Coulomb corrections in non-relativistic scattering

Nuclear Physics B60 (1973) 443 .477. North-tlolland Publishing Company

COULOMB CORRECTIONS

IN N O N - R E L A T I V I S T I C S C A T T E R I N G

J. ttAM1LTON and I. OVERBO Nordita, Copenhagen B. TROMBORG Niels Bohr Institute, Copenhagen Received 16 May 1973 Abstract: Coulomb scattering is put in dispersion theoretic form by using a small photon mass h. By carefully separating out the In ?, terms a dispersion relation for the Coulomb corrections in hadronic scattering is set up and solved. This is done for repulsive and attractive Coulomb interactions and for a two-channel problem. The techniques are designed so that they can be extended to the relativistic problem.

1. Introduction In order to find pure hadronic scattering amplitudes from experimental data it is necessary to make the Coulomb corrections. Thus the measured 'nuclear' phase shifts in pp scattering or in 7r+p scattering are not identical with the pure hadronic phase shifts which we would have if there were no Coulomb interaction. The difference arises because the phase shifts are non-linear functions of the interactions which produce them. Moreover the determination of the non-linear terms coming from the mixing of the hadronic and electromagnetic interactions is not trivial. Dispersion theory is the only precise method for dealing with hadronic interactions at low and medium energies at present. That makes it necessary in order to solve our problem to put the electromagnetic interactions in dispersion relation form, and this is not easy to do on account of the vanishing mass of the photon which causes essential singularities in the scattering amplitudes. We are obliged to work with photons having a small mass ~,, corresponding to Yukawa interactions of very long range X- 1 ; at the end of the calculation we let ?, go to zero. This limiting procedure can be tricky and to carry it out requires a fairly detailed knowledge of the structure of the theory. In the present paper we try out the use of dispersion relations for electromagnetic phenomena and the techniques for letting ), vanish, by calculating the Coulomb corrections in non-relativistic potential scattering. The difference from most other treatments of Coulomb corrections is in the use of dispersion theory methods throughout: at the end we have an easy-to-solve dispersion relation for the Coulomb

444

J. Itamilton et al., Coulomb correction,

correction itself (for quantum mechanical treatments see refs. [ll and for dispersion theory methods see refs. [2, 31). The low energy pp scattering formula [4] of Landau and Smorodinsky is a simple example of the method. Also in the attractive Coulomb case (eg. 7r- p) the effects of the bound states are readily included. The same technique makes it possible to solve a two-channel problem of the type n - p -+ n - p, n - p -+ n°n, n°n ~ n°n. A model calculation is given to demonstrate the usefulness of several approximations to the Coulomb corrections. In a later paper similar methods will be used for relativistic meson-baryon scattering where we have to deal with the infrared radiation phenomena and with measurement problems before we can determine the Coulomb or electromagnetic corrections. 1. I. Definition o f the C o u k ) m b correction

In this paper we only consider non-relativistic potential scattering. Suppose that a pair of particles scatter each other in a potential

V(r) = VH(r ) + o~/r ;

(I)

VH(r ) is a short range hadronic potential and a/r (a = ~ )i is the Coulomb repulsion. The scattering amplitude can be written (see for example chap. 11 of ref. [5])

fc(0)+f (0),

(2)

where f c ( O ) = - 5¢ [20 sin21012 - 1 exp [ - i 7 In (sin 2 ½0) + 2io0]

(2a)

is the pure Coulomb amplitude, and 2i61 f ' ( 0 ) = ~ (21+ 1) e 2i°l e 2 i q -- 1 P l ( c ° s O ) " 1=0

(2b)

Here q, O are the momentum and the scattering angle in the c.m.s., and o I is the pure Coulomb phase shift, ol = a r g p ( l +

1+i7),

(3a)

with 7 = ma/q = l/(qaB),

(3b)

where m is the reduced mass and a B is the Bohr radius. In eq. (2b), 61 is the hadronic phase in the presence of the Coulomb interaction, and 61 - 0 for VH = 0. We can determine 61 from the measured differential cross section do" : ifc(O) + f , ( O ) l 2 dr/

J. ttarnilton et aL, Cbulomb corrections

445

However 61 is not identical with the pure hadronic phase shift (6H) / which is produced by the potential VH(r) alone. The difference

A/=61- (6H)I

(4)

is the Coulomb correction, and it is the object of this paper to calculate it by dispersion methods. We shall first treat the Coulomb repulsive case and later discuss the modifications required in the attractive case.

2. Gorshkov's theorem and the function

Y(s)

The analytic structure of scattering amplitudes is well known for finite range potentials. Thus instead of eq. (1) we use e

- hr

(5)

V(r) = Vii(r) + a -.-r

where a exp ( - k r ) / r is a repulsive Yukawa potential arising from the exchange of photons of mass X. We shall use the following partial-wave amplitudes (p.w.a.) each with orbital angular momentum I: (fH)/, the strong interaction amplitude derived from Vit(r); 0'(x)l, the amplitude derived from a exp ( - M ) / r : (J~:llx)/, the amplitude derived from V ( r ) [eq. (5)]. We also use the 'pure Coulomb' amplitude 2io I

OCC)1 _ e -- 1 • 2iq '

for0~q-<~

.

(We shall mostly omit the index/). These amplitudes are functions of q, or of s=q

2 .

It has been shown by Gorshkov and by others [6] that

(6)

1 + 2 i q f c H x = exp [--iy ln(4q2/X2+ 1) + 2 & C ] [l + 2/q f + O(M] ,

(C = Euler's constant = 0 . 5 7 7 . . . ) where on the physical cut 0 ~< s ~
f ( s +) - e 2iq

-1

(7)

Also we have

f=fC +f'

'

(7a)

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J. ltamilton et aL, 6bulomb corrections

where 2io I e 2 i h l -

1" (s +) = e

1

(0-.< s < oo).

2iq

(7b)

(Eq. (6) is in agreement with the conjecture of Dalitz [7] that the In ?, divergence can be factored out). 2.1. H o l o m o r p h y regions

The hadronic amplitude f n is holomorphic in the q-plane cut along ia <<.q <<.i oo, - i °° <<.q <<.- ia. i lere a is of the order of the mass of the lightest hadron which can be exchanged between the scattered particles. The first of these cuts is on the physical s-sheet (Im q ) 0) while the other is on the second s-sheet (reached by passing through the physical cut in the s-plane). The amplitude fH can have resonance poles on the second s-sheet and bound state poles on the physical sheet. We assume for simplicity that)'ii has no bound state poles. The amplitudesfcit x a n d f c ~ are holomorphic in the q-plane cut along ½ t~ <~q <~ i oo, _ ioo <~q ~<- 2-t?,. 1. These cuts reach close to q = 0 because of the long range X- 1 of the interaction c~ exp ( - L r ) / r . llowever fcm,-fox only has cuts along ia <~q <<.i oo, - i oo <~q <~ _ iI iX, because the longer range parts of the potentials giving these p.w.a, are identical. By eqs. (6), (7a) fCIIX -Y('x = exp [ - i v In (4q2/X 2 + 1) + 2 i v C ] [ [ ' + O(?,)1 •

(8)

The function In (4q2/X 2 + 1) is defined in the q-plane cut "along ½ iX <~ q ~< i oo, i oo <~ q ~<_ ~ iX so that it is real and positive for 0 < s < oo on the physical s-sheet. Thus the factor exp [ -/3, In (4q2/X 2 + 1)]

(9)

is regular in the same cut q-plane and its phases on the cuts are shown in fig. I. This factor has been chosen to be real analytic in the s-plane. Since-fcII x - (Ca is regular on ~ iX <~q <~ ia the term ( f ' + O (X)) in eq. (8) must have a cut along ~ iX <~q <~ ia to compensate the cut in eq. ('9). We can assume in dealing with p.w.a.'s that as X ~ 0 this compensating cut is solely i n ( ' . (The assumption is justified by the agreement with the result of Cornille and Martin [8] in eq. (17) below. See also re(. [9]). So we have the result that the amplitude f defined by

f (q)

=

f ' e n~

for

Re q ~, 0 ,

f,

for

Re q < 0

e ~

(lO)

J. ltamilton et aL, Coulomb corrections

447

e - ~T e ~T q-plane

liX/2 ............

+ ............ 0

e~T~ e -~T I"~. 1. Phases of exp [-i~' In (4q2/h 2 + l)l in tile q-plane cut along (21-ih, i o~) and (- i oo ~ix). is holomorphic in the q-plane cut along ia ~
2.2. The fwwtion Y(s) On the physical s h e e t f can also be defined by f (s)= lira exp [--. X~0 where l_; Y (S) = - a B T r

Yfs)-2iTC](fCH x - ] C x ) ,

(10a)

In(4sl/~ 2) d s ' . X / ~ S ' - s)

(10b)

o

The function Y(s) is regular in the s-plane cut along 0 ~< s appendix I of ref. [10]), Y(s+)=--rrT-iTln(4q2/~2),

f o r 0 < s ~ < ,,o.

~< oo

and (see for example

(10c)

It is obvious from eqs. (lOa), (lOb) that the function.f is regular in the s-plane cut "along - oo ~
J. Hamilton et aL, Coulomb co~Tections

448

3. The effective function Fl(S) We wish to derive an amplitude which is given by the phase 61 (and not ol) on the physical cut, and in addition has useful analytic properties. Eqs. (7b) and (10) show that on the physical cut e

- 2io I

f(s+)

-

e

2i~ 1

- 1

2iq

en'r,

(O~
(11)

However the factor exp ( - 2 i o t ) spoils the useful analytic properties o f f ( s ) . In the repulsive Coulomb case exp ( - 2 i o l ) has zeros at i7 = - (n +/), (n = 1, 2 . . . . ); these are on the second s-sheet and they have an accumulation point at s = 0. Also exp ( - 2 i o l ) has poles at i7 = + (n +/), (n = 1, 2 . . . . ) which are on the physical s-sheet. These poles are removed by multiplying the function in eq. (1 1) by (l!) 1 [ r (l + 1 + iv) r ( t + I - i7)] - 1 This is chosen to be real analytic in s and it tends to unity as s -~ oo (3' -~ 0). It has zeros at i3' = -+ (n +/), (n = 1, 2 . . . . ) and it is regular except at s = O. Also, we multiply by q - 21 to get a reduced partial-wave amplitude; this will not introduce any poles at the threshold. The resulting function

Fl(S )

=~f

e-2iOl(l!)2 q-21

[

r ( t + 1)

[1-'(1 +

=f Lq'r(i~-i +i~)

1 + i3') 1-'(1+ 1 - i 3 ' ) ] - I

] 2

(12)

(12a)

is regular and real analytic on the physical sheet of the s-plane cut along -

o o < ~ s < - a 2, O<~s<~oo. On the physical cut

e Fl(S+)-

2i6 I

-1

2/q 21+1 ~p(s),

(13)

(0
where 2n~t ~o(s) - e

- 1 2~'7 2rr'~

~O(s) - e

forl=O,

(13a)

1

- 1 [--[ (1 + ~ 2 / m 2 ) - 1 2"rt3' m=1

for 1 > O.

( 13 b)

£ ltamilton et aL, Coulomb ~orrections

449

In deriving these equation we have used n"y 7) P ( / + l +iT) r ( / + 1 - - i 7 ) - sinh(n"

1 1]

(m 2 +7 2 ") "

(13c)

m = 1

We call Fl(S) the effective function. It plays a basic role in describing the hadronic interaction in the presence of Coulomb interactions.

3.1. The effective range equation (S-wave} Consider now an S-wave. The factor following f on the right of eq. (12a) has no zeros or poles on the physical sheet of the s-plane. Thus we expect that (apart from possible zeros in Fo(s ) due to the hadronic interaction) the function F~ ! (s) is regular and real analytic on the physical sheet of the s-plane cut along - oo ~< s ~< --a 2, 0 ~< s ~< ,~. On the physical cut F . o l ( s + ) _ 27r c o t S - i a B e 2~7 -- 1 '

(0
(14)

The function h(s)=---2 f

.d_dt_

__

(15)

"o (t2-'r2)(e 2,'t- I) is regular in the physical s-plane cut along 0 ~< s ~< ~ . On the cut Re h(s) = Re ff (i3') - In 7 (0
I m h (s +) -


(16)

r r e 2rr't - 1

where

(z) =d7 In r (z). The expression for Re h (s) is given by Binet's result [ 1 1 ] for in F (z). Eqs. (14) and (16) show that F~ 1 (s) and ( - 2 / a B ) h(s) have the same discontinuity across 0 ~< s < oo. Therefore the real analytic function

K (s) = F O 1 (s) + (2]a B) h (s)

(17)

is holomorphic in the s-plane cut along - oo ~< s ~< - a 2. It might have isolated poles due to zeros o f F 0 ( s ) in the cut s-plane. However, it must be regular in a neighbourhood o f s = 0. This has also been proved by Cornille and Martin [8]. The effective range relation comes from the real part of eq. (17):

J. Ilamilton et at, Coulomb corrections

450

g (3') + y - c o t 8

_

1

e2n't i-- - ~ aB K (s), 1

(0 ~< s ~< oo),

(18)

where g (3') -= Re h (s) = Re ~k (i7) -- In 3'.

(18a)

The function K(s) has a useful Taylor expansion about s = 0. Eq. (18) is the formula of Landau and Smorodinsky [4] which was originally given for low-epergy proton-proton scattering (see also ref. [! 2] for a dispersion theoretic application of eq. (17)). We notice that Re ~ (i3') = 7 2

~

1

C.

(19a)

m= 1 m(m 2 + 72) Also, for large 3' (3' >> 1), g (3') _

1 + O (3'-4). 123,2

(19b)

At very low energies 3' is large and positive. We shall call the energy range where 3' ~> 1, the atomic energy region, in zrN and pp scattering 13'1= 1 for q = 0.89 MeV/e and 3.4 MeV/e, respectively. Eq. (18) shows that as the energy decreases, the effective hadronic phase/5 goes to zero like exp (-- 2n3'). This behaviour arises from the Coulomb repulsion which strongly suppresses the wave function when the two particles are at hadronic distances from each other. The pres'~,~ce of the exponential factor exp (2n3') in eq. (18) shows that the amplitude/~0 (s) has an essential singularity at s = 0. However, it follows from eq. (17) that F 0 (s) is bounded on the physical sheet in a neighbourhood of s = 0. On tile physical cut,

F o ( S + ) . [ K. ( s ).- "~q2 . a. B

21ri

e-2n/qaB]-I

ass~0.

(19c)

aB On the second ;heet F 0 (s) has the cut - i oo ~
3.2. Effective range relation fi~r other partial waves For I > 0, F]- 1(s) is also holomorphic in the cut s-plane, and on the physical cut

F! (s) is given by eq. (13). Therefore the function

J. tlamilton et aL, Coulomb corrections

451

I 2 q2l [ I ( 1 +3'2/m2) h(s) aB m =1 has tile same discontinuity across the physical cut as F/-- I

(s). ltence the function

l

KI(S ) = F i l ( s ) + - 2 - q 21 [7 (1 + T 2 / m 2 ) h ( s ) aB

(20)

m= i

is real analytic and holomorphic in the physical s-plane cut along - oo ~< s ~< - a 2, and it is regular in a neighbourhood of s = 0. On taking the real part o f e q . (20) we find the effective range relation forl~> I [13]

lI

m=l

2 +

.cot l

;1 a

g(3') + . . . . . . e 2Try. lJ

g' a B K l ( s ) ,

(21)

with g(3') given by eq. (18a). if we let ~-+ 0 then a B -+ ~ , and for fixed q > 0, 3 ' ~ 0. By eq. (19a) we see that eq. (21) then gives q21+l cot 6 H = l i m K l ( s ) ,

(21 a)

0~ - + 0

which is the usual form for the hadronic effective range equation.

4. Dispersion relation for the Coulomb correction We can set up a dispersion relation for the Coulomb correction A [eq. use the reduced pure hadronic p.w.a. (FII)! given by

(4)[.

We

(FH(S))I = q-21 fll (s), and on the physical cut FH(s+ ) -

e

2i8 H

-! 2£/21+1

,

(O
The function F ( s ) - Flt(S ), where F(s) is given by eq. (12), is a natural function to use since IF(s) -- FH(S)I ~ 0 as ct -+ 0. We write

F(s) - Fll(S) H(s) -

A (s)--

'

(22)

where A (s) is regular except for the cut 0 ~< s ~< oo and on this cut A (s) = IA(s)l e 2/5 H. We w*ite

452

J. Hamilton et aL, Coulomb corrections

A(s) =exp

(s - :~ ) . 0

(-s;- s-)(s'--s)

'

(22a)

where ~- is real and ,~- < 0. Our results are independent of g-; this will be seen from the fact that they depend on the ratio o f A (s) at different values ofs. The functions F ( s ) and Fll(s) are both regular in the physical s-plane cut along _ oo ~< s ~< a 2, 0 ~< s < oo. Thus t f ( s ) is regular in the s-plane cut along _ oo <~s <, -. a 2, 0 <. s <~oo. On the left-hand cut we write hnH(s+)=#(s)/A(s),

(_oo~
(23)

On the physical cut 0 ~< s ~< oo, H(s+)=

[(~p- 1) sinSil e •

1 q21+ 1 IA(s)l

i811

+ ¢sinA

eiA]

(24)

where ~0 is given by eqs. (13a), (13b). The essential singularity in F(s) at s = 0 gives an essential singularity in H(s) at s = 0. However, IF(s)l is bounded as s ~ 0 on the physical sheet (except for the special case that the hadronic interaction is such that K(0) = 0), so H(s) is also bounded as s ~ 0. By eqs. (19c) and (20) Im F ( s +) = O (exp (-27rT)) as s ~ 0 along the physical cut. So by eq. (22) f

lmH(s+)

~ (all- 2all)a H q2l+l/A(O)

for atomic energies. Here arl is the scattering length of the pure hadronic amplitude fH(s) and a'lt = F ( 0 ) = ( K ( 0 ) ) - I. Also, Re H ( 0 ) = (a'lt

all)/A(0 ) . i

From eq. (21a) it follows that a H - a H goes to zero ifo~ vanishes. Schwinger [ 14] has given a simple wave mechanical formula for calculating (a{l)- I _ (all)- 1 for S-waves. Thus on the physical sheet the essential singularity is harmless and H(s) satifies the simple dispersion relation _a

H(s)=lzr f Im/-/(s'+)d,'+ 1f s' - s ~0

_

2

oo

__;(s')ds' A ( s ' ) ( s ' s)

+E(s),

(25)

where E (s) is regular in the whole plane, and we assume it is a real polynomial. We also assume that no subtractions are needed in eq. (25). For 0 ~< s ~< oo eq. (24) gives

J. ltamilton et al. Coulomb corrections

453

"sin 2A = (so- l _ i) sin 26 H + 2 9 - 1 q2/+l IA (s)l Re H ( s ) .

(26)

4. I. Asymptotic behavhmr We shall examine the a s y m p t o t i c form of the various terms in eq. (26) as s ~ oo. By eqs. (13) SO(s) = I + zr7 + O (3,2) as s ~ oo.

(27a)

We assume that

[VH(r)ldr
and

Vll(r)=O(r-2+c)

asr~0.

C

where c and e are positive constants. Then, if there are no bound states [ 15], 611 (s) = O ( q - e ) + O ( l n q / q ) as s ~ oo.

(27b)

This implies (el. appendix 1 of ref. [101) that IA(s)l tends to a constant as Isl-+ co. The correction A would vanish if the phase shifts were linear functionals of the potentials. The Born a p p r o x i m a t i o n is good at high energies, so here the phase shifts are a p p r o x i m a t e l y linear in the potentials. Therefore we expect that A will go to zero faster than the larger of 611 and o I as s ~ o o . This can be seen explicitly from the a s y m p t o t i c expression for A given by Auvil [ 1], In addition one can show that q A(s)-+0

as s ~ °° .

(27c)

F r o m eqs. (26) and (27) we get

d +1 R e H ( s ) - + 0

(28)

as s - . o o

By eq. (24) lm H(s+) = -

1

q2l+ 1 IA(s)[

[SOsin 2 A -- (SO -- 1) sin 2 611 ]

(0 ~< s ~< oo ) .

(29)

Thus, as s ~ oo, I

[ - n T s i n 2 6H + sin2 A] , lm H(s+) ~ q21~- - 1 [A(s)/

(29a)

and by eqs. (27)

S/+ l

lm

H(s+) = O(s e) + O(s- ½) as

s ~

oo,

(29b)

where e is defined above. We shall also assume that fl+lp(s)=O(isl where e I > 0 .

Ej)

as s ~ - o o ,

(29c)

£ Hamilton et aL, Coulomb corrections

454

By

the same method as is used for deriving unitary sum rules [161 it now follows from eqs. (25), (28), (29b), (29c) that E (s) - O,

and __a 2

A(s) s~'ds + --~

,

hnH(s+)ds = 0

(30)

0

for n = 0, 1. . . . .

1. These sum rules are useful in the practical evaluation of A.

5. Evaluation and properties of A (s) If we write L ( s ) = q21+l so 1 ]A(s)I , then eqs. (25), (26), (29) give the integral equation for A: sin2A=(~;l

l)sin281!

+~L(s) l'; sin2A+(~-l-l)sin 261! 0

LC)C

s)

(31)

2

+ 2_TrL (s) f a

ds'

; (s')~ A~r~s)(s,

d s'

= I 1 + 12 + 13 • (The integral in 12 is a principal value). We assume that the left-hand cut term O(s) (eq. (23)) in 13 is known (cf. sect. 6). If 6ti(s ) (0 ~< s ~< oo) is known then eq. (31) gives A (s). In practice 8(s) (eq. (4)) is determined by the experiments and A(s) is then found by iteration in eq. (31); in the first step 6(s) is inserted in place of 6H(S ) and interation is continued till a stable solution A is obtained. Eq. (31) is non-linear since 12 contains A, but this does not cause practical difficulties. Except at atomic energies (i.e. 7 ~ 1), A(s) = O(171). Therefore in Im H (s+) (eq. (29)) the term ~0 sin 2 A can be ignored compared with (~0 - i) sin28H except at atomic energies where these two terms tend to cancel. Because ~(s) = O(e 2~r7) for 7 ~ 1, it follows from eq. (31) that =~-I arc sinl I ,

with

11 =(q9 -1 - 1 ) sin 26 H ,

(32)

£ llamilton et at, Coulomb corrections

455

is a good approximation to A in the atomic energy range. At higher energies A is O(1~'1). Thus a good approximation to A(s) is obtained by using A for A in 12 in eq. (31). Practical calculations show that further iterations do not alter A(s) noticeably (cf. sect. 7). If the hadronic potential Vil(r) is weak so that Isin 51fl ,¢ 1 for 0 ~
R (s) = (FH(s))2/A(s)

(33a)

is regular in the s-plane cut along - oo ~< s ~< - a 2, and we expect R(s) to vary smoothly in the physical region. Also

R(s) =

sin 2 611 for 0 -G
s 21-~1 IA(s)I

(33b)

so for large s, R (s) decreases at least as fast as s- 21- 1. Near a resonance IA(s)l will then vary like sin 2 8ti. Also, for 7 '~ 1, by eq. (20)

I m H(s+)

~ .

1r7 sin2 811 .

.

.

q2l+ 1 IA(s)l

7r s/R (s)

,

(33c)

aB

so lmH(s+) is negative and slowly varying in the physical region above atomic energies. It then follows from eq. (31) that 1121 and 1131will have maxima near the resonance position s R (arising from the maximum in ]A(s)l) and this will in general produce an extremum in A(s) near the resonance position.

5.1. Sauter's equation Sauter's method [3] (see also ref. [2]) is somewhat similar to ours although its range of usefulness is strictly limited. We show the connection between his equation (eq. (5) ref. [3a]) and our result. Consider the function

M(s) = 1

7r, aBs~

/711(s) A(s) '

where s~ is defined in the s-plane cut along -- oo ~< s ~< 0 so that it is real and positive l on 0 ~< s ~< oo. Just above the cut - oo ~< s ~< 0, s~ = ils~ I. The dispersion relation for M(s) is

J. Hamilton et al-, Coulomb corrections

456

ReM(s)=lPf

I ,mM(s'_) ds' + . . 1

7r

o

s-s

%

j~ _~

Im[FH(S')(s')-[}ds' A(s')(s'

(34)

s)

On the physical cut 0 ~< s ~< oo sin 251[ Re M (s) = ½lr7

(34a)

q2l÷ 1 [A(s)l

..nTsin2~H lm M (s+) -

(34b)

q21+ 1 Im(s)l

On the physical region where 7 "~ 1, eqs. (33c), (34b) give ImM(s+) ~ ImH(s+), and eqs. (26), (27a), (34a) give sin 2A Rc (n(s) --M(s)) ~ ....... 2q 21+ 1 IA(s)[ We use the dispersion relation for ( H ( s ) - M(s)), and for 3' '~ 1 we can to a good approximation ignore the dispersion integral over the region of atomic energies. Then we have, for 3' "~ I, sin2A~2q2i+llA(s)l

_~

X ( ~

a B _~

A(s')(s'-s)

" ds

.

(35) This is Sauter's equation. The difficulty is that to use eq. (35) we must know R e F H (s) on the unphysical region - oo ~< s ~< 0, and in general Re Fil(S) is not well known on that region except near s = 0. Thus in practice eq, (35) can only be used in a small energy band above the atomic energy range.

6. The left-hand discontinuity of H(s) The effective function F(s) (eq. (12)) does not depend on X so the function H(s) (eq. (22)) has no In X divergence. The same must be true for the left-hand cut discontinuity lm H ( s ) ( - oo < s <<,- a2), and this shows that there are interesting cancellations between graphs involving hadronic plus Coulomb interaction and pure hadronic graphs. In relativistic theory the analogous situation is that all the infrared

J. Hamilton et aL, Coulomb corrections

II

457

11111

photcm line

Fig. 2. The graphs which give the left-hand cut terms to order ~. The Born term in fC~ is)d~.h) divergences cancel in the combination of hadronic and photon exchange processes which give the left-hand cut of H(s). By eqs. (8), (10), (12)

F(s) = lim h~ 0

Hx-fch) "

sinh 7r3' 7r7

'(

,),

1] q2+ m =1 m2 a 2

(36) × exp[2h, lln(21ql/k)-- 2i(3"C + Ol)]l /

for -- oo ~< s ~< 0. Here q = ilql, so

3" = I/(qa B) =-il3'1. The In k term is cancelled by a similar term in (fCHX - fcx)" In eq. (36) the factor multiplying (fCHX2 - / ' C a ) is real, and the only left-hand singularity of F(s) is the cut - ~ ~< s ~< - a coming from the cut in (fCHX - fcx)" On this cut 13'1is small, and we expand in 13'1 using

3"C+Ol=3"

I ~ 1 +O(13'13). m=l m

Thus lmF(s+) = h~01im{Im(fcHx-fcx)q-2lexpI213",(ln

2~__~1__m=, ~ 1)]

+O(i.).12)} ' (36a)

for - oo ~< s ~< - a 2. Also

where fH'r is the mixed hadronic-photon amplitude of order a. The contributing graphs are shown in fig. 2. By eqs. (22), (23)

458

J. llamilton et al., Coulomb corrections

(t)

.(2)

(a)

fill

(b)

---F-7 + - F - 7 Icl

f(21 =

+

(d) )( I'~dron line photon line

Fig. 3. The graphs which give the various left-hand cut terms lnl~l) ..... lm~l~) u,~d in sect. 7.

p(s)=lim

q-2t

171 In ~ - -

x~0

I m f l t + I m f H . r+O(13,12

,

m= I

(_ oo~
(36b)

The cancellation means that l m f H . Y - 21"/I InN I m f H is independent of In h. In a model calculation we now show how I m f H . r is found on the nearby left-hand cut, and how the cancellation comes about.

7. The model calculations of S-and P-wave corrections We consider first an S-wave with repulsive Coulomb effect and the hadronic potenti',d of eq. (1) given by VH(r) = G e - u r / r ,

(37)

where/a- 1 is of hadronic range. The hadronic amplitude fH has the left-hand cut - oo~
lmil )(s+)_

2s '

( - oo~
(38a)

The discontinuity I m rJ(Hl )? comes from the graphs of order a G (Fig. 3c). It is evaluated in appendix A. For k ,~/z,

J. Hamilton et aL, Coulomb corrections

459

(38b) Therefore Im H(s+) = 171 - s -

A(s)-

, In (2. . . . . 1)1

+ O (17[2) , ( - ~2 ~< s ~< --~,u 2) . (38c)

On ~, ~< s ~< _ ~ 2 we also have contributions from the graphs of order ~G 2 (fig. 3d). The method of Martin [17] gives im f~?)(s+) = - - 7rm2G2- In (2lql.~__ 1), Iql 3

(-~

s ~<--/,12) .

(30)

On evaluating Imr(2) ~1t~. we find that 2,71 In (-2--I) lm f(H2) + hn r~(2) -"H7 does not diverge as ), ~ 0. This gives Im H(2)(s+) on - ~ ~< s ~< - ~2. The result is in appendix A; there we "also give the corresponding P-wave results for hn ft~), l m H (1). I m / ~ ) and lm/_/2).

Z 1. Example o f the dispersion theoretic calculations The hadronic interaction in eq. (37) is used with the reduced mass m = 100 MeV and/a = 197.3 MeV. Thus 7 = 1 forq = 0.73 MeV/c. We consider the cases/= 0 and l = 1 and use G = - 1 for the S-wave and G = - 7 . 5 for the P-wave. (The P-wave coupling constant is chosen large so as to give a resonance). In figs. 4 and 5 we show the results of a Schrodinger theory calculation of 6H(S ) and the correction A(s) in the Coulomb repulsive case. In.fig. 4 we also give A (s) for the S-wave in the Coulomb attractive case. In figs. 6 and 7 we show the evaluation of the dispersion relation terms for sin 2A in the S- and P-wave Coulomb repulsive cases. The curve S denotes the SchrOdinger calculation of sin 2,5 and D is the result of the dispersion theoretic calculation using eq. (31). The other curves show the various terms in eq. (3 I). They are: 1:(~0 1_ 1) sin 261_I. This is seen to be a fair approximation except near and above the resonance in the P-w ave. II: The physical integral denoted by 12 in eq. (31). Notice that it gives a large contribution near the resonance in the P-wave. This is due to the factor sin28H in

460

£ Hamilton et aL, Coulomb corrections

30*

S - wave

Aa• r X 10 20 =

10 =

5

10

50

100

500

q[MeV/c] Fig. 4. Schr6dinger calculation of the S-wave hadronic phase 6 H of sect. 7 and the corrections Aattr and Are_ in the Coulomb attractive and repulsive cases. The corrections are on a scale magnified by ~'0.

IA(s)[ (eq. (33b)). We have used A (eq. (32)) for A in lm H(s+), (0 <~s <~oo). Further iterations will change the results for sin 2A by about 10 - 4 . !1I: The contribution of order a G to 13 (eq. (31)). IV: The contribution o f o r d e r a G 2 t o l 3. D is the sum I + II + II1 + IV. In the S-wave case the curves S and D coincide within 2 - 1 0 - 4 and they are shown as one curve. In the P-wave case the agreement between D and S is not perfect because the coupling constant G is large and higher orders in G will also contribute to the left-hand cut of H(s). (Contributions 111 and IV are almost the same size in the P-wave case).

8. Effective range relation for Coloumb attraction We now consider the case that the Coulomb interaction is attractive. This is a little more complicated than the Coulomb repulsive case because the p.w.a.'sfcH x a n d f c h now have Coulomb bound state poles on the physical sheet. Because the Coulomb effect is attractive, in place of eq. (3b) we have 7 = -l/(qaB).

(40)

J. tlamilton et at, Coulomb corrections

,

100"

,

D

. . . .

461

I

e

....

P-wave

BO°

60*

ko*

20*

- A r e P X/*0

0e I0

S0

100

S00

I000

q [ M eVlc ]

Fig. 5. Schr6dinger calculation of the P-wave hadronic phase ~H o f sect. 7 and the correction in the C o u l o m b repulsive case. The correction is on a scale magnified by 40.

I

"

'

'

' I

l

'

I

~

'

'

'I

0.02 m"

-0.0&

S,D -0-01

'

a

5

'

'

I

II

10

I

I

q[MeVlc]

*

50

,

,,I

100

Fig. 6. Dispersion t h e o r y c a l c u l a t i o n o f sin 2 A in the S-wave C o u l o m b repulsive case.

" 500

J. Hamilton et aL, Coulomb corrections

462

t

"

"

'

'1

w

0.01

-0.01

- 0.02 ~-0.03

L

-o.o~. ~-

I

]I --

-oo.

"

\',//

lo

5o

lo0

soo

looo

q[MeV/c ]

Fig. 7. Dispersion theory calculation of sin 2A in the P-wave Coulomb repulsive case. The C o u l o m b phase shift o I is again given by eq. (3a) and exp (2iol) has poles at the points 7 = i(n + / ) ( n = 1, 2 . . . . ) on the physical sheet; these are the C o u l o m b b o u n d states. It also has a set o f zeros at 3' = --i(n + / ) ( n = 1 , 2 . . . . ) on the second sheet. The hadronic interaction causes only a small change in the energies of the b o u n d states. This is because the Bohr radius a B is much greater than the range of the hadronic interactions and the hadronlc scattering length. We define f u n c t i o n s . f , ~0, F, and H as in the C o u l o m b repulsive case, the only change being that 7 is now given by eq. (40). The 13'1 of sects. 6 and 7 is to be replaced by -13'1 in the case o f attraction. The function F(s) n o w has a set of zeros at 3' = i(n +/), (n = 1, 2 . . . . ) and a set o f poles at nearby points ( b o t h sets being on the physical sheet). Also, F(s) has cuts ia <~q <~ i o% i oo < q <<,O. The function F - 1(s) has a set of poles at 7 = i(n +/), (n = 1, 2 . . . . ) and a set o f zeros at nearby points.

8.1. S-waves We consider first the case l = 0. At the points 3' = in, (n = 1, 2 . . . . ), by eqs. (7), (10),

£ Hamilton et aL, 6bulomb corrections

.If 7r7

e - 2iao = I f (q)

e2i°°-1.] e-2ia o en~ - a B ( - l ) n 2/q

n7

2rri

463

(41)

because f(q) e- 2ioo vanishes at 3' = in. Therefore the function F - 1 (s)-- 2n/" coth (7r7) aB

(42)

has no poles at 3' = in, (n = 1, 2 . . . . ). On the physical cut 0 < s ~< oo (where 7 < 0), F-I(s+)-

21ri coth Or7) =21r aB aB

[- c o t ' i _] 1 __fi-~r~ + e - 2~r'r_ 1 "

(42a)

This function has the same discontinuity across the physical cut as (2/aB)h(s), where h(s) is given in eq. (15). Notice that because 7 < 0 on the physical cut, in place ofeq. (16) we have, for 0~
7r e-- 2~rv_ 1

Therefore the function

K(s)=F_l(s)

2hi c o t h ( r r 3 ' ) - 2

h(s)

(44)

is holomorphic in the s-plane cut along - oo ~< s -<<- a 2 and is regular near s = 0. This gives the effective range relation 7r cot 6 I - e 2~r'r

g

(13'1)-~-a B -' K(s),

(45)

w i t h g defined in eq. (18a). Except for 13'1>~ 1 this form differs little from the effective range relation for hadronic interactions. By eq. (19b)

L a 2Bq 2 g(lT[) ~ 12

for

h'l >> 1

•

(45a)

the re fore 7r c o t , ~ 12 aa K(0)

as q -~ 0 ,

(45b)

so ' -~ ' (0) # 0 as q ~ 0. This is very different from the behaviour of the S-wave observed phase in the repulsive case, where ~ = O[exp(-27r/aBq) ] as q ~ 0.

J. Hamilton et aL, Coulomb corrections

464

8.2. Other waves For l > 0 the same line of argument is used, starting from the f u n c t i o n fe-2i°l/~[ evaluated at 3' = i(n +/), (n = 1, 2 . . . . ) in place o f e q . (41). We have, instead o f e q . (45), the form =

nc°tfil ......... 1 - e 2n'r

g(13'l)

[m- '[l (q + 1a~B) - 1 2 ~1 a B K(s) =1 m2

(46)

As the energy decreases to low values the observed phase fit(s) decreases like q21+l so long as 13'1 "~ 1. In the atomic energy range the behaviour is different, and eq. (46) shows that fi/(s) ~ 6/(0) ¢ 0 as q ~ 0.

8.3. The bound states contribution Except in the a t o m i c energy region the contribution of the b o u n d states to the function F(s) is very small. Partly this is because F(s) contains the difference of the pure C o u l o m b and the modified C o u l o m b bound states, and partly because of certain factors in the definition of F(s). For the pure C o u l o m b interaction the b o u n d state positions are given b y 3" = i(n +/), (n = 1, 2 . . . . ). In the presence o f strong interactions these positions become 7 =/(neff + / ) , where neff is the effective q u a n t u m number. The function

lal(n ) = n - n e f is k n o w n as the q u a n t u m defect. It varies slowly with n [18]. We shall now examine the S-wave case. By eq. (44) the effective function is

F(s) =

(s) + 2 h (s) + 2zri coth (zr7 aB

,

(47a)

aB

and for - o~ ~< s <~ 0 this gives

F(s) = sin (zrl3'l)

[(

o

)

K(s) + a 2 h(s) sin 0th, I) + 2 n cos (Trh, I aB

1-1 ,

(47b)

since here 3' = ih'l. The poles are at 13'1 = neff, corresponding to s = sn. The pole positions are given by

aB

cot ( ~ n e f ) = - cot ( ~ ( n ) ) = - ~

K(Sn) -

h(sn).

(48)

F r o m eq. (15), for 7 = iJ3'l we have [11]

h(s) = - In 13'1 + 1/(213'1) + qJ(13'l), so for 13'1>> I,

h(s) = - 1/(1213,12) + O(13'1- 4 ) .

(49)

465

£ ttamilton et at, Coulomb corrections

Therefore in eq. (48)h(s n )-+ 0 as s n ~ 0, and comparison with eq. (45b) gives cot A (0) = -!-la K(0) = cot (Tr/a (oo)), 2rr B

i.e. A (0) = 7rta(oo)= arccot (~Bn K(0)) .

(50)

It is easy to see that also Al(0) = 7rlal(OO) for l > 0. This result has been found by Seaton [19]. Since A(0) is related to the quantum defect it may be of importance to determine A (s) also at atomic energies. Near a pole eq. (47b) gives

f(s)

-

sin(~'neff) B(neff)

1 13'1 - neff

+O(1),

(51a)

where 1

R(Jvl)-- %3`13

d K ( s ) , dh(s) +2_ ds "-dlXF-Tr21 sin0rl3`l)+[X(s)aBh(S)]rrcos(zr,T[).

For 13'1~> 1 the term 1 dK(s) + dh(s) dl3`l aB[3`l 3 . ds is much smaller than lr 2 and for 13'1>~ 1 it is very small (cf. eq. (49)). So using eq. (48) B(neff) ~ _

2zr2 1 , a B sin (nneff) '

and eq. (51a) shows that near a pole F(s) ~ _ ~ aB (sin ~ ( n ) ) 2

1 + O(1) = [sin 7r/a(n)]2 + O(1).(51b) [71 -- neff n3 aB rr2(s _ Sn )

In order to estimate the contribution of the infinite set of poles to F(s) we take tt(n) ~/a (oo), and summing the terms like eq. (51b) we get

1 (sin rr

(f(S))p°les"~--aB

~(oo)

)2

n=|~ n - 3 [ s + ( n a B ) - 2 ] - I

(52) = - - a B (sin rt~(oo))2 ½[if(iT)+ ff(_iT) + 2 C ] .

.It. Hamilton et al, Coulomb corrections

466

On the physical cut the last factor in eq. (52) is (ReqJ(iT) + C), and using eq. (19a) we find for 171 "~ 1 on the physical cut (F(S))ltx.)les ~ -aB(/d(oo))2 1.2 7:2 •

(52a)

This is O((x72) and can be neglected above the atomic energy region. In the last step we }lave replaced sin (nla(oo))/~ by ~(oo) since/a(oo) ~ 2atl/a B ,4 1, (eq. (50)), where a H is the hadronic scattering length.

8. 4. Behaviour near the threshold For s ~ 0 on the physical cut, (F(s))poles diverges logarithmically because Re ff 0"7) ~ In 171 for 7 ~ oo. Eq. (44) shows that ReF(s) and Im F(s)go to finite non-zero limits as s ~ 0 on the physical cut, and by eqs. (13), (13a) aB lim ImF(s+) = ~ sin 2 A (0). s~O+ As a consequence 1 p f 7r- 0

ImF(s'+) aB sin 2 A ( 0 ) lns a s s ~ 0 + . s-s ds' ~ - - -27r2

This divergent behaviour near s -- 0 in the dispersion integral for ReF(s) can only be cancelled by the In s dependence in (F(s))pole s as s ~ 0+. The cancellation is ensured by eq. (50).

8.5. Other waves Analogous arguments show that for I > 0 the poles contribution is

,,,,,>>,o,.,--,,',"'

(:'si: U.>),.,, n=l+ 1

In(.'+.,,")

,

]

rl (l-m2/n 2)

m =1

For 171 ~ 1, this gives

(F(S))poles = - a2l+ l (fl sin 2n#l(°°) 7 r )

lnlTI + 0 ( 1 ) .

For 171 "¢ 1, (F(s))pole s is of the order 72. The poles contribution is therefore negligible above the atomic energy region. Again the cancellation of the logarithmic divergence is ensured by the relation At(0 ) = 7r/at(oo).

J. Hamilton et aL, Coulomb corrections

467

8. 6. Dispersion relation for the Coulomb correction The dispersion relation for the Coulomb correction A(s) has the same form as in the case of Coulomb repulsion, namely eq. (26). The dispersion relation for Re H(s) is ; ReH(s) = (H(S))pole s +-1- p rr 0

~0sin 2 A - ( 9 1) sin 2 611 ds' ............. (q,)2l+ t iA(s,)l(s, _ s) (54)

_a 2

+1 f

Im H(s'+) , s'- S ds .

Since (H(s))pole s ~ A(0)- 1 (F(s))pole s this is of order 2"2 for 17[ '~ 1. It follows from eqs. (13) and (26) that

Al(O) = Zrlal(OO) ~ 27r a B 21- 1 ( l ! ) 2 [(aH)/+ A(0) Re H I (0)l ,

(55)

where (all) l are the pure hadronic scattering lengths. Notice that A(0) Re HI(O)~ (a H)l is of the order a. The term in lm H(s+), (0 <. s <. oo) containing sin2A does not vanish at s = 0, and it gives the In s term in the physical integral which cancels the In s term in (H(s))_. Ks" Because of this cancellation near threshold, and because the term containing sin2~ " in lm H(s+) only contributes in the atomic energy region, we get a good approximation to A(s) by ignoring both the (H(S))poles and sin 2 A terms in eq. (54).

8. Z Example We have calculated the correction A(s) for an S-wave with Coulomb attraction and the same attractive hadronic potential (G = - 1) as in sect. 7 above. The Schrodinger calculation of A is shown in fig. 4. At the higher energies it is equal and opposite to A for Coulomb repulsion. The dispersion theoretic calculation of sin 2A is shown in fig. 8 and the various curves are labelled as in figs. 6 and 7. In fig. 8 D=I+II+III+IV+V, where the new curve V shows the bound states contribution; its divergence as s ~ 0 is counteracted by curve II. In the calculation we have replaced A(s) by A(s) of eq. (32) in lm H (s+), (0 ~< s ~< oo). Then putting ~r/a(oo) = A(0) in (H(S))poles makes the Ins terms cancel. The curves S and D are shown as one because they coincide within 3 • 10- 4. If the term in Im H(s+), (0 <~s <~oo) containing sin 2 A and the term (H(s))pole s

468

J. Hamilton et aL, Coulomb corrections

0.12

,

i

i

,

i

i

]

i

i

i

i

,

i

i

I

X 0.10

~\\ S,D

0.08

0.06

004

0.02

\\

11I

-0.02 i

l

i

I

5

,

,

i

i

I

,

i

10

,

I

50

,

,

i

i

I

100

i

i

500

q + I [MeVlc ]

Fig. 8. Dispersion theory calculation of sin 2 n in the S-wave Coulomb attractive case. are ignored the agreement with the curve S is within 0.6 per cent near threshold (and it is much better elsewhere).

9. A two-channel problem So far only elastic two-body scattering has been treated. However, our dispersion technique is easily extended to apply to two-channel two-body scattering. We consider the simple example involving rr-p elastic and charge exchange scattering, i.e. the processes

469

J. Hamilton et aL, Coulomb corrections

lr-p--*Tr-p, 7r-p--*Tr°n, 7r°n~Tr°n.

(56)

Photon emission and the mass differences within the isospin multiplets will be ignored Our main purpose here is to set up the basic scheme which will in a later paper be applied to the relativistic case. The method follows very directly from what we have done above. We have here to derive dispersion relations for the corrections to the phases and for the mixing parameter which describes the transitions from one isospin state to the other. The notation used for the channels is somewhat like that of Auvil [20]. (For other Schr6dinger theory treatments see also Oades and Rasche ]1 ] and Oades [21]. See ref. [20] for others). Let Sclbx be the 2 X 2S-matrix for the processes in eq. (56) for given or~;ital angular m o m e n t u m I and total angular m o m e n t u m J . SCtl~" is given on th= basis of the observed charged channels (lTr p>, 17r°n>}. The matrix which transforms this _3 }is into the conventional basis of isospin states { br = ~-), II -~-) U=~3=

1

When a matrix is given in the isospin basis it will be labelled by U, thus [22] U = UScttx U - 1 SCH~, The pure hadronic S-matrix is S H . In the isospin basis it is diagonal and SU

( e 2i6 h \

0

0

)

e 2/6~I- '

1 and h3 phase shifts. where ~ ! , 531 are the pure hadronic I = hIn order to remove the long range Coulomb effects we define the matrices G and l~ by G = ( exp[~i3'ln(4s~20 + 1)-i'),C]

f__(

1! exp [~ Y(s) -- &CI r f t + , + i3,) 0

?),

0) 1

'

(58a)

(58b)

where Y(s) is given by eq. (10b) and 7 by eq. (40). Thus in place ofeq. (10c) we have Y(s+) = rr7 +/3' In (4s/X 2)

for 0 ~< s ~< oo.

The exponential function in G has the cut structure shown in fig. 1. By Gorshkov's theorem (sect. 2 above) the matrix SCH = lim h--,0

GScrtxG

(58c)

J. ltamilton et aL, Coulomb corrections

470

is finite. SCH is unitary and symmetric since SCItX and G are unitary and symmetric. If there are no hadronic interactions then (cf. eq. (7))

SCH ~ SC - (e20 °'

~)

9. I. The correction matrix D and the effective matrix F We define a matrix D by (59)

D =SH½ S c ½ S c l ISC{SH ½ .

Clearly D is unitary and symmetric and D - 1 if either there is no hadronic or no Coulomb interaction. The most general form of D can be written l

DU = ((1

-

e2)ge 2*at \ ieei(A 1+A 3)

ieei(At + zx3) ]

.

(I - c2)½ e 2ia3]

'

(59a)

A 1 , A 3 are the phase shift Coulomb corrections and e is the (real) mixing parameter. In Auvil's notation [20] these are d/ll,d33,1 and .~v~dtl3, respectively. The effective matrix F is defined as in eqs. (10a) and (12a) by

2iq21+l F(s) = lim I7(SCH x

Scx) Iy , (60) x--+0 where SCX is the S-matrix when there is no hadronic interaction. By the same arguments as in sects. 2, 3 above we see that F(s) has only the hadronic cuts (the process rr- p -+ n7 gives rise to a radiative cut which is ignored in the present work, but will be included in the relativistic treatment). On the physical cut

(0 ~< s -<<,,o),

G-l ~=Sc 1, ~ , where

0)

and ~p (s) is given in eqs. (13a), (13b). Therefore eq. (60) can be written

2iq21+ 1 F(s) = IYG- 1 (ScH _ SC ) G - 1 i) 1

1

1

1

--~. o~--~,

(0<~oo1.

(61)

J. Hamilton et al., Coulomb corrections

471

Because eq. (61) is of the form 2iq21+ 1 F = ~½ (M -- I)(o ½ ,

where M is unitary and symmetric, we have the result im ( F - 1 ) = __q21+ 1 ~ - 1 ,

(0 < s • oo) "

(62)

This is obviously the unitarity relation. If there were no Coulomb interaction then

~b=L The arguments relating to eq. (42) show that the function

'

F - l ( s ) - 2hi coth (~7) l-~m=l aB =

')(; 0) m2a 2

is regular in the region of the Coulomb bound state poles near s = 0. Also, with h(s) given by eq. (15), l

lmI27h(s+)

m=lI-I(l+72/m2)q 2l+1 -21r--~/ ( T r Tcoth )aB

I m=ll-I(q2 + ~ ]

( 0 < s
=q2t+l¢-l,

where ~0 is given by eqs. (13a), (13b). Therefore the matrix K(s) defined by 2hi coth(TrT)

K(s)=F-l(s) -

a~

]7

m=l

q2+ m2

I + q21+ 1 (27h(s)

£

(1 +72/m 2) 1 0

0 i

)

o°) (63)

has no physical cut since lm K = 0 for 0 ~
We define F H by 2/q 2/+1 F H =S n - 1 , and H(s) by

472

Z !

ltamilton et aL, Coulomb corrections 1

H(s) = A- ~ (F.. FI.I) A- ~ ,

(64a)

where (64b) and A~.(s)=exp

(2~s n ~ s'(s' 6iH(-s'2s) - ds'),

(i= 1,3).

(64c)

Now we write

H U = ( Hll

H13]

~HI3

H33]

and 1

~1 =-~(1 + 2:~) 2,

I

~3 = ~(2 + :r)2.

Then Oi = 1 + 003'[), (i = 1,3) for 13'1'~ 1. The defining relations now show that on the physical cut 0 ~< s ~< o%

q

2/+1

l

[AilReHii(s) = (1 - e 2 ) ~ i s i n 2A i + (~i

_

1) sin 26~i + Rii(s), (i = 1,3), (65a)

q2l+llAil ImHii(s+ ) = (1 - e2)~ 0 i sin 2 A i - (0 i - 1) sin 2 ~ t + ½1ii(s)' (i= 1,3), (65b) where Rii(s ) and lii(s ) are given in appendix B. For ]3'1 '~ 1, Rif(s ) and Also, for 0 ~< s ~<~, 1

Iii(s ) are 0(3'2).

1

2q21+ 1 IA 1A317 ReH13 (s) = ~ e (4 + ~02-+ 4so)

+ o~V/-2(I --~p½)(1 -- E2)½ [(1 + 2~p½) sin 6 h cos63 + (2 + ~p~-)sin ~4 COS~ 1 ] + R !30),

(66a)

2q 2l+1 ]A1A3I~- ImH13(s+) = - z3x/2"(1 - ~oXl - e2) ½ sin 61 sin 63H + 113(s), (66b) where R 13(s), 1130) are given in appendix B. They are 0(3 ,2) for 13'1"~ 1.

J. Hamilton et al., Coulomb corrections

473

The dispersion relations are

1 f .lmH0"(s'+) f ......., ds' + lit

Re Hii(s ) = (His(S))pole s + ~-- P

0

s -- s

I m Hij (s'+) d s' (67)

t

LHC

S

-

S

for i, j = 1,3 and 0 ~< s ~< ~ . LHC denotes the hadronic left-hand cuts. For 13'1"~ 1 the termsRij, lip and (HisJS))pole s can be ignored. The equations for A 1 , ,53 are then the same form as in the single channel case. They are, to order 13'1, sin 2A i

2rr7 3(i--1)r

-

[1

X

~

P

sin 25~i

+ 2q2/+l IAi(s)I

( l - < j / ) s i n 2 ' H i ds'

I

+~r f (q')21+llAi(s')l(s'- s) LIIC

lm'Hii(s'+) .... s - s

1 ds

, (i = 1 , 3 ) ,

(68)

and for the mixing parameter, to order 13'1, e = ~V/2-rl'7 sin (61H + 63 ) + 2q 21+1 IAl(S)A3(s)l I (69) ×

[~P~ Vc2"(~°-l)sinSlsinS~lds' +-flrl 3(q') 21+11AI(S')A3(s')II(s '- s)

LtlC

Im~/3(s'+)_tdsl s -. s

These are not coupled equations. In order to get a good approximation for 13'1~> 1 the terms Rij in eqs. (65a), (66a) must be retained and the coupled system of equations now has to be solved. However, we can still leave out the terms 16 and (Hi/)pole s because the contributions from these terms tend to cancel near s = 0, just as in the single channels case. The determination of the left-hand cut terms lm Hij(s ) will be left until we discuss the fully relativistic problem. We are grateful to Drs. E. Sauter and L.E. Lundberg for very helpful discussions.

Appendix A

Evaluation of left-hand cut terms Consider the S-wave with the interaction

V(r) = a e-Xr/r + VH(r ) = a e-Xr/r + Ge-Ur/r,

(A.I)

J. flamilton et al., Coulomb corrections

474

where p >> X. We shall use the method of A. Martin [17]. Let V(r) have the Laplace transform

2m V(r) = ;

C(/3)e-3r d3.

The the left-hand cut discontinuity of the S-wave amplitude f(s) is given by 2K---h

4K2lm.r(s+)=C(2,0+f 7r

C(2K -/3) pK (/3) d/3

(A.2)

C(/3 -/3') p,~ (/3') d/3'.

(A.2a)

h

f o r g1 X ~ < ~ o % w h e r e q = i K a n d 3-h /3(/3 -- 2K)O,c(/3)= C(/3) + f h

We take C(/3) = 0 for 13< X. For the potential V(r) above, X) +

c(/3) = 2m~0(/3-

2maO(/3

- U) .

Putting a ; 0 and using the term of order G in eq. (A.2) gives lmfl~) in eq. (38a). Iterating eqs. (A.2) amd (A.2a) gives 2K--l,

--~-4K2lm f(s÷) = C(2K) + f

C( 2K~3___273)C(/3) )d/3

?.

(A.31

+f

C(2K - 3)

C(3 - 7)C(3') d

d3 + (higher order terms).

- s,e(1) The amplituae ~ in " fig. 3c is given by the graphs of order aG. We have for 1

rl~

Im'f(Hly) ( s+) = 2nKm2 2 a G f2~-x

2nm2aG 2K-u

f k

d3

o(2K - / 3 - ~ ) 0 ( 3 - x) d~

_ nmGh'l[ln(2~l_l)

~ 3 - 2K)

using eq. (A.3). This gives eq. (38b).

s

A4,

J. tlamilton et aL, Coulombcorrections

475

Similarly the term of order G 2 (fig. 3b) is

lqt3

'

(A.5)

.

This gives eq. (39). The first and last graphs in fig. 3d contribute to lm f(2) the amount JH'), 47rm3~(; 2

~-

~-

~h

~ 0(2K .- ~ - ~ )

0(~

-~¢-- 2~)

~(~- 2K)

d~

3'(3'- 2K)

h

= 2_lrm3c~G2K 3 [ /2~.-~ h+~z

3' - / J ) 0 ( 3 ' -- ),) d

d / 3 + ~ In

,(x + .)

(A.6a) The centre graph in fig. 3d gives similarly

27rm3ctG2 /K--X O(2K--~---IJ) I~-X K2

0(~-

--- ~(-[3 - 2~)

7rm3c~G2/'2'¢-u

3' - x ) 0 ( 3 ' - u)

3'(3'- 2K)

-] d3,_l d/3

_In (2K~_-/3)~~' 2K/~_/j) d/3 (A.6b)

I

lm f(2) is the sum of eqs. (A.6a) and (A.6b). JH3' From eqs. (A.5), (A.6) we get for - o~ ~
]n {2_~_-4 ÷

21

K3 The dependence on In ~ has disappeared. In the P-wave case with the same strong potential eq. (37) we have, for -o=~
1 +~-

' JH'T

]

~ In--(2K -/J)(2K - /~ -X)J

J. llamilton et aL. Coulomb corrections

476

- mnGITI . ' s L\I ! s+ and, for-oo~
_

Imf~?)(s+)

In /12,

.re:C: 4r ' 2K3

lira h--*0

13'1 n - - ~ - - 1

nm2G21"yl K2

-

.....

Z+y/2.0

dy

J4~2 [y(v- 4~u2 -/.t4/s)] ~ '

Imf~?)(s+)+,mJH,(S+ (1+-~-)

In ( { 3 - la)2(2K - / a X 2 K - 3)_)d3 /3(2K - ~)

/a

where t=(2K + u-

~)2 ~ ( 2 K . - u )

3(2K -.- ~)

"

These results can be obtained by the same technique as used in the S-wave case. It is, however, simpler to project out the partial wave amplitude from a double spectral representation of the scattering amplitude. The double spectral function can be determined from a reeursion relation (cf. sect. 11.4 in ref. [23]).

Appendix B

The functionsRjk(S ) and Ijk(S ) (/; k = 1,3) used in sect. 9 are given by

Rjj +iljj=~ ,v/2-e(1 -~o~)1(1 +j) +(1 + k)~o½] e i(Sk - J H + al+A3) + i~(1 -- ~0{)2 e - 2i6jH [ 1 -- (1 -- e2) } e2i(6k+ Ak )] + ie2q~//[l + where we have used j= l , k = 3 o r / = 3, k = 1.

(l-e2)~l,

477

J. ttamilton et aZ. Coulomb corrections

Also, _2

,'~

i

.

1

3

R13 + i 1 1 3 - ~x,/2~(l -- ~o~)(1 -- 62) ~ 1(1 + 2~p~) sin A 1 e t ( S H - 6 H + ~ ' ) 1

•

3

1

+ (2 + ~o~) sin A 3 e t ( ~ H - ~ H + z x 3 ) ] + ~ 6(4 + ¢~ + 4S0)(e'•('xl + 4 3 )

I

3

_ 1) + i~ X/~-( 1 - ¢) e2 e _i(~H+~tl)/[l

1

+(l_e2)~ ].

References [1l L. van Hove, Phys. Rev. 88 (1952) 1358; J. Hamilton and W.S. Woolcock, Phys. Rev. 118 (1960) 291; H.J. Schnitzer, Nuovo Cimento 28 (I 963) 752; P.R. Auvil, Phys. Rev. 168 (1968) 1668; G.C. Oades and G. Rasche, Helv. Phys. Acta 44 (1971) 5. [21 R.F. Dashen and S.C. Frautschi, Phys. Rev. 135 (1964) B 1190, 1196; 137 (1965) B 1318. [31 (a) E. Sauter, Nuovo Cimento 61A (1969) 515; (b) 6A (1971) 335. [41 L.D. Landau and J. Smorodinsky, J. Phys. Acad. USSR 8 (1944) 154;" H.A. Bethe, Phys. Rev. 76 (1949) 38. [5] A. Messiah, Quantum Mechanics vol. 1 (North-Holland, Amsterdam, 1961). [6] V.G. Gorshkov, JETP (Soy. Phys.) 13 (1961) 1037; E. Brezin, C. ltzykson and J. Zinn-Justin, Phys. Rev. DI (1970) 2349; J. Hamilton, lecture notes (unpublished). [71 R.H. Dalitz, Proc. Roy. Soc. A206 (1951) 509. [81 H. Cornille and A. Martin, Nuovo Cimento 26 (1962) 298. [9] Y.L. Mentkovsky, Nucl. Phys. 65 (I965) 673. [ 10J J. Hamilton and B. Tromborg, Partial wave amplitudes and resonance poles (Oxford University Press, 1972). [11 ] E.T. Whittaker and G.N. Watson, Modern Analysis, chap. 12. (Cambridge University Press, 1920) 3rd ed. [121 D.Y. Wong and H.P. Noyes, Phys. Rev. 126 (1962) 1866. [13] T. Teichmann, Phys. Rev. 83 (1951) 141. 114] J. Schwinger, Phys. Rev. 78 (1950) 135. [ 15] F. Calogero, Variable phase approximation to potential scattering (Academic Press, N.Y., 1967). [16] A. Donnachieand J. Hamilton, Phys. Rev. 138 (1965) B678. [171 A. Martin, Nuovo Cimento 15 (1960) 99; 19 (1961) 1257. [18] N.F. Mott and It.S.W. Massey, Theory of atomic collisions (3rd edition) (Oxford University Press, 1965) chap. 27. [191 M.J. Seaton, Compt. Rend. 240 (1955) 1319. 1201 P.R. Auvil, Phys. Rev. D4 (1971) 240. [21] G.C. Oades, Springer Tracts in Modern Physics, 55 (1970) 61. [22] H.Y. Chiu, Phys. Rev. 111 (1958) 1170. [231 X. De Alfaro and T. Regge, Potential .scattering (North-Holland, 1965).