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A non-analytic S-matrix
Nuclear P h y s i c s B12 (1969) 281-300.North-Holland Publ. Comp., A m s t e r d a m
A NON-ANALYTIC S-MATRIX R. E. C U T K O S K Y *, P . V . L A H D S H O F F , D . I . O L I V E a n d J . C. P O L K I N G H O R N E **
Depa~rnent o/Applied Mathematics and Theoretical Physics, University of Cambridge, England Received 23 June 1969 A b s t r a c t : A study is made of t h e o r i e s having the unusual analyticity p r o p e r t i e s r e cently p r o p o s e d by Lee and Wick. A p r e s c r i p t i o n is given for setting up a c o v a r iant p e r t u r b a t i o n - t h e o r y expansion of the s c a t t e r i n g amplitudes, based on F e y n man graphs. It is found that the p r e s e n c e of a complex pole in the upper half plane of the p h y s i c a l sheet leads to points of non-analyticity in the physical region, such that the values of the amplitude to e i t h e r side of the point a r e not r e l a t e d by a n a l ytic continuation. It is shown how this is compatible with unitarity. The nature of the non-analyticity is not fully d e t e r m i n e d by unitarity. Neither, in the c a s e of the m o r e complicated graphs, is it fully d e t e r m i n e d by the p e r t u r b a t i o n - t h e o r y p r e scription, and some e x t r a c o n s t r a i n t must be imposed on the theory to r e m o v e the ambiguity. It is shown that the p r e s c r i p t i o n of Lee and Wick has an exactly s i m i l a r ambiguity, but for their p r e s c r i p t i o n different r e s u l t s a r e obtained in different Lorentz f r a m e s . An e s t i m a t e is mhde of the extent to which the theory violates causality, and is found to be too s m a l l to m e a s u r e .
1. I N T R O D U C T I O N L e e a n d W i c k h a v e r e c e n t l y r e k i n d l e d i n t e r e s t in t h e o r i e s w i t h u n c o n v e n t i o n a l a n a l y t i c i t y p r o p e r t i e s [1, 2]. T h e y h a v e p r e s e n t e d d e t a i l e d c a l c u l a t i o n s , in s e v e r a l s i m p l e m o d e l s , i l l u s t r a t i n g how u n i t a r i t y c a n b e p r e s e r v e d d e s p i t e t h e i n t r o d u c t i o n of p a r t i c l e s t h a t i n t e r a c t v i a n o n - H e r m i t i a n c o u p l i n g s . In t h e i r m o d e l s s u c h a p a r t i c l e l e a d s to t h e e x i s t e n c e of a p a i r of c o m p l e x - c o n j u g a t e p o l e s on t h e p h y s i c a l s h e e t of t h e s c a t t e r i n g a m p l i t u d e . T h e e v o l u t i o n o p e r a t o r U(t, to) i s not u n i t a r y a n d , m o r e o v e r , it b e c o m e s u n b o u n d e d a s t -~ ± co. H o w e v e r , t h e p h y s i c a l s t a t e s a r e d e f i n e d s o as to exclude the complex-mass particles, and contain only real-mass particles. The S-matrix is obtained by subtracting from U the terms that grow exponentially, and is shown to be unitary. O u r own i n t e r e s t in t h e s e m o d e l s a r i s e s f r o m t h e o p p o r t u n i t y t h a t t h e y provide for studying the interplay between analyticity and unitarity; they s e r v e t o e m p h a s i z e t h e s i g n i f i c a n c e of t h e a s s u m p t i o n s m a d e in t h e u s u a l S-matrix theory. * On leave of absence f r o m C a r n e g i e - M e l l o n U n i v e r s i t y (1968-1969). ** On leave of absence at the Lawrence Radiation Laboratory, Berkeley.
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One a s s u m p t i o n that is p a r t i c u l a r l y i m p o r t a n t in the usual t h e o r y is that the s c a t t e r i n g a m p l i t u d e is an a n a l y t i c function in the p h y s i c a l region, except f o r c e r t a i n ' a n a l y t i c s i n g u l a r i t i e s ' . By an a n a l y t i c s i n g u l a r i t y we shall m e a n that a path of a n a l y t i c continuation which m a k e s a s m a l l detour a r o u n d it r e l a t e s the v a l u e s of the p h y s i c a l a m p l i t u d e to e i t h e r side of it. In this p a p e r we shall conclude that the introduction of the u n p h y s i c a l c o m p l e x m a s s p a r t i c l e s r e q u i r e s that, if u n i t a r i t y is to be s a t i s f i e d , t h e r e be ' n o n analytic s i n g u l a r i t i e s ' in the p h y s i c a l region. F o r t h e s e , v a l u e s of the p h y s i c a l a m p l i t u d e to e i t h e r side of t h e m a r e not r e l a t e d by an a n a l y t i c continuation. Such s i n g u l a r i t i e s a r e f a m i l i a r enough in the study of u n i t a r i t y i n t e g r a l s [4] but t h e i r o c c u r r e n c e in the a m p l i t u d e i t s e l f is an e n t i r e l y new f e a t u r e . T h e s i m p l e s t n o n - a n a l y t i c s i n g u l a r i t y is a s s o c i a t e d with an u n p h y s i c a l s t a t e of two c o m p l e x - m a s s p a r t i c l e s , at the r e a l p h y s i c a l v a l u e s = (M+M*) 2 of the e n e r g y v a r i a b l e . In the usual S - m a t r i x t h e o r y such a ' n o r m a l t h r e s h o l d ' is a s s o c i a t e d with a t w o - p a r t i c l e i n t e r m e d i a t e s t a t e in the u n i t a r i t y equation. H e r e , this s t a t e , b e i n g u n p h y s i c a l , does not a p p e a r in u n i t a r i t y , and we shall show in s e c t . 3 that it m u s t c o n s e q u e n t l y b e a s s o c i a t e d with a nona n a l y t i c s i n g u l a r i t y . The d i s c u s s i o n in that s e c t i o n is b a s e d on u n i t a r i t y , which by i t s e l f allows c o n s i d e r a b l e f r e e d o m a s to the p r e c i s e c h a r a c t e r of the n o n - a n a l y t i c s i n g u l a r i t y . A p a r t i c u l a r e x a m p l e of a m o r e p r e c i s e s p e c ification is d e s c r i b e d in s e c t . 2, w h e r e we give a p o s s i b l e p r e s c r i p t i o n f o r c a l c u l a t i o n of the a m p l i t u d e in t e r m s of c o v a r i a n t F e y n m a n i n t e g r a l s . Our F e y n m a n - i n t e g r a l m o d e l g i v e s r e s u l t s s i m i l a r to t h o s e of L e e and Wick. A p a r t i c u l a r p r o b l e m with the L e e - W i c k p r e s c r i p t i o n is that it is f o r m u l a t e d in a given fixed r e f e r e n c e f r a m e , and is c o n s e q u e n t l y not m a n i f e s t l y c o v a r i a n t . L e e [2] h a s e x a m i n e d a s i m p l e F e y n m a n g r a p h with two c o m p l e x m a s s e s , and c h e c k e d that f o r this the L e e - W i c k p r e s c r i p t i o n does give an i n v a r i a n t r e s u l t . H o w e v e r , a s we d i s c u s s in appendix A, the p r o b l e m is m o r e c o m p l i c a t e d when t h e r e a r e m o r e c o m p l e x - m a s s p a r t i c l e s involved, and we find that, beyond the t h r e s h o l d f o r p r o d u c i n g four c o m p l e x m a s s p a r t i c l e s , t h e i r p r e s c r i p t i o n l e a d s to n o n - c o v a r i a n t t e r m s . T h e s a m e g r a p h s a l s o c a u s e difficulty in our p r e s c r i p t i o n , a s we show in sect. 4. H o w e v e r , f o r us the p r o b l e m is that the a m p l i t u d e , while it is c o v a r i a n t , is not fully s p e c i f i e d by the r a t h e r g e n e r a l f o r m u l a t i o n we h a v e given. S o m e e x t r a , m o r e detailed, r e q u i r e m e n t s m u s t be i m p o s e d in o r d e r to p r o d u c e a unique amplitude. As r e m a r k e d by L e e and Wick, t h e r e is a r i s k that the t h e o r y is nonc a u s a l . In appendix B we give a m o r e s u c c i n c t v e r s i o n of an e s t i m a t e they h a v e m a d e of the magnitude of one of the n o n - c a u s a l e f f e c t s , and a g r e e with t h e i r conclusion that it is too s m a l l to o b s e r v e . H o w e v e r , the l a c k of caus a l i t y does have a profound effect on t h e o r e t i c a l a s p e c t s of the s t r u c t u r e of s c a t t e r i n g a m p l i t u d e s , a s we d i s c u s s in s e c t . 2. 2. P E R T U R B A T I O N THEORY In this s e c t i o n we c o n s i d e r s o m e s i m p l e p e r t u r b a t i o n - t h e o r y g r a p h s , m a k i n g the s i m p l e s t p o s s i b l e m o d i f i c a t i o n of the u s u a l F e y n m a n t h e o r y that
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is e x p l i c i t l y c o v a r i a n t . F o r t h e s e g r a p h s our p r e s c r i p t i o n g i v e s r e s u l t s equivalent to the r a t h e r m o r e c o m p l i c a t e d f o r m a l i s m of L e e and Wick. All p a r t i c l e s will b e s u p p o s e d to h a v e z e r o spin. F o r o r d i n a r y , r e a l m a s s p a r t i c l e s we u s e the usual p r o p a g a t o r , and f o r the u n p h y s i c a l c o m p l e x - m a s s p a r t i c l e s we u s e the p r o p a g a t o r A A* A(P2) - p2 - M 2 + p2 - M*2 "
(2.1)
The couplings a r e a s s u m e d r e a l ; the n o n - H e r m i t i c i t y of the o r i g i n a l H a m i l tonian is r e p r e s e n t e d by the n o n - r e a l m a s s e s a p p e a r i n g in the p r o p a g a t o r [ 1]. The F e y n m a n i n t e g r a l is then c a l c u l a t e d a s the u s u a l i n t e g r a l using, f o r r e a l e x t e r n a l f o u r - m o m e n t a b e l o w all t h r e s h o l d s , the W i c k - r o t a t e d contour. Since the couplings a r e r e a l , a s well a s the p r o p a g a t o r A(p2) a s a whole, the s c a t t e r i n g a m p l i t u d e c a l c u l a t e d in t h i s way will be H e r m i t i a n . Also, if the e n e r g y v a r i a b l e s a t i s f i e s s < (2 R e M ) 2 , the c o m p l e x p o l e s do not p r o duce any n o n - a n a l y t i c i t y and n e i t h e r do they c o n t r i b u t e to the discontinuity a c r o s s the r e a l a x i s , so the u s u a l p e r t u r b a t i o n t h e o r y d i s c u s s i o n s of unit a r i t y f o r r e a l p a r t i c l e s a r e not a f f e c t e d in this region.
Fig. 1. The simplest Feynman graph that leads to non-analyticity. The wavy lines denote complex-mass particles, and the straight lines denote r e a l - m a s s particles. C o n s i d e r in p a r t i c u l a r the g r a p h of fig. 1, w h e r e the wavy l i n e s r e p r e s e n t c o m p l e x - m a s s p a r t i c l e s and the s t r a i g h t l i n e s a r e p h y s i c a l p a r t i c l e s . L e t k b e the loop i n t e g r a t i o n m o m e n t u m and w o r k in the c e n t r e - o f - m a s s f r a m e . In fig. 2a, we d r a w , f o r s < ( 2 R e M ) 2 , the p o s i t i o n in the kO p l a n e of the c o m p l e x - m a s s p o l e s r e l a t i v e to the p a t h of i n t e g r a t i o n , a s given by the p r e s c r i p t i o n above. (The c r o s s e s r e p r e s e n t the p o l e s of one of the p r o p a g a t o r s , the c i r c l e s t h o s e of the other.) If we r e g a r d the t h r e e - d i m e n s i o n a l loop i n t e g r a t i o n to be a l r e a d y c a r r i e d out t h e s e p o l e s would i n t e g r a t e into b r a n c h points with a t t a c h e d cuts. As s is i n c r e a s e d along the r e a l a x i s , the c r o s s e s m o v e t o w a r d s the right, until at s = (M+M*) 2 the l e f t - h a n d p a i r of c r o s s e s p i n c h e s with the r i g h t - h a n d p a i r of c i r c l e s . We now give a p r e s c r i p t i o n f o r e v a l u a t i n g the a m p l i t u d e f o r l a r g e r v a l u e s of s. T h i s p r e s c r i p t i o n will, a s we show in s e c t . 3, p r e s e r v e H e r m i t i c i t y and u n i t a r i t y , but we m u s t s t r e s s that it is by no m e a n s the only p r e s c r i p t i o n with t h e s e p r o p e r t i e s . F o r e x a m p l e , any a m p l i t u d e whose K - m a t r i x is r e a l will s a t i s f y t w o - p a r t i c l e u n i t a r i t y . We r e g a r d the t h e o r y as b e i n g a m a t h e m a t i c a l l i m i t of a situation w h e r e the m a s s e s on the two i n t e r n a l l i n e s have d i f f e r e n t i m a g i n a r y p a r t s . T h u s the u p p e r p r o p a g a t o r h a s p o l e s at M 2 and M~ 2, and the l o w e r one h a s p o l e s at M ~ and M~ 2, w h e r e
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F i g . 2. I n t e g r a t i o n p a t h s in t h e k o p l a n e f o r the d i a g r a m of fig. 1 f o r t h r e e d i f f e r e n t v a l u e s of t h e e n e r g y v a r i a b l e a s d e s c r i b e d in t h e t e x t . T h e c r o s s e s d e n o t e s i n g u l a r i t i e s of one p r o p a g a t o r , t h e c i r c l e s d e n o t e t h o s e of t h e o t h e r .
ml- M2
= i8,
(2.2)
5 being taken real, small and positive. Then the pinches no longer occur at a real value of s, and they no longer occur simultaneously in the upper and lower b_alves of the k o plane. That is, the continuation of the amplitude originally specified by the contour of fig. 2a has two branch points, at s = (M 1 + ~ ) 2 , s = (A~ +M2)2 , each displaced from the real axis as drawn in fig. 3. (There are additional branch points at (M 1 + M 2 ) 2 and (M] +M~.)2.) For finite 5 we can continue the amplitude along the real axis into the region between the cuts, and it will remain real. For this continuation, the path of integration in the k o plane is dragged into the shape shown in fig. 2b. W e now let 5 ---0. Then the values of the amplitude defined in this way to the right of the two branch points that have pinched down on the real axis cannot any longer be obtained by analytic continuation from those on the lea. The only continuation that can be m a d e from the lea is then either over
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CM,.
X Fig. 3. Displaced branch points in the s-plane for the Feynman graph of fig. 1. both pinching b r a n c h points, into the r e g i o n m a r k e d l , or u n d e r t h e m into the r e g i o n m a r k e d 1. The f i r s t of t h e s e continuations, f o r example, would r e s u l t in the k ° plane contour of fig. 2c. On the other hand ff we continue to the left the amplitude f o r the p a r t i c u l a r g r a p h of fig. 1 as defined at the right of the pinching b r a n c h points, we find that it is actually analytic at s = (M+M*)2; it m a y be shown that in this p a r t of the amplitude the e f f e c t s of the two b r a n c h points happen to c a n c e l when they pinch t o g e t h e r . N e v e r t h e l e s s , the v a l u e s of the amplitude so continued into s < (M+M*)2 a r e not supposed to be p h y s i c a l l y significant. We extend this p r e s c r i p t i o n to m o r e c o m p l i c a t e d graphs. The extension to o t h e r n o r m a l t h r e s h o l d s , involving any n u m b e r of r e a l - m a s s p a r t i c l e s and one p a i r of c o m p l e x - m a s s p a r t i c l e s , is i m m e d i a t e . As a l e s s s t r a i g h t f o r w a r d example, we now c o n s i d e r the g r a p h s of fig. 4. T h e s e a r e given by the s a m e F e y n m a n i n t e g r a l , the only d i f f e r e n c e being that f o r fig. 4b the v a r i a b l e t, which would be fixed at s o m e negative value in o r d e r to lie in the p h y s i c a l r e g i o n f o r fig. 4a, is fixed at ~~ . In e i t h e r c a s e , fig. 5a shows, in the r e a l (s, ~) plane, the n o r m a l t h r e s h o l d s and the Landau c u r v e f o r the t r i a n g l e s i n g u l a r i t y c o r r e s p o n d i n g to m a s s (M+M*) in the l e f t - h a n d side of the d i a g r a m . The dotted c u r v e applies only f o r the c a s e of fig. 4b, for which it m a r k s the b o u n d a r y of the p h y s i c a l region; it is the s e c o n d - t y p e s i n g u l a r i t y c u r v e [5], which t o u c h e s the Landau c u r v e at the point X. (Our notation c o r r e s p o n d s to that of L O P [6].)
Fig. 4. Feynman graphs leading to physical-region triangle singularities involving complex-mass particles. In the usual t h e o r y , only the a r c AB of the Landau c u r v e , c o r r e s p o n d i n g to p o s i t i v e v a l u e s of the F e y n m a n u - p a r a m e t e r s , is s i n g u l a r in the p h y s i c a l sheet. T h i s p r o p e r t y is c l o s e l y r e l a t e d to r e q u i r e m e n t s of c a u s a l i t y [7]: that the flight t i m e s of the i n t e r n a l p a r t i c l e s be positive. In the p r e s e n t
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(.~.) F i g . 5. P l o t s of the L a n d a u c u r v e f o r the t r i a n g l e s i n g u l a r i t i e s of the F e y n m a n g r a p h of fig. 4b. In (a) the L a n d a u c u r v e i s p l o t t e d in the (Re s , Re(~) p l a n e f o r M 1 = M 2. The s t r a i g h t l i n e s t a n g e n t to the c u r v e at A and B a r e the s and (~ n o r m a l t h r e s h o l d s . In (b) it is s h o w n how the l o c u s of the s i n g u l a r i t y m o v e s in the s - p l a n e w h e n Re (~ is i n c r e a s e d b e y o n d its t h r e s h o l d (point B) and I m a i s f i x e d at a v e r y s m a l l and p o s i t i v e value, f o r e a c h of the two t e r m s in which the l o c u s c o l l a p s e s to r e a l v a l u e s when M 1 = M 2. The c r o s s e s (point A) d e n o t e the d i s p l a c e d n o r m a l - t h r e s h o l d b r a n c h p o i n t s . The s o l i d p o r t i o n s of c u r v e s i n d i c a t e t h a t the s i n g u l a r point i s on the e x p o s e d s h e e t , and the d a s h e d p a r t of t h e c u r v e s i n d i c a t e s t h a t the s i n g u l a r point i s r e a c h e d by a c o n t i n u a t i o n r o u n d the c o r r e s p o n d i n g n o r m a l - t h r e s h o l d b r a n c h point.
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t h e o r y the a n a l y t i c i t y p r o p e r t i e s usually a s s o c i a t e d with c a u s a l i t y have alr e a d y been violated, by the introduction of poles in the u p p e r - h a l f plane on the p h y s i c a l sheet and by the r e s u l t i n g p r e s e n c e of n o n - a n a l y t i c s i n g u l a r i ties in the p h y s i c a l region. We shall now e n c o u n t e r f u r t h e r such unusual p r o p e r t i e s ~ we find that the whole a r c of the Landau c u r v e is s i n g u l a r in the p h y s i c a l region, including the p a r t s ooA and B °o that c o r r e s p o n d to s o m e of the a - p a r a m e t e r s being negative, and so to negative flight t i m e s for the int e r n a l p a r t i c l e s [7]. It is, however, an analytic singularity. To a n a l y s e the g r a p h s , introduce different c o m p l e x m a s s e s M 1 and M2, and take the s a m e l i m i t as b e f o r e . E a c h c o m p l e x - m a s s p r o p a g a t o r is the s u m of two t e r m s (2.1), so that the F e y n m a n integral is the sum of four t e r m s . Of these, the two t e r m s r e s p e c t i v e l y involving M1, M~ and M~, M 2 a r e those that a r e of i n t e r e s t , b e c a u s e they p r o d u c e a r e a l Landau c u r v e in the limit 5 -~ 0. F o r 5 > 0, so that Im (M~ +M2) < 0, the situation f o r the s e c o n d of t h e s e t e r m s is just the usual one. The p h y s i c a l limit is evaluated on the u p p e r side of the cuts attached to the n o r m a l t h r e s h o l d s a = 4 p 2 and s = (M~ + M 2 + + ~)2, so that only the a r c AB is singular. F o r r e a l ~ this a r c is d e p r e s s e d into the l o w e r s - p l a n e , so that one a l s o p a s s e s above this s i n g u l a r i t y to evaluate the p h y s i c a l limit. This is i l l u s t r a t e d in fig. 5b. F o r the f i r s t t e r m one p a s s e s above the a = 4p2 cut as usual, but below the s = (M 1 +M~ + ~)2 cut, s i n c e this b r a n c h point a p p e a r s in the upper half s - p l a n e f o r ~ > 0. Thus the a r c AB is n o n - s i n g u l a r for this t e r m , but ~oA and Boo a r e singular, as is a l s o i l l u s t r a t e d in fig. 5b. F o r r e a l a the a r c s XA and Boo a r e r a i s e d into the upper half of the s - p l a n e when 5 > 0, so one p a s s e s below the cuts a t t a c h e d to t h e s e a r c s in the s - p l a n e to evaluate the p h y s i c a l limit. The a r c X~o (which is only p r e s e n t in the c a s e of fig. 4b) is d e p r e s s e d into the lower half s - p l a n e , so the cut a t t a c h e d to this a r c r u n s below the unphysical limit. The c h a n g e - o v e r at the point X is a c c o u n t e d for by noting that the s e c o n d - t y p e c u r v e is s i n g u l a r along the p a r t of it above X when it is a p p r o a c h e d t h r o u g h r e a l v a l u e s of s, a f r o m outside the p h y s i c a l region, though not when a p p r o a c h e d f r o m inside the p h y s i c a l r e g i o n (being then shielded by the cut attached to the a r c X ~ of the Landau curve). The r e s u l t , when we add t o g e t h e r the two t e r m s in the limit 5 ~ 0, is that the a r c s AB, Xoo have cuts a t t a c h e d to them running below the p h y s i c a l limit in the s - p l a n e , while the a r c s XA, B~o have cuts attached running above the p h y s i c a l limit. In each c a s e the v a r i a b l e a is r e g a r d e d as fixed at a r e a l value, evaluated above the a = 4 p 2 cut. In sect. 3 we show how t h e s e p r e s c r i p t i o n s a r e a c c o m m o d a t e d in the u n i t a r i t y equations. We may s u m m a r i z e as follows our p r e s c r i p t i o n for continuing past the (M+M*) n o n - a n a l y t i c i t y point: we change the c o m p l e x m a s s by s m a l l a m o u n t s i 5M. The change is to be different in each different c o m p l e x p r o p a g a t o r . Then each s i n g u l a r i t y involving two c o m p l e x - c o n j u g a t e m a s s e s is r e p l a c e d by a p a i r of t h r e s h o l d s , one lying above, and one lying below, the r e a l axis. The n o r m a l t h r e s h o l d s a r i s i n g f r o m a s e q u e n c e of i t e r a t e d bubbles or l a d d e r s (for example) a r e then r e p l a c e d by a conjugate p a i r of c l u s t e r s of b r a n c h points. The path of analytic continuation is defined to be the r e a l axis. This is in exact analogy to the F e y n m a n p r e s c r i p t i o n of
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giving each internal r e a l m a s s an addition -iE (where ~ may a l s o be different for each p r o p a g a t o r ) . In continuing past a t h r e s h o l d for s e v e r a l p a r t i c l e s , two of which have c o m p l e x m a s s e s , we m u s t r e q u i r e that e << 5 = = 15M1- 5M21 ; o t h e r w i s e , the r a t i o s of the different m a s s changes b e c o m e i m m a t e r i a l in the limit in which all m a s s changes vanish. This p r e s c r i p t i o n also implicitly defines a way to continue past o t h e r Landau s i n g u l a r i t i e s . In o r d e r for the r e s u l t to be s e l f - c o n s i s t e n t , it is sufficient that the s i n g u l a r i t y c u r v e always r e t r e a t f r o m r e a l values of the v a r i a b l e s when the m a s s e s a r e altered. This is not obvious, but is is e a s i l y checked for the t r i a n g l e s i n g u l a r i t i e s c o n s i d e r e d in this p a p e r , and it is t r u e except when t h e r e is a r e a l point of contact with the s e c o n d - t y p e sing u l a r i t y curve. In this c a s e (see above) the rule is that in continuing along a path with r e a l v a l u e s of s, t and a, the path is not to encounter the s e c o n d type singularity. The t h r e s h o l d for four c o m p l e x - m a s s p a r t i c l e s is obviously m o r e delicate, b e c a u s e it is p o s s i b l e for the t h r e s h o l d to o c c u r at a r e a l point even when all four c o m p l e x m a s s e s a r e given different i m a g i n a r y additions. It will be shown, in sect. 4, that a s e r i o u s ambiguity does, in fact o c c u r on this account. But this is an ambiguity r a t h e r than an i n c o n s i s t e n c y ; an additional p r e s c r i p t i o n would be needed c o m p l e t e l y to define the t h e o r y .
3. UNITARITY Consider first the 'normal threshold' at s = (M+M*) 2. In sect. 2 we gave a particular prescription for dealing with the single Feynman graph of fig. I, but this by itself is too simple to give a complete picture of the p r o p e r t i e s of the t h r e s h o l d . As s u g g e s t e d at the end of sect. 2, one has to s u m an infinite n u m b e r of i t e r a t i o n s of this bubble graph, and also include bubble g r a p h s with r e a l - m a s s internal lines. One r e s u l t of this will be that the b r a n c h points at (M 1 +M~2)2 and (M1* +M2) 2 , which pinch in the limit 5 --* 0, a p p e a r on both s h e e t s of the s = 4~ 2 cut. That is, they o c c u r both in the p h y s i c a l r e g i o n for the p h y s i c a l (+) amplitude, and for the H e r m i t i a n - c o n j u g a t e (-) amplitude. This we show in fig. 6a and in fig. 6b, w h e r e the dotted lines show the paths of continuation we must follow to keep r e s p e c t i v e l y with the (+) and (-) amplitudes. As 5 ~ 0, each path is pinched by the p a i r of b r a n c h points, and in the limit the analytic continuation is no l o n g e r possible. Suppose that < R e M < 2~ .
(3.1)
Then to either side of s = (M+M*) 2 the p h y s i c a l u n i t a r i t y equation r e a d s (see E L O P [3])
Unlike the f a m i l i a r situation with n o r m a l t h r e s h o l d s , no e x t r a t e r m a p p e a r s above s = (M+M*) 2, and we show now how the p r e s c r i p t i o n of sect. 2 is c o m p a t i b l e with this.
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OC-}
Fig. 6. The dashed lines show the paths of analytic continuation for the (+) and (-) amplitudes, in relation to the displaced thresholds for unphysical particles. T h i s is v e r y s i m p l e . We continue eq. (3.2) f r o m the left of s = (M+M*)2, p a s s i n g a b o v e the b r a n c h point at s = (M 1 +M~.) 2 and b e l o w the b r a n c h point a t (M1 +M~) 2. Along t h i s path of continuation, a c c o r d i n g to the two diag r a m s of fig. 6, the (+) a m p l i t u d e r e m a i n s the (+) a m p l i t u d e and the (-) a m plitude r e m a i n s the (-) a m p l i t u d e . T h u s eq. (3.2) a s a whole r e m a i n s unchanged, a s is r e q u i r e d . T h u s the n o n - a n a l y t i c i t y , which a p p e a r s when we t a k e the l i m i t 5 ~ 0, is such a s to allow u n i t a r i t y . We can a l s o show that n o n - a n a l y t i c i t y is a c t u a l l y a n e c e s s a r y f e a t u r e , if the a m p l i t u d e is to obey both u n i t a r i t y and h e r m i t icity. F o r to a v o i d n o n - a n a l y t i c i t y , the p a t h of continuation for the (+) a m plitude in fig. 6a would h a v e had to p a s s to the s a m e side of the two b r a n c h p o i n t s , e i t h e r a b o v e t h e m both of below t h e m both, to a v o i d b e i n g pinched in the l i m i t 8 ~ 0. T o p r e s e r v e h e r m i t i c i t y , the p a t h of continuation f o r the (-) a m p l i t u d e m u s t be the m i r r o r i m a g e in the r e a l axis of that f o r the (+) a m p l i t u d e . T h u s the continuation of eq. (3.2) that p r e s e r v e d the (+) a m p l i tude would not p r e s e r v e the (-) a m p l i t u d e . Our d e m o n s t r a t i o n of the c o m p a t i b i l i t y with u n i t a r i t y h a s r e q u i r e d m e r e ly the a s s u m p t i o n t h a t the t h e o r y can b e r e g a r d e d a s a m a t h e m a t i c a l l i m i t of one containing two d i f f e r e n t c o m p l e x m a s s e s . T h i s a s s u m p t i o n , without f u r t h e r s p e c i f i c a t i o n of the n a t u r e of the limit, is by no m e a n s enough to d e t e r m i n e the n a t u r e of the n o n - a n a l y t i c i t y . To do t h i s , one m u s t p o s t u l a t e equations giving the d i f f e r e n c e b e t w e e n the p h y s i c a l a m p l i t u d e as defined f o r s > (M+M*)2, and the continuation e i t h e r a b o v e o r b e l o w s = (M+M*)2 of the p h y s i c a l a m p l i t u d e a s defined f o r s < (M+M*)2. B e f o r e the l i m i t 5 --* 0 is taken, t h e s e d i f f e r e n c e s a r e just the d i s c o n t i n u i t i e s a c r o s s the (M1 +M~) and (M~ + M2) cuts.
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The s i m p l e s t postulate for t h e s e discontinuties is that they a r e s i m i l a r in f o r m to the usual n o r m a l - t h r e s h o l d discontinuities. T h i s is the c a s e for our p e r t u r b a t i o n - t h e o r y p r e s c r i p t i o n . Thus
T h e notation u s e d in eqs. (3.3a, b) is that the upper wavy line r e p r e s e n t s e i t h e r M 1 or M~I, and the lower e i t h e r M 2 or M~; the * shows which of the p a i r is the conjugated m a s s . Along with eqs. (3.3a, b) we have the H e r m i t Jan- conjugate equations:
H e r e the labels (+ T), (+ ~), (-T), and (- 1) a r e defined in fig. 6, with the l i m i t M 2 ~ M 1 being understood. T h e s e equations involve the amplitudes f o r the unphysical p r o c e s s e s tt + g. ~
M +M*
(3.4a)
and have no content until the p r o p e r t i e s of t h e s e amplitudes a r e specified. This we do by following again the suggestion of our p e r t u r b a t i o n t h e o r y , which is to postulate that they obey equations just like eqs. (3.3), with app r o p r i a t e changes of the e x t e r n a l p a r t i c l e s . The s a m e applies to the amplitude f o r M +M* - - M +M* ,
(3.4b)
which will then also e n t e r in the equations. We shall also suppose that discontinuities c o r r e s p o n d i n g to the (M +M*) s i n g u l a r i t i e s in s u b e n e r g y v a r i a b l e s of m u l t i p a r t i c l e amplitudes take analogous f o r m s . One might p e r h a p s be unhappy about defining his t h e o r y as a m a t h e m a t i cal limit in which the m a s s e s of the p a r t i c l e s a r e v a r i e d , in an a r t i f i c i a l way, and in fact once we have postulated eqs. (3.3) this idea is no longer n e c e s s a r y (though it is still convenient). T o see this, we now show that if we a s s u m e that the physical u n i t a r i t y equation (3.2) is valid for s < (M+M*) 2, t o g e t h e r with the analogous equations f o r the p r o c e s s e s (3.4), and that we also have eqs. (3.3) t o g e t h e r with the analogous equations for the a m p l i t u d e s (3.4), then the physical u n i t a r i t y equation (3.2) is also valid for s > (M+M*) 2. Suppose then that eq. (3.2) is valid f o r s < (M+M*) 2, and continue it to s > (M+M*) 2, via the l o w e r - h a l f plane. T h i s gives
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Add t h i s to eq. (3.3b). E l i m i n a t e the (+ 1) a m p l i t u d e s f r o m the r i g h t - h a n d s i d e of the r e s u l t by u s i n g eq. (3.3b) a g a i n and a l s o the analogue of eq. (3.5) obtained b y r e p l a c i n g the l e f t - h a n d s t a t e by (M+M*). The r e s u l t is
Now u s e eq. (3.3a*) to e l i m i n a t e the (- 1) amplitude:
We now h a v e to show that the r i g h t - h a n d side of eq. (3.7) v a n i s h e s . T h i s we do by r e p e a t i n g all the c a l c u l a t i o n s with the r i g h t - h a n d s t a t e r e p l a c e d by ( M + M * ) . The a n a l o g u e of eq. (3.7) thus r e a d s : 0 T h e s e c o n d f a c t o r d o e s not Vanish; in f a c t its i n v e r s e can be found f r o m the analogue of eq. (3.3a*), and so we c o m p l e t e the proof. We e m p h a s i z e that eqs. (3.3) could be r e p l a c e d by quite d i f f e r e n t a s s u m p t i o n s , with u n i t a r i t y still being m a i n t a i n e d . The i c e c r e a m - c o n e
T h e i c e c r e a m - c o n e s i n g u l a r i t y of fig. 4b, f o r the c a s e w h e r e all the p a r t i c l e s a r e ' o r d i n a r y ' , h a s b e e n a n a l y s e d in detail in L O P [6]. We now study how that s i n g u l a r i t y fits in with u n i t a r i t y in the p r e s e n t t h e o r y . We a s s u m e a knowledge of the t e c h n i q u e s of L O P , or of t h e i r f o r m a s s i n c e d e v e l o p e d [4]. We c o n s i d e r the s i n g u l a r i t y of fig. 4a (now r e g a r d e d a s a L a n d a u diag r a m r a t h e r than a F e y n m a n graph), u s i n g the u n i t a r i t y equation SS + = 1 f o r the p r o c e s s ~ + p + ~ ~ ~ + p + ~, below the 4 ~ t h r e s h o l d . F o r s i m p l i c i ty, we a s s u m e s o m e s o r t of c o n s e r v a t i o n law that e l i m i n a t e s 2~ s t a t e s f r o m the equation. It will a g a i n be c o n v e n i e n t to r e g a r d the t h e o r y a s a m a t h e m a t i c a l l i m i t of one with two d i f f e r e n t c o m p l e x m a s s e s . The d i s c u s sion is on the s a m e l i n e s a s that of L O P , e x c e p t that f o r 5 > 0 t h e r e is the (M 1 + M~2) ' n o r m a l - t h r e s h o l d ' o c c u r r i n g on the unusual side of the path of continuation that p r e s e r v e s the (+) a m p l i t u d e , and the (M~I + M2) s i n g u l a r i t y on the unusual side f o r the (-) a m p l i t u d e . I n t r o d u c e a v a r i a b l e ~ that m e a s u r e s d i s p l a c e m e n t s along the inward n o r m a l to the i c e c r e a m - c o n e L a n d a u c u r v e L: a s we c r o s s the c u r v e , d s = C a 1 d~?, da = C a 2 dT?.
(3.9)
(This definition of 77 d i f f e r s f r o m that in L O P by the f a c t o r C in eqs. (3.9), and c o r r e s p o n d s to the definition u s e d in l a t e r work. It will be r e c a l l e d that C is p o s i t i v e on the a r c s XA, AB, B °o of fig. 5a, and is n e g a t i v e on ooX when the point X is on the a r c . ) We shall c h o o s e to c r o s s L via a r o u t e f o r which
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(3.10)
ImT? < 0 .
Suppose that, on analytic continuation by such a route, from outside L to inside L, ,
(3.11) That is, at this stage, we are not assuming that the continuation preserves either the (+) amplitude of the (-) amplitude, though it may do so. Standard methods then show that, on the arc AB, the only t e r m in the unitarity equation that generates the singularity is the f i r s t t e r m in fig. 7.
Fig. 7. As one continues along the route (3.10), the integration in the intermediate state passes to the wrong side of the (M~ +M2) threshold in a subenergy of the left-hmld bubble, and so the result differs from the natural continuation of that t e r m by a t e r m involving the discontinuity across the ( ~ +M2) cut. According to the assumptions we have made above, this t e r m is
~
-
(,3.12)
~
Hence if, following the usual procedure, we take account also of regeneration mechanisms, and subtract the resulting continued unitarity equation from the physical unitarity equation that operates inside L, we obtain
+
(3.13)
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S i m i l a r m a n i p u l a t i o n s applied to the a r c Boo show that on that a r c the r i g h t hand side of eq. (3.13) m u s t be r e p l a c e d by
+ (3.14) On the a r c s coX, XA the s e c o n d t e r m of fig. 7 a l s o g e n e r a t e s the s i n g u l a r i t y , and a f t e r s o m e m a n i p u l a t i o n we find that on t h o s e a r c s the r i g h t - h a n d side of eq. (3.9) b e c o m e s
(3.15) F o r the o r d i n a r y i c e c r e a m - c o n e s i n g u l a r i t y we get z e r o i n s t e a d of eqs. (3.14), so that in f a c t t h e r e is no s i n g u l a r i t y on the a r c s coA, Boo. In the p r e s e n t c a s e t h e r e m u s t be s o m e d i f f e r e n t s o r t of b e h a v i o u r on t h e s e a r c s , but the a b o v e r e s u l t s by t h e m s e l v e s do not tell us w h e t h e r it is a s i n g u l a r ity or n o n - a n a l y t i c i t y . A c c o r d i n g to the p r e s c r i p t i o n s of s e c t . 4, in this c a s e we a c t u a l l y e x p e c t a s i n g u l a r i t y , and in f a c t our deductions f r o m unit a r i t y a r e in a g r e e m e n t with t h o s e p r e s c r i p t i o n s . The p r e s c r i p t i o n s s a y that on the a r c s AB, B °O
and then the deduction f r o m u n i t a r i t y is that the discontinuity b e t w e e n the (+) a m p l i t u d e and the (1) a m p l i t u d e is g i v e n on t h o s e two a r c s r e s p e c t i v e l y by the r i g h t - h a n d side of eq. (3.13) and by eq. (3.14), hut with the r i g h t m o s t (-) b u b b l e s in t h o s e t e r m s r e p l a c e d by (+) bubbles. On the a r c XA, the p r e s c r i p t i o n s s a y that
(The d i f f e r e n c e b e t w e e n eqs. (3.16) and (3.17) a r i s e s b e c a u s e ~? is m e a s u r e d along the i n w a r d n o r m a l to L, so that on XA the p o s i t i v e ~? d i r e c t i o n c o r r e sponds to the n e g a t i v e s direction). T h u s on that a r c the discontinuity b e t w e e n the (-) a m p l i t u d e and the (2) a m p l i t u d e is given by eq. (3.15), but with the l e f t - m o s t bubble r e p l a c e d by a (-) bubble. On the a r c coX, C h a s changed sign and so t h e r e is an e x t r a m i n u s sign in the r e l a t i o n b e t w e e n 77 and s; this a c c o u n t s f o r the e f f e c t s of the contact at X of L with the s e c o n d type s i n g u l a r i t y , a s d e s c r i b e d in s e c t . 2. (The point X is only p r e s e n t f o r fig. 4b, but the a n a l y s i s of that s i n g u l a r i t y is v e r y s i m i l a r . ) T h i s d e s c r i p t i o n h a s b e e n f o r a continuation via I m ~ < 0 of the S S + = 1
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u n i t a r i t y equation. It may be v e r i f i e d that a continuation via Im ~? > 0 gives the s a m e r e s u l t , and so also does the S+S = 1 f o r m of unitarity.
4. F U R T H E R DIAGRAMS We now apply p r e s c r i p t i o n s s i m i l a r to those in sect. 2 to two f u r t h e r F e y n m a n g r a p h s , those in figs. 8a and 8b (The g r a p h of fig. 8a is the s a m e as that of fig. 4a, but we a r e now c o n c e r n e d with a different p h y s i c a l region).
Fig. 8a. The same Feynman graph as shown in fig. 4a, but in a different physical region obtained by giving the variables different values. Fig. 8b. A Feynman graph with four internal complex-mass particles. We find that for t h e s e the Landau c u r v e is a c u r v e of n o n - a n a l y t i c s i n g u l a r ity. We also find that in the c a s e of fig. 8b, the p r e s c r i p t i o n s a r e not c o m plete; f u r t h e r p o s t u l a t e s a r e n e c e s s a r y if the amplitude is to be fully det e r m i n e d . As with the s i m p l e r s i n g u l a r i t i e s that we have a l r e a d y considered, we f i r s t d i s c u s s the p e r t u r b a t i o n - t h e o r y p r e s c r i p t i o n s and then des c r i b e how they fit in with unitarity. C o n s i d e r the F e y n m a n g r a p h of fig. 8a. I n t r o d u c e two complex m a s s e s a s in sect. 2, s a t i s f y i n g eq. (2.2). The r e a l Landau c u r v e a r i s e s again f r o m the combinations (M 1 +M~) and (M~ +M2) of complex m a s s e s . F o r 5 > 0 the analytic p r o p e r t i e s of the t e r m involving the l a t t e r combination a r e just as usual: a p h y s i c a l - r e g i o n s i n g u l a r i t y only on the a r c AB of fig. 5a (which is now to be r e g a r d e d as depicting the new Landau c u r v e c o r r e s p o n d i n g to fig. 8a), which for r e a l ~ lies below the r e a l axis in the s - p l a n e , on the upper (physical) side of the n o r m a l - t h r e s h o l d cut attached to s = (~ +M~ +M2)2. F o r the other t e r m , the p r e s c r i p t i o n f o r a p p r o a c h i n g the ' n o r m a l - t h r e s h o l d ' cut attached to e = (p + M 1 +M~2)2 is the opposite f r o m the usual one, and so in the s - p l a n e the s i n g u l a r i t y a p p e a r s (again only on AB) in the s - p l a n e on the lower side of the cut attached to s = (p + M 1 +M~2)2 , which is just the p h y s i c a l side of this cut. F o r r e a l a the s i n g u l a r i t y is above the r e a l axis in the s - p l a n e . The situation is drawn in fig. 9; h e r e X and Y denote the two s i n g u l a r i t i e s , and the dotted line is the path of continuation that p r e s e r v e s the (+) amplitude. As 6 ~ 0, X and Y pinch the path of continuation and so lead to a n o n - a n a l y t i c singularity. In fig. 9 we have a l s o drawn the cut attached to s = 9~ 2, which is p r e s e n t in the c o m p l e t e amplitude. If we sweep this cut into the upper half plane, an exactly s i m i l a r set of s i n g u l a r i t i e s is r e v e a l e d , with the (-) b o u n d a r y value
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Fig. 9. Locus of singularities and branch cuts in the s-plane, and the path of analytic
continuation between the displaced singularities, for the Feynman graph of fig. 7. r e p l a c i n g the (+) ( c o m p a r e figs. 6a and 6b). That is, if we continue the (+) amplitude along the path shown so that it r e m a i n s the (+) amplitude (a continuation that is no l o n g e r p o s s i b l e in the l i m i t 5 ~ 0), the (-) amplitude continued along the s a m e path will r e m a i n the (-) amplitude. In o r d e r f o r t h e s e p r e s c r i p t i o n s to be in a g r e e m e n t with u n i t a r i t y , it is n e c e s s a r y that this continuation be n a t u r a l for all the t e r m s in which the s i n g u l a r i t y might be g e n e r a t e d , and an investigation of t h e s e shows that this is indeed the c a s e . This would not be t r u e f o r continuations along other paths. The F e y n m a n g r a p h of fig. 8b contains two p a i r s of c o m p l e x - m a s s lines. A r e a s o n a b l e p r e s c r i p t i o n s e e m s to be to i n t r o d u c e different m a s s e s on e a c h of t h e s e lines, M 1 and M2 on the upper p a i r s a t i s f y i n g eq. (2.2), and M 1' and M 2' on the l o w e r p a i r s a t i s f y i n g ~I
- Mg. = i5' .
(4. I )
Take 5, 5' > 0, and then c o n s i d e r the l i m i t in which 5 ~ 0, 5' ~ 0. The int e g r a l contains 24 different t e r m s , but in only four of t h e s e do the s i n g u l a r ities a p p r o a c h the r e a l axis in the limit. It t u r n s out that the r e s u l t s of this p r e s c r i p t i o n depends on the r e l a t i v e m a g n i t u d e s of 5 and 5' as they tend to z e r o . The location of s i n g u l a r i t i e s in the s - p l a n e f o r 5 > 5' is shown in fig. 10a. T h e r e is one p a i r of s i n g u l a r i t i e s that p i n c h e s on each of the a r c s ~oA, AB and B °O in fig. 5a (that f i g u r e now being r e g a r d e d as the Landau c u r v e f o r fig. 8), so p r o d u c i n g a n o n - a n a l y t i c s i n g u l a r i t y on each a r c . The fact that the s i n g u l a r a r c Aoo l e a v e s the r e a l s , a plane f o r n o n - z e r o 5, 5', with 5 > 5', depends on the following inequality which can be deduced f r o m the Landau equation valid on A¢o:
(M+M*)a 1 > (M' + M ' * ) a 3 . F o r 5 < 5', t h e r e a r e two p a i r s of s i n g u l a r i t i e s (as shown in fig. 10b), that pinch on the a r c AB, so p r o d u c i n g n o n - a n a l y t i c i t y t h e r e , but on the a r c s ¢oA and B ¢o the amplitude is r e g u l a r - that is, t h e r e is not even an analytic singularity. The two p r e s c r i p t i o n s a r e c l e a r l y v e r y different in t h e i r effects, and an examination of u n i t a r i t y v e r i f i e s that either, or a combination of both, is allowed. In o r d e r fully to d e t e r m i n e the t h e o r y s o m e f u r t h e r c o n s t r a i n t , that is, s o m e e x t r a d y n a m i c s , m u s t be imposed. In appendix A we show
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M"..,-
*.0 L
V,
(S, +N, * ML+ Jq,.)
V,
Pie.@IVt% J
Fig. 10. Loci of singularities and branch cuts in the s - p l a n e for the graph of fig. 8, when cy is real. Case (a) a r i s e s where Im (M1 - M2) > Im (M1 ' - M2'), and case (b) in the opposite situation. The c r o s s e s denote the displaced normal thresholds, and the points A and V the displaced triangle singularities; A denotes a point on the ar c AB of fig. 4, and V denotes a point on the arc Boo. It should be noted that we have drawn on one figure the singularities a r i s i n g f r o m four individual t e r m s ; thus the points in situation (a) a r e not shielded by the branch cuts lying between them and the r e a l axis (shown as a dashed line). t h a t in t h e t h e o r y of L e e a n d W i c k , t h e p r e s c r i p t i o n of L e e [2] l e a d s s o m e t i m e s t o o n e of o u r t w o p r e s c r i p t i o n s , s o m e t i m e s to t h e o t h e r , a c c o r d i n g to which L o r e n t z f r a m e is c h o s e n for its f o r m u l a t i o n . W e a r e p l e a s e d to a c k n o w l e d g e a n i n t e r e s t i n g d i s c u s s i o n w i t h P r o f e s s o r T. D. L e e .
APPENDIX A
The Lee-Wick prescription L e e and Wick u s e a t h r e e - d i m e n s i o n a l
formalism;
it is e q u i v a l e n t t o a
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f o u r - d i m e n s i o n a l one in which the i n t e g r a t i o n path o v e r p o avoids the p o l e s at p o = + E (where E 2 = M 2 + 2 i v M + p 2) a c c o r d i n g to a definite p r e s c r i p t i o n . T h i s path is equivalent, f o r e x t e r n a l f o u r - m o m e n t a v a l u e s that a r e r e a l and b e l o w all t h r e s h o l d s , to a W i c k - r o t a t e d path. H o w e v e r , L e e and Wick a l s o c h o s e a s p e c i a l f r a m e in which all m o m e n t u m c o m p o n e n t s P o a r e r e a l , while E o is c o m p l e x . Then, in a n o t h e r f r a m e , m o v i n g with r e s p e c t to the fixed one with v e l o c i t y Vo, the m o m e n t u m p h a s an i m a g i n a r y p a r t : Imp = vo(1-v
ImE o,-~vMvo(1-v
ReEol- .
(A.1)
The inverse transformation gives
2-1ReE o = (ReE- Rep • Vo)(1-v ) z
(A.2)
and t h e r e f o r e , to f i r s t o r d e r in V, I m p = My vo ( R e E - v o • R e p ) - 1
(A.3)
L e t us define v= Rep/ReE
(A.4)
.
T o f i r s t o r d e r in ~, we m a y i n t e g r a t e o v e r r e a l p , p r o v i d e d we add I m p a s given by eq. (A.3) in c a l c u l a t i n g all functions of p . In p a r t i c u l a r , E = [M2+2iMv(1
- v" Vo)
-1
± +p212 ;
(A.5)
so to w r i t e the i n t e g r a l in an a r b i t r a r y f r a m e , we give the following v e l o c i t y - d e p e n d e n t addition to the i m a g i n a r y p a r t of the m a s s : 8 = v v.
Vo/(1-
v. V o ) .
(A.6)
To e x a m i n e L o r e n t z i n v a r i a n c e , we s e e what happens when a s m a l l change is m a d e in v o. T o l o w e s t o r d e r in V, t h e r e will be no change in the i n t e g r a l u n l e s s changing v o c a u s e s s o m e s i n g u l a r i t y to j u m p a c r o s s a path of i n t e g r a t i o n . T h i s need not happen u n l e s s the p a t h is a c t u a l l y b e i n g p i n c h e d b e t w e e n s i n g u l a r i t i e s . T h e r e f o r e , b e l o w all t h r e s h o l d s , or in any o t h e r r e g i o n r e a c h e d by a n a l y t i c continuation, the i n t e g r a l is i n v a r i a n t . Next we look at the n o n - a n a l y t i c i t y points a s s o c i a t e d with the two m a s s e s M ± iV. In this c a s e , we f i r s t c o n s i d e r the i n t e g r a l s in the c . m . f r a m e f o r t h e s e two p a r t i c l e s . T h e i r m o m e n t a a r e ± p , but v o is a l s o an i n t e g r a t i o n v a r i a b l e if t h e s e a r e additional p a r t i c l e s . T h e i n t e g r a l will contain e n e r g y d e n o m i n a t o r s in which the i m a g i n a r y p a r t s of the m a s s e s a r e changed by 5~: =V v.
V o / ( l ~ : v . v o) ;
(A.7)
f u r t h e r m o r e , f o r e v e r y t e r m in which eq. (A.7) a p p e a r s , t h e r e will b e a s e c o n d (coming f r o m i n t e r c h a n g e of M and M*) in which 5± is r e p l a c e d by -5±. With a r e d e f i n i t i o n of the two t e r m s a c c o r d i n g to the sign of v . Vo, we can r e p l a c e eq. (A.7) by
v. Vol/(l
V. Vo ) .
(A.8)
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We now c a l c u l a t e the i m a g i n a r y p a r t of an e n e r g y d e n o m i n a t o r containing both M+i7 and M - i7 in the original f r a m e , but e x p r e s s it in t e r m s of the c.m. v a r i a b l e s : I m D = ( 1 - v )oz x I m ( E + + E _ )
2 -~ , = M6E-l(1-Vo)~
(A.9)
where
6
=
8++8_
= 71
v. V o [ / ( 1 - ( v . %)9.)
(A.10)
(again, t h e r e will be another t e r m with -6). Now c o n s i d e r an analytic continuation through r e a l values of the e x t e r n a l m o m e n t a . Then, except for I v. Vol ~ ~/7 (where E << 7 gives the conventional d i s p l a c e m e n t of p h y s i c a l m a s s e s f r o m r e a l values) t h e r e is no singul a r i t y in the r e g i o n of integration. F u r t h e r m o r e , e v e r y s i n g u l a r i t y is disp l a c e d in the s a m e s e n s e as it would be if 6 w e r e constant, and 6+ = ½6. In sect. 2, the continuation was defined by introducing a constant 8, taking the l i m i t 6 4 0 at the end. In that c a s e as well as h e r e , t h e r e w e r e two t e r m s , one with 5 and one with -6, so the LW p r e s c r i p t i o n and o u r s coincide, at l e a s t to the lowest o r d e r in Y. To t r e a t virtual s t a t e s with four c o m p l e x m a s s e s , M+i7 and M±iT', we introduce the quantities v, v ' , Vo, Vo', 8±, 8±', etc. by the s a m e method as before. In addition to e n e r g y d e n o m i n a t o r s whose i m a g i n a r y p a r t s contain e i t h e r 6 or 5' alone, as in eq. (A.9), t h e r e will be new ones, containing for e x a m p l e : ImD2 =
M E-1(1-V2o)½-M'5'E'-l(1-V'o2) ½ .
(A. 11)
No'ce that a t e r m containing, say +i6, in one e n e r g y d e n o m i n a t o r , n e v e r contains - i 6 in another although for e v e r y t e r m containing +i5 t h e r e will be another with -i6. Unlike eq. (A.9), ImD 2 has different signs in different p a r t s of the r e gion of integration; the d i s p l a c e m e n t of the s i n g u l a r i t y is not the s a m e for all p a r t s of m o m e n t u m space. F u r t h e r m o r e , the s e n s e of the d i s p l a c e m e n t depends on the choice of the r e f e r e n c e f r a m e through Vo and Vo'. In o r d e r to c h e c k that the a p p a r e n t n o n - i n v a r i a n c e does not always disa p p e a r when v o and v o' a r e i n t e g r a t e d over, we look at the s i n g u l a r p a r t of the integral for fig. 9. N e a r the t r i a n g l e s i n g u l a r i t y , v o and v o' a r e d e t e r mined by the Landau equations. They depend on the velocity of the o v e r - a l l c e n t e r of m a s s , and we can choose, e.g. either ]Vol >> I Vo'l or vice v e r s a . This m e a n s we can obtain either 6 >> 6' for m o s t of the i m p o r t a n t r e g i o n of integration, or the r e v e r s e . W e h a v e shown, in sect. 4, that t h e s e two possibilities lead to different behaviour at the t r i a n g l e s i n g u l a r i t y , so the a m plitude as c o n s t r u c t e d by the LW p r e s c r i p t i o n is not invariant. APPENDIX
B
Causality F o r c o m p l e t e n e s s , we p r e s e n t h e r e a calculation of the advanced signal
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a s s o c i a t e d with the type of n o n - a n a l y t i c i t y at the ' n o r m a l t h r e s h o l d ' s = (M+M*) 2. Our g e n e r a l a p p r o a c h follows that of Eden and Landshoff [9]. W r i t e W = 4-s. C o n s i d e r the n o n - a n a l y t i c t e r m 1
Fn.a.(W) = O ( - W + 2 R e M ) ( 2 R e M - W )
1
2 (ReM) -~ F a .
(B.1)
This is to be c o n s i d e r e d in conjunction with a t e r m Fa(W) , which is analytic in a p a r t of the u p p e r half plane c l o s e to the r e a l axis and such that Fa(2 Re M) = F a. Take an initial wave packet ~I,( W, 2 Re M, A), which is c e n t r e d a r o u n d W = 2 R e M with a width A; ~I, is infinitely differentiable and v a n i s h e s for W - 2 R e M I > A. We take it to be n o r m a l i s e d , so that ' 1 ~I,(2 ReM) ~ A-~. The s c a t t e r e d amplitude, f o r t i m e delay t = -T < 0, is
,~(~) = f dW e i W T ~I,(W)[Fa(W) +Fn.a.(W) ] .
(B.2)
F o r T >> A we have [ 10] • (r) ~ Fa: e 2 i r ( R e M ) (ReM)½ A½ r~ '
(B.3)
w h e r e u n i m p o r t a n t n u m e r i c a l f a c t o r s a r e dropped. The contribution of F a d r o p s off f a s t e r than any p o w e r of 7. F r o m eq. (B.3) we obtain the r e l a t i v e intensity of an a d v a n c e d signal between r and r + dr: d~ P ( r ) d r ~ (ReM)AT 3 "
(B.4)
T h i s can be i n t e g r a t e d to give the r e l a t i v e p r o b a b i l i t y of r > t, c o m p a r e d with the total f o r w a r d s c a t t e r i n g : P(t)
1 (ReM)At2 .
(B.5)
An equivalent d i s c u s s i o n , leading to the s a m e r e s u l t , has been given by L e e [2]. He, also, has noted that when v a l u e s of M, A, and t a p p r o p r i a t e to conventional e x p e r i m e n t s a r e i n s e r t e d into eq. (B.5), one gets a r i d i c u l o u s ly s m a l l n u m b e r . Under a p p r o p r i a t e conditions - p e r h a p s , for example, in a c o s m o l o g i c a l p r o b l e m involving a v e r y high density (p ~ M3) and t e m p e r a t u r e ( k T ~ M ) - the advanced signal might give a quantitatively significant effect.
REFERENCES
[1] T.D. Lee and G.C.Wick, Nucl. Phys. B9 (1969) 209. [2] T.D. Lee, to be published in Quanta (University of Chicago Press, 1969}. [3] R.J. Eden, P.V. Landshoff, D.I.Olive and J. C. Polldnghorne, The analytic Smatrix (Cambridge University Press, 1966} (called ELOP below}, sect. 4.4; D. I. Olive, Phys. Rev. 135B (1964) 745. [4] M.Bloxham, D.I.Olive and J.C.Polkinghorne, J. Math. Phys. 10 (1969) 494, 545.
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[5] ELOP, Sect. 2.10. [6] P.V.Landshoff, D.I.Olive and J . C . P o l k i n g h o r n e , J. Math. Phys. 7 (1966) 1600. [7] ELOP, sect. 2.5; S. Coleman and R. E. Norton, Nuovo Cimento 38 (1965) 438. [8] ELOP, Chapter 4. [9] R . J . E d e n and P.V.Landshoff, Ann. of Phys. 31 (1965) 370. [10] M.J. Lighthill, Introduction to F o u r i e r analysis and genera!ised functions (Cambridge University P r e s s , 1959).
A NON-ANALYTIC S-MATRIX R. E. C U T K O S K Y *, P . V . L A H D S H O F F , D . I . O L I V E a n d J . C. P O L K I N G H O R N E **
Depa~rnent o/Applied Mathematics and Theoretical Physics, University of Cambridge, England Received 23 June 1969 A b s t r a c t : A study is made of t h e o r i e s having the unusual analyticity p r o p e r t i e s r e cently p r o p o s e d by Lee and Wick. A p r e s c r i p t i o n is given for setting up a c o v a r iant p e r t u r b a t i o n - t h e o r y expansion of the s c a t t e r i n g amplitudes, based on F e y n man graphs. It is found that the p r e s e n c e of a complex pole in the upper half plane of the p h y s i c a l sheet leads to points of non-analyticity in the physical region, such that the values of the amplitude to e i t h e r side of the point a r e not r e l a t e d by a n a l ytic continuation. It is shown how this is compatible with unitarity. The nature of the non-analyticity is not fully d e t e r m i n e d by unitarity. Neither, in the c a s e of the m o r e complicated graphs, is it fully d e t e r m i n e d by the p e r t u r b a t i o n - t h e o r y p r e scription, and some e x t r a c o n s t r a i n t must be imposed on the theory to r e m o v e the ambiguity. It is shown that the p r e s c r i p t i o n of Lee and Wick has an exactly s i m i l a r ambiguity, but for their p r e s c r i p t i o n different r e s u l t s a r e obtained in different Lorentz f r a m e s . An e s t i m a t e is mhde of the extent to which the theory violates causality, and is found to be too s m a l l to m e a s u r e .
1. I N T R O D U C T I O N L e e a n d W i c k h a v e r e c e n t l y r e k i n d l e d i n t e r e s t in t h e o r i e s w i t h u n c o n v e n t i o n a l a n a l y t i c i t y p r o p e r t i e s [1, 2]. T h e y h a v e p r e s e n t e d d e t a i l e d c a l c u l a t i o n s , in s e v e r a l s i m p l e m o d e l s , i l l u s t r a t i n g how u n i t a r i t y c a n b e p r e s e r v e d d e s p i t e t h e i n t r o d u c t i o n of p a r t i c l e s t h a t i n t e r a c t v i a n o n - H e r m i t i a n c o u p l i n g s . In t h e i r m o d e l s s u c h a p a r t i c l e l e a d s to t h e e x i s t e n c e of a p a i r of c o m p l e x - c o n j u g a t e p o l e s on t h e p h y s i c a l s h e e t of t h e s c a t t e r i n g a m p l i t u d e . T h e e v o l u t i o n o p e r a t o r U(t, to) i s not u n i t a r y a n d , m o r e o v e r , it b e c o m e s u n b o u n d e d a s t -~ ± co. H o w e v e r , t h e p h y s i c a l s t a t e s a r e d e f i n e d s o as to exclude the complex-mass particles, and contain only real-mass particles. The S-matrix is obtained by subtracting from U the terms that grow exponentially, and is shown to be unitary. O u r own i n t e r e s t in t h e s e m o d e l s a r i s e s f r o m t h e o p p o r t u n i t y t h a t t h e y provide for studying the interplay between analyticity and unitarity; they s e r v e t o e m p h a s i z e t h e s i g n i f i c a n c e of t h e a s s u m p t i o n s m a d e in t h e u s u a l S-matrix theory. * On leave of absence f r o m C a r n e g i e - M e l l o n U n i v e r s i t y (1968-1969). ** On leave of absence at the Lawrence Radiation Laboratory, Berkeley.
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One a s s u m p t i o n that is p a r t i c u l a r l y i m p o r t a n t in the usual t h e o r y is that the s c a t t e r i n g a m p l i t u d e is an a n a l y t i c function in the p h y s i c a l region, except f o r c e r t a i n ' a n a l y t i c s i n g u l a r i t i e s ' . By an a n a l y t i c s i n g u l a r i t y we shall m e a n that a path of a n a l y t i c continuation which m a k e s a s m a l l detour a r o u n d it r e l a t e s the v a l u e s of the p h y s i c a l a m p l i t u d e to e i t h e r side of it. In this p a p e r we shall conclude that the introduction of the u n p h y s i c a l c o m p l e x m a s s p a r t i c l e s r e q u i r e s that, if u n i t a r i t y is to be s a t i s f i e d , t h e r e be ' n o n analytic s i n g u l a r i t i e s ' in the p h y s i c a l region. F o r t h e s e , v a l u e s of the p h y s i c a l a m p l i t u d e to e i t h e r side of t h e m a r e not r e l a t e d by an a n a l y t i c continuation. Such s i n g u l a r i t i e s a r e f a m i l i a r enough in the study of u n i t a r i t y i n t e g r a l s [4] but t h e i r o c c u r r e n c e in the a m p l i t u d e i t s e l f is an e n t i r e l y new f e a t u r e . T h e s i m p l e s t n o n - a n a l y t i c s i n g u l a r i t y is a s s o c i a t e d with an u n p h y s i c a l s t a t e of two c o m p l e x - m a s s p a r t i c l e s , at the r e a l p h y s i c a l v a l u e s = (M+M*) 2 of the e n e r g y v a r i a b l e . In the usual S - m a t r i x t h e o r y such a ' n o r m a l t h r e s h o l d ' is a s s o c i a t e d with a t w o - p a r t i c l e i n t e r m e d i a t e s t a t e in the u n i t a r i t y equation. H e r e , this s t a t e , b e i n g u n p h y s i c a l , does not a p p e a r in u n i t a r i t y , and we shall show in s e c t . 3 that it m u s t c o n s e q u e n t l y b e a s s o c i a t e d with a nona n a l y t i c s i n g u l a r i t y . The d i s c u s s i o n in that s e c t i o n is b a s e d on u n i t a r i t y , which by i t s e l f allows c o n s i d e r a b l e f r e e d o m a s to the p r e c i s e c h a r a c t e r of the n o n - a n a l y t i c s i n g u l a r i t y . A p a r t i c u l a r e x a m p l e of a m o r e p r e c i s e s p e c ification is d e s c r i b e d in s e c t . 2, w h e r e we give a p o s s i b l e p r e s c r i p t i o n f o r c a l c u l a t i o n of the a m p l i t u d e in t e r m s of c o v a r i a n t F e y n m a n i n t e g r a l s . Our F e y n m a n - i n t e g r a l m o d e l g i v e s r e s u l t s s i m i l a r to t h o s e of L e e and Wick. A p a r t i c u l a r p r o b l e m with the L e e - W i c k p r e s c r i p t i o n is that it is f o r m u l a t e d in a given fixed r e f e r e n c e f r a m e , and is c o n s e q u e n t l y not m a n i f e s t l y c o v a r i a n t . L e e [2] h a s e x a m i n e d a s i m p l e F e y n m a n g r a p h with two c o m p l e x m a s s e s , and c h e c k e d that f o r this the L e e - W i c k p r e s c r i p t i o n does give an i n v a r i a n t r e s u l t . H o w e v e r , a s we d i s c u s s in appendix A, the p r o b l e m is m o r e c o m p l i c a t e d when t h e r e a r e m o r e c o m p l e x - m a s s p a r t i c l e s involved, and we find that, beyond the t h r e s h o l d f o r p r o d u c i n g four c o m p l e x m a s s p a r t i c l e s , t h e i r p r e s c r i p t i o n l e a d s to n o n - c o v a r i a n t t e r m s . T h e s a m e g r a p h s a l s o c a u s e difficulty in our p r e s c r i p t i o n , a s we show in sect. 4. H o w e v e r , f o r us the p r o b l e m is that the a m p l i t u d e , while it is c o v a r i a n t , is not fully s p e c i f i e d by the r a t h e r g e n e r a l f o r m u l a t i o n we h a v e given. S o m e e x t r a , m o r e detailed, r e q u i r e m e n t s m u s t be i m p o s e d in o r d e r to p r o d u c e a unique amplitude. As r e m a r k e d by L e e and Wick, t h e r e is a r i s k that the t h e o r y is nonc a u s a l . In appendix B we give a m o r e s u c c i n c t v e r s i o n of an e s t i m a t e they h a v e m a d e of the magnitude of one of the n o n - c a u s a l e f f e c t s , and a g r e e with t h e i r conclusion that it is too s m a l l to o b s e r v e . H o w e v e r , the l a c k of caus a l i t y does have a profound effect on t h e o r e t i c a l a s p e c t s of the s t r u c t u r e of s c a t t e r i n g a m p l i t u d e s , a s we d i s c u s s in s e c t . 2. 2. P E R T U R B A T I O N THEORY In this s e c t i o n we c o n s i d e r s o m e s i m p l e p e r t u r b a t i o n - t h e o r y g r a p h s , m a k i n g the s i m p l e s t p o s s i b l e m o d i f i c a t i o n of the u s u a l F e y n m a n t h e o r y that
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is e x p l i c i t l y c o v a r i a n t . F o r t h e s e g r a p h s our p r e s c r i p t i o n g i v e s r e s u l t s equivalent to the r a t h e r m o r e c o m p l i c a t e d f o r m a l i s m of L e e and Wick. All p a r t i c l e s will b e s u p p o s e d to h a v e z e r o spin. F o r o r d i n a r y , r e a l m a s s p a r t i c l e s we u s e the usual p r o p a g a t o r , and f o r the u n p h y s i c a l c o m p l e x - m a s s p a r t i c l e s we u s e the p r o p a g a t o r A A* A(P2) - p2 - M 2 + p2 - M*2 "
(2.1)
The couplings a r e a s s u m e d r e a l ; the n o n - H e r m i t i c i t y of the o r i g i n a l H a m i l tonian is r e p r e s e n t e d by the n o n - r e a l m a s s e s a p p e a r i n g in the p r o p a g a t o r [ 1]. The F e y n m a n i n t e g r a l is then c a l c u l a t e d a s the u s u a l i n t e g r a l using, f o r r e a l e x t e r n a l f o u r - m o m e n t a b e l o w all t h r e s h o l d s , the W i c k - r o t a t e d contour. Since the couplings a r e r e a l , a s well a s the p r o p a g a t o r A(p2) a s a whole, the s c a t t e r i n g a m p l i t u d e c a l c u l a t e d in t h i s way will be H e r m i t i a n . Also, if the e n e r g y v a r i a b l e s a t i s f i e s s < (2 R e M ) 2 , the c o m p l e x p o l e s do not p r o duce any n o n - a n a l y t i c i t y and n e i t h e r do they c o n t r i b u t e to the discontinuity a c r o s s the r e a l a x i s , so the u s u a l p e r t u r b a t i o n t h e o r y d i s c u s s i o n s of unit a r i t y f o r r e a l p a r t i c l e s a r e not a f f e c t e d in this region.
Fig. 1. The simplest Feynman graph that leads to non-analyticity. The wavy lines denote complex-mass particles, and the straight lines denote r e a l - m a s s particles. C o n s i d e r in p a r t i c u l a r the g r a p h of fig. 1, w h e r e the wavy l i n e s r e p r e s e n t c o m p l e x - m a s s p a r t i c l e s and the s t r a i g h t l i n e s a r e p h y s i c a l p a r t i c l e s . L e t k b e the loop i n t e g r a t i o n m o m e n t u m and w o r k in the c e n t r e - o f - m a s s f r a m e . In fig. 2a, we d r a w , f o r s < ( 2 R e M ) 2 , the p o s i t i o n in the kO p l a n e of the c o m p l e x - m a s s p o l e s r e l a t i v e to the p a t h of i n t e g r a t i o n , a s given by the p r e s c r i p t i o n above. (The c r o s s e s r e p r e s e n t the p o l e s of one of the p r o p a g a t o r s , the c i r c l e s t h o s e of the other.) If we r e g a r d the t h r e e - d i m e n s i o n a l loop i n t e g r a t i o n to be a l r e a d y c a r r i e d out t h e s e p o l e s would i n t e g r a t e into b r a n c h points with a t t a c h e d cuts. As s is i n c r e a s e d along the r e a l a x i s , the c r o s s e s m o v e t o w a r d s the right, until at s = (M+M*) 2 the l e f t - h a n d p a i r of c r o s s e s p i n c h e s with the r i g h t - h a n d p a i r of c i r c l e s . We now give a p r e s c r i p t i o n f o r e v a l u a t i n g the a m p l i t u d e f o r l a r g e r v a l u e s of s. T h i s p r e s c r i p t i o n will, a s we show in s e c t . 3, p r e s e r v e H e r m i t i c i t y and u n i t a r i t y , but we m u s t s t r e s s that it is by no m e a n s the only p r e s c r i p t i o n with t h e s e p r o p e r t i e s . F o r e x a m p l e , any a m p l i t u d e whose K - m a t r i x is r e a l will s a t i s f y t w o - p a r t i c l e u n i t a r i t y . We r e g a r d the t h e o r y as b e i n g a m a t h e m a t i c a l l i m i t of a situation w h e r e the m a s s e s on the two i n t e r n a l l i n e s have d i f f e r e n t i m a g i n a r y p a r t s . T h u s the u p p e r p r o p a g a t o r h a s p o l e s at M 2 and M~ 2, and the l o w e r one h a s p o l e s at M ~ and M~ 2, w h e r e
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.
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F i g . 2. I n t e g r a t i o n p a t h s in t h e k o p l a n e f o r the d i a g r a m of fig. 1 f o r t h r e e d i f f e r e n t v a l u e s of t h e e n e r g y v a r i a b l e a s d e s c r i b e d in t h e t e x t . T h e c r o s s e s d e n o t e s i n g u l a r i t i e s of one p r o p a g a t o r , t h e c i r c l e s d e n o t e t h o s e of t h e o t h e r .
ml- M2
= i8,
(2.2)
5 being taken real, small and positive. Then the pinches no longer occur at a real value of s, and they no longer occur simultaneously in the upper and lower b_alves of the k o plane. That is, the continuation of the amplitude originally specified by the contour of fig. 2a has two branch points, at s = (M 1 + ~ ) 2 , s = (A~ +M2)2 , each displaced from the real axis as drawn in fig. 3. (There are additional branch points at (M 1 + M 2 ) 2 and (M] +M~.)2.) For finite 5 we can continue the amplitude along the real axis into the region between the cuts, and it will remain real. For this continuation, the path of integration in the k o plane is dragged into the shape shown in fig. 2b. W e now let 5 ---0. Then the values of the amplitude defined in this way to the right of the two branch points that have pinched down on the real axis cannot any longer be obtained by analytic continuation from those on the lea. The only continuation that can be m a d e from the lea is then either over
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CM,.
X Fig. 3. Displaced branch points in the s-plane for the Feynman graph of fig. 1. both pinching b r a n c h points, into the r e g i o n m a r k e d l , or u n d e r t h e m into the r e g i o n m a r k e d 1. The f i r s t of t h e s e continuations, f o r example, would r e s u l t in the k ° plane contour of fig. 2c. On the other hand ff we continue to the left the amplitude f o r the p a r t i c u l a r g r a p h of fig. 1 as defined at the right of the pinching b r a n c h points, we find that it is actually analytic at s = (M+M*)2; it m a y be shown that in this p a r t of the amplitude the e f f e c t s of the two b r a n c h points happen to c a n c e l when they pinch t o g e t h e r . N e v e r t h e l e s s , the v a l u e s of the amplitude so continued into s < (M+M*)2 a r e not supposed to be p h y s i c a l l y significant. We extend this p r e s c r i p t i o n to m o r e c o m p l i c a t e d graphs. The extension to o t h e r n o r m a l t h r e s h o l d s , involving any n u m b e r of r e a l - m a s s p a r t i c l e s and one p a i r of c o m p l e x - m a s s p a r t i c l e s , is i m m e d i a t e . As a l e s s s t r a i g h t f o r w a r d example, we now c o n s i d e r the g r a p h s of fig. 4. T h e s e a r e given by the s a m e F e y n m a n i n t e g r a l , the only d i f f e r e n c e being that f o r fig. 4b the v a r i a b l e t, which would be fixed at s o m e negative value in o r d e r to lie in the p h y s i c a l r e g i o n f o r fig. 4a, is fixed at ~~ . In e i t h e r c a s e , fig. 5a shows, in the r e a l (s, ~) plane, the n o r m a l t h r e s h o l d s and the Landau c u r v e f o r the t r i a n g l e s i n g u l a r i t y c o r r e s p o n d i n g to m a s s (M+M*) in the l e f t - h a n d side of the d i a g r a m . The dotted c u r v e applies only f o r the c a s e of fig. 4b, for which it m a r k s the b o u n d a r y of the p h y s i c a l region; it is the s e c o n d - t y p e s i n g u l a r i t y c u r v e [5], which t o u c h e s the Landau c u r v e at the point X. (Our notation c o r r e s p o n d s to that of L O P [6].)
Fig. 4. Feynman graphs leading to physical-region triangle singularities involving complex-mass particles. In the usual t h e o r y , only the a r c AB of the Landau c u r v e , c o r r e s p o n d i n g to p o s i t i v e v a l u e s of the F e y n m a n u - p a r a m e t e r s , is s i n g u l a r in the p h y s i c a l sheet. T h i s p r o p e r t y is c l o s e l y r e l a t e d to r e q u i r e m e n t s of c a u s a l i t y [7]: that the flight t i m e s of the i n t e r n a l p a r t i c l e s be positive. In the p r e s e n t
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(.~.) F i g . 5. P l o t s of the L a n d a u c u r v e f o r the t r i a n g l e s i n g u l a r i t i e s of the F e y n m a n g r a p h of fig. 4b. In (a) the L a n d a u c u r v e i s p l o t t e d in the (Re s , Re(~) p l a n e f o r M 1 = M 2. The s t r a i g h t l i n e s t a n g e n t to the c u r v e at A and B a r e the s and (~ n o r m a l t h r e s h o l d s . In (b) it is s h o w n how the l o c u s of the s i n g u l a r i t y m o v e s in the s - p l a n e w h e n Re (~ is i n c r e a s e d b e y o n d its t h r e s h o l d (point B) and I m a i s f i x e d at a v e r y s m a l l and p o s i t i v e value, f o r e a c h of the two t e r m s in which the l o c u s c o l l a p s e s to r e a l v a l u e s when M 1 = M 2. The c r o s s e s (point A) d e n o t e the d i s p l a c e d n o r m a l - t h r e s h o l d b r a n c h p o i n t s . The s o l i d p o r t i o n s of c u r v e s i n d i c a t e t h a t the s i n g u l a r point i s on the e x p o s e d s h e e t , and the d a s h e d p a r t of t h e c u r v e s i n d i c a t e s t h a t the s i n g u l a r point i s r e a c h e d by a c o n t i n u a t i o n r o u n d the c o r r e s p o n d i n g n o r m a l - t h r e s h o l d b r a n c h point.
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t h e o r y the a n a l y t i c i t y p r o p e r t i e s usually a s s o c i a t e d with c a u s a l i t y have alr e a d y been violated, by the introduction of poles in the u p p e r - h a l f plane on the p h y s i c a l sheet and by the r e s u l t i n g p r e s e n c e of n o n - a n a l y t i c s i n g u l a r i ties in the p h y s i c a l region. We shall now e n c o u n t e r f u r t h e r such unusual p r o p e r t i e s ~ we find that the whole a r c of the Landau c u r v e is s i n g u l a r in the p h y s i c a l region, including the p a r t s ooA and B °o that c o r r e s p o n d to s o m e of the a - p a r a m e t e r s being negative, and so to negative flight t i m e s for the int e r n a l p a r t i c l e s [7]. It is, however, an analytic singularity. To a n a l y s e the g r a p h s , introduce different c o m p l e x m a s s e s M 1 and M2, and take the s a m e l i m i t as b e f o r e . E a c h c o m p l e x - m a s s p r o p a g a t o r is the s u m of two t e r m s (2.1), so that the F e y n m a n integral is the sum of four t e r m s . Of these, the two t e r m s r e s p e c t i v e l y involving M1, M~ and M~, M 2 a r e those that a r e of i n t e r e s t , b e c a u s e they p r o d u c e a r e a l Landau c u r v e in the limit 5 -~ 0. F o r 5 > 0, so that Im (M~ +M2) < 0, the situation f o r the s e c o n d of t h e s e t e r m s is just the usual one. The p h y s i c a l limit is evaluated on the u p p e r side of the cuts attached to the n o r m a l t h r e s h o l d s a = 4 p 2 and s = (M~ + M 2 + + ~)2, so that only the a r c AB is singular. F o r r e a l ~ this a r c is d e p r e s s e d into the l o w e r s - p l a n e , so that one a l s o p a s s e s above this s i n g u l a r i t y to evaluate the p h y s i c a l limit. This is i l l u s t r a t e d in fig. 5b. F o r the f i r s t t e r m one p a s s e s above the a = 4p2 cut as usual, but below the s = (M 1 +M~ + ~)2 cut, s i n c e this b r a n c h point a p p e a r s in the upper half s - p l a n e f o r ~ > 0. Thus the a r c AB is n o n - s i n g u l a r for this t e r m , but ~oA and Boo a r e singular, as is a l s o i l l u s t r a t e d in fig. 5b. F o r r e a l a the a r c s XA and Boo a r e r a i s e d into the upper half of the s - p l a n e when 5 > 0, so one p a s s e s below the cuts a t t a c h e d to t h e s e a r c s in the s - p l a n e to evaluate the p h y s i c a l limit. The a r c X~o (which is only p r e s e n t in the c a s e of fig. 4b) is d e p r e s s e d into the lower half s - p l a n e , so the cut a t t a c h e d to this a r c r u n s below the unphysical limit. The c h a n g e - o v e r at the point X is a c c o u n t e d for by noting that the s e c o n d - t y p e c u r v e is s i n g u l a r along the p a r t of it above X when it is a p p r o a c h e d t h r o u g h r e a l v a l u e s of s, a f r o m outside the p h y s i c a l region, though not when a p p r o a c h e d f r o m inside the p h y s i c a l r e g i o n (being then shielded by the cut attached to the a r c X ~ of the Landau curve). The r e s u l t , when we add t o g e t h e r the two t e r m s in the limit 5 ~ 0, is that the a r c s AB, Xoo have cuts a t t a c h e d to them running below the p h y s i c a l limit in the s - p l a n e , while the a r c s XA, B~o have cuts attached running above the p h y s i c a l limit. In each c a s e the v a r i a b l e a is r e g a r d e d as fixed at a r e a l value, evaluated above the a = 4 p 2 cut. In sect. 3 we show how t h e s e p r e s c r i p t i o n s a r e a c c o m m o d a t e d in the u n i t a r i t y equations. We may s u m m a r i z e as follows our p r e s c r i p t i o n for continuing past the (M+M*) n o n - a n a l y t i c i t y point: we change the c o m p l e x m a s s by s m a l l a m o u n t s i 5M. The change is to be different in each different c o m p l e x p r o p a g a t o r . Then each s i n g u l a r i t y involving two c o m p l e x - c o n j u g a t e m a s s e s is r e p l a c e d by a p a i r of t h r e s h o l d s , one lying above, and one lying below, the r e a l axis. The n o r m a l t h r e s h o l d s a r i s i n g f r o m a s e q u e n c e of i t e r a t e d bubbles or l a d d e r s (for example) a r e then r e p l a c e d by a conjugate p a i r of c l u s t e r s of b r a n c h points. The path of analytic continuation is defined to be the r e a l axis. This is in exact analogy to the F e y n m a n p r e s c r i p t i o n of
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giving each internal r e a l m a s s an addition -iE (where ~ may a l s o be different for each p r o p a g a t o r ) . In continuing past a t h r e s h o l d for s e v e r a l p a r t i c l e s , two of which have c o m p l e x m a s s e s , we m u s t r e q u i r e that e << 5 = = 15M1- 5M21 ; o t h e r w i s e , the r a t i o s of the different m a s s changes b e c o m e i m m a t e r i a l in the limit in which all m a s s changes vanish. This p r e s c r i p t i o n also implicitly defines a way to continue past o t h e r Landau s i n g u l a r i t i e s . In o r d e r for the r e s u l t to be s e l f - c o n s i s t e n t , it is sufficient that the s i n g u l a r i t y c u r v e always r e t r e a t f r o m r e a l values of the v a r i a b l e s when the m a s s e s a r e altered. This is not obvious, but is is e a s i l y checked for the t r i a n g l e s i n g u l a r i t i e s c o n s i d e r e d in this p a p e r , and it is t r u e except when t h e r e is a r e a l point of contact with the s e c o n d - t y p e sing u l a r i t y curve. In this c a s e (see above) the rule is that in continuing along a path with r e a l v a l u e s of s, t and a, the path is not to encounter the s e c o n d type singularity. The t h r e s h o l d for four c o m p l e x - m a s s p a r t i c l e s is obviously m o r e delicate, b e c a u s e it is p o s s i b l e for the t h r e s h o l d to o c c u r at a r e a l point even when all four c o m p l e x m a s s e s a r e given different i m a g i n a r y additions. It will be shown, in sect. 4, that a s e r i o u s ambiguity does, in fact o c c u r on this account. But this is an ambiguity r a t h e r than an i n c o n s i s t e n c y ; an additional p r e s c r i p t i o n would be needed c o m p l e t e l y to define the t h e o r y .
3. UNITARITY Consider first the 'normal threshold' at s = (M+M*) 2. In sect. 2 we gave a particular prescription for dealing with the single Feynman graph of fig. I, but this by itself is too simple to give a complete picture of the p r o p e r t i e s of the t h r e s h o l d . As s u g g e s t e d at the end of sect. 2, one has to s u m an infinite n u m b e r of i t e r a t i o n s of this bubble graph, and also include bubble g r a p h s with r e a l - m a s s internal lines. One r e s u l t of this will be that the b r a n c h points at (M 1 +M~2)2 and (M1* +M2) 2 , which pinch in the limit 5 --* 0, a p p e a r on both s h e e t s of the s = 4~ 2 cut. That is, they o c c u r both in the p h y s i c a l r e g i o n for the p h y s i c a l (+) amplitude, and for the H e r m i t i a n - c o n j u g a t e (-) amplitude. This we show in fig. 6a and in fig. 6b, w h e r e the dotted lines show the paths of continuation we must follow to keep r e s p e c t i v e l y with the (+) and (-) amplitudes. As 5 ~ 0, each path is pinched by the p a i r of b r a n c h points, and in the limit the analytic continuation is no l o n g e r possible. Suppose that < R e M < 2~ .
(3.1)
Then to either side of s = (M+M*) 2 the p h y s i c a l u n i t a r i t y equation r e a d s (see E L O P [3])
Unlike the f a m i l i a r situation with n o r m a l t h r e s h o l d s , no e x t r a t e r m a p p e a r s above s = (M+M*) 2, and we show now how the p r e s c r i p t i o n of sect. 2 is c o m p a t i b l e with this.
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OC-}
Fig. 6. The dashed lines show the paths of analytic continuation for the (+) and (-) amplitudes, in relation to the displaced thresholds for unphysical particles. T h i s is v e r y s i m p l e . We continue eq. (3.2) f r o m the left of s = (M+M*)2, p a s s i n g a b o v e the b r a n c h point at s = (M 1 +M~.) 2 and b e l o w the b r a n c h point a t (M1 +M~) 2. Along t h i s path of continuation, a c c o r d i n g to the two diag r a m s of fig. 6, the (+) a m p l i t u d e r e m a i n s the (+) a m p l i t u d e and the (-) a m plitude r e m a i n s the (-) a m p l i t u d e . T h u s eq. (3.2) a s a whole r e m a i n s unchanged, a s is r e q u i r e d . T h u s the n o n - a n a l y t i c i t y , which a p p e a r s when we t a k e the l i m i t 5 ~ 0, is such a s to allow u n i t a r i t y . We can a l s o show that n o n - a n a l y t i c i t y is a c t u a l l y a n e c e s s a r y f e a t u r e , if the a m p l i t u d e is to obey both u n i t a r i t y and h e r m i t icity. F o r to a v o i d n o n - a n a l y t i c i t y , the p a t h of continuation for the (+) a m plitude in fig. 6a would h a v e had to p a s s to the s a m e side of the two b r a n c h p o i n t s , e i t h e r a b o v e t h e m both of below t h e m both, to a v o i d b e i n g pinched in the l i m i t 8 ~ 0. T o p r e s e r v e h e r m i t i c i t y , the p a t h of continuation f o r the (-) a m p l i t u d e m u s t be the m i r r o r i m a g e in the r e a l axis of that f o r the (+) a m p l i t u d e . T h u s the continuation of eq. (3.2) that p r e s e r v e d the (+) a m p l i tude would not p r e s e r v e the (-) a m p l i t u d e . Our d e m o n s t r a t i o n of the c o m p a t i b i l i t y with u n i t a r i t y h a s r e q u i r e d m e r e ly the a s s u m p t i o n t h a t the t h e o r y can b e r e g a r d e d a s a m a t h e m a t i c a l l i m i t of one containing two d i f f e r e n t c o m p l e x m a s s e s . T h i s a s s u m p t i o n , without f u r t h e r s p e c i f i c a t i o n of the n a t u r e of the limit, is by no m e a n s enough to d e t e r m i n e the n a t u r e of the n o n - a n a l y t i c i t y . To do t h i s , one m u s t p o s t u l a t e equations giving the d i f f e r e n c e b e t w e e n the p h y s i c a l a m p l i t u d e as defined f o r s > (M+M*)2, and the continuation e i t h e r a b o v e o r b e l o w s = (M+M*)2 of the p h y s i c a l a m p l i t u d e a s defined f o r s < (M+M*)2. B e f o r e the l i m i t 5 --* 0 is taken, t h e s e d i f f e r e n c e s a r e just the d i s c o n t i n u i t i e s a c r o s s the (M1 +M~) and (M~ + M2) cuts.
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The s i m p l e s t postulate for t h e s e discontinuties is that they a r e s i m i l a r in f o r m to the usual n o r m a l - t h r e s h o l d discontinuities. T h i s is the c a s e for our p e r t u r b a t i o n - t h e o r y p r e s c r i p t i o n . Thus
T h e notation u s e d in eqs. (3.3a, b) is that the upper wavy line r e p r e s e n t s e i t h e r M 1 or M~I, and the lower e i t h e r M 2 or M~; the * shows which of the p a i r is the conjugated m a s s . Along with eqs. (3.3a, b) we have the H e r m i t Jan- conjugate equations:
H e r e the labels (+ T), (+ ~), (-T), and (- 1) a r e defined in fig. 6, with the l i m i t M 2 ~ M 1 being understood. T h e s e equations involve the amplitudes f o r the unphysical p r o c e s s e s tt + g. ~
M +M*
(3.4a)
and have no content until the p r o p e r t i e s of t h e s e amplitudes a r e specified. This we do by following again the suggestion of our p e r t u r b a t i o n t h e o r y , which is to postulate that they obey equations just like eqs. (3.3), with app r o p r i a t e changes of the e x t e r n a l p a r t i c l e s . The s a m e applies to the amplitude f o r M +M* - - M +M* ,
(3.4b)
which will then also e n t e r in the equations. We shall also suppose that discontinuities c o r r e s p o n d i n g to the (M +M*) s i n g u l a r i t i e s in s u b e n e r g y v a r i a b l e s of m u l t i p a r t i c l e amplitudes take analogous f o r m s . One might p e r h a p s be unhappy about defining his t h e o r y as a m a t h e m a t i cal limit in which the m a s s e s of the p a r t i c l e s a r e v a r i e d , in an a r t i f i c i a l way, and in fact once we have postulated eqs. (3.3) this idea is no longer n e c e s s a r y (though it is still convenient). T o see this, we now show that if we a s s u m e that the physical u n i t a r i t y equation (3.2) is valid for s < (M+M*) 2, t o g e t h e r with the analogous equations f o r the p r o c e s s e s (3.4), and that we also have eqs. (3.3) t o g e t h e r with the analogous equations for the a m p l i t u d e s (3.4), then the physical u n i t a r i t y equation (3.2) is also valid for s > (M+M*) 2. Suppose then that eq. (3.2) is valid f o r s < (M+M*) 2, and continue it to s > (M+M*) 2, via the l o w e r - h a l f plane. T h i s gives
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Add t h i s to eq. (3.3b). E l i m i n a t e the (+ 1) a m p l i t u d e s f r o m the r i g h t - h a n d s i d e of the r e s u l t by u s i n g eq. (3.3b) a g a i n and a l s o the analogue of eq. (3.5) obtained b y r e p l a c i n g the l e f t - h a n d s t a t e by (M+M*). The r e s u l t is
Now u s e eq. (3.3a*) to e l i m i n a t e the (- 1) amplitude:
We now h a v e to show that the r i g h t - h a n d side of eq. (3.7) v a n i s h e s . T h i s we do by r e p e a t i n g all the c a l c u l a t i o n s with the r i g h t - h a n d s t a t e r e p l a c e d by ( M + M * ) . The a n a l o g u e of eq. (3.7) thus r e a d s : 0 T h e s e c o n d f a c t o r d o e s not Vanish; in f a c t its i n v e r s e can be found f r o m the analogue of eq. (3.3a*), and so we c o m p l e t e the proof. We e m p h a s i z e that eqs. (3.3) could be r e p l a c e d by quite d i f f e r e n t a s s u m p t i o n s , with u n i t a r i t y still being m a i n t a i n e d . The i c e c r e a m - c o n e
T h e i c e c r e a m - c o n e s i n g u l a r i t y of fig. 4b, f o r the c a s e w h e r e all the p a r t i c l e s a r e ' o r d i n a r y ' , h a s b e e n a n a l y s e d in detail in L O P [6]. We now study how that s i n g u l a r i t y fits in with u n i t a r i t y in the p r e s e n t t h e o r y . We a s s u m e a knowledge of the t e c h n i q u e s of L O P , or of t h e i r f o r m a s s i n c e d e v e l o p e d [4]. We c o n s i d e r the s i n g u l a r i t y of fig. 4a (now r e g a r d e d a s a L a n d a u diag r a m r a t h e r than a F e y n m a n graph), u s i n g the u n i t a r i t y equation SS + = 1 f o r the p r o c e s s ~ + p + ~ ~ ~ + p + ~, below the 4 ~ t h r e s h o l d . F o r s i m p l i c i ty, we a s s u m e s o m e s o r t of c o n s e r v a t i o n law that e l i m i n a t e s 2~ s t a t e s f r o m the equation. It will a g a i n be c o n v e n i e n t to r e g a r d the t h e o r y a s a m a t h e m a t i c a l l i m i t of one with two d i f f e r e n t c o m p l e x m a s s e s . The d i s c u s sion is on the s a m e l i n e s a s that of L O P , e x c e p t that f o r 5 > 0 t h e r e is the (M 1 + M~2) ' n o r m a l - t h r e s h o l d ' o c c u r r i n g on the unusual side of the path of continuation that p r e s e r v e s the (+) a m p l i t u d e , and the (M~I + M2) s i n g u l a r i t y on the unusual side f o r the (-) a m p l i t u d e . I n t r o d u c e a v a r i a b l e ~ that m e a s u r e s d i s p l a c e m e n t s along the inward n o r m a l to the i c e c r e a m - c o n e L a n d a u c u r v e L: a s we c r o s s the c u r v e , d s = C a 1 d~?, da = C a 2 dT?.
(3.9)
(This definition of 77 d i f f e r s f r o m that in L O P by the f a c t o r C in eqs. (3.9), and c o r r e s p o n d s to the definition u s e d in l a t e r work. It will be r e c a l l e d that C is p o s i t i v e on the a r c s XA, AB, B °o of fig. 5a, and is n e g a t i v e on ooX when the point X is on the a r c . ) We shall c h o o s e to c r o s s L via a r o u t e f o r which
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(3.10)
ImT? < 0 .
Suppose that, on analytic continuation by such a route, from outside L to inside L, ,
(3.11) That is, at this stage, we are not assuming that the continuation preserves either the (+) amplitude of the (-) amplitude, though it may do so. Standard methods then show that, on the arc AB, the only t e r m in the unitarity equation that generates the singularity is the f i r s t t e r m in fig. 7.
Fig. 7. As one continues along the route (3.10), the integration in the intermediate state passes to the wrong side of the (M~ +M2) threshold in a subenergy of the left-hmld bubble, and so the result differs from the natural continuation of that t e r m by a t e r m involving the discontinuity across the ( ~ +M2) cut. According to the assumptions we have made above, this t e r m is
~
-
(,3.12)
~
Hence if, following the usual procedure, we take account also of regeneration mechanisms, and subtract the resulting continued unitarity equation from the physical unitarity equation that operates inside L, we obtain
+
(3.13)
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S i m i l a r m a n i p u l a t i o n s applied to the a r c Boo show that on that a r c the r i g h t hand side of eq. (3.13) m u s t be r e p l a c e d by
+ (3.14) On the a r c s coX, XA the s e c o n d t e r m of fig. 7 a l s o g e n e r a t e s the s i n g u l a r i t y , and a f t e r s o m e m a n i p u l a t i o n we find that on t h o s e a r c s the r i g h t - h a n d side of eq. (3.9) b e c o m e s
(3.15) F o r the o r d i n a r y i c e c r e a m - c o n e s i n g u l a r i t y we get z e r o i n s t e a d of eqs. (3.14), so that in f a c t t h e r e is no s i n g u l a r i t y on the a r c s coA, Boo. In the p r e s e n t c a s e t h e r e m u s t be s o m e d i f f e r e n t s o r t of b e h a v i o u r on t h e s e a r c s , but the a b o v e r e s u l t s by t h e m s e l v e s do not tell us w h e t h e r it is a s i n g u l a r ity or n o n - a n a l y t i c i t y . A c c o r d i n g to the p r e s c r i p t i o n s of s e c t . 4, in this c a s e we a c t u a l l y e x p e c t a s i n g u l a r i t y , and in f a c t our deductions f r o m unit a r i t y a r e in a g r e e m e n t with t h o s e p r e s c r i p t i o n s . The p r e s c r i p t i o n s s a y that on the a r c s AB, B °O
and then the deduction f r o m u n i t a r i t y is that the discontinuity b e t w e e n the (+) a m p l i t u d e and the (1) a m p l i t u d e is g i v e n on t h o s e two a r c s r e s p e c t i v e l y by the r i g h t - h a n d side of eq. (3.13) and by eq. (3.14), hut with the r i g h t m o s t (-) b u b b l e s in t h o s e t e r m s r e p l a c e d by (+) bubbles. On the a r c XA, the p r e s c r i p t i o n s s a y that
(The d i f f e r e n c e b e t w e e n eqs. (3.16) and (3.17) a r i s e s b e c a u s e ~? is m e a s u r e d along the i n w a r d n o r m a l to L, so that on XA the p o s i t i v e ~? d i r e c t i o n c o r r e sponds to the n e g a t i v e s direction). T h u s on that a r c the discontinuity b e t w e e n the (-) a m p l i t u d e and the (2) a m p l i t u d e is given by eq. (3.15), but with the l e f t - m o s t bubble r e p l a c e d by a (-) bubble. On the a r c coX, C h a s changed sign and so t h e r e is an e x t r a m i n u s sign in the r e l a t i o n b e t w e e n 77 and s; this a c c o u n t s f o r the e f f e c t s of the contact at X of L with the s e c o n d type s i n g u l a r i t y , a s d e s c r i b e d in s e c t . 2. (The point X is only p r e s e n t f o r fig. 4b, but the a n a l y s i s of that s i n g u l a r i t y is v e r y s i m i l a r . ) T h i s d e s c r i p t i o n h a s b e e n f o r a continuation via I m ~ < 0 of the S S + = 1
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u n i t a r i t y equation. It may be v e r i f i e d that a continuation via Im ~? > 0 gives the s a m e r e s u l t , and so also does the S+S = 1 f o r m of unitarity.
4. F U R T H E R DIAGRAMS We now apply p r e s c r i p t i o n s s i m i l a r to those in sect. 2 to two f u r t h e r F e y n m a n g r a p h s , those in figs. 8a and 8b (The g r a p h of fig. 8a is the s a m e as that of fig. 4a, but we a r e now c o n c e r n e d with a different p h y s i c a l region).
Fig. 8a. The same Feynman graph as shown in fig. 4a, but in a different physical region obtained by giving the variables different values. Fig. 8b. A Feynman graph with four internal complex-mass particles. We find that for t h e s e the Landau c u r v e is a c u r v e of n o n - a n a l y t i c s i n g u l a r ity. We also find that in the c a s e of fig. 8b, the p r e s c r i p t i o n s a r e not c o m plete; f u r t h e r p o s t u l a t e s a r e n e c e s s a r y if the amplitude is to be fully det e r m i n e d . As with the s i m p l e r s i n g u l a r i t i e s that we have a l r e a d y considered, we f i r s t d i s c u s s the p e r t u r b a t i o n - t h e o r y p r e s c r i p t i o n s and then des c r i b e how they fit in with unitarity. C o n s i d e r the F e y n m a n g r a p h of fig. 8a. I n t r o d u c e two complex m a s s e s a s in sect. 2, s a t i s f y i n g eq. (2.2). The r e a l Landau c u r v e a r i s e s again f r o m the combinations (M 1 +M~) and (M~ +M2) of complex m a s s e s . F o r 5 > 0 the analytic p r o p e r t i e s of the t e r m involving the l a t t e r combination a r e just as usual: a p h y s i c a l - r e g i o n s i n g u l a r i t y only on the a r c AB of fig. 5a (which is now to be r e g a r d e d as depicting the new Landau c u r v e c o r r e s p o n d i n g to fig. 8a), which for r e a l ~ lies below the r e a l axis in the s - p l a n e , on the upper (physical) side of the n o r m a l - t h r e s h o l d cut attached to s = (~ +M~ +M2)2. F o r the other t e r m , the p r e s c r i p t i o n f o r a p p r o a c h i n g the ' n o r m a l - t h r e s h o l d ' cut attached to e = (p + M 1 +M~2)2 is the opposite f r o m the usual one, and so in the s - p l a n e the s i n g u l a r i t y a p p e a r s (again only on AB) in the s - p l a n e on the lower side of the cut attached to s = (p + M 1 +M~2)2 , which is just the p h y s i c a l side of this cut. F o r r e a l a the s i n g u l a r i t y is above the r e a l axis in the s - p l a n e . The situation is drawn in fig. 9; h e r e X and Y denote the two s i n g u l a r i t i e s , and the dotted line is the path of continuation that p r e s e r v e s the (+) amplitude. As 6 ~ 0, X and Y pinch the path of continuation and so lead to a n o n - a n a l y t i c singularity. In fig. 9 we have a l s o drawn the cut attached to s = 9~ 2, which is p r e s e n t in the c o m p l e t e amplitude. If we sweep this cut into the upper half plane, an exactly s i m i l a r set of s i n g u l a r i t i e s is r e v e a l e d , with the (-) b o u n d a r y value
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Fig. 9. Locus of singularities and branch cuts in the s-plane, and the path of analytic
continuation between the displaced singularities, for the Feynman graph of fig. 7. r e p l a c i n g the (+) ( c o m p a r e figs. 6a and 6b). That is, if we continue the (+) amplitude along the path shown so that it r e m a i n s the (+) amplitude (a continuation that is no l o n g e r p o s s i b l e in the l i m i t 5 ~ 0), the (-) amplitude continued along the s a m e path will r e m a i n the (-) amplitude. In o r d e r f o r t h e s e p r e s c r i p t i o n s to be in a g r e e m e n t with u n i t a r i t y , it is n e c e s s a r y that this continuation be n a t u r a l for all the t e r m s in which the s i n g u l a r i t y might be g e n e r a t e d , and an investigation of t h e s e shows that this is indeed the c a s e . This would not be t r u e f o r continuations along other paths. The F e y n m a n g r a p h of fig. 8b contains two p a i r s of c o m p l e x - m a s s lines. A r e a s o n a b l e p r e s c r i p t i o n s e e m s to be to i n t r o d u c e different m a s s e s on e a c h of t h e s e lines, M 1 and M2 on the upper p a i r s a t i s f y i n g eq. (2.2), and M 1' and M 2' on the l o w e r p a i r s a t i s f y i n g ~I
- Mg. = i5' .
(4. I )
Take 5, 5' > 0, and then c o n s i d e r the l i m i t in which 5 ~ 0, 5' ~ 0. The int e g r a l contains 24 different t e r m s , but in only four of t h e s e do the s i n g u l a r ities a p p r o a c h the r e a l axis in the limit. It t u r n s out that the r e s u l t s of this p r e s c r i p t i o n depends on the r e l a t i v e m a g n i t u d e s of 5 and 5' as they tend to z e r o . The location of s i n g u l a r i t i e s in the s - p l a n e f o r 5 > 5' is shown in fig. 10a. T h e r e is one p a i r of s i n g u l a r i t i e s that p i n c h e s on each of the a r c s ~oA, AB and B °O in fig. 5a (that f i g u r e now being r e g a r d e d as the Landau c u r v e f o r fig. 8), so p r o d u c i n g a n o n - a n a l y t i c s i n g u l a r i t y on each a r c . The fact that the s i n g u l a r a r c Aoo l e a v e s the r e a l s , a plane f o r n o n - z e r o 5, 5', with 5 > 5', depends on the following inequality which can be deduced f r o m the Landau equation valid on A¢o:
(M+M*)a 1 > (M' + M ' * ) a 3 . F o r 5 < 5', t h e r e a r e two p a i r s of s i n g u l a r i t i e s (as shown in fig. 10b), that pinch on the a r c AB, so p r o d u c i n g n o n - a n a l y t i c i t y t h e r e , but on the a r c s ¢oA and B ¢o the amplitude is r e g u l a r - that is, t h e r e is not even an analytic singularity. The two p r e s c r i p t i o n s a r e c l e a r l y v e r y different in t h e i r effects, and an examination of u n i t a r i t y v e r i f i e s that either, or a combination of both, is allowed. In o r d e r fully to d e t e r m i n e the t h e o r y s o m e f u r t h e r c o n s t r a i n t , that is, s o m e e x t r a d y n a m i c s , m u s t be imposed. In appendix A we show
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M"..,-
*.0 L
V,
(S, +N, * ML+ Jq,.)
V,
Pie.@IVt% J
Fig. 10. Loci of singularities and branch cuts in the s - p l a n e for the graph of fig. 8, when cy is real. Case (a) a r i s e s where Im (M1 - M2) > Im (M1 ' - M2'), and case (b) in the opposite situation. The c r o s s e s denote the displaced normal thresholds, and the points A and V the displaced triangle singularities; A denotes a point on the ar c AB of fig. 4, and V denotes a point on the arc Boo. It should be noted that we have drawn on one figure the singularities a r i s i n g f r o m four individual t e r m s ; thus the points in situation (a) a r e not shielded by the branch cuts lying between them and the r e a l axis (shown as a dashed line). t h a t in t h e t h e o r y of L e e a n d W i c k , t h e p r e s c r i p t i o n of L e e [2] l e a d s s o m e t i m e s t o o n e of o u r t w o p r e s c r i p t i o n s , s o m e t i m e s to t h e o t h e r , a c c o r d i n g to which L o r e n t z f r a m e is c h o s e n for its f o r m u l a t i o n . W e a r e p l e a s e d to a c k n o w l e d g e a n i n t e r e s t i n g d i s c u s s i o n w i t h P r o f e s s o r T. D. L e e .
APPENDIX A
The Lee-Wick prescription L e e and Wick u s e a t h r e e - d i m e n s i o n a l
formalism;
it is e q u i v a l e n t t o a
S-MATRIX
297
f o u r - d i m e n s i o n a l one in which the i n t e g r a t i o n path o v e r p o avoids the p o l e s at p o = + E (where E 2 = M 2 + 2 i v M + p 2) a c c o r d i n g to a definite p r e s c r i p t i o n . T h i s path is equivalent, f o r e x t e r n a l f o u r - m o m e n t a v a l u e s that a r e r e a l and b e l o w all t h r e s h o l d s , to a W i c k - r o t a t e d path. H o w e v e r , L e e and Wick a l s o c h o s e a s p e c i a l f r a m e in which all m o m e n t u m c o m p o n e n t s P o a r e r e a l , while E o is c o m p l e x . Then, in a n o t h e r f r a m e , m o v i n g with r e s p e c t to the fixed one with v e l o c i t y Vo, the m o m e n t u m p h a s an i m a g i n a r y p a r t : Imp = vo(1-v
ImE o,-~vMvo(1-v
ReEol- .
(A.1)
The inverse transformation gives
2-1ReE o = (ReE- Rep • Vo)(1-v ) z
(A.2)
and t h e r e f o r e , to f i r s t o r d e r in V, I m p = My vo ( R e E - v o • R e p ) - 1
(A.3)
L e t us define v= Rep/ReE
(A.4)
.
T o f i r s t o r d e r in ~, we m a y i n t e g r a t e o v e r r e a l p , p r o v i d e d we add I m p a s given by eq. (A.3) in c a l c u l a t i n g all functions of p . In p a r t i c u l a r , E = [M2+2iMv(1
- v" Vo)
-1
± +p212 ;
(A.5)
so to w r i t e the i n t e g r a l in an a r b i t r a r y f r a m e , we give the following v e l o c i t y - d e p e n d e n t addition to the i m a g i n a r y p a r t of the m a s s : 8 = v v.
Vo/(1-
v. V o ) .
(A.6)
To e x a m i n e L o r e n t z i n v a r i a n c e , we s e e what happens when a s m a l l change is m a d e in v o. T o l o w e s t o r d e r in V, t h e r e will be no change in the i n t e g r a l u n l e s s changing v o c a u s e s s o m e s i n g u l a r i t y to j u m p a c r o s s a path of i n t e g r a t i o n . T h i s need not happen u n l e s s the p a t h is a c t u a l l y b e i n g p i n c h e d b e t w e e n s i n g u l a r i t i e s . T h e r e f o r e , b e l o w all t h r e s h o l d s , or in any o t h e r r e g i o n r e a c h e d by a n a l y t i c continuation, the i n t e g r a l is i n v a r i a n t . Next we look at the n o n - a n a l y t i c i t y points a s s o c i a t e d with the two m a s s e s M ± iV. In this c a s e , we f i r s t c o n s i d e r the i n t e g r a l s in the c . m . f r a m e f o r t h e s e two p a r t i c l e s . T h e i r m o m e n t a a r e ± p , but v o is a l s o an i n t e g r a t i o n v a r i a b l e if t h e s e a r e additional p a r t i c l e s . T h e i n t e g r a l will contain e n e r g y d e n o m i n a t o r s in which the i m a g i n a r y p a r t s of the m a s s e s a r e changed by 5~: =V v.
V o / ( l ~ : v . v o) ;
(A.7)
f u r t h e r m o r e , f o r e v e r y t e r m in which eq. (A.7) a p p e a r s , t h e r e will b e a s e c o n d (coming f r o m i n t e r c h a n g e of M and M*) in which 5± is r e p l a c e d by -5±. With a r e d e f i n i t i o n of the two t e r m s a c c o r d i n g to the sign of v . Vo, we can r e p l a c e eq. (A.7) by
v. Vol/(l
V. Vo ) .
(A.8)
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R.E. CUTKOSKY et al.
We now c a l c u l a t e the i m a g i n a r y p a r t of an e n e r g y d e n o m i n a t o r containing both M+i7 and M - i7 in the original f r a m e , but e x p r e s s it in t e r m s of the c.m. v a r i a b l e s : I m D = ( 1 - v )oz x I m ( E + + E _ )
2 -~ , = M6E-l(1-Vo)~
(A.9)
where
6
=
8++8_
= 71
v. V o [ / ( 1 - ( v . %)9.)
(A.10)
(again, t h e r e will be another t e r m with -6). Now c o n s i d e r an analytic continuation through r e a l values of the e x t e r n a l m o m e n t a . Then, except for I v. Vol ~ ~/7 (where E << 7 gives the conventional d i s p l a c e m e n t of p h y s i c a l m a s s e s f r o m r e a l values) t h e r e is no singul a r i t y in the r e g i o n of integration. F u r t h e r m o r e , e v e r y s i n g u l a r i t y is disp l a c e d in the s a m e s e n s e as it would be if 6 w e r e constant, and 6+ = ½6. In sect. 2, the continuation was defined by introducing a constant 8, taking the l i m i t 6 4 0 at the end. In that c a s e as well as h e r e , t h e r e w e r e two t e r m s , one with 5 and one with -6, so the LW p r e s c r i p t i o n and o u r s coincide, at l e a s t to the lowest o r d e r in Y. To t r e a t virtual s t a t e s with four c o m p l e x m a s s e s , M+i7 and M±iT', we introduce the quantities v, v ' , Vo, Vo', 8±, 8±', etc. by the s a m e method as before. In addition to e n e r g y d e n o m i n a t o r s whose i m a g i n a r y p a r t s contain e i t h e r 6 or 5' alone, as in eq. (A.9), t h e r e will be new ones, containing for e x a m p l e : ImD2 =
M E-1(1-V2o)½-M'5'E'-l(1-V'o2) ½ .
(A. 11)
No'ce that a t e r m containing, say +i6, in one e n e r g y d e n o m i n a t o r , n e v e r contains - i 6 in another although for e v e r y t e r m containing +i5 t h e r e will be another with -i6. Unlike eq. (A.9), ImD 2 has different signs in different p a r t s of the r e gion of integration; the d i s p l a c e m e n t of the s i n g u l a r i t y is not the s a m e for all p a r t s of m o m e n t u m space. F u r t h e r m o r e , the s e n s e of the d i s p l a c e m e n t depends on the choice of the r e f e r e n c e f r a m e through Vo and Vo'. In o r d e r to c h e c k that the a p p a r e n t n o n - i n v a r i a n c e does not always disa p p e a r when v o and v o' a r e i n t e g r a t e d over, we look at the s i n g u l a r p a r t of the integral for fig. 9. N e a r the t r i a n g l e s i n g u l a r i t y , v o and v o' a r e d e t e r mined by the Landau equations. They depend on the velocity of the o v e r - a l l c e n t e r of m a s s , and we can choose, e.g. either ]Vol >> I Vo'l or vice v e r s a . This m e a n s we can obtain either 6 >> 6' for m o s t of the i m p o r t a n t r e g i o n of integration, or the r e v e r s e . W e h a v e shown, in sect. 4, that t h e s e two possibilities lead to different behaviour at the t r i a n g l e s i n g u l a r i t y , so the a m plitude as c o n s t r u c t e d by the LW p r e s c r i p t i o n is not invariant. APPENDIX
B
Causality F o r c o m p l e t e n e s s , we p r e s e n t h e r e a calculation of the advanced signal
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299
a s s o c i a t e d with the type of n o n - a n a l y t i c i t y at the ' n o r m a l t h r e s h o l d ' s = (M+M*) 2. Our g e n e r a l a p p r o a c h follows that of Eden and Landshoff [9]. W r i t e W = 4-s. C o n s i d e r the n o n - a n a l y t i c t e r m 1
Fn.a.(W) = O ( - W + 2 R e M ) ( 2 R e M - W )
1
2 (ReM) -~ F a .
(B.1)
This is to be c o n s i d e r e d in conjunction with a t e r m Fa(W) , which is analytic in a p a r t of the u p p e r half plane c l o s e to the r e a l axis and such that Fa(2 Re M) = F a. Take an initial wave packet ~I,( W, 2 Re M, A), which is c e n t r e d a r o u n d W = 2 R e M with a width A; ~I, is infinitely differentiable and v a n i s h e s for W - 2 R e M I > A. We take it to be n o r m a l i s e d , so that ' 1 ~I,(2 ReM) ~ A-~. The s c a t t e r e d amplitude, f o r t i m e delay t = -T < 0, is
,~(~) = f dW e i W T ~I,(W)[Fa(W) +Fn.a.(W) ] .
(B.2)
F o r T >> A we have [ 10] • (r) ~ Fa: e 2 i r ( R e M ) (ReM)½ A½ r~ '
(B.3)
w h e r e u n i m p o r t a n t n u m e r i c a l f a c t o r s a r e dropped. The contribution of F a d r o p s off f a s t e r than any p o w e r of 7. F r o m eq. (B.3) we obtain the r e l a t i v e intensity of an a d v a n c e d signal between r and r + dr: d~ P ( r ) d r ~ (ReM)AT 3 "
(B.4)
T h i s can be i n t e g r a t e d to give the r e l a t i v e p r o b a b i l i t y of r > t, c o m p a r e d with the total f o r w a r d s c a t t e r i n g : P(t)
1 (ReM)At2 .
(B.5)
An equivalent d i s c u s s i o n , leading to the s a m e r e s u l t , has been given by L e e [2]. He, also, has noted that when v a l u e s of M, A, and t a p p r o p r i a t e to conventional e x p e r i m e n t s a r e i n s e r t e d into eq. (B.5), one gets a r i d i c u l o u s ly s m a l l n u m b e r . Under a p p r o p r i a t e conditions - p e r h a p s , for example, in a c o s m o l o g i c a l p r o b l e m involving a v e r y high density (p ~ M3) and t e m p e r a t u r e ( k T ~ M ) - the advanced signal might give a quantitatively significant effect.
REFERENCES
[1] T.D. Lee and G.C.Wick, Nucl. Phys. B9 (1969) 209. [2] T.D. Lee, to be published in Quanta (University of Chicago Press, 1969}. [3] R.J. Eden, P.V. Landshoff, D.I.Olive and J. C. Polldnghorne, The analytic Smatrix (Cambridge University Press, 1966} (called ELOP below}, sect. 4.4; D. I. Olive, Phys. Rev. 135B (1964) 745. [4] M.Bloxham, D.I.Olive and J.C.Polkinghorne, J. Math. Phys. 10 (1969) 494, 545.
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[5] ELOP, Sect. 2.10. [6] P.V.Landshoff, D.I.Olive and J . C . P o l k i n g h o r n e , J. Math. Phys. 7 (1966) 1600. [7] ELOP, sect. 2.5; S. Coleman and R. E. Norton, Nuovo Cimento 38 (1965) 438. [8] ELOP, Chapter 4. [9] R . J . E d e n and P.V.Landshoff, Ann. of Phys. 31 (1965) 370. [10] M.J. Lighthill, Introduction to F o u r i e r analysis and genera!ised functions (Cambridge University P r e s s , 1959).