QUANTUM FIELD THEORY in .NET

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QUANTUM FIELD THEORY
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vanishes at infinity. Then we can write
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dv' 1m Y(v') ( 1 + -,-) -,-1
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(5-177)
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If Y(v) would grow like a power of v a similar representation would hold for a function obtained by dividing v by a real polynomial in v. We must emphasize that only physically measurable quantities enter the above representation. The absorptive part 1m Y(v) is related to the total cross section through the optical theorem, Eq. (5-156), while the possible pole terms have as residues products of coupling constants. Indeed, formulas of the type (5-177) enable us to measure such coupling constants.
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As an illustration, consider the time-honored example of pion-nucleon scattering. For arbitrary charge we write the two independent amplitudes in the form (5-178) with ii(P2), u(pd the on-shell Dirac spinors describing the final (momentum P2) and initial (momentum PI) nucleons. The incoming pion has momentum ql and the final one q2. The kinematical invariants are
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+ ql)2
(5-179)
t = (ql - q2)2 = _2q2(1 - cos 8)
u = (PI - q2)2
+ 2Jl2 -
S- t
Here M and Jl are respectively the nucleon and pion mass, 8 is the center of mass scattering angle, q is the pion three-momentum in this frame, and W is the total center of mass energy in such a way that (5-180) The differential cross section is, according to Sec. 5-1-1, (5-181) and in (5-178) the functions A and B only depend on the invariants. Anticipating the discussion of symmetries (Chap. 11) we know that strong interactions are isospin invariant and that the nucleon has isospin i and the pion isospin 1. Therefore all the channels are described in terms of two reduced matrix elements corresponding to total isospin i or ~, that is, A and B are combinations of A I, A 3 and BI, B3 respectively for any process with definite charges. This gives the following table:
(5-182)
The last is called a charge exchange reaction. Since only two reduced amplitudes describe three measurable processes, this implies triangular inequalities among cross sections. It is convenient to
ELEMENTARY PROCESSES
introduce the combinations
(5-183)
respectively even or odd under S+->U exchange. From the explicit structure of Eq. (5-178) and using the variable
(5-184)
odd under crossing (note that for t follows that
0 it coincides with the laboratory incoming pion energy), it
A( >(-v,t)=
A( )(v,t)
(5-185)
B( )(-v,t) = +B( )(v,t)
At t = 0 the amplitudes are analytic (in v) in a plane cut along the real axis from the elastic threshold [s = (M + Jl)2, v = JlJ to infinity (the s cut) and from minus infinity to the u channel threshold [u = (M + Jl)2, v = -Jll Furthermore, isolated poles occur along the real axis corresponding to the nucleon intermediate state in the sand u channels. We may write the effective pseudoscalar pion-nucleon interaction (5-186) with r. being the nucleon isospin matrices and n. (IX the pion field. Thus the pole contributions are
or 2 PI Y pole = g.NNu(P2)rpr.ys 2
1,2,3) the three hermitian components of
+ 41 + M
Pl ql
YSU(pl)
+ g.NNu(p2)r.rpys
PI Jl 2
41 + M
Pl ql
YSU(Pl)
(5-187)
Now 2Pl ql
= 2Mv - t/2;
we may use the Dirac equation satisfied by on-shell spinors to write
(5-188)
We recognize that the poles only contribute to the B amplitudes with
B~+) =
g;;NN (_1~ 2M vp - v
g;;NN B (_) -~p - 2M vp - v
__+ 1~) (1- + -+-) 1 vp v vp v
(5-189)
Note that the location of the poles is t dependent. Inserting the value of vp at t = 0 and ignoring for the moment the question of subtractions, we find the forward dispersion relations in the form
A( )(v, 0) = -1 f<Xl dv' 1m A( )(v', 0) ( 1- -1 ) 1t
v'-v
v'+v
(5-190)
v' - v v'
2 B( >(v,O)=g NN ( 1 2M vp - v
1 1 --+ -1 ) +- f<Xl dv' 1m B( >(v', 0) (--+--) - 1
QUANTUM FIELD THEORY
In fact analyticity in s may be proved for fixed (negative) t in a finite range -tM (5-190) may be generalized to
A( )(v, t) = -1
t ~ 0, so that
~+t/4M
dv' 1m A( )(v', t) ( 1 - -,-) -, 1
g;NN B( )(v,t)=-- ( 1
vp - v
---+ -1 ) +- f'" - 1
(5-191) 1 -,-+ -,-)
dv'lmB( )(v',t) ( 1 n ~+t/4M v -
These relations may be used in conjunction with a partial wave analysis along the lines of Sec. 5-3-1. For this purpose, we rewrite the quantity (Mj4nW)ff" appearing in the cross section (5-181) as (5-192) where Al and A2 are the initial and final nucleon helicities and X, that [compare with (2-37)]
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