Scattering functions with crossing symmetry and their applications to the problems of meson-nucleon scattering

Nuclear

Physics 5

(1958)

1--10; (~)North-Holland

Publishing Co., A m s t e r d a m

Not to be reproduced by photoprint or microfilm without written permission from the publisher

SCATTERING FUNCTIONS WITH CROSSING SYMMETRY AND THEIR

APPLICATIONS TO THE PROBLEMS NUCLEON SCATTERING

OF

MESON-

NING HU

Joint Institute o] Nuclear Research, Dubna, USSR R e c e i v e d 11 J u n e 1957 Abstract: I t is s h o w n t h a t t h e general f o r m of s c a t t e r i n g f u n c t i o n s s a t i s f y i n g a p p r o p r i a t e c r o s s i n g c o n d i t i o n s are g e n e r a l s o l u t i o n s of t h e e q u a t i o n s of C h e w a n d Low. T h e s e s c a t t e r i n g

f u n c t i o n s fall into t w o classes. I n one class t h e y a r e a n a l y t i c f u n c t i o n s (usually double v a l u e d ) of t h e m o m e n t u m v a r i a b l e k w i t h o n l y s i m p l e poles a n d zeros, while in t h e o t h e r class t h e y h a v e a line of d i s c o n t i n u i t y o n t h e i m a g i n a r y k-axis b e t w e e n k = - - i a n d h= --ico.

1. I n t r o d u c t i o n Recently it has been shown by m a n y authors 1) t h a t the principle of causality leads to restrictions on the analytic behaviour of Heisenberg's scattering functions S (co, k); ~o, k being, respectively, energy and momentum variables of the scattered particle. In the case of meson-nucleon collision, broadly speaking, these restrictions amount to the requirement t h a t S(oJ, k) should have no poles above the real axis in the complex ~o-plane. Alternatively, Chew and Low obtained a set of equations for the phase shifts which require also t h a t S(o, k) should have no poles in the complex a~-plane except on the real axis between o~ = --1 and o~ = + 1 (here and in the following we shall use the natural units /~ = c = h = 1). Both developments bring out an interpretation of the scattering functions when the energy variable is negative. It is the aim of the present paper to investigate the general analytic properties of the scattering functions which satisfy certain crossing conditions. Our starting point is the theory of the S-matrix as formulated by Heisenberg. Doubt as to the usefulness of this formulation was raised after it was discovered that the S-matlix as an analytic function m a y possess redundant poles. However, it was shown that these poles will not appear when the interaction is cut off at great distances. This means that we can always choose an S-matrix which represents the scattering phenomena quite accurately and at the same time is free from redundant poles. This representation of the S-matrix will be used in the present theory. January 1958

1

NING H U

2. A n a l y t i c P r o p e r t i e s of S (oJ, k) w h e n oJ is p o s i t i v e on the Effective Range Approximation

"radial"

The asymptotic wave function of a scattered particle when it is very far from the scatterer m a y be written as

1 ~v(r) , ~ - - [ e - ' ~ + S ( a ~ , k)e~], o~ = ~¢/l+k ~ ,

(1)

r

where we have suppressed the indices representing angular momentum states as well as total spin and total isobaric spin states if they exist. We shall now make the ansatz t h a t S should be an analytic function of k in the whole complex k-plane. In doing this, however, we must introduce one further requirement that all results which are obtained from (1) by analytic continuation and which satisfy the physical boundary conditions at infinity must represent a true solution of the system. Thus, if we replace k by --k in (1) we obtain ~v(r) --~ 1 [eaSt+ S (oJ, --k)e -a~] r

: ls(°~'r - - k ) [ e - ' ~ +

1

S(o~, --k)ea"l"

(2)

Now the function (2) satisfies the same boundary condition at infinity as (1), therefore it must represent a true solution according to our requirement. But (1) a,nd (2) must also be the same solution; consequently

1

s(o, k) =

S(o,--k)"

(3)

The unitary condition for S(o~, k) is

s(o ,

k)=

or

s(o, k ) =

1

k)'

(4)

where S*(o~, k) denotes the complex conjugate of S(~o, k) when oJ and k are both considered as real. Since o is a double valued function of k, S(oJ, k) is therefore a double valued function of k or ~o. In order to confine ourselves only to the region where the real part of o~ is always positive, cuts must be introduced in the complex k-plane along the imaginary axis from k = --ioo to k = - - i and from k = + i to k : + i oo. In the following we shall call the cut plane the upper Riemann sheet and the other plane connected to it through the cut-lines the lower Riemann sheet. After the introduction of the cuts, S(o~, k ) i n t h e upper Riemann sheet m a y simply be written as is in general discontinuous across the cuts. From (3) and (4)

S(k). S(k)

SCATTERING FUNCTIONS WITH CROSSING SYMMETRY

3

we see t h a t w h e n e v e r S(k) has a pole at k = k n, it m u s t also h a v e a pole at k = --kn* and two zeros at k = --kn a n d k = k**. The general f o r m of S(k) m a y therefore be w r i t t e n as

S(k) = s

(k)H

(k +ik) II (0 < k < 1);

(5)

k, represents a b o u n d s t a t e a n d k n represents a m e t a s t a b l e state of the system, whereas Sl(k) is a function of k which is u n i t a r y on the real kaxis and m a y b e c o m e discontinuous along the two cuts i n t r o d u c e d above. According to results o b t a i n e d from causality considerations and from the t h e o r y of Chew a n d Low, S(k) should h a v e n6 poles or a n y o t h e r kind of singularity a b o v e the real k-axis (except, of course, the cut on the positive i m a g i n a r y axis i n t r o d u c e d above). T h e cut is i n t r o d u c e d o n l y as a consequence of the fact t h a t ~o is a double v a l u e d function of k. One would e x p e c t t h a t if S (k) is c o n t i n u e d analytically across the cut to the lower R i e m a n n sheet, a function of similiar analytic properties would be obtained. Before the crossing condition was known, the physical m e a n i n g of the analytic c o n t i n u a t i o n was not quite clear, a n d consideration was confined to those cases in which S(~o, k) was of the same t y p e of function in the lower R i e m a n n sheet as it was in the u p p e r R i e m a n n sheet 3). I n the following section we shall show t h a t there also exist i m p o r t a n t cases in which the crossing condition requires the p r o p erties of S (o~, k) in the lower R i e m a n n sheet to be quite different from those in the u p p e r sheet. In the non-relativistic limit, the cut in (5) will go to infinity and we obtain 3)t

S(k) = e*a'II (k,-- ik) (kn*--k) (k,,+k) (k,+ik) I I (k,~--k)(k,~*+k)"

(6)

T h e formula for S(k) given a b o v e a n d also for S(o~, k) o b t a i n e d in the following section would be of little practical value if the poles were n u m e r o u s a n d clustered t o g e t h e r in such a w a y t h a t it would be difficult to d e t e r m i n e t h e m f r o m experiments. T h e question m a y t h e n arise w h e t h e r each pole possesses an unequivocal m e a n i n g as a m e t a s t a b l e state. In the case of nuclear i n t e r a c t i o n in which the range of force is v e r y small, however, the t The possible a p p e a r a n c e of the exponential factor e x p (ick) (which equals u n i t y if c = 0) follows f r o m t h e condition for S(k) a t infinity, derived f r o m the completeness condition (cf. ref. a)): f/" S(k) etkO"W) dk = 0, which holds for large values of r+r'. The p a t h of integration is an infinite semi-circle above t h e real k-axis. I t i s s e e n t h a t if t h e a b o v e condition is satisfied b y a n S(k) w i t h o u t the exponential factor exp (ick), it will also be satisfied if S(k) is multiplied b y this factor since thv effect of i n t r o d u c i n g this factor is equivalent to replacing r+r' b y r+r'+c.

4

NING HU

poles are found to be far apart. This is due to the fact t h a t with decreasing range of force, the energy intervals between successive metastable levels increase. Hence when k is v e r y small, we m a y introduce in (6) the following substitution

(k,,--k)(k,~*+k)

. 1

for all n. Furthermore, we know t h a t for the proton-neutron system there is only one b o u n d state near k = 0; therefore we have

K--ik S(k) = e '°~ - - . KWik

(7)

Since c can only be of the order of a small length representing the range of force, we m a y write approximately

l +½ick L--ik e' ~ ~ - -1 - - ½ic k L + ik

(8)

where L ---- 2/c. Eq. (7) t h e n becomes

L--ik K--ik S(k) = L + i k K + i k "

(9)

Inserting this in .S(k)+l cot 0(k) = ~ S ( k ) - - 1 where 0(k) denotes the phase shift, we obtain immediately k cot 0 (k) --

KL g+~

1 + K-~

ks.

(10)

This is just the usual effective range formula in which k 2 is the value of the deuteron binding energy. The same expression (9) has been given b y Jost and K o h n 4). These authors consider k = iL as a r e d u n d a n t pole of S(k). F r o m our derivation we see t h a t this is actually a false pole introduced b y the approximation (8).

3. T h e Case of M e s o n - N u c l e o n

Scattering

The crossing condition m a y be stated as follows: where q, p represent m o m e n t a of the scattered particle in the initial a n d final states respectively, a n d t o ( o ) is connected to the complete S-matrix b y the following relation: ( p l S l q ) = 0n -

2~i0 (Eq-- Ep)tqp(o).

(12)

SCATTERING FUNCTIONS WITH CROSSING SYMMETRY

On the one-meson approximation, the complete S-matrix simply becomes the scattering function considered in the last section with p = q ---- k. We shall first consider the case of a neutral scalar meson. In this case
k) = S(k).

k) =

(13)

The general form of S (o~, k) satisfying the above condition has been derived in ref. a); it is

S(k) = e'*']-[ Ka-ik ] l (k,,*--k)(k,,+k) ~ K a + i k %- (k,,--k)(k,*+k)

(14)

where K a represents the energy value of a bound state and k, represents a metastable state. Next we consider the case of a charged scalar meson. We m a y put 2

tpq(W) = X A,(p, q)h,,(oJ, k) q,--1

(15)

A1 (P' q) : TPl Tql~- TP2 "rq2 A2(P, q) = ~pq--~(TplTql-~Tp2Tq2).

(16)

where

The S functions for the states ~ = 1, 2, are given by

S=(~, k ) = 1--2~ih~(m, k)oak.

(17)

Application of the condition (11)leads immediately to the following result: Sd--~

) =

S2(,o),

S2(--OJ) :

Sl(O) ).

(18)

This means t h a t the two scattering functions corresponding to two different isobaric states together form a single analytic function of co and k defined in both Riemann sheets of the complex k-plane. It was shown in ref. 3) t h a t the general form of this function is Sl(O) , k ) =

S2(--(D , k)

o~--l--ib,k

= e'° IJ o,--l+ib,k (Wn 9" =

kn2-+-l,

(ks+k~*)(e°--c°"*)--(°Jn*--e°")(k--kn*) (19)

b, = real)

where b,, ks are arbitrary constants to be determined by the detailed structure of the system. Each b, will give rise to a bound state, and each pair of o~, and k s determines a metastable state of the system. If the imaginary parts of the k,'s in (15) and (20) are all negative, then these two functions will have no pole above the real k-axis except on the imaginary axis between k : 0 and k : i. It can easily be seen t h a t t h e y are the general solutions of the equations of Chew and Low for the cor-

6

NING HU

responding cases t. Our results agree with the conclusion obtained b y Castillejo, Dalitz and Dyson that the general solutions of the Chew and Low equations in the two preceding cases contain an infinite number of adjustable constants. We now come to the case of S-wave scattering for an interaction "symmetrical" with respect to charged and neutral mesons. The following investigation will show that the scattering functions for this case are essentially of a new type. The matrix element tpq(eO) m a y be written as S

tqp(OJ, k ) =

• A.(p, q)h.(oJ, k)

(20)

ct--1

where Al(P, q) = rp" rq ---- TplTql+Tp2Tq2+Tp3Tq 3,

A2(P, q)

(21)

---- 6pq--{Tp • rq.

The crossing condition (11) leads immediateiy to the following conditions for S 1 and S, corresponding respectively to the singlet and triplet states: S,(--o~, k ) + 2 S d - - o , , k ) = Sl(~, k)+2S,.(~, k),

s~(-o~, k ) - S d - ~ , k ) = --Sd~, k)+S,(~o, k).

(22)

According to the requirement expressed b y the dispersion relation as well as b y the equation of Chew and Low, S(oJ, k) and $2(o~, k) should be continuous as k passes from one Riemann sheet of the complex k-plane to the other through the cut between k = i and k = ioo. Let k = iK--e, where K is real and positive and greater than unity; eqs. (22) then become

SI (iK--e) + 2S~(iK--e ) = SI (iK + e) + 2S2(iK + e ) SI (iK--e)-- S~(iK--e ) = --SI (iK + e) + S~(iK + e )

(23)

(1 =< K < oo) where we have omitted the variable ¢o ~ ~v/l+k ~ since all functions in (23) are given in the upper Riemann sheet of the complex k-plane. Our problem now reduces to finding two functions Sl(k ) and S,(k) defined in the upper Riemann sheet of the complex k-plane and satisfying the conditions (3) and (4), and whose boundary values along the cut on the positive imaginary axis satisfy (23). After such Sl(k ) and S2(k ) are obtained, they can immediately be continued analytically to the lower Riemann sheet of the complex k-plane through the cut between k = i and k = i oo to get the complete functions SI(o, k) and S~(o, k). We can easily see that the last two functions satisfy (22). If S(k) have no poles above the real axis except on the imaginary axis, then Sl(m, k) and $2(o~, k) will have the same t I t will be n o t e d t h a t in o r d e r to s a t i s f y t h e e q u a t i o n of C he w a n d Low, S(eo, k) m u s t s a t i s f y c e r t a i n c o n d i t i o n s a t eo = 0 a n d eo = oo. H o w e v e r , t h e s e c o n d i t i o n s o n l y g i v e ri s e t o t w o r e l a t i o n s a m o n g t h e i n f i n i t e n u m b e r of a d j u s t a b l e c o n s t a n t s .

SCATTERING

FUNCTIONS

WITH

CROSSING SYMMETRY

property in the lower Riemann sheet. In other words $1(o~, k) and $2(co, k) satisfy Chew and Low's equation. It should be pointed out, however, that except for the trivial and unphysical case $1(~o, k) ~ $2(~o, k), the condition (23) cannot be satisfied also b y values of k on the lower cut between k--- --i and k : - - i oo. This can be seen b y noticing that

S~(--iK--e) =

1

S.(iK +,)

Eqs. (23) would then become, for k = iK--e, 1

Sl(ig--e ) 1

S~(iK--e)

+

2

1

S2(iK--e )

Sx(ig+e)

1

--1

+

2

S2(iK+e ) 1

(24)

S2(iK--, ) -- S~(iK+e) + Sz(iK+e) (1 ~ K < oo).

The above relations cannot be satisfied simultaneously with (23) except in the unphysical case $1(o~, k) -----S, (eo, k). This means that S~,(w, k) must change discontinuously as k passes through the cut on the negative imaginary axis from one Riemann sheet to the other. We see further that S~(o), k) are unitary only on the real axis in the upper Riemann sheet. In the lower' Riemann sheet, as can be seen from (22), they become linear combinations of two unitary functions (see (24) below) and therefore cannot be unitary themselves. This m a y be understood as a consequence of the presence of the line of discontinuity. The non-unitarity of S(o~, k) for negative values of o~ can also be seen physically from the following consideration. From (22) we obtain S , ( - - ~ , k ) = ~Sl(~, k ) + ½ S d ~ , k). The right-hand side of (24) is just the scattering function of direct scattering of a negative meson b y a proton. This scattering function is obviously not unitary since the same collision also leads to charge-exchange scattering with a neutral meson as an outgoing particle. This consideration together with (18) also brings out a connection between negative values of eo and charge conjugation. We have also considered the case of F-wave meson-nucleon scattering under symmetrical pseudo-vector coupling. There the crossing relations m a y be written

S i(iK--e)+2Sa(iK--e) = Sx(iK+e)+2Sa(iK+e) S~(iK--e)+ ½Sa(iK--e ) = S,(iK +e)+~Sa(iK +e ) Sx (iK--e) + Sz(iK--e ) - 2Sa(iK . e ) = --Sx (iK +~ ) - Sz(iK + e) + 2Sa(iK + e) (1 ___K < co).

8

NING HU

We found t h a t the scattering functions are of the same t y p e as those in the last case.

4. Effective Range E x p a n s i o n of P h a s e Shifts F r o m the results of the last section we see t h a t it will be convenient to e x p a n d S~(~o, k) into a power series with respect to the branch point k : i in the following form:

k)=

A':'

8':'

~=0

= k--i:

(28)

n=*0

or, alternatively, we m a y use one of the following series t: oo

k)/k*'+' = X n-----0

n----0

(27) n*=0

n.=0

(l = angular m o m e n t u m variable), where A~~), B~~1, a~~), b~~1, ~ 1 and /,~) are complex constants. Owing to the presence of the line of discontinuity from k - - - - - - i to k : - - i oo, the circle of convergence of the above series will be I~l - - 2 , if preceding functions contain no poles inside this circle. In the case when there are poles, the same circle of convergence m a y still be obtained b y a suitable choice among (27) and (28), since k z~+l cot ~ and ~/k az+x do not have poles at the same position. We notice t h a t the circle of convergence includes a portion of the real k-axis from k = -- ~/3 to k = + ~/3. The corresponding energy interval is --2 < oJ < + 2 which just corresponds to the lower limit where the effect of meson production would set in. Therefore the circle of convergence actually includes the whole interval of the real k-axis in which the one-meson approximation is strictly valid. F r o m results of last section, we see t h a t the expressions (27) and (28) must satisfy the following requirements: i) t h e y m u s t be real only on the real axis in the upper R i e m a n n sheet of the complex k-plane, b u t not on the real axis in the lower R i e m a n n sheet; if) t h e y m u s t satisfy b o u n d a r y conditions such as (24) and (25) along the cut on positive i m a g i n a r y axis from ~ : 0 to ~ = 2i (i.e., from k ---- i to k =

3i);

iii) t h e y m u s t be odd functions of k on the real axis in the upper R i e m a n n sheet of the complex k-plane (this is a consequence of (3)); T h e c o n n e c t i o n b e t w e e n S a n d ~ is S = e x p 2/~ = (cot~+i)/(cotcS--i). T h e u n i t a r y condition for S requires that ~ and cot ~ must be real on the real k-axis in the upper Riemann sheet.

S C A T T E R I N G F U N C T I O N S W I T H CROSSING S Y M M E T R Y

9

iv) their values in the interval 1 ~ IwI ~ 2 must agree with their observed values from the scattering experiments. Our expansion m a y be considered as an improvement over the usual effective range expansion k8 cot ~(o, k) =

a+bw+cw2+

• • •

(29)

which also uses k = i as the centre of expansion. However(29) inevitably leads to the result t h a t S~(w, k) must be unitary on the real axes of both Riemann sheets of the complex k-plane, which is in contradiction with the crossing condition. We believe t h a t our series (27) or (28) form a better basis for extrapolating the experimental values for w > 1 to w = 0 to obtain the value of the interaction constant g.

5. Metastable States of the K-Capture Type As we have stated in the introduction, the dispersion relation requires t h a t S(w, k) should have no poles above the real axis in the upper Riemann sheet of complex w-plane in which the imaginary part of k is positive, or in other words, S(w, k) should have no poles above the real axis in both Riemann sheets of the complex k-plane except on the imaginary axis between k = 0 and k = i. In this section we shall show t h a t this requirement is violated by the very existence of ze-mesic atoms. For the sake of simplicity, we shall first assume t h a t the meson interacts with the proton only through a Coulomb field. Then there will exist a series of bound states between the negative meson and the proton. These bound atomic states will be represented b y poles of S(w, k) on the real axis between w ---- --1 and w = + 1 in the upper Riemann sheet of the complex w-plane. Now let us switch on the nuclear interaction between the meson and the proton. It has been shown by Deser e l a l . ~) that the effect of the nuclear interaction is to displace these poles by complex values Aw = ~+i/~. From conditions (3) and (4) it can easily be seen t h a t the complex conjugate displacement Aw* = ~--i~ must give rise to another pole. These two poles will correspond respectively to metastable states of K-capture and K-emission types 8). This means t h a t the requirement imposed by the dispersion relation as stated above is now violated. If we look into the cause of the failure of the dispersion relation, we shall find t h a t it is closely connected with the fact t h a t in the derivation of this relation, the interaction of meson and nucleon with other fields was neglected. Indeed, the proton could never capture a meson from the K-orbit if it were not allowed to emit y-rays or other lighter particles, and in the above example the atomic orbits of mesons would not be there at all if the Coulomb field were neglected.

10

NING H U

From the above consideration we m a y conclude that in order to decide whether S(~o, k) has poles above the real axis in the upper sheet of the complex w-plane, one must take into account interactions of mesons and nucleons with other fields. As the usual dispersion relation was derived b y considerations which only take into account the interaction between mesons and nucleons, it cannot say anything about poles which represent metastable states of the K-capture type. The displacements of levels A~o, mentioned above, are actually very small quantities. One m a y think, perhaps not without justification, that the dispersion relation is after all still correct when these displacements can be neglected. To this we wish only to add that our main objection is to the implication that poles representing metastable states of the K-capture type violate the causality principle. References 1) M. L. Goldberger, Phys. Rev. 97 (1955) 508; 99 (1955) 979; F. J. Dyson, Phys. Rev. 100 (1955) 344; A. Klein, Phys. Rev. 104 (1956) 1131; R. Oehme, Phys. Rev. 100 (1955) 1503; 102 (1956) 1174 2) G. F. Chew a n d F. E. Low, Phys. Rev. 101 (1956) 1570 3) N. Hu, Phys. Rev. 74 (1948) 131 4) R. Jost and W. Kohn, Phys. Rev. 87 (1952) 977; Mat. Fys. Medd. Dan. Vid. Selsk. 27 (1953) no. 9 5) L. CastiUejo, R. H. Dalitz and F. J. Dyson, Phys. Rev. 101 (1956) 453 6) G. Feldman and P. T. Matthews, Phys. Rev. 101 (1956) 1212 7) S. Deser, M. L. Goldberger, K. B a u m a n n a n d W. Thirring, Phys. Rev. 9b (1954) 774 8) W. Heitler and N. Hu, Nature 159 (1947) 776