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(10-44)
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(10-46)
we obtain X and i in momentum space
= __ 1
fd qo pOf(qo,qOp,+ iB + _1 fd qo pOg(qo,qOp,-P)iB P) _ 2in _ fd qo pO _ qOp,+P)iB + _1_ fd qo f*(qo, 2in
(10-47) g*(qo, p, P) pO _ qO - iB
- (P) = __1_ Xp 2in
The relation between X and i is such that their discontinuities are conjugate. A rotation to the imaginary axis will be allowed providedf and g have a suitable support in the variable qo. This follows from their definition by inserting intermediate states f(q, P) =
~ <0
<Pi (0) n) <nJ <pz(O) p)
e-i(P"-~lP).
= I (2n)4 b4(q - Pn + '11 P) <0 <Pi (0) n) <n <pz(O)J p)
(10-48)
g(p, P)
(2n)4 b4(q
+ Pn -
'1zP) <0 <Pz(O)J n) <n <Pi (0) p)
J J J
The stability of particle 1 requires in the first expression p~ ~ mi, p~ > O. This means that f(q, P) vanishes unless ('11P + q)Z ~ mr, '11 Po + qo > O. Likewise in the second expression p~ ~ m~, p~ > 0, hence g(q, P) vanishes unless ('1z P _ q)2 ~ m~, '1zPo - qo > O. In the representation (10-47), the integral of f (or f*) extends from w+ to + 00, the one of g (or g*) from - 00 to W-, with
W+ =
W_ =
Jmi + ('11 P + p)Z - Jm~
'11 Po
+ ('1z P -
+ '1z Po
(10-49)
A rotation to the imaginary axis without encircling singularities will be possible if w_ < w+. For a stable bound state Po < J(ml + mz + pZ. If we choose the center of mass frame P = 0 and '11,Z = ml,Z/(ml + mz), then and
mzPo ml + mz
J mz + p z < 0 z
Consequently, we have a gap between the two cuts in the integral representations (10-47). We may now return to the Bethe-Salpeter equation and perform the Wick rotation using these results. For simplicity we retain the center of mass frame and the previous assignments for '11,Z' Furthermore, we limit ourselves to the
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
so-called ladder approximation, i.e., the kernel V is evaluated in the Born approximation V= ig 1g2 (10-50) (p - p,)Z _ J1z + is The equation takes the form
[(ml ml m2 Po + po)Z _ p2 _ miJ[(ml m2 m2 Po + +
- ig 1 g 2 fd 4p'
(2n)4
po)Z -
pZ - m~Jxp(p)
+ is
(10-51)
Xp(P') (p - p, - J12
Consider the analytic continuation po -+ po e iO with () varying from 0 to n12. For the first term in (10-51) we use the representation (10-47) and examine separately the cases pO> 0 and po < O. Using the properties w- < 0 < w+ we see that no singularity is encountered in the process of analytic continuation. For the second term we proceed to a simultaneous rotation po -+ po e iO, p'o -+ p'o e i8, and for the same reason we do not encounter any singularity from Xp(p') nor from the denominator. Using abusively for () = nl2 the same notations for the functions of the euclidean argument the final result assumes the form
[(po -
ml iml m2 PO)2 + pZ + mrJ[(po + ml im2mz PO)2 + pZ + m~Jxp(p) + +
- (2n)4
Xp(P') glg2 fd4 , p (p _ p' + J12
0) (1 -52
The metric used in (10-52) is the euclidean one (p - p, = L:=o (Pi - pi)2.
The extension of the preceding method to higher kernels requires a more elaborate analysis in several variables. A similar study may also be carried out for the scattering case where part of the integration contour remains pinched between the two cuts. We leave it as an exercise to the reader to examine these cases.
10-2-3 Scalar Massless Exchange in the Ladder Approximation
We present the solution obtained by Wick and Cutkosky for the equation (10-52) when J12 = 0 as an illustration of the methods and problems encountered in studying the relativistic bound-state equations. This choice is not completely arbitrary from the physical point of view since in spite of the oversimplifications resulting in part from the neglect of spin (particularly of the exchanged field) the situation has some analogy with the realistic cases such as positronium. Furthermore, it allows an almost analytic solution and exhibits peculiarities which may arise in more complex models. The set of ladder diagrams looks at first as a natural generalization of nonrelativistic potential theory. However, such an approximation overlooks essential features of a relativistic quantum theory. By neglecting the crossed-ladder dia-
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