**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)

**vQn**IT

**vQn**

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) ,***5*

**(C+D)***, the double spectral representation obtained by the heuristic method in sect. 4-2 is the same as that*

**(E+F)**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****)I****pi**4**p****b >**

= *- 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**

= **(pA-pC 2**

**tpD-p**B)2

are two independent kinematical invariants.
**tpD-p**

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