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The interpretation of the sum rule is still complicated because of its dependence on Po. Instead of choosing the nucleon rest frame, where Po = m, it is more clever to use a limiting one obtained by letting Po go to infinity with q2 held fixed: Po, [P [ -+ 00 P. q = 0 (11-96) q2 -+ _ q2 fixed
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If we are allowed to interchange limit and integration we find that
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This is a severe constraint in view of (11-91) and (11-92), since in particular it implies that the left-hand side is independent of q2. It tests the local character of current algebra. There are several applications of this type of sum rule. We quote only a few of them. The Cabibbo-Radicati sum rule is obtained when applying Eq. (11-97) to the isospin vector currents Vi +i2, V i - i2 . It reads V 1 dv 2 dF (0) = [FV (0)J2 + -2-----} _2_ - (20"f/2 - 0"~/2) (11-98) dq 2m 2n IX Vthresh V
The notation Ff(q2), Fi(q2) stands for the isovector nucleon form factors, i.e., such that the matrix element of the electromagnetic current reads
< [JI' Pi >=-[YI' (Ff+Ff'[3) + iO"l'v qv(F~+Fr'[3)JUi P2 'em(o)[ U2 2 2m 2
and O"f/2' 0"~/2 are the total hadronic cross sections for the scattering of "isovector" photons off nucleons in the channels of total isospin i or i respectively. The
Figure 11-8 Inclusive process v + N
+ X.
experimental values for the two sides of the sum rule are in good agreement. In the unit of the inverse square pion mass one finds
dFf -2 2 dq2 (0) = 0.132m" F,{(0)]2 [ -2m
1 + - 2 - fro
Vthresh V
v V (20'1/2 - 0'3/2)
-2 0.126m"
The inclusive cross section for the process neutrino (v) or antineutrino (N) -4 lepton (I) + X, where X is an unidentified hadronic state (Fig. 11-8), is expressed in terms of the structure functions Wr, W2, W3 appearing in the expression (11-93) as
+ nucleon
dO'(v,V) d 1 q21 dv
= -- -
G2 E' (
2nm E
2wlv'V)sin 2 - + wi"'V) cos 2 - =+= W~v,V) - - - sin 2 22m 2
+ E'
(11-101) The kinematical variables v and q2 are related to the incident neutrino energy
E, final lepton energy E', and laboratory scattering angle () through
p. q
m(E - E')
The lepton mass has been neglected for high-energy experiments. This explains the disappearance of W4 , Ws, W6 in (11-101) and the simple expression for the momentum transfer. Such processes will be studied in more detail in Chap. 13. For fixed q2 and v, when E (hence E') grows, () tends to zero and the cross section may be approximated by the W2 contribution alone. Therefore
dO'(v,V) G2 lim - d = 2E-+ro q nm
1 21
dv wi"'V)(v, q2)
Crossing symmetry implies
W;(V)(v, l) = - W;(V)( - v, q2)
Current algebra then allows us to write the Adler sum rule
l:! ro dl q2 1- dlq21
( dO'(v)
dO'(V) )
2 G[
~ 2T3 cos
3Y) . 2 ]
with T3 the third component of isospin, Y the hypercharge on the target, and (}c the Cabibbo angle. An analogous result applies to the inclusive inelastic electron scattering e + N --+ e + X using the vector current -V 1 +i2. Since current algebra does not apply to the isoscalar part of the electromagnetic current the prediction obtained by Bjorken is only an inequality for the sum over neutron and proton targets: (11-106)
11-3-2 Approximate Conservation of the Axial Current and Chiral Symmetry
An SU(2) x SU(2) Lie algebra is generated by the commutation rules of vector and axial charges. Assuming that axial currents are approximately conserved, the resulting symmetry is called chiral symmetry. It is implemented in the Goldstone mode with the pions as massless particles. This hypothesis enables us to deduce new consequences from current algebra in the form of sum rules and low-energy theorems. The generalization to SU(3) x SU(3) including strangeness changing currents appears more questionable. Consider the axial current matrix element between a pion state and the vacuum
<01 At(x) Ink(p)
iPIlJjk I" e- ip . x
This amplitude determines the rate of the decay n
G2 2J,2( 2 r " .... Il v -_ mil"4 m"3 -
p - vas
2 ()
2)2 mil
Experimentally, I" is measured as
I" ~ 93 MeV
From (11-107) the matrix element of all AIl(x) is
<OloIlAt(x)lnk(p) = m;JjJ<I"e- ip . x
Therefore current conservation implies I"m; = O. We have the choice between two possibilities. Either I" or is zero, in contradiction to experimental facts. Nevertheless, let us boldly imagine a world with massless pions as a tentative approximation. If this SU(2) x SU(2) group is implemented with an invariant vacuum we are led to an unrealistic world with parity degenerate multiplets and corresponding selection rules. On the other hand, a Goldstone realization is in agreement with the fact that pions have a mass much smaller than other mesons. Let us therefore explore the consequences of setting all All = 0 and m; = o. Compute the axial current matrix element between nucleon states
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