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+ ~2~3PI + ~3~IP~ + ~2 + ~3
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~lmI - ~2m~ - ~3m~
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and the Landau equations, after elimination of ~1 = 1 - ~2 - ~3, yield, for ~ = ~2 = ~3 (0 < ~ < since 0 < ~1 < 1), ~2(4p2 - p~) = m~ { (4p2 _ p~)(~2 _ ~) = m2 _ p2 The solution ~ lies between 0 and i provided
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p2 > m 2 + m~
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(6-120)
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This expresses condition (6-119) in that special case. The anomalous threshold takes place at
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If we consider a physical instance such as the electromagnetic form factor of the nucleon, it seems that we are in trouble sinc-e an anomalous threshold might occur below the normal one. Indeed, the lowest state coupled to the photon is the two-pion system, m = ml = m2 2: m., and the lowest one coupled to the on-shell nucleon (p2 = PI = p~ = M&) is the pion-nucleon state; hence m + m3 2: m. + M N. If we take m3 = m = (m. + MN)/2, all the previous conditions are fulfilled; Eq. (6-120) reduces to l)MN 2: m. that is,
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and is obviously satisfied by the experimental masses. Fortunately, this reasoning has not taken into account the various conservation laws, e.g., the conservation of baryonic charge which tells us that any state coupled to the nucleon must contain at least one baryon, of mass larger than M N Thus the condition (6-120) cannot be fulfilled and we are safe. Similarly, anomalous thresholds are not present in the form factors of pions or kaons, but they appear for hyperons or deuterons.
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QUANTUM FIELD THEORY
---- k
mZ m3 m4
Pi -P3 fk
Figure 6-33 The box diagrams in the s - t channel (a) and in the crossed u - t channel (b).
The analyticity properties of the scattering amplitude depicted in Fig. 6-33a, the box diagram, may be studied along the same lines:
d4k -2 ( n)4 [(P 1
+ k)2 -
2J m2 [ (Pi -
+ k)2 -
2J 2J 2 m4 [ (P2 - k) 2 - ml (k 2 - m3)
(6-121)
In the s channel, we find a normal threshold at S = (ml + m2)2 [in the t channel at t The discontinuity across the corresponding cut may be evaluated:
TG(s
+ m4)2J.
+ ie, t) -
TG(s - ie, t)
= 2i~s TG(s, =
[(ac - bd cos 8) + Dl/2] [ . 2s.le 1/2(S, mI, m~) 21 In 8 s - (ml 4nDl/2 (ac - bd cos 8) - Dl/2
+ m2)
(6-122)
where 8 stands for the center of mass scattering angle cos 8 =
.Ie 1/2(S, pI, pn.le 1/2(S, p~, pi)
[ - u + -"-----'----'--"------=--t ( P I - p~)(p~ - pi)]
The expressions of a and b have been given in (6-118), and c and d are obtained from a and b by changing Pi into P3, P2 into P4, m3 into m4' Finally, These expressions simplify in the case of equal masses mI = m~ = m 2, m~ = mi = m'2, to
PI = p~ =
(6-123)
p~ = pi = p2
The discontinuity ~s TG considered as a function of t has itself a discontinuity in the t channel, called the double spectral function Pst:
Pst(s, t)
1 2i [~s TG(s, t
+ ie) -
~s TG(s, t - ie)J
2s.le 1/2(S, mI, m~)
8[s - (ml
+ m2) J8[t -
+ m4) J8(D)
(6-124)
Therefore, TG(s, t) satisfies fixed s or fixed t dispersion relations, or even an analytic representation in the set of variables sand t:
PERTURBA nON THEORY
TG(s, t)
A, TG(s', t) ds' ---'---,,---'--'--'S S
(ml+m2)2
(6-125a)
(s' d 'd' p,t, t') S t (S' _ S)(t' - t)
The second expression in (6-125a) involving the double discontinuity P" is a special case of a Mandelstam representation. The box diagram considered here has vanishing double spectral functions P,. or Pt . This is not the case of the crossed diagram of Fig. 6-33b, for which Pt. only is nonvanishing, and which therefore satisfies the fixed s dispersion relation
(6-125b)
or the Mandelstam representation
TG(t, u) = -1 1(2
dt'du'
(t' Pt., u') (t' - t)(u' - u)
(6-125c)
All this is not completely correct. In the foregoing, we have assumed that all external particles are stable against decay: PI < (m2 + m3)2, etc., and that no anomalous threshold occurs. The precise conditions for this may be studied in the same way as in the vertex case. It has been conjectured that, more generally, in favorable cases such as pion-nucleon scattering, the physical scattering amplitude satisfies a general Mandelstam representation of the form
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