G(S1+) + M(Slt) = 0 Thus #(£) in eq (3-88) can be put in the form

In document The analytic properties of the triangle and box diagram amplitudes (Page 91-96)

E ( £ ) =

__________ [G(S)-«(S)j 2__________

e

~2

f

~2

g

2

(2fffs+£2+F2) (s-slt) (e-q_) so that

#(£) % (Positive valued function of £)(£~£^+) X Hence

disc, [arg £(£)] = -disc, [arg (£-£1+)] = ~2tt , and

(3-105)

disc, [log #(£)] = -27T

i

, (3-106) where the discontinuity is again computed as

[(value below cut)-(value above cut)] .

Strictly, (-2tt£) is the limit of the left hand side of eq. (3-106) as £ approaches the real axis from below, and as

C

is deformed

to the segment of the real axis from to +°° with the interval

[_C^r ,

l\

being traversed

in both directions.

Finally, from e q s .

(3-87), (3-104) and (3-106) we have

disc.U(S)] = -- --- = ICS) , (3-107)

vQnIT

where Y(£) is given in eq. (3-68).

and transition cases 1 (t ) is given by eq. (3-63) for all

Tq j; [1, +°°) and for the anomalous threshold case j(Tq} is given by

eq. (3-64) for all £ [£ , +°°) . The explicit forms of the weight

functions are as given in sects. 3-5 and 3-6 for the various cases given in eq. (3-46).

In the following two chapters, the methods used in this chapter

are extended to apply to the box diagram amplitude. In chapter 4, we

extend the method of direct transformation to obtain a number of

different spectral representations for the box diagram amplitude. We

find that (for the range of s and t considered) the box diagram

amplitude can be written either in the form of a double dispersion

integral in s and t or as the sum of a double dispersion integral

in s and t and a single dispersion integral in s , depending on

the values of the masses of the particles in fig. 4-1. In this chapter,

we also discuss a heuristic method of obtaining the weight function in an approximate normal threshold dispersion relation for the two-particle

two-particle scattering amplitude. Then, in chapter 5, we start with

a particular normal threshold double dispersion relation obtained in

chapter 4 and continue in the external masses and then in s and t .

In this way, we obtain a spectral representation for the box diagram

amplitude for both real and complex s and t . In particular, a

spectral representation is obtained for all s in the upper half

4-1. Introduction

In this chapter we start our investigation of the analytic

properties of the box diagram amplitude 2j-j(s, £) associated with the Feynman diagram shown in fig. 4-1. This amplitude describes the

collision process

AB

-+

CD

where the intermediate particles in the process are

E, F

,

G

and

H

as shown in fig. 4-1. Our main object is to obtain a spectral representation of 2q (8,

t)

for all masses satisfying the stability conditions

A < (E+G)

,

B < (F+G)

,

C

<

(E+H)

,

D < (F+H)

. We also obtain an approximation to the scalar invariant scattering amplitude T(s,

t)

for the process

AB

-*■

CD

and write this approximation in the form of a double

dispersion relation. Then we show that when (A+B) 5 (E+F) ,

(C+D) <

(E+F)

this double spectral representation is the same as that obtained for 2g(s,

t)

.

We begin in sect. 4-2 by writing down the scalar invariant scattering amplitude T(s,

t)

for the process

AB ■+ CD

. Then we obtain an approximation to T(s,

t)

in the form of a double spectral representation by using unitarity to evaluate the contribution to ImT from a two-particle intermediate state

(EF)

and making t-channel single pole approximations to the amplitudes for the processes

AB + EF

,

CD + EF .

The rest of this chapter is then devoted to a study of the box diagram amplitude 2q (s ,

t)

. In section 4-3 we transform the box

diagram amplitude from its Feynman parametrised form to a more

convenient form and discuss the implication of the stability conditions

A < (G+E)

,

B <

(

G+F)

,

C <

(

H+E)

,

D

< (

H+F)

for the vertices of the diagram. The boundary of the region of integration in the triple

Fig. 4-1.

integral form obtained in section 4-3 is studied in section 4-4 and in section 4-5 we obtain some results necessary for reversing the order of integration in the triple integral.

The order of integration is then reversed in sections 4-6 and 4-7 and two of the integrations are carried out to obtain a single spectral representation of the box diagram amplitude. We also show that under certain circumstances the single dispersion integral can be written as a double dispersion integral in s and t .

Finally in section 4-8 we study the region in which the double spectral function is non-zero. For a restricted range of s and t

(or x and y ) we find relations between the masses of the interacting particles for which the double spectral representation is valid and give the explicit form of the spectral representation for all allowed mass configurations, including cases for which the double spectral representation breaks down. In particular, we show that for the case when (A+B) < (E+F) , (C+D) 5 (E+F) , the double spectral representation obtained by the heuristic method in sect. 4-2 is the same as that

obtained by the direct transformation of the Feynman parametrisation of the box diagram amplitude.

4-2, Approximation to the i n v a r i a n t amplitude

As discussed in chapter 2, the method of approximating the amplitudes on the right hand side of the unitarity relation and then obtaining a dispersion relation is well known (Mandelstam 1959b). We now give a slightly modified version of Mandelstam’s heuristic method and then in sects. 4-6 and 4-8 compare the result obtained in this way with that obtained from the transformation of the Feynman

parametrisation of the box diagram amplitude.

<p(:pd|(s-i)Ipi4pb >

= - U 2 T l f 2 &<- ^ ) {pc +pD- p A - p B) ^ { E AEBEc EDy ^ T l . s , t ) (4-1)

where

3 = (P^+P S )2 = >

* =

(pA-pC 2

=

tpD-pB)2

are two independent kinematical invariants.

Suppose that there is a two-particle state (EF) with the same internal quantum numbers as (AB) and (CD) , and that (A+B) 5 (E+F)

(C+D) < (E+F) . Then, as discussed in sect. 2-3 , (see also Rasche and Woolcock, 1970), the contribution of this state to

Im ^AB-+CD^S ’ ^ ( s , H • n ' )) in the unitarity relation is

In document The analytic properties of the triangle and box diagram amplitudes (Page 91-96)