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P I (2n)4 p2 _ m2
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PERTURBATION THEORY
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Figure 6-4 Diagrams with a short-circuited propagator, in configuration space (a) or in momentum space (b).
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Let us illustrate these rules to lowest order for the two- and four-point Green functions. Factoring out the global momentum conservation [) function, the contributions of the diagrams of Fig. 6-5 to the two-point function G2 (p, - p) read
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Similarly, the contributions from the diagrams of Fig. 6-6 to G4(Plo P2, P3, P4) (with Pi + P2 + P3 + P4 = 0) are
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(a) i2(2n)4 [ . [)4(Pi + P2) (pi - m2 + is)(p~ - m2 + is)
[)4(Pi + P3) (pi - m2 + is)(p~ - m2 + is) [)4(Pi
+ (pi
p-k i -k 2
- m2
+ P4) ] + is)(p~ - m2 + is)
-- (]) k2
Figure 6-5 Low-order contributions to the two-point function in momentum space.
QUANTUM FIELD THEORY
- - PI
- PI
- P2
- - PI
p,X p ,
Figure 6-6 Low-order contributions to the four-point function.
(Pi -
m 2 + ie)(p~ - m 2 + ie)(p~ - m2 + ie)(p~ - m 2 + ie)
2 j= 1
PI -
i m2
+ ie
(-iAl
d4k i2 (2n)4 (k 2 _ m 2 + ie)[(pl + P2 - k)2 - m 2 + ie]
(d) or (e)
Same as (c), with P2 +--*P3 or P2 +--*P4
In the computation of the corresponding S-matrix elements the external propagators i(Pf - m 2 + ie)-l are just compensated by the i(Dx, + m2) operator of the reduction formulas (5-28). In the latter, the last integration over Xi identifies the incoming momentum of the Green function as defined in (6-20) with the physical on-shell momentum of the S-matrix element. Therefore, to get the contribution of a given Feynman diagram to the S matrix, we omit the external propagators and put the external momenta on the mass shell. As we shall see later, this rule must be supplemented by mass and wave-function renormalizations. When a diagram is disconnected, the Green function G still contains (j functions expressing the conservation of a partial sum of external momenta. In Chap. 5, we have defined the connected Green functions Gc by the recursive expression (6-22)
where 1== {i, ... , n}, XI = {x[, ... , xn}, together with the initial condition G(xt, X2) = Gc(x[, X2) (for the cp4 theory). The diagrams without the vacuum subdiagram
PERTURBATION THEORY
Figure 6-7 A "tadpole" diagram.
involved in the perturbative expansion of G(xt, X2) are (topologically) connected. More generally, it will be shown by induction that only connected diagrams contribute to Gc(Xl, ... , X2n)' Let us assume that this is true up to the (2n - 2)point function, and let us rewrite (6-22) as
G(Xt, ... ,X2n) = Gc(xt, ... ,X2n)
uJ(I.=I 1,<[> 1
I TI Gc(XIJ
(6-23)
After substitution of the diagrammatic expansion for G(x t, ... , X2n) and for each
Gc in the sum of the right-hand side, each contribution from a disconnected
diagram to G(x t, ... , X2n) may be identified with a term of the sum, and vice versa. This shows that the algebraic definition of connectivity [Eq. (6-22)J may be identified with the topological one. Let us return to the role of the normal ordering of the interaction lagrangian in (6-14a). Had we kept the ordinary product 2"m! = - A/4! <p4(X), new diagrams would have emerged, namely, those with contractions of fields pertaining to the same vertex. For instance, the "tadpole" diagram of Fig. 6-7 would appear in the two-point function. In certain circumstances, it may be more convenient to keep the ordinary product and to deal with tadpole diagrams. This is, for instance, what happens when we want to perform a local inhomogeneous transformation on the field (a translation, say), while keeping some symmetry properties. For the cp4 theory, passing from one ordering to the other simply amounts to a change of the mass term. The latter change is actually infinite, but may be included in the mass renormalization that we shall study soon. After this derivation of the Feynman rules for a self-interacting scalar field, we shall now do-more rapidly-the same job for more elaborate and physically more interesting theories, namely, for spinor and scalar quantum electrodynamics. Other cases involving fields with an internal symmetry will be also encountered in the following chapters. At any rate, the general method for deriving Feynman rules should be clear. In any doubtful situation, in particular when symmetry factors are dubious, it may be safer to return to the starting point, that is, Eq. (6-10) and Wick's theorem.
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