Loopwise Expansion in VS .NET

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6-2-1 Loopwise Expansion
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The loopwise perturbative expansion, i.e., the expansion according to the increasing number of independent loops of connected Feynman diagrams, may be identified with an expansion in powers of h. By definition, the number of independent loops is nothing but the number of independent internal four-momenta in the diagrams, when conservation laws at each vertex have been taken into account. For a connected diagram, with I internal lines and V vertices, we have V c5 4 functions expressing this conservation, and after extracting the conservation of the incoming momenta, we are left with V - 1 constraints. Therefore, the number of independent momenta or loops is
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L=I-(V-l)
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(6-69)
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Notice that L is not the number of faces or closed circuits that may be drawn
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QUANTUM FIELD THEORY
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Figure 6-22 The tetrahedron diagram has only three independent loops.
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on the internal lines of the diagram. For instance, the tetrahedron diagram of Fig. 6-22 has four closed circuits but only three independent loops. To find the connection between L and the power of h, we collect all factors h. We leave aside the factor h that gives the mass term a correct dimension. In other words, the Klein-Gordon equation should read [o~ + (mc/hf] cp = 0, indicating that the mass term is of quantum origin. This phenomenon is disregarded in the sequel. There are thus two origins of such factors. First the commutation (or anticommutation) relations imply a factor h, for example, [cp(x), n(y)] = iMP(x - y), which leads to a factor h in each propagator
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<Tcp(x)cp(y)
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e,k.(x-y)
+ u;
Second, the evolution operator e- iHt/ h contains h explicitly, and so does exp [(i/h) J 2 int (CPin) d4 zJ. Thus, we have a factor h for each propagator and h- 1 for each vertex. The total power for a diagram with E external lines is hI+ E - V = h E -1+ L. For a fixed numbh of external lines, i.e., for a given Green function, the announced result follows.
In units where c = 1, a scalar field cp has dimension [cp] ~ (energY/length)1/2, the coupling constant Aof Sf,n! = Acp4 has dimension [n] -1 ~ (energy x length)-I, and a spinor field has dimension [1/1] ~ (energy)1 /2/length. This gives the right dimension [n] to the actions
d z G(8CP
+ Acp4]
~(i~ - ~)I/I
Of course, the lowest-order diagrams in h are the Born diagrams without any loop, such as those computed in the preceding section. The reader may then wonder why this topological loopwise expansion coincides with the expansion according to powers of the coupling constant, in a theory with a single coupling constant. This is because in such a theory there exist auxiliary relations between V, the number of vertices (the power of A), and L. Take, for example, the cp4 theory; counting the total number of incident lines at each vertex tells us that
4V= E +21
PERTURBATION THEORY
for a diagram with E external lines. Eliminating I with the help of (6-69), we get L - 1 + E/2 = V(E is even).
6-2-2 Truncated and Proper Diagrams
We introduce some terminology that will prove useful in the sequel. The truncated functions are defined through the multiplication of Green functions in momentum space (without the c5 4 function of total energy momentum) by the inverse two-point functions pertaining to each external line :
n> 2
(6-70)
The two-point function G(2) is referred to as the complete propagator. For p2 '" m 2, we hav~ [G(2)(p, _p)]-1 '" (iZ)-1(p2 - m 2) where Z is the wave function renormalization introduced in Chap. 5. Hence, up to powers of Z, the on-shell values of these truncated functions are the quantities entering the reduction formulas. For instance, the connected part of the matrix element of Eq. (5-28) reads . P1, ,Pn OU t q!, ,qm III )c - z(n+m)/2 G(n+m) ( -Pb"" -Pm q!, ,qm) I 2_ 2_ 2 t
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