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foo dW,2 1m f2(W,2) = ~ 2
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_(W,2) - (J +(W,2) W,2 - w 2
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Here (J (w2 ) refer to the total cross section for a circularly polarized photon with its spin parallel or anti parallel to the spin of the target. The t-channel exchange implied by the helicity flip gives credence to the fact that the integral over the difference of cross sections stands a good chance of converging, hence justifying our assumption of an unsubtracted dispersion relation. From this relation we derive that (11-126) For protons the left-hand side is equal to 205 ,ub while the right-hand side is between 200 and 270 ,ub according to the treatment of high-energy data. Let us apply similar techniques to amplitudes involving axial currents. This
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QUANTUM FIELD THEORY
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will give us some information on low-energy pion-nucleon scattering. We introduce the axial vector matrix element between initial nucleon and final pionnucleon states:
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(2n)4(j4(p1
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+ ql
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- pz - qz)T/
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fd4xe-,ql x<nj(qz)N(pz)IA~(x)IN(pI)
(11-127) According to (11-117) this is related to the pion-nucleon amplitude .oJ"N' To see this we contract both sides of (11-127) with ql and get
(2n)4(j4(p1
+ ql
- pz - qz)qfT/
__ .
mIT - ql
fd4xe-iql'X<nj(qz)N(pz)18I'A~(x)IN(pd) f
Z j"
x e -iql'x
Hence (11-128) When ql goes to zero the left-hand side vanishes unless T/ has a singularity. The only one arises from the nucleon pole (Fig. 11-11). But contrary to what happened in the Compton case this singularity is compensated by the vanishing of the numerator and qfTdk ~ qI/ql PI -+ 0 when ql -+ O. This leads to the Adler compatibility condition for the extrapolated amplitude lim .oJ~'iv(Pz, qz; PI, ql) = 0 ql-->Q (11-129)
This soft pion limit is of course unphysical. An alternative derivation assumes directly mIT = 0 and 81'AI' = O. From q!TIl = 0 we find (11-129) by isolating the pion pole from other singularities. If we stick to this slightly unrealistic world with zero-mass pions, we can write a Ward identity similar to (11-89) for the amplitude
Tj~ =
iq d4x e . x <H(pz) IT
A~(x)A~(O) IH(Pd)
(11-130)
with H an arbitrary hadronic state.
N(Pl)
Figure 11-11 Nucleon pole contribution to the matrix element
<nNIAIN).
SYMMETRIES
It follows that
qllTj~ =
i f d4x
eiq.X<H(P2)1<5(xO)[Ab(x),A~(0)J IH(PI)
= _e jk1 f d4xeiq.X<54(x)<H(P2)1 V;(O)IH(PI) jk1 = -e (Pl
+ pz)TJ
(11-131)
We have used current algebra and the fact that V~ is the isospin current, so that TJ stands for the isospin of the hadron. When PI = P2, TIlV has a double pole at q2 = 0, with a residue proportional to the nH scattering amplitude at threshold Tjk=iqllqVj,2yjk + ... (11-132) IlV (q2 " "H Consequently, the threshold value of Y"H (in the real world when P" q is given by
or !:1"H
m"mH)
threshold
m"mH ~ -~j,2
< 2T
(11-133)
If T stands for the total isospin in the s channel we have the relation
(11-134)
This result may be reexpressed in terms of the s-wave scattering lengths in the various isospin channels defined as
Y~H Ithreshold =
a3j2
8n(mH
+ m,,)aT
(11-135)
For pion-nucleon scattering this yields the values
0.083m;;- I
(11-136)
in surprising agreement with the measured values
a~i~
= (0.171
0.005)m;;-1
a~i~ = -(0.088
0.004)m;;-1
(11-137)
Similar successful calculations may be performed for pion-pion or pion-kaon scattering. We can also reexpress these low-energy theorems in the form of sum rules using dispersion relations. We split the pion-nucleon amplitude in even or odd parts under crossing (Sec. 5-3-4):
(11-138)
and use the variables v = P" q and transfer t. Phenomenological considerations predict the behavior of these amplitudes for large vat t = 0:
V--+oo V--+oo
QUANTUM FIELD THEORY
Therefore it is likely that g-- (v, O)/v satisfies an unsubtracted dispersion relation. The sum rule derived below will then provide a consistency check of this assumption. With the pole contribution of the nucleon intermediate state at 2v + = 0, the forward dispersion relation reads 2 2 g--(v 0) m""NN + _ g 2 1m g--(v /) _----'---'--'- = (11-139) 4 /2 2 V v2 - m n /4 n m.,ptN V _ v
m~ in the Born term denominator):
foo dv' _,-;;--_'-;;-'From (11-133) we obtain the value at threshold (neglecting m; as compared to
g;NN
2 + - foo
n m~v
2" Im.'Y
+ 0 (m;) 2
(11-140)
Finally, we express 1m g- in terms of the total cross sections with the help of the optical theorem and use the Goldberger-Treiman relation to substitute GA/G V for j,,2. This leads to the Adler-Weisberger sum rule
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