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Discontinuities across cuts in the complex angular momentum plane
Volume 4, number 1
PHYSICS
H a n s Bethe and to M r . B e n j a m i n Day f o r m a n y v a l u a b l e d i s c u s s i o n s , and to t h a n k P r o f e s s o r Bethe f o r h i s a d v i c e on the p r e p a r a t i o n of t h i s note. I a l s o wish to t h a n k P r o f e s s o r G. E. B r o w n for i n f o r m i n g m e of his r e m o v a l e n e r g y a n a l y s i s .
References 1) K.A.Brueckner, J . L . G a m m e l and H.Weitzner, Phys. Rev. 110 (1958) 431. 2) K.A. Brueckner and D.T.Goldman, Phys. Rev. 116 (1959) 424. 3) K.A. Brueckner, A.M. Lockett and M. Rotenberg, Phys. Rev. 121 (1961) 255.
THE
LETTERS
1March1963
4) K. S. Masteron Jr. and A. M. Lockett, to appear in Phys. Rev. 5) H.A. Bethe, B.H. Brandow and A. G. Petschek, Phys. Rev. 129, no. 1 (1963). 6) R.Rajaraman, Phys. Rev. 129, no. 1 (1963). 7) C. Bloch and J. Horowitz, Nuclear Phys. 8 (1958) 91. 8) See also: R. J. Eden, V.J. Emery and S. Sampanthar, Proc. Roy. Soc. A 253 (1959) 177, 186. H. J. Mang and W.Wild, Z. Phys. 154 (1959) 182. J.F.Dawson, I. Talmi and J.D.Walecka, Ann. Phys. 18 (1962) 339. H.S.K/Shler, Nuclear Phys. 32 (1962) 661. 9) K.A. Brueckner, J . L . G a m m e l and J . T . K u b i s , Phys. Rev. 118 (1960) 1438. 10) Private communication from Professor G. E. Brown.
DISCONTINUITIES ACROSS CUTS COMPLEX ANGULAR MOMENTUM
IN PLANE
R. G. NEWTON * Istituto di Fisica dell'Universit/t di Roma Received 6 February 1963
It has r e c e n t l y b e e n shown by the a u t h o r 1) that if, i n the n o n - r e l a t i v i s t i c t h r e e - b o d y p r o b l e m , the total angular m o m e n t u m j is made a continuous c o m p l e x v a r i a b l e in a " n a t u r a l " way, t h e n the p a r t i a l wave a m p l i t u d e s have i n f i n i t e l y m a n y e n e r g y d e p e n d e n t b r a n c h p o i n t s . The a r g u m e n t i s b a s e d on the o b s e r v a t i o n that the i n v e r s e G(~; x, x') of a (matrix) integral operator
In the c a s e of p r e s e n t p h y s i c a l i n t e r e s t the i n t e g r a l o p e r a t o r F m a y be t a k e n to b e the d e n o m i n a t o r (1 + iK) i n the S - m a t r i x , K b e i n g the c u s t o m a r y K - m a t r i x , and
g= l + iKD .
H e r e K D i s the d i a g o n a l p a r t of the K - m a t r i x , due to the d i s c o n n e c t e d d i a g r a m s ; x i s the e n e r g y d i s t r i b u t i o n b e t w e e n the p a r t i c l e s ; ,~ m a y be e i t h e r the F(~; X , X ' ) = g(Ot, X) 5(X-X') + M ( X , X ' ) , t o t a l a n g u l a r m o m e n t u m , when the t o t a l e n e r g y i s c o n s i d e r e d a s a f u n c t i o n of % h a s b r a n e h p o i n t s . If f i x e d , o r it m a y be the t o t a l e n e r g y , when the t o t a l the x - i n t e g r a t i o n s a r e S t i e l t j e s i n t e g r a l s with w e i g h t s a n g u l a r m o m e n t u m i s fixed. In e i t h e r c a s e we get do(x) so that c u t s whose o r i g i n i s s e e n to be i n effect with the i t e r a t i o n of t w o - b o d y p o l e s v i a the d i s c o n n e c t e d graphs t ~n an 5 ( x - x n) , x < 0 do(X)dx - I ¢ If ~ is the total energy, we get cuts not only from the 1 , 0 ~ X < oo , real two-body poles of the S-matrix, but also from the complex ones. Their onsets lead to the "wooly cusps" by 5 ( x - x ' ) we m e a n a D i r a c 5 - f u n c t i o n f o r x , x ' > O, of Nauenberg and Pais 2) and of Baz' 3). a K r o n e c k e r 5 if x , x ' < 0, and z e r o o t h e r w i s e ; and The generality of this argument shows that cuts should M ( x , x ' ) c o n t a i n s no 5 - f u n c t i o n s ; then the c u t s a r e appear in the j - p l a n e no matter how the other angular a l o n g c u r v e s C g i v e n by momenta are handled, provided only that the continuation is such as to lead to poles in the two-body ampliC: detg(ev,x)=0 , O ~
Volume 4, number 1
PHYSICS
T h e d e m o n s t r a t i o n of r e f . 1) l e a v i n g o p e n the r e m o t e p o s s i b i l i t y t h a t the b r a n c h p o i n t s show c a n c e l out, we now c a l c u l a t e e x p l i c i t l y the d i s c o n t i n u i t i e s across the cuts. W h e n ~ l i e s on C we e x p a n d g - 1 a b o u t i t s p o l e : g ' l ( , % x ) : [ x - Xo(¢~)]-I R(~) + . . . . A s i s w e l l known, e x c e p t in t h e c a s e of d e g e n e r a c y R will be factorable:
Rd ) : When ~ is shifted at right angles to C, x o changes by a purely imaginary amount Xo(~)
Xo(~) + i c ,
-
provided that dXo(~)_ d~
0 det g/~ J ~ d e t g/~ x ]x=x o
¢
LETTERS
t i n u i t y g e n e r a l i s e s the w e l l known f a c t o r i s a t i o n of the r e s i d u e a t a p o l e . It should not b e o v e r l o o k e d t h a t G a l s o h a s p o l e s on the cut s w h o s e p o s i t i o n s d e p e n d on x and x ' . T h e s e m a n i f e s t t h e m s e l v e s a s 5 - f u n c t i o n s in (D), b e i n g e x p l i c i t l y v i s i b l e in t h e f i r s t t h r e e t e r m s of t h e a l t e r n a t i v e f o r m of (D). AG = A g -1 - G~MAg-1 - Ag-IMG~ + G+MAg-IMG:r If, f o r e x a m p l e , x and x' a r e both n e g a t i v e , the e n t i r e c o n t r i b u t i o n to t h e d i s c o n t i n u i t y a c r o s s a cut c o m e s f r o m the l a s t t e r m . In a d d i t i o n , G m a y h a v e fixed ( i . e . , x and x' i n d e p e n d e n t ) p o l e s w h e r e the F r e d h o l m d e t e r m i n a n t of F v a n i s h e s . Simple familiar examples are furnishes by G r e e n ' s f u n c t i o n s . On a b a s i s of the u n p e r t u r b e d energy, x =Eo,
g-1 = Go(E,Eo) =
O.
T h e d i f f e r e n c e b e t w e e n t h e v a l u e s of g - 1 on the two s i d e s of C i s t h e r e f o r e A g - 1 : 2fri 6(X-Xo) R .
g-1 M
,
N(R) = 1 - M
IE> = [1 - (E-Ho+i¢)- I H'] -1
g-1 N(P~
for
N(L) = (I + g - i M ) - i ,
I N(_R) : N(L) , g - I N(+R)
(D)
where the subscripts + and - indicate the values above and below the cut, respectively. Written out explicitly, this means that
IE> o .
It is also possible to express the discontinuity in terms of "principal value" quantities. The result is AG = C(L) Ag-1 C~R) ,
N(R) = (i + M g ' l ) - I
to calculate the discontinuity of G = N(L)g-1. After some simple algebra we obtain the "generalised unitarity"
G : NiL)
1 E-E o
and M i s the i n t e r a c t i o n H a m i l t o n i a n . Eq. (D) then gives us directly a familiar formula for the exact, normalised wave functions, namely
W e now u s e the i n t e g r a l e q u a t i o n s N(L) : 1 - N(L)
1 M a r c h 1963
where C(L) : N(CL) [1 - i*tR(MN(cL))X_--x,:Xo1-1 ,
C(R) = [I + i~(Nc(R)M)x=_x,=xo R]-1 /~CR), Nc(L) : (1 + p g - I M ) - I Nc(R) : (1 + MPg-1) -1
and pg-1 s t a n d f o r the p r i n c i p a l v a l u e , i . e . , the a v e r a g e of the v a l u e s of g - 1 a b o v e and b e l o w C. The a u t h o r i s i n d e b t e d to D r . D. Z w a n z i g e r f o r stimulating discussions.
The result first of all shows that provided only that the zero of d e t g on C is simple, both as a function of x and as a function of a, the endpoint singuiarities produced by the vanishing of g are always branch points, even though their precise nature depends on the remaining integrand near the endpoints. In addition, the explicit form of the discon-
12
References 1) R.G. Newton, The total angular momentum plane in the non-relativistic three-body problem, to be published in Nuovo Cimento. 2) Nauenberg and Pals, Phys. Rev. 120 (1962) 360. 3) A . B a z ' , J. Exptl. Theoret. Phys. (USSR) 40 (1961) 1511 translation: Soviet Physics JETP 13 (1961) 1058. See also: D. Zwanziger, preprint (August, 1962).
PHYSICS
H a n s Bethe and to M r . B e n j a m i n Day f o r m a n y v a l u a b l e d i s c u s s i o n s , and to t h a n k P r o f e s s o r Bethe f o r h i s a d v i c e on the p r e p a r a t i o n of t h i s note. I a l s o wish to t h a n k P r o f e s s o r G. E. B r o w n for i n f o r m i n g m e of his r e m o v a l e n e r g y a n a l y s i s .
References 1) K.A.Brueckner, J . L . G a m m e l and H.Weitzner, Phys. Rev. 110 (1958) 431. 2) K.A. Brueckner and D.T.Goldman, Phys. Rev. 116 (1959) 424. 3) K.A. Brueckner, A.M. Lockett and M. Rotenberg, Phys. Rev. 121 (1961) 255.
THE
LETTERS
1March1963
4) K. S. Masteron Jr. and A. M. Lockett, to appear in Phys. Rev. 5) H.A. Bethe, B.H. Brandow and A. G. Petschek, Phys. Rev. 129, no. 1 (1963). 6) R.Rajaraman, Phys. Rev. 129, no. 1 (1963). 7) C. Bloch and J. Horowitz, Nuclear Phys. 8 (1958) 91. 8) See also: R. J. Eden, V.J. Emery and S. Sampanthar, Proc. Roy. Soc. A 253 (1959) 177, 186. H. J. Mang and W.Wild, Z. Phys. 154 (1959) 182. J.F.Dawson, I. Talmi and J.D.Walecka, Ann. Phys. 18 (1962) 339. H.S.K/Shler, Nuclear Phys. 32 (1962) 661. 9) K.A. Brueckner, J . L . G a m m e l and J . T . K u b i s , Phys. Rev. 118 (1960) 1438. 10) Private communication from Professor G. E. Brown.
DISCONTINUITIES ACROSS CUTS COMPLEX ANGULAR MOMENTUM
IN PLANE
R. G. NEWTON * Istituto di Fisica dell'Universit/t di Roma Received 6 February 1963
It has r e c e n t l y b e e n shown by the a u t h o r 1) that if, i n the n o n - r e l a t i v i s t i c t h r e e - b o d y p r o b l e m , the total angular m o m e n t u m j is made a continuous c o m p l e x v a r i a b l e in a " n a t u r a l " way, t h e n the p a r t i a l wave a m p l i t u d e s have i n f i n i t e l y m a n y e n e r g y d e p e n d e n t b r a n c h p o i n t s . The a r g u m e n t i s b a s e d on the o b s e r v a t i o n that the i n v e r s e G(~; x, x') of a (matrix) integral operator
In the c a s e of p r e s e n t p h y s i c a l i n t e r e s t the i n t e g r a l o p e r a t o r F m a y be t a k e n to b e the d e n o m i n a t o r (1 + iK) i n the S - m a t r i x , K b e i n g the c u s t o m a r y K - m a t r i x , and
g= l + iKD .
H e r e K D i s the d i a g o n a l p a r t of the K - m a t r i x , due to the d i s c o n n e c t e d d i a g r a m s ; x i s the e n e r g y d i s t r i b u t i o n b e t w e e n the p a r t i c l e s ; ,~ m a y be e i t h e r the F(~; X , X ' ) = g(Ot, X) 5(X-X') + M ( X , X ' ) , t o t a l a n g u l a r m o m e n t u m , when the t o t a l e n e r g y i s c o n s i d e r e d a s a f u n c t i o n of % h a s b r a n e h p o i n t s . If f i x e d , o r it m a y be the t o t a l e n e r g y , when the t o t a l the x - i n t e g r a t i o n s a r e S t i e l t j e s i n t e g r a l s with w e i g h t s a n g u l a r m o m e n t u m i s fixed. In e i t h e r c a s e we get do(x) so that c u t s whose o r i g i n i s s e e n to be i n effect with the i t e r a t i o n of t w o - b o d y p o l e s v i a the d i s c o n n e c t e d graphs t ~n an 5 ( x - x n) , x < 0 do(X)dx - I ¢ If ~ is the total energy, we get cuts not only from the 1 , 0 ~ X < oo , real two-body poles of the S-matrix, but also from the complex ones. Their onsets lead to the "wooly cusps" by 5 ( x - x ' ) we m e a n a D i r a c 5 - f u n c t i o n f o r x , x ' > O, of Nauenberg and Pais 2) and of Baz' 3). a K r o n e c k e r 5 if x , x ' < 0, and z e r o o t h e r w i s e ; and The generality of this argument shows that cuts should M ( x , x ' ) c o n t a i n s no 5 - f u n c t i o n s ; then the c u t s a r e appear in the j - p l a n e no matter how the other angular a l o n g c u r v e s C g i v e n by momenta are handled, provided only that the continuation is such as to lead to poles in the two-body ampliC: detg(ev,x)=0 , O ~
Volume 4, number 1
PHYSICS
T h e d e m o n s t r a t i o n of r e f . 1) l e a v i n g o p e n the r e m o t e p o s s i b i l i t y t h a t the b r a n c h p o i n t s show c a n c e l out, we now c a l c u l a t e e x p l i c i t l y the d i s c o n t i n u i t i e s across the cuts. W h e n ~ l i e s on C we e x p a n d g - 1 a b o u t i t s p o l e : g ' l ( , % x ) : [ x - Xo(¢~)]-I R(~) + . . . . A s i s w e l l known, e x c e p t in t h e c a s e of d e g e n e r a c y R will be factorable:
Rd ) : When ~ is shifted at right angles to C, x o changes by a purely imaginary amount Xo(~)
Xo(~) + i c ,
-
provided that dXo(~)_ d~
0 det g/~ J ~ d e t g/~ x ]x=x o
¢
LETTERS
t i n u i t y g e n e r a l i s e s the w e l l known f a c t o r i s a t i o n of the r e s i d u e a t a p o l e . It should not b e o v e r l o o k e d t h a t G a l s o h a s p o l e s on the cut s w h o s e p o s i t i o n s d e p e n d on x and x ' . T h e s e m a n i f e s t t h e m s e l v e s a s 5 - f u n c t i o n s in (D), b e i n g e x p l i c i t l y v i s i b l e in t h e f i r s t t h r e e t e r m s of t h e a l t e r n a t i v e f o r m of (D). AG = A g -1 - G~MAg-1 - Ag-IMG~ + G+MAg-IMG:r If, f o r e x a m p l e , x and x' a r e both n e g a t i v e , the e n t i r e c o n t r i b u t i o n to t h e d i s c o n t i n u i t y a c r o s s a cut c o m e s f r o m the l a s t t e r m . In a d d i t i o n , G m a y h a v e fixed ( i . e . , x and x' i n d e p e n d e n t ) p o l e s w h e r e the F r e d h o l m d e t e r m i n a n t of F v a n i s h e s . Simple familiar examples are furnishes by G r e e n ' s f u n c t i o n s . On a b a s i s of the u n p e r t u r b e d energy, x =Eo,
g-1 = Go(E,Eo) =
O.
T h e d i f f e r e n c e b e t w e e n t h e v a l u e s of g - 1 on the two s i d e s of C i s t h e r e f o r e A g - 1 : 2fri 6(X-Xo) R .
g-1 M
,
N(R) = 1 - M
IE> = [1 - (E-Ho+i¢)- I H'] -1
g-1 N(P~
for
N(L) = (I + g - i M ) - i ,
I N(_R) : N(L) , g - I N(+R)
(D)
where the subscripts + and - indicate the values above and below the cut, respectively. Written out explicitly, this means that
IE> o .
It is also possible to express the discontinuity in terms of "principal value" quantities. The result is AG = C(L) Ag-1 C~R) ,
N(R) = (i + M g ' l ) - I
to calculate the discontinuity of G = N(L)g-1. After some simple algebra we obtain the "generalised unitarity"
G : NiL)
1 E-E o
and M i s the i n t e r a c t i o n H a m i l t o n i a n . Eq. (D) then gives us directly a familiar formula for the exact, normalised wave functions, namely
W e now u s e the i n t e g r a l e q u a t i o n s N(L) : 1 - N(L)
1 M a r c h 1963
where C(L) : N(CL) [1 - i*tR(MN(cL))X_--x,:Xo1-1 ,
C(R) = [I + i~(Nc(R)M)x=_x,=xo R]-1 /~CR), Nc(L) : (1 + p g - I M ) - I Nc(R) : (1 + MPg-1) -1
and pg-1 s t a n d f o r the p r i n c i p a l v a l u e , i . e . , the a v e r a g e of the v a l u e s of g - 1 a b o v e and b e l o w C. The a u t h o r i s i n d e b t e d to D r . D. Z w a n z i g e r f o r stimulating discussions.
The result first of all shows that provided only that the zero of d e t g on C is simple, both as a function of x and as a function of a, the endpoint singuiarities produced by the vanishing of g are always branch points, even though their precise nature depends on the remaining integrand near the endpoints. In addition, the explicit form of the discon-
12
References 1) R.G. Newton, The total angular momentum plane in the non-relativistic three-body problem, to be published in Nuovo Cimento. 2) Nauenberg and Pals, Phys. Rev. 120 (1962) 360. 3) A . B a z ' , J. Exptl. Theoret. Phys. (USSR) 40 (1961) 1511 translation: Soviet Physics JETP 13 (1961) 1058. See also: D. Zwanziger, preprint (August, 1962).