Unitarity and Partial Wave Decomposition in .NET framework

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5-3-1 Unitarity and Partial Wave Decomposition
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Unitarity of the S matrix reflects the fundamental principle of probability conservation. Even though we have to introduce in certain instances the artificial device of an indefinite metric in Hilbert space, the physical quantities always refer to states with positive norm, preserved through the time evolution. Formal developments should not obscure this important fact, reflected in a number of relations, the prototype of which is the optical theorem of Bohr, Peierls, and Placzek. Let us therefore look at the restrictions implied by the unitarity of the S matrix written as
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sts = I = I + i(T -
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(5-152)
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in accordance with Eq. (5-6). Inserting momentum conservation
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<II T Ii> = (2n)4J4(Pf
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(5-153)
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QUANTUM FIELD THEORY
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we obtain the nonlinear relation among amplitudes:
( ji - Yif) = i I (2n)4J4(Pn - P;)Y~fg;,i
(5-154)
involving a sum over all possible intermediate states In) coupled to Ii) and If). In practical cases the scattering process is initiated by a two-body state. Let us concentrate on two-body elastic scattering. Generalizing this definition we may assume that the particles exchange spin or internal quantum numbers, in agreement with the corresponding conservation laws, provided they remain within the same symmetry multiplet (see Chap. 11). For simplicity we ignore the case of identical particles. First choose If) = Ii) in Eq. (5-154). This corresponds to forward scattering with spin and internal variables equal in the initial and final configuration. We have to assume here the absence oflong-range forces. The left-hand side of Eq. (5-154) reduces then to 2i 1m ii' The right-hand side is related to the total cross section (}tot(i) up to a flux factor contributed by the initial state. To be specific, denote by (ma, Sa) and (mb' Sb) the masses and spins of the initial particles. For a given spin state i and total center of mass energy squared s we derive..' from Eq. (5-13),
(}tot(i) = 2Al/2( 1 2 2) I (2n)4J4(Pn s, ma, mb n
Pi)Y~ig;,i
(5-155)
(5-155a)
where
and we have used in (5-155) a normalization of states appropriate to an invariant phase space element d 3 p/2E(2n The common center of mass three-momentum Ipi of particles A and B is related to s through 4Sp2 = A(s,m;,m~). Using (5-155) we find the optical theorem in the form (5-156) The amplitude ii enters in the expression of the forward elastic cross section. Assume the polarizations to be such that the initial state is invariant under rotations around the incident momentum. The elastic cross section can then be integrated over the azimuthal angle and expressed in terms of the momentum transfer t, instead of the cosine of the scattering angle. In the process A + B -+ A + B call (Pa, Pb) and (p~, pI,) respectively the initial and final momenta satisfying energy momentum conservation Pa + Pb = p~ + pl,. The Mandelstam variables are defined as
+ Pb =
+ Pb
t = (Pa - p~ = (Pb - Pb)2
+t +u=
2(m;
+ m~)
(5-157)
= (Pa - Pb
(Pb - p~
The differential elastic cross section can be written as (5-158)
ELEMENTARY PROCESSES
Forward scattering is characterized by t
0 and 3(8,0) is what we denoted as
:Yii above. Consequently,
dUel ( 0):"- _1_ [ (Re 3{i f S, 2 2 dt 16n A(S, ma , mb)
+ U 2 (.)] > U~t(i) tot I 16n
(5-159)
In very high-energy collisions it may happen that the imaginary part of the forwardscattering amplitude dominates over the real one. In this case Eq. (5-159) may serve to normalize the differential elastic cross section, assuming the total cross section to be known. It is possible to extract further consequences from the unitarity condition if we succeed in diagonalizing the 3 operator, at least partially. For two-body scattering this is achieved by using as a basis the eigenstates of the total angular momentum. The helicity formalism of Jacob and Wick is best suited to perform the corresponding projection. Call Aa and Ab (A~, AI,) the helicities of the initial (final) particles in the center of mass frame where Pa + Pb = p~ + Ph = O. For a twobody state in this frame let () and <p be the polar angles of the relative threemomentum P with respect to a fixed system of axis. Consider R o,,,,, the product of a rotation of angle () around the y axis followed by a rotation of angle <p around the z axis. This rotation transforms the unit vector z into the vector pil P I. A state of total angular momentum J with projection M along zis obtained as IJ, M, Aa, Ab) =
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