n d e p e n d e n t of x, eq. (A.3) ~ i v e s the a s y m p t o t i c b e h a v i o u r of S~)"" exp'licitly.~ If a ~< -1, a + / 3 < -1, S~'~" c o n v e r g e s u n i f o r m l y in x and g o e s like x/3 f o r l a r g e x. It r e m a i n s to put the s e r i e s in eq. (A.1) into this f o r m . We do this by c o m b i n i n g p a i r s of t e r m s and e x a m i n i n g the l e a d i n g b e h a v i o u r .
n =2N-1
(-1)nna(n+ x)/3 = { - ( 2 N - 1 ) a ( 2 N - l + x)/3 +(2N)a(2N+ x)/3} - . . . .
(A. 5)
The quantity in c u r l y b r a c k e t s m a y be e x p a n d e d in p o w e r s of 2N and 2N+ x, thus:
( } = a(2N)a-l(2N+x)fl+
/3(2N)a(2g+x)fl+ . . . ,
and t h e r e f o r e , to l e a d i n g o r d e r in n and x, we m a y w r i t e :
n =2N-1
(-1)n na(n+ x)/3 = a s(aN__1 ,fl(x) + /3 s(aN~_ 1 (x).
(A.6)
T h e l e a d i n g a s y m p t o t i c b e h a v i o u r in x is d e t e r m i n e d by the f i r s t t e r m on the r i g h t of (A.6):
S(a~l '/3(x) = O(x fl) = O(x a'+/3)
f o r a ~< O, f o r a > 0.
R e m e m b e r i n g what a and /3 a r e , f o r /V/~fKf, K we note that a+/3 d o e s not d e pend on s. Setting
204
P. HORWITZ
÷ ~ = ½ ÷ .~
- A - B -- - ½ l ,
(A.7)
we o b t a i n the a p p r o p r i a t e B j o r k e n l i m i t b e h a v i o u r f o r a c e r t a i n r a n g e in s: _!l
Bj
x -½1+s-A+B
(s >i A - B ) .
We n o t e that the s e r i e s of eq. (15) c o n v e r g e s f o r a+ fl--< °1, w h i c h r e q u i r e s l >/ 2. S i n c e the known c o m m u t a t i o n r e l a t i o n s p r e d i c t l >/ 3, the a m p l i t u d e i s f i n i t e f o r a l l the c a s e s of i n t e r e s t .
REFERENCES [1] G. Veneziano, Nuovo Cimento 57A (1968) 190. [2] M. Ademollo, G. Veneziano and S.Weinberg, Phys. Rev. Letters 22 (1969) 83. [3] G. P. Canning, Nucl. Phys. B14 (1969) 437; F.Takagi, Prog. Theor. Phys. 41 (1969) 1555. [4] J.A. Cronin and K. Kang, Phys. Rev. Letters 23 (1969) 1004. [5] M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195. [6] H. Osborn, Nucl. Phys. B17 (1970) 141. [7] J.D. Bjorken, Phys. Rev. 148 (1966) 1467. [8] F. Cooper, Phys. Rev. D1 (1970) 1140; R. C.Brower and J . H . W e i s , Phys. Rev. 188 (1969) 2495. [9] K. Kawarabayashi, S. Kitakado and H. Yabuki, Phys. Letters 28B (1969) 432; Phys. Rev. 184 (1969) 1956. [10] M. Bander, Nucl. Phys. B13 (1969) 587. [11] N.H. Fuchs and T. K. Kuo, Phys. Rev. D1 (1970) 1357. [ 12] D.W. McKay, W. F. P a l m e r and W. W. Wada, Phys. Rev. D2 (1970) 742. [13] H. Osborn, Nuovo Cim. Letters 1 (1969) 513. [14] S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. 41 (1969) 531. [15] F. Csikor, Phys. Letters 31B (1970) 141.
Nuclear Physics B27 (1971) 193-204. North-Holland Publishing Company
OFF SHELL ~rK AMPLITUDES CONSISTENT WITH CHIRAL SYMMETRY BREAKING AND THE BJORKEN LIMIT Paul RORWITZ * Department o f P h y s i c s and Institute of Theoretical Science, University of Oregon, Eugene, Oregon Received 19 October 1970
A b s t r a c t : We p r e s e n t a t w o - p a r a m e t e r family of YK --07rK amplitudes with two ~" or a ~r and a K off the m a s s shell. They d e s c r i b e scattering with a r b i t r a r y daughter states and reduce to a finite s e r i e s of Veneziano-like t e r m s (satisfying the Adlerz e r o condition) when at l e a s t one momentum is on a p a r t i c l e or daughter pole. They p o s s e s s Bjorken l i m i t behaviour in a finite region of the s t u plane. With one ~r or one K soft, they a r e consistent with u^U + c u 8 s y m m e t r y breaking. The exact Bjorken l i m i t behaviour d e t e r m i n e s one of the p a r a m e t e r s . A modified KSFR relation, and calculations of c a n d f k / f + ( O ) f ~ make possible a comparison with experiment.
1. I N T R O D U C T I O N T h e V e n e z i a n o a m p l i t u d e [1] f o r f o u r - p o i n t f u n c t i o n s i s an e x t r e m e l y e l e g a n t m o d e l e m b o d y i n g n a r r o w r e s o n a n c e s l y i n g on l i n e a r t r a j e c t o r i e s , t o g e t h e r w i t h d u a l i t y in t h e s t r i c t m a t h e m a t i c a l s e n s e t h a t t h e p o l e s c o r r e s p o n d i n g to t h e r e s o n a n c e s e x c h a n g e d in o n e c h a n n e l s u m to g i v e R e g g e a s y m p t o t i c b e h a v i o u r in t h e c r o s s e d c h a n n e l . Any a t t e m p t to c o n n e c t t h e m o d e l w i t h t h e p r e d i c t i o n s m a d e by c u r r e n t a l g e b r a t e c h n i q u e s n e c e s s a r i l y i n v o l v e s t h e e x t r a p o l a t i o n of the a m p l i t u d e to u n p h y s i c a l v a l u e s of o n e o r more external masses. The simplest possible ansatz for this extrapolation i s , of c o u r s e , t h a t t h e V e n e z i a n o a m p l i t u d e i t s e l f c o n t a i n s , t h r o u g h i t s d e p e n d e n c e on t h e M a n d e l s t a m v a r i a b l e s , a l l the n e c e s s a r y d e p e n d e n c e on ext e r n a l m a s s e s . Such an a s s u m p t i o n l e a d s , f o r t h e c a s e of p s e u d o s c a l a r m e s o n e l a s t i c a m p l i t u d e s , to r e a s o n a b l e m a s s r e l a t i o n s [2] (with the e x c e p t i o n of t h e ~7 ( r e f . [3])), w h e n t h e A d l e r z e r o c o n d i t i o n i s i m p o s e d . Such a condition, however, cannot give information concerning purely multiplicat i v e f u n c t i o n s , s u c h a s f o r m f a c t o r s . I n d e e d , f o r the c a s e of 7rK s c a t t e r i n g i t i s known [4] t h a t the a b s e n c e of s u c h f o r m f a c t o r s l e a d s to a v i o l a t i o n of c o n s i s t e n c y r e q u i r e m e n t s i m p o s e d by s y m m e t r y b r e a k i n g of the t y p e p r o * Work supported in p a r t by the US Atomic Energy Commission under contract AT(45-1) -2041.
194
P. HORWITZ
posed by Cell-Mann, Oakes and Renner [5]. F u r t h e r m o r e , i n s i s t e n c e on these consistency conditions leads to f o r m f a c t o r s which explicitly exhibit poles at points c o r r e s p o n d i n g to the m a s s e s of daughter p a r t i c l e s [6]. It is quite easy to show, however, that the m e r e introduction of a single multiplicative function of the off-shell m o m e n t a cannot r e a s o n a b l y (we shall be m o r e explicit below) give r i s e to yet another r e q u i r e m e n t f r o m field theory: the Bjorken limit [7]. In o r d e r to i n c o r p o r a t e this feature one is f o r c e d to introduce an infinite number of satellite t e r m s . This has the effect, of c o u r s e , of i n c r e a s i n g the a r b i t r a r i n e s s of the amplitude. However, the imposition of the c o n s t r a i n t s a r i s i n g f r o m assumption of the (3,3*) ~9 (3",3) nature of the s y m m e t r y breaking m e c h a n i s m , r e f e r r e d to above, r e d u c e s this a r b i t r a r i n e s s . In this p a p e r we p r e s e n t a two p a r a m e t e r family of amplitudes for the reaction 7r+K- --* ~+K- with the final p a r t i c l e s off their m a s s shells. In cons t r u c t i n g these amplitudes we have not been concerned with "deriving" them, e.g. f r o m the truncation of higher n-point functions [8]; r a t h e r we have s p e cifically chosen an amplitude for which c u r r e n t a l g e b r a i m p o s e s as many c o n s t r a i n t s as possible, and have endeavored to satisfy these c o n s t r a i n t s while maintaining such d e s i r a b l e c h a r a c t e r i s t i c s as duality for the o n - s h e l l amplitudes, absence of ghosts and a n c e s t o r s , f a c t o r i z a t i o n , Adler z e r o s for the amplitudes involving daughter p a r t i c l e s , and as far as Oossible a s y m m e t r i c t r e a t m e n t of external and internal off-shell p a r t i c l e s . Our amplitudes do not satisfy the Bjorken limit e v e r y w h e r e on the st u plane, but the b r e a k down is consistent with, though not r e q u i r e d by, the derivation of the Bjorken limit. Although we make no claim for the uniqueness of the amplitudes we have constructed, we have, w h e r e v e r possible, c o m p a r e d their p r e d i c t i o n s to experimental data. To this end we r e q u i r e a slight generalization to the case where the two off-shell p a r t i c l e s a r e pions. This generalization turns out to be r a t h e r elegant, w h e r e a s the equivalent one to two kaons off shell is not. We also d i s c u s s briefly the case of off-shell p a r t i c l e s whose combined quantum n u m b e r s a r e exotic. The amplitudes which a r i s e in this case vanish f a s t e r than any power in the Bjorken limit, a r e s u l t which is at l e a s t c o n s i s tent with the Weltanschaung e x p r e s s e d by the Veneziano model.
2. CONSTRAINTS IMPOSED BY ( 3 , 3 * ) • (3",3) SYMMETRY BREAKING We consider the p r o c e s s 7T+(#i) + K-(ki) --" ~+(pf) + K ' ( k f ) , with
? 2 Pt = mTr '
k.2 = m 2 . 1
The relevant amplitude is, a s s u m i n g PCAC for pions and kaons:
OFF SHELL ffK AMPLITUDES ~/r~rK (2~)4 54{pi+ki-Pf-kf),,%rfKfX
= i(2~)3(4p°k°)~
195
i s t.~,2 k2~
, ,vf, fJ
fd4xd4y e ipfx e ~hfy
<01T(D;(x)D~-(y))I~+(pi) K-(ki)>,
(1)
where D~ - ~ A ~ a n d the subscripts denote the particles which are off shell.2 We define the Mandelstam invariants s = (Pi+ki) 2, t = (Pi _pf)2, u = ~i -kf) . Here u is an exotic channel, so that the on-shell amplitude is described in the Veneziano model by a single term [9]
ATrK(s,t) -- -fiV(s,t) - -fl
F(1 - aK,(S)) F(1 - ap(t)) F(1 - aK,(S) - ap(t))
(2)
T h e n o r m a l i z a t i o n of M in eq. (1) is fixed by the r e q u i r e m e n t m K-k
~K
2
2
M~rfKf(s,t;p ,k ) p2=m 2 =A~g(s,t).
(3)
k2---~ T a k i n g the l i m i t Pf/x -~ 0 in eq. (1) and i n t e g r a t i n g by p a r t s , we obtain 1
~K 2 = M#fKf(s,mTr;O,s)
(2~)3(4pTkT)~<0I~+K-(0)I~+(pi)K - (ki)),
(4)
where we have defined
(5) and s i m i l a r I y , in the l i m i t k f ~ -~ 0 1
M~f~f(s, ~ m~; s, 0) : (2~)3(4~%°)~<01~-~+(0) l~+~i) K-@i)>-
(6)
If, following Gell-Mann, Oakes and Renner [5], we make the assumption that the SU(3) ® SU(3) symmetry of the world is broken by a term that transforms like uo + cu8, we can relate the matrix elements in eqs. (4) and (6), the ratio of which becomes [4] simply a constant, dependent on c:
<°]%+K-L~> ~ + c n ~ <0~K_ + I~K>=~ .
(7)
We must, then, impose on our functions M the requirement that M(p~~ O) be proportional to M(kp-~0), the constant of proportionality then leading to a determination of c. This requirement is not met by the Veneziano form, eq. (2), alone [4, 6]. We shall have occasion below, in comparing our results to experiment, to make use of the relation
196
P. HORWITZ
I
-
--/abc <8
]
(8)
c ]~ c where D V - @ V~. This relation zs easily proved, starting from the equations of the current algebra: a b [Ap(x),Ao(Y)]6(x o _yO) = i/ab c vC (x) 64(x _ y ) + s . t . ,
[Aa(x),db(y)]
5(xo _yO) = //abe
vCp.(x) 64(x -Y)
+ s.t.,
(9)
(10)
w h e r e "s.t." stands for "Schwinger t e r m s " which we take to be in the form of gradients of delta functions. Operating on eq. (9) by ~/axg and on eq. (10) by a/ayg, f r o m the left, and taking the difference of the resulting equations, the Schwinger t e r m s and all other t e r m s involving d e r i v a t i v e s of delta functions cancel out and we a r e left with
{(Aa(x),Db(y)] _ [Ab(y),D~(x)]} 5(x o _ yO) = if abcD~(x) 8 4 ( x . y ) ,
(11)
f r o m which eq. (8) follows upon taking m a t r i x elements.
3
CONSTRUCTION O r
TT~
The s i m p l e s t explicit f o r m for the off-shell extrapolation of the Veneziano amplitude is simply a multiplicative function of the two off-shell m o menta. The n u m b e r of linearly independent L o r e n t z s c a l a r s which can be f o r m e d f r o m pf~ and kf~ is three, and we choose them as p2, k2, s. We have, then
~K
2 k 2) =A~K(s,t)
M~fKf(S , t; p f ,
2
2
FufKf(S; pf, kf ).
(12)
However, if we r e q u i r e that M be m e r o m o r p h i c in all v a r i a b l e s (which is physically r e a s o n a b l e , since we a r e working in a n o n - u n i t a r y a p p r o x i m a tion) we see that F must, in fact, be independent of s. If it were not it would produce either unwanted poles,, in s n°r a n c e s t o r s (polynomials in s). T h e r e f o r e F is a function o n l y of p~ and k~. But in this c a s e it is quite im possible for M to satisfy the Bjorken limit. In this limit we have Pfo --' i°° , kfo ~ -i~o, with the r e s u l t that -t ~ _p2 ._. _kf2 ~ +x ~ + o% up to t e r m s of o r d e r x~. In this limit we r e q u i r e ~K
2
2
1
M~fKf(s,t;p f , k f ) ~ qo l = x --~l ,
(13)
w h e r e q stands for either pf o r kf, and l is s o m e positive integer, independent of s, in principle d e t e r m i n a b l e f r o m the commutation p r o p e r t i e s of the fields and their time derivatives. In the case at hand the exact value of l is not in fact known, but all we need here is that l is independent o f x . Since we have
OFF SHELL 7rK AMPLITUDES
A~K(s,t} B~
197
xaK*(S)'
(14)
eqs. (12) and (13) a r e c l e a r l y incompatible. As the next m o s t c o m p l i c a t e d a n s a t z we c o n s i d e r adding s a t e l l i t e V e n e ziano t e r m s to eq. (12). The f o r m - f a c t o r functions a r e then allowed to be finite p o l y n o m i a l s in s. C o n s i d e r a t i o n s s i m i l a r to those above indicate, h o w e v e r , that even with this amount of latitude no finite s e r i e s can a c c o m m o d a t e the r e q u i r e m e n t eq. (13). In the c a s e of an infinite s e r i e s we can a r r a n g e for it to c o n v e r g e n o n - u n i f o r m l y in x and exploit this to e l i m i n a t e the s - d e p e n d e n c e which the Regge b e h a v i o u r has bequeathed us. Such s e r i e s have been written down, e.g. by Bander [10], and o u r investigation of the a s y m p t o t i c b e h a v i o u r (see appendix) is b a s e d in p a r t on his. We c o n s i d e r the function M~'K~fKfis"
t;p2'k2f)
=
r(1-aK*(s))r(t+n-ap(t)) r(l+n-aK*(s)-ap(t))
(-1) n "Y~TKn= 0 F i n + B )
X
r(n-%(P2)) r(n-ag(k2)) r(n+A-s) r(n+A-p
)
r(n + A -
tea-s)
(15)
'
w h e r e A and B a r e a r b i t r a r y c o n s t a n t s ; a~T(X) = Up(X) - ½, aK(X) = aK*(X) - ½, a r e the t r a j e c t o r i e s which p a s s through the n and K r e s p e c t i v e l y . Eq. (3) l e a d s to
We note f i r s t of all, that r ( n + m 2) Fin + ½)
M~-K ~ f K f"t s , m 2 ", 0 ' s ) = - ~ K f ( s ) n ~= 0 (-1)n F(n+A)F(n+B) '
~ (-1)n r(n+m2)r(n+X+m v2- m K2)
~rK ( ,m2K;S,0) = -Y~K f ( s ) MTrfKf s n=0
F(n +A) F(n + B)
(17)
(18) '
where, ,f(s) - F(1 - aK*(S))/F(A -s) and we a r e using m a s s units for which 2 ( m ~ - r n ~ ) = 1. In the two z e r o - m a s s l i m i t s the s - d e p e n d e n c e f a c t o r s out of the s e r i e s and is identical, leaving the ratio independent of s , as it m u s t be in o r d e r to s a t i s f y eq. (7). This function is thus c o n s i s t e n t with a (3,3*) • (3",3) s y m m e t r y b r e a k i n•g m e c h a n i s m . yK We next point out that the reszdues of poles in MTrfKf at finite values of p2 and k 2 a r e finite s u m s , the b e h a v i o u r of which f o r l a r g e values of s o r t is s t r a i g h t f o r w a r d l y seenn to ben consistent, with the Regge hypothesis. The r e s i d u e of the pole at p~ = m~, k~ = m 2, in p a r t i c u l a r , is s i m p l y A 7TK Only when both of the " o f f - s h e l l " legs a r e on d a u g h t e r m a s s e s do we get m o r e than one t e r m in the s e r i e s , and in this c a s e the f a c t o r
198
P. HORWIT Z
F(n +A - s ) / F ( A - s ) e n s u r e s the c o r r e c t a n g u l a r m o m e n t u m f o r the t - c h a n n e l p o l e s in t h e s e s a t e l l i t e t e r m s . T h e o n - s h e l l a m p l i t u d e d e s c r i b i n g the r e a c tion ~K ~ urnKn - w h e r e u m i s the s c a l a r d a u g h t e r of the ruth r e c u r r e n c e on t h e u t r a j e c t o r y , and s i m i l a r l y f o r K n - i s a s e r i e s c o n t a i n i n g m i n ( m , n ) t e r m s . T h i s i s s u f f i c i e n t to e n s u r e t h a t t h e A d l e r c o n d i t i o n c o n t i n u e s to h o l d , e v e n when o n e of the p a r t i c l e s i s a d a u g h t e r . T h e p r e s e n c e of the f a c t o r F(1 - a K . ( S ) ) w i l l e n s u r e t h a t the t h r e e - p o i n t f u n c t i o n d e s c r i b i n g the d e c a y of K* into two o f f - s h e l l p a r t i c l e s h a s the s a m e B j o r k e n l i m i t b e h a v i o u r a s the full a m p l i t u d e . T h i s m e a n s , h o w e v e r , t h a t w e m u s t b e c a r e f u l n o t to c h o o s e a v a l u e f o r A w h i c h would c a u s e t h e f a c t o r F ( A - s ) in t h e d e n o m i n a t o r to c a n c e l the K* p o l e . In p a r t i c u l a r , the s y m m e t r i c f o r m f o r the a m p l i t u d e o b t a i n e d when A - s = 1 - aK*(S) i s to b e a v o i d e d f o r t h i s r e a s o n . (It a l s o g i v e s f ( s ) = 0 in eqs. (17) and (18)). F r o m eq. (8) w e s e e t h a t the m a t r i x e l e m e n t of D V w i l l o b e y an u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n f o r A > m 2 + ½. A s s u m i n g o n l y t h e c u r r e n t c o m m u t a t i o n r e l a t i o n s and the s u p p r e s s i o n of e x o t i c s t a t e s , l F u c h s and Kuo [11] h a v e shown t h a t the K I 3 f o r m f a c t o r s a r e b o u n d e d by s - L T h i s i m p l i e s A >/ m 2 + 1 in our model. T h e e x a m i n a t i o n of the b e h a v i o u r of M in the B j o r k e n l i m i t ( d e f i n e d a s p~J . -~ k~J . -~ t -~ - x - ~ - ~ ) i s r e l e g a t e d to an a p p e n d i x . T h e r e i t i s shown t h a t the s e r i e s of eq. (15) c o n v e r g e s in a l l c a s e s of p h y s i c a l i n t e r e s t , and ~K M~fKf ~
XY'
where ~' = m a x ( ½ + m 2 - 2 A + s , ½+ m 2 _ A _ B ) . Setting A + B = ½(1+ Z) +
(19)
w e h a v e the d e s i r e d r e s u l t , eq. (13), f o r s --< 2A - m 2 - ½ ( l + 1). F o r s l a r g e r than t h i s v a l u e the a m p l i t u d e f a l l s l e s s r a p i d l y than w o u l d be p r e d i c t e d by t h e B j o r k e n l i m i t . But in t h i s c a s e the d e r i v a t i o n of the B j o r k e n l i m i t b r e a k s down b e c a u s e t h e f a c t o r ( p o ) - l c a n n o t b e r e m o v e d f r o m the i n t e g r a l o v e r Im M, w h i c h c e a s e s to c o n v e r g e u n i f o r m l y . T h e b e h a v i o u r of o u r a m p l i t u d e i s t h u s c o n s i s t e n t w i t h t h e d e r i v a t i o n of the B j o r k e n l i m i t , though i t i s u n n a t u r a l , f r o m t h e p o i n t of v i e w of f i e l d t h e o r y , t h a t the e x i s t e n c e of t h e m a t r i x e l e m e n t of a p r o d u c t of two o p e r a t o r s s h o u l d d e p e n d on the e x t e r n a l s t a t e s 5. T h u s , " w e a k " B j o r k e n l i m i t b e h a v i o u r , s u c h a s we h a v e h e r e , i s m a t h e m a t i c a l l y c o n s i s t e n t , b u t at v a r i a n c e w i t h c o m m o n l y a c c e p t e d i d e a s of f i e l d t h e o r y . W e s h a l l b e c o n t e n t w i t h i t , h o w e v e r , in t h i s p a p e r . 4. G E N E R A L I Z A T I O N S O F T H E A M P L I T U D E W e h a v e c o n s t r u c t e d a s c a t t e r i n g a m p l i t u d e w h i c h s a t i s f i e s a l l the c o n :~ I am indebted to J. Weis and K. Wilson for conversations on this point.
OFF S H E I L ~'K AMPLITUDES
199
s t r a i n t s r e q u i r e d by (3",3) $ (3,3*) s y m m e t r y b r e a k i n g and p o s s e s s e s a c o r r e c t B j o r k e n l i m i t in the l i m i t e d s e n s e d e s c r i b e d above. U n f o r t u n a t e l y , it d e p e n d s on two p a r a m e t e r s w h i c h a r e in p r i n c i p l e u n d e t e r m i n e d , a l t h o u g h eq. (19) would p r o v i d e one r e l a t i o n b e t w e e n t h e m if we knew l. C e r t a i n g e n e r a l i z a t i o n s , h o w e v e r , a r e p o s s i b l e without the addition of new p a r a m e t e r s and t h e s e m a y be u s e d e.g. to fix the n o r m a l i z a t i o n of the a m p l i t u d e in t e r m s of the p-width. T h e m o s t u s e f u l g e n e r a l i z a t i o n is that w h i c h i n v o l v e s t a k i n g the two p i o n s off t h e i r m a s s s h e l l s . F o r this we c h o o s e , in a r a t h e r o b v i o u s f a s h i o n , the function
7rK
t;p2,p2)=
M~fgi(s '
~
(-1) n n =0 F ( n + B )
r(l+n-az*(s))V(i-%(t)) F ( l + n - aK*(S ) -
up(t))
× r ( n - %(p2)) r ( n - %(p2}} r ( n +A - t )
r(n+A_p2) r(n+A_p2)
(20)
r(A-t)
T h e n o r m a l i z a t i o n condition a n a l o g o u s to (3) g i v e s
~,~ = ~f2m4F(B)[F(A-m2)] 2 . T h e p a r a m e t e r A m u s t be i d e n t i c a l f o r M~ K and M .nK b e c a u s e of the r e q u i r e m e n t M~JK(p 2 =m 2) = M~K(k 2 =m2); B is t"hen deter"mined by the B j o r k e n l i m i t co'h'~lition (n'~te tha"t'~n this c a s e the a p p r o p r i a t e l i m i t is s -~ _oo, t finite); and a s s u m i n g that l is the s a m e f o r the nn and nK c a s e s , B is a l s o the s a m e . If we t r y to take the two k a o n s off the m a s s shell, we m u s t u s e a function of the f o r m of eq. (20) with an r e p l a c e d by a K. In this c a s e , h o w e v e r , the B j o r k e n l i m i t b e h a v i o u r is d i f f e r e n t f r o m that o b t a i n e d by u s e of eq. (19); ~K by the s a m e c o n s i s t e n c y c o n s i d e r a A m u s t be the s a m e f o r M ~ as f o r M~K tions a s a b o v e , but now we a r e f o r c e d to v a r y B in o r d e r even to a c h i e v e an i n t e g r a l p o w e r law in the B j o r k e n limit. In p a r t i c u l a r , if we c h o o s e :
M~ K ts ÷.k2 k2~ KfKiX ,~, f ,
ij
~ (-1) n r(l+n-aK*(s))F(1-ap(t)) =n=oF(n+B,) F(l+n-aK.(s)-ap(t)) r(nx C(n+A-k 2)
+A-t) r(n+A-k
2)
r(A-t)
'
(21)
the r e l a t i o n a n a l o g o u s to (19) is A + B ' -- ½(1+/') + 2m 2 - m 2
7T'
(22)
w h e r e l ' is the i n t e g e r a p p r o p r i a t e to the m a t r i x e l e m e n t of a c o m m u t a t o r of n o n - s t r a n g e o p e r a t o r s b e t w e e n kaon s t a t e s . In p r i n c i p l e , l ' could d i f f e r f r o m l, but m e r e l y r e q u i r i n g that l ' be an i n t e g e r f o r c e s B' to d i f f e r f r o m B.
200
P. HORWITZ
This is an unnatural feature of the model. In c o m p a r i s o n s with experiment, t h e r e f o r e , we shall avoid taking two kaons off the m a s s shell. This r e d u c e s the n u m b e r of quantities which we can calculate, but at l e a s t we shall be using a s e l f - c o n s i s t e n t model. The p r o b l e m encountered here is a puzzling one which may be related to a s i m i l a r difficulty which afflicts the standard Veneziano amplitude for KK s c a t t e r i n g [6, 12]. F o r amplitudes like M~.Ku~o, where the off-shell p a r t i c l e s a r e in an exotic n 1~I channel, the c o m m u t a t o r s "of the fields with their v a r i o u s time d e r i v a t i v e s all have exotic quantum n u m b e r s and if the theory has no local exotic o p e r a t o r s they must all vanish. The Bjorken p r o c e d u r e then p r e d i c t s that such amplitudes must fall off f a s t e r than any power. The Veneziano amplitude and all its satellites have p r e c i s e l y this behaviour, so p r a c t i c a l l y any f o r m we choose for the extrapolated amplitude will do. A p a r t i c u l a r l y s y m m e t r i cal choice is
~K k2,p2) MniKf(s,t;
~ (-1) n = -~nK n=O ~
F(l+n-aK*(S))F(l+n-ap(t)) F ( l + n - aK*(S) - ap(t))
×
r(n+A -p2i ) r(n+A -k2)
,
(23)
w h i c h falls off exponentially in the B j o r k e n limit (-s -~ -t -~ _p2 _~ _k 2 _, x -
+cO )o
5. COMPARISON TO E X P E R I M E N T Since the p r e s e n t model r e d u c e s to the standard Veneziano f o r m when all p a r t i c l e s a r e on their (lowest) m a s s shells, all quantities derivable f r o m the o n - s h e l l amplitude (phase shifts, s c a t t e r i n g lengths) will be identical to those p r e d i c t e d by the Veneziano amplitude. There a r e t h r e e ways, however, in which we can use the extrapolated amplitude to make contact with experiment: (i) the A d l e r - W e i s b e r g e r relation [13] fixes the n o r m a l i z a tion of the amplitude, providing a modified KSFR relation; (ii) through eq. (7), we can d e r i v e a value for c, which can also be e s t i m a t e d , e.g. through m a s s relations [5]; (iii) eq. (8) enables us to d e r i v e information concerning certain KI3 p a r a m e t e r s . If we knew l, we would have only one f r e e p a r a m e t e r in M ~ a n d / W ~ ~. Since the amplitudes we a r e c o n s i d e r i n g r e f e r only to connected d i a g r a m s , the l we a r e seeking is the s m a l l e s t i n t e g e r for which l-1 + -
-
oum
o
,
Udxo2 and we a s s u m e that the s a m e l applies to the 7rn case. F r o m strong PCAC and the canonical commutation relations it is c l e a r that l >/ 3. We shall make the c o m p a r i s o n with e x p e r i m e n t using I = 3,4, 5, 6. We fix the n o r m a l i z a t i o n of f n by
OFF SHELL yK AMPLITUDES 3
201
1
(2y)Y(2qo)~
1
2
2
(24)
in which case the experimentalvalue is fy = 0.67 rn~. The Adler-Weisberger relation [13] for this amplitude is
a ~(~, o;o,o)i
i
(25)
s=m 2 = 5 '
a--~-
w h i c h f i x e s the v a l u e of yv~. In t e r m s of A and l, t h e n , we d e t e r m i n e the p r o d u c t ~ f 2 , /3 i n t u r n m a y b e d e t e r m i n e d f r o m the o b s e r v e d K* width, w h i c h g i v e s /3/4~ = 2.24. F o r the p a r t i c u l a r c a s e of i n t e r e s t eq. (8) g i v e s , a f t e r an i s o s p i n r o t a t i o n :
<0 I%+g- - ~K-~+ ]K-~+> = i <0[DK+IK- o>.
(26)
We define the K13 form factors by
~[&(s)(k-p)~ + f_(s)(k+p).] --(4k°F)~
/+(0) =
2 rM~_K ,,, 2 M~_K ,^ 2 ^ 0)] A m 2 t ufKftU, m u ; 0, 0) ~fKfLU, i n K ; u, ,
(28)
w h e r e A m 2 - m 2 - m~. 2 The m o s t a c c e s s i b l e e x p e r i m e n t a l quantity is fK/(f+(O) fu), f o r w h i c h the l a t e s t v a l u e i s 1.22 ~ 0.06 (ref. [14]). F i n a l l y , we can c a l c u l a t e c t h r o u g h eq. (7). T h i s q u a n t i t y i s r o u g h l y e s t i m a t e d to b e a b o u t - 1 . 2 5 (ref. [5]). We p r e s e n t the r e s u l t s of the c a l c u l a t i o n s i n t a b l e 1. F o r e a c h v a l u e of l, A h a s b e e n r o u g h l y c h o s e n to m a x i m i z e a g r e e m e n t f o r a l l t h r e e q u a n t i t i e s . We i n c l u d e in the t a b l e the v a l u e of A w h i c h r e s u l t e d f r o m t h i s p r o c e d u r e ; c i s v e r y i n s e n s i t i v e to A o v e r a wide r a n g e ; i3 and f+(0) a r e q u i t e s t r o n g l y d e p e n d e n t on A. T h e c a l c u l a t i o n s w e r e p e r f o r m e d on a c o m p u t e r , s i n c e s o m e of the s e r i e s i n v o l v e d do n o t c o n v e r g e very rapidly. Table 1 Values of/3/4/r,
fK/f+(O)f~ and c corresponding to different choices of l.
l
A
fl/4y
fK/f+(O)f~
3
0.95
2.67
0.74
-1.18
4
1.4
2.32
1.00
-1.18
5
2.1
2.24
1.19
-1.18
6
2.8
2.25
1.29
-1.18
The p a r a m e t e r A has been chosen in each case to give the best overall fit, and the value so arrived at is that displayed in the table.
202
P. HORWITz
6. DISCUSSION AND CONCLUSION F r o m table 1 it is evident that values of I > 3 a r e p r e f e r r e d . The a g r e e ment with e x p e r i m e n t is p a r t i c u l a r l y good for l = 5 and l = 6. F o r l = 3 the p r e f e r r e d value for A violates the Fuchs-Kuo bound [11], and in any case the fit to e x p e r i m e n t is quite poor. The value for c is essentially independent of l and is close to that which is obtained f r o m s i m p l e r f o r m - f a c t o r models
[6,151. The model amplitude proposed in this p a p e r is not unique, and no attempt has been made here to "derive" it, o r something like it, f r o m a dual n-point function. Its a g r e e m e n t with experiment, where c o m p a r i s o n is possible, is very good, if one is willing to allow some latitude in the choice of l. The app a r e n t inability of the model to d e s c r i b e , in a s y m m e t r i c fashion, p r o c e s s e s with two off-shell kaons, is a s e r i o u s drawback, which may point the way to an i m p r o v e m e n t of the model. At p r e s e n t it r e p r e s e n t s the only attempt known to the author to combine l o w - e n e r g y chiral c o n s t r a i n t s with Bjorken limit behaviour a r i s i n g out of a sum of poles in external m a s s variables. We feel that r e q u i r i n g "good" behaviour at low and high e n e r g i e s is likely to be important in exploring the relation between chiral s y m m e t r y breaking, c u r r e n t algebra, and the d u a l - r e s o n a n c e model.
APPENDIX In this appendix we shall derive the Bjorken limit behaviour of the a m p l i tude M~n~Kf, which we r e p r o d u c e below: nK 2 kf2) M~fKf(S, t;pf,
~
(-1) n
= -~'nK n= 0 F---(-~B)
x
F(1-aK*(S))F(l+n-ap(t)) F(1 - aK.(S) - ap(t))
F(A -s)
F(n+A_pf2)r(n+A-f)k 2
(15)
We break the series into two parts: M Bj-~ 19 F(1 -aK*(S)) 2 ( ~ 1 1 (_l)n r(n+Ar(n+B)-S)+ ~ F(A - s ) n=0 n =2N-1
a =A - B - s ,
(_l)nna(n+x)fl, (A. 1)
i 2 _ 2A + s and we have a s s u m e d that fl =-~+mTr
x(--- -t ~ _p2 ~ _k 2 >> 2N) is l a r g e enough for the Sterling approximation to be a r b i t r a r i l y good for the F-functions in which it a p p e a r s , and s i m i l a r l y for n >/ 2 N - 1. The finite s e r i e s obviously behaves like xfl; the infinite one, however, does not converge uniformly for a > -1 and must be examined separately. Consider the integral oo
Ia~=- f O
(277)a(21?+l)fid~?,
OFF SHELL 7rK AMPLITUDES
203
w h i c h c o n v e r g e s of and only if, a > -1, a+/3 < -1. If this condition is s a t i s fied we have
M lim lim ( 2 rln ) a ( 2 rln + 1) fl A rln , Ia/3 = lira 77N--' 0 M--' oo ZX77n--,0 n =N
(A.2)
where lim
r/n = ~o,
A~?n --- ~?n+l - rTn.
n----~ oo
We m a y , without l o s s of g e n e r a l i t y , c h o o s e ~n ~ n/x, in w h i c h c a s e ZXrln = 1/x, and we h a v e :
S(aN)~( x) x ~ - ~ xa+/3+lIa /3
f o r a > -1, /3 < - 1 ,
(A.3)
w h e r e we have u s e d the notation
S(~fl(x) - ~
n=N
(2n)a(2n+x)/3.
(A.4)
S~nce I,~a is i n d e p e n d e n t of x, eq. (A.3) ~ i v e s the a s y m p t o t i c b e h a v i o u r of S~)"" exp'licitly.~ If a ~< -1, a + / 3 < -1, S~'~" c o n v e r g e s u n i f o r m l y in x and g o e s like x/3 f o r l a r g e x. It r e m a i n s to put the s e r i e s in eq. (A.1) into this f o r m . We do this by c o m b i n i n g p a i r s of t e r m s and e x a m i n i n g the l e a d i n g b e h a v i o u r .
n =2N-1
(-1)nna(n+ x)/3 = { - ( 2 N - 1 ) a ( 2 N - l + x)/3 +(2N)a(2N+ x)/3} - . . . .
(A. 5)
The quantity in c u r l y b r a c k e t s m a y be e x p a n d e d in p o w e r s of 2N and 2N+ x, thus:
( } = a(2N)a-l(2N+x)fl+
/3(2N)a(2g+x)fl+ . . . ,
and t h e r e f o r e , to l e a d i n g o r d e r in n and x, we m a y w r i t e :
n =2N-1
(-1)n na(n+ x)/3 = a s(aN__1 ,fl(x) + /3 s(aN~_ 1 (x).
(A.6)
T h e l e a d i n g a s y m p t o t i c b e h a v i o u r in x is d e t e r m i n e d by the f i r s t t e r m on the r i g h t of (A.6):
S(a~l '/3(x) = O(x fl) = O(x a'+/3)
f o r a ~< O, f o r a > 0.
R e m e m b e r i n g what a and /3 a r e , f o r /V/~fKf, K we note that a+/3 d o e s not d e pend on s. Setting
204
P. HORWITZ
÷ ~ = ½ ÷ .~
- A - B -- - ½ l ,
(A.7)
we o b t a i n the a p p r o p r i a t e B j o r k e n l i m i t b e h a v i o u r f o r a c e r t a i n r a n g e in s: _!l
Bj
x -½1+s-A+B
(s >i A - B ) .
We n o t e that the s e r i e s of eq. (15) c o n v e r g e s f o r a+ fl--< °1, w h i c h r e q u i r e s l >/ 2. S i n c e the known c o m m u t a t i o n r e l a t i o n s p r e d i c t l >/ 3, the a m p l i t u d e i s f i n i t e f o r a l l the c a s e s of i n t e r e s t .
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