A note on dispersion relations

Nuclear Phys,cs 15 (1960) 518--518, ( ~ North-Holland Pubhsh,ng Co, Amsterdam Not to be reproduced by photoprmt or mmrofflm vnthout written pernussmn from the publisher

A NOTE ON DISPERSION RELATIONS ALLADI R A M A K R I S H N A N , N R KA.NGANATHAN t and R VASUDEVAN tt

Department o/ Physws, Umvers~ty o/ Madras and S K. SRINIVASAlq

Department o/ Appl,ed Mathematzcs, Indzan Instztute of Technology, Gutndy, Madras Received 26 October 1959 Dmperslon theory is examined on the basis of a reciprocal relationship between the real and imaginary p a r t of the scattering a m p h t u d e and it is shown t h a t the knowledge of the absorptive part over the unphyslcal region leads to a linear integral equation for the dispersive part

Abstract:

The theory of dispersion relation arose in an attempt to meet the inadequacies of the perturbation theory. It is based upon the possibility of writing the real part of the scattering amplitude as an integral over the imaginary part as a consequence of the analytic properties of the matrix element. In a corresponding manner, the imaginary part can be related to the real part. If in the former relation we substitute the latter, we are naturallyled to an identity which merely expresses the fact that one set of relations implies the other. However, we wish to point out that there are circumstances under which the substitution leads to a meaningful integral equation for the real or the imaginary part. Such a case arises if the imaginary part is known over a partial range. In our opinion, the principle used in this note is of general value. However, to demonstrate it, we confine ourselves to the special case of forward scattering of a charged pion by a nucleon. The application of dispersion relations to pion-nucleon scattering phenomena becomes a little involved owing to the charge states of pions and nucleons (see for example, Goldberger 1)). From the forward scattering amplitudes F+ and F_ of positive and negative pions by protons, which m a y be written Fe(o~) = De(m)+iAe(oJ)

(1)

in the system in which the proton is at rest (co denotes the energy of the pion), the following linear combinations can be formed: FE(o~) = ½{F_(co)+F+(co)},

(2)

Fo(o~ ) = ½{F_ (co) -- F+ (co) }.

(3)

* Indian Atomic Energy Commission Junior Research Fellow tt Government of India Senior Research Scholar. Now Assistant Research Physlcmt at the University of California, La Jolla 516

A NOTE ON DISPERSION RELATIONS

517

F r o m the general t h e o r y of scattering of a pseudoscalar boson b y a nucleon, Goldberger 1) has shown t h a t FE* (o~) = FE(--o0),

(4)

Fo*(O) = --Fo(--eo).

(5)

I t is easy to write down the dispersion relations b y using equations (1) to (5) (see Goldberger et al. *)):

D~(eo)_DE(I~) =

2kZp

oJ'A~(oJ')dw'

o k'~(o,"-o,') '

(6)

2o, P I°°A0(o,')do, '

D°(°~) = --Z-

Jo

-o~'2--co ------~ '

(7)

w h e r e / , is the meson mass and k its m o m e n t u m The absorptive parts A~ and A o can be evaluated explicitly In the range 0 < ~o < # in the usual way. Thus (6) and (7) can be written in the form 2k 2 P foo o,AE(w,)do/

12(0 2

Ds(e°)--D~('u) =MEo*--(1/2M)23 + -~-- 3~, ~

Do(w)-oJ2

21~o

2o

(1/2M)2 + - P ~

'

(8)

foo/, ofz_w6 Ao(o/)da/"

(9)

At this stage we observe t h a t inverse relations can be written down expressing A E or A o as an integral over D~ or D O, as the case m a y be. The arguments which have led to (5) and (6) yield AE(~°) _

2aJp (oo DE(oj, ) d '

Jo o P - c o " A o ( o ) --

o,,

(10)

2 P t'l°° o ' D o ( o / ) d o '

(11)

:71:

,d 0

(D t 2 - - (D 2

We have assumed t h a t no subtraction is necessary for the absorptive parts We now observe t h a t if (10) and (11) are fed into the right h a n d side of (8) a n d (9) respectively, we obtain integral equations for D E and D o • D ~ ( w ) - - D E ~ ) -~

Do(o ) =

12o,"4k* ~.~o. o,'do; f°°DE(~o")do," M[~02 (1/2M)2] ~ P j ~ , k , 2 ( o / 2 0j2)3 ° g , ~ , (12) 2] 2o oJ2 - (1/2M) 2

4o, f °° ~2

do/

P (Dt2--C92

f°°o,"Do(o,") 0

COt t 2 - - COP2

do/'

(13)

518

ALLADI RAMAKRISHNANet ~g~.

We hope that this method of viewing the dispersive part as the solution of an integral equation might sometimes be helpful in understanding physical phenomena. Investigations based on (12) and (13) are in progress and will be reported in due course. The procedure adopted here amounts to using our knowledge of the functions over the range (0, p) to obtain the functions in the residual range. References 1) M L Goldberger, Phys Rev. 99 (1955) 979 2) M L Goldberger, H Mlyazawa and R Oehme, Phys. Rev 99 (1955) 956