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N(P2)
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Figure 11-9 Pion contribution to the axial current matrix element between nucleon states.
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with q =P2 - Pl. Comparing with (11-69) we find
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gA(O) = GA/G V
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(11-112)
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Current conservation implies
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2mgA(q2)
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+ q2hA(q2) =
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(11-113)
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from which it is not wise to conclude that gA(O) = O. Indeed, the form factor h(q2) has a pole at q2 = 0 corresponding to pion exchange (Fig. 11-9). The corresponding contribution reads
if" q q~
ig"NNU(P2)Y5 rju (Pl)
(11-114)
with g"NN the pion-nucleon effective coupling constant (Sec. 5-3-4) approximately given by
g"NN 4n
(11-115)
From (11-113) and (11-114) we obtain at zero transfer the Goldberger-Treiman relation (11-116) in agreement with the experimental data, within a 10 percent error [the righthand side of (11-116) is 1.34 to be compared with 1.22]. This is a quite remarkable result since it relates quantities from strong (g"NN) and weak (j", GA/G V ) interactions. In the real world with massive pions we would like to have some control This is achieved by over the extrapolation from the value q2 = 0 to q2 = assuming a partial conservation law (PCAC)
o"At(x) = m;j"nj(x)
(11-117)
which identifies the divergence of the current with a smooth interpolating pion field. We know already that o"A" has the correct quantum numbers. From Eq. (11-110) it may indeed be used to generate the asymptotic pion states if j" =1= O. Equation (11-117) is supplemented by an hypothesis of a smooth extrapolation of form factors beyond the mass shell. This means that matrix elements of 0" A" are dominated by the pion pole for small values of the transfer
QUANTUM FIELD THEORY
momentum [q2 [ ;5
m;; :
(11-118)
where C is a residue to be determined. This is what is meant in practice by partial conservation of the axial current.
11-3-3 Low-Energy Theorems and Sum Rules
With current algebra and PCAC, we are in a position to derive low-energy theorems. The situation is analogous to the one prevailing in electrodynamics. We shall therefore begin with a study of low-energy Compton scattering. To simplify we write the amplitude for photon scattering on a spin less target of unit charge (Fig. 11-10):
(2n)4t5 4(pl
+ kl
- P2 - k2)!T !T
2 d4x d4y 2 d4x cqcl
c~cr
e ik
ei(k"x-k!"y)
<P2 [Tjll(x)jV(y) [Pi) (11-119)
,. x
<P2 [ TMx)jv(O) [Pi)
Choosing for convenience the polarization vectors such that Cl kl = C2 k2 and taking Lorentz and time-reversal invariances into account, we write the general form of Tllv as
+ BPIlP v + C(Pvklll + P ll k 2v) + Dk 11l k 2v (11-120) Here P stands for the average target momentum, P = (Pi + P2)/2, and the scalar
Tllv = Agllv
amplitudes A, B, C, and D are free of kinematical singularities. From current conservation it follows that k~ Tllv = O. Therefore
+ p. k 2C + kl . k2D = P k 2B + kl k 2C =
(11-121) 0
In the low-energy limit kb k2 -+ 0, the amplitudes B, C, and D are dominated by their dynamical pole terms given by Born terms with renormalized residues.
Figure 11-10 Compton amplitude and Born terms.
SYMMETRIES
An elementary computation leads to the value of the amplitude A at threshold lim A = 2 (11-122)
In this limit, but without any perturbative approximation, we therefore find for the cross section
kl.k,..-+O
- = 2( 1' 2)
(11-123)
in agreement with the classical evaluation of Chap. 1 and the lowest-order calculation (Sec. 5-2-1). Low, Gell-Mann, and Goldberger who first developed these arguments obtained from the same hypothesis the next term linear in k1 and k2 for spin i targets. The amplitude ' for forward scattering is written (with w the common laboratory energy w = P' kim) (11-124) The polarizations are now taken to have vanishing time components. The lowenergy theorem then states
f1(0)
= --
f2(0) = - 8m 2 (g -
(11-125)
The amplitude f2 involves the anomalous part of the magnetic moment of the target (g - 2 = 3.58 for the proton). The first relation is of course equivalent to (11-122). This prediction is difficult to test directly. It is better to transform it into a sum rule as suggested by Drell and Hearn. We use an unsubtracted dispersion relation for f2(W 2) [the corresponding one for f1(W 2) would require at least one subtraction] to write
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