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A) = ----;=:P,=+=M= (X(A))
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With E denoting the nucleon center of mass energy this yields
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(5-193)
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The 2 x 2 matrix appearing in (5-192) admits an angular decomposition of the following form:
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/;.,,)., =
+ i) f1,').1 f.& i:,).,(cp, 8, 0)
(5-194)
where J runs over half-integer values. Invariance under parity yields fl~2,1/2 = f! 1/2,-1/2 and f02,-1/2 = f! 1/2,1/2. Time reversal does not give any new information. Using the spinors of definite helicities we have, at cp = 0,
fl/2,1/2 fl/2,-1/2
f-l/2,-1/2 -f-l/2,1/2
= ( 1
til 2 f + f) X2(-Z)xlC2-) =
8 + f2) cOS:2
(5-195)
= (/1 -
fz)x1(i)Xl(-i)
= (/! -
sin~
(5-196)
As was already mentioned, parity diagonalizes the matrix f).I0).2 in such a way that
/!~2,1/2
f! 1/2,1/2 = 2e i
J .
sin i5 J ,
The phase shifts 15 J, are real below the inelastic threshold and correspond to the scattering in the channel of total angular momentum J and parity ( _1)1+ 1/2. In nonrelativistic notation these states would be obtained from well-defined orbital angular momentum I with J = I i so that the corresponding phase shifts are also traditionally denoted 15,,+ and i5l+ 1,- respectively. Let us now use the explicit formula for the Wigner f.& functions;
+ !)d{/2,1/2 =
cos:2 (Pi+ 1/2
Pi-l/2)
(5-197)
+ !)d-'-I/2,1/2 =
8 sin:2 (Pi+ 1/2
+ Pi-l/2)
ELEMENTARY PROCESSES
With the I notation ((jJ,+ == (jl,+, (jJ,- == (jl+ 1,-) this yields the following expansions
fl(cos 8)
Pi+ I(COS 8)ji,+ - Pi-I(cos 8)ji,_ Pi(cos 8)(ji,+ - ji,-)
(5-198)
h(cos 8) =
e ib /, sin bz, ji, =---q
Of course, for I = 0 the only surviving amplitude isfo,+. The question of subtractions in the dispersion relations should now be investigated. We postpone some general statements until the next subsection except to say that in pion-nucleon scattering it can be shown that the only amplitude requiring a subtraction is A( +). A classical application of Eqs. (5-190) is to the determination of the effective coupling constant. The principle of the method is to evaluate accurately from low-energy phase shift analysis the B amplitude for n+ -proton elastic scattering (it is, in fact, dominated by the J = i, T = i resonance) and to compare its real part in the forward direction with the value given by the dispersion integral. The latter is also dominated by its low-energy contribution so that the approach is consistent. The value quoted by Hamilton and Woolcock is
4n \ 2M
~ (gnNNfl)2 =
+ 0.002
(5-199)
Dispersion integrals playa leading role in disentangling the various phase shifts at intermediate energies. As an exercise we suggest extending the formalism to the t-channel reaction nn ..... N N.
5-3-5 Momentum Transfer Analyticity
In the sequel, we disregard the spin of the external particles. Even though it is not a straightforward
matter to include it, this can be done without modifying the results. Instead of using as a starting point of our discussion a reduction formula involving a particle in both the initial and final states, as we did in Sec. 5-3-2, let us rather reduce two initial particles in an elastic process with momenta PI and ql and associated fields t/J and <p. Thus we would initially write the connected S-matrix element
(P2,q2[S[Phql\ = -
dx 4 d4y e-i(p,.x+q,.y) (Ox
+ ml)(Oy + m;)(P2,q2[ Tt/J(x)<p(y) [0),
(5-200)
Returning again to the derivation of this formula, we observe that we might as well have used a retarded commutator instead of a chronological product in the physical region of the elastic process. Dropping inessential contact terms, we may therefore consider the replacement (with j and j", the sources of the t/J and <p fields)
+ mlj(Oy + m;) (P2, q2[ Tt/J(x)<p(y) [0), ..... (pz, q2[ 8(xO + q2 PI - ql)i5!i
yO) [j (x), j",(y)] [0 \
(5-201)
Taking momentum conservation into account, we obtain
(P2, q2[ S [Ph ql \ = (2n)4 (j4(P2 :TIi = i
d4x ei(q,- p,) x/2 (P2, q2[ 8(XO{j G} j",(
-~)]10),
(5-202)
This representation shows that :T is analytic in the variable q = (ql - PI)/2, as follows from the Jost-Lehmann-Dyson formula, and we may repeat an analysis similar to the one given for forward scattering. We take the total incoming momentum P = PI + ql as a time axis. Thus we shall work in the center of mass frame and exploit (5-202) to obtain analytic properties in the cosine of the
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