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1. Draw all distinct diagrams with 2n external points Xl, ... , X2n and p vertices Yl, ... , YP' and sum all the contributions according to the following rules.
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Figure 6-1 Examples of diagrams with a symmetry factor: in case (a), S = 2!; in case (b), S = 3!.
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QUANTUM FIELD THEORY
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Figure 6-2 Low-order contributions to the two-point function in cp4 theory.
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2. To each vertex attach a factor - iA. 3. To each line between Zi and Zj attach a factor cP(Zi)cP(Zj) given by (6-16). 4. Each diagram has to be divided by a symmetry factor S. As an example, let us examine the two-point function (- iA)P <01 Tcp(Xl)CP(X2) : cp4(Yl) :/4! : cp4(yp) :/4! 10) to low orders. For p = 0, we get cP(Xl)<P(X2), while there is no p = 1 contribution, since we forbid contractions cP(y) <b(y). For p = 2, there are three contributions, depicted in Fig. 6-2
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(a) (b) (c)
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( - iA.f r;---;-, r;---;-, 4 -4-!- CP(Xl)CP(X2) [CP(Yl)CP(Y2)]
The same with Yl +-+ Y2
Let us emphasize the importance and the meaning of the word "distinct" in rule 1 above. In configuration space, diagrams are considered as distinct when they are topologically different, all the points Xi and Yj being fixed. For instance, diagrams (b) and (c) of Fig. 6-2 are distinct, whereas diagrams (a) and (b) of Fig. 6-3 are not; in the latter case only one of them has to be taken into account. Let us divide the diagrams into two classes. The first class contains no vacuum-vacuum subdiagram, i.e., subdiagrams (like the one of Fig. 6-2a) which are not connected to the external points; the second class includes such subdiagrams. To each diagram of the first class, we may associate a set of secondclass diagrams, by addition of vacuum-vacuum subdiagrams. Denoting by the
Figure 6-3 Example of indistinguishable diagrams; only one of them has to be taken into account.
PERTURBATION THEORY
superscript (1) the first-class contribution to the Green function, we may write
G(x1o ... ,X2n)<01 Texp
{i fd4x 2'int[<P(X)]} 10) = p~op! fd4Yl"'d4YPkf:O (p) <01 T<p(Xl)'" <p(X2n)2'int(Yl)"'2'int(Yk) 10)(1) k
x <OIT2'int(Yk+l)"'2'int(Yp)IO)
(6-18)
since the (p - k)th-order vacuum-vacuum subdiagram corresponds to the vacuum expectation value of the time-ordered product of P - k interaction lagrangians. Moreover, the combinatorial factor
p) p! ( k = k!(p - k)!
results from the fact that, after integration over Y 10 .. , YP' all possible permutations of the Y give rise to equal contributions. It is easy to see that the vacuum amplitudes <01 Texp [if d4x 2'int(X)] 10) factorizes on both sides of Eq. (6-18) and
G(X1o"" X2n) =
p=o p.
d4Yl'" d4yp <01 T<p(Xl)'" <p(X2n)2'int(yd'" 2'int(Yp) 1 )(1)
(6-19)
Only diagrams without vacuum subdiagrams contribute to the right-hand side ofEq. (6-19). It may be simpler to give the Feynman rules in momentum space, i.e., to consider the Fourier transform G:
G(P1o ',P2n) =
d4Xl" d4x 2n exp
(-i j~l
(6-20)
Pj" Xj) G(X1o" "X2n)
With our conventions, all the P are incoming momenta. Translation invariance entails momentum conservation Iin Pj = 0, which implies G(Pl"",P2n)= (2n)4t5 4(pl + ... + P2n)G(P1o"',P2n) (6-21) For simplicity we use the same symbol in x or P space. We have to integrate over all space-time points x and Y to get G from (6-19) and (6-20). Using the Fourier representation (6-16) of the elementary contraction, the integration over Xi produces a contribution i(Pf - m 2 + is)-l for each external line. We are left with the integrations over the Yj and the k/, which are the momenta running through the internal lines. Let us concentrate on the contribution attached to a given configuration of lines and vertices for fixed internal momenta and given external configuration arguments x 10 , X2n' Any permutation of the labels Y1o"" YP of the internal vertices yields the same contribution. However, not all
QUANTUM FIELD THEORY
of the p! permutations of the y need generate topologically distinct diagrams in the original configuration space. If this were the case we would get a factor p! compensating the factor (p!) - 1 appearing in Eq. (6-19). It may happen for some diagrams that two (or more) vertices play identical roles (this is the case in the example of Fig. 6-3a where Y3 and Y4 are such vertices). If the p vertices are divided into groups of Vb V2,"" Vs vertices playing the same role (Vl + ... + Vs = p), the integration over the y variables yields a degeneracy factor p !/Vl ! ... vs!. This compensates only partially the (p !)-1 factor of Eq. (6-19) and leaves a vertex symmetry factor I/Vl!'" vs!, which multiplies the line symmetry factor discussed previously. For instance, the diagram of Fig. 6-3a has, in momentum space, a global symmetry factor i xi. Finally, every y integral is of the form
d4Yj e- iYj ' qj = (2n)4J4(qj)
where qj denotes the sum of all external or internal momenta flowing into the vertex Yj. Momentum is conserved at each vertex, and is thus globally conserved. Feynman rules in momentum space may now be written, for the Green function G(Pb ... , P2n), as follows:
1. Draw all topologically distinct diagrams with 2n external lines of incoming momertta Pl," ',P2n and without vacuum subdiagrams. For each diagram, denote by k b . .. , kJ the momenta of internal lines. In a scalar theory without
derivative coupling, the choice of an orientation ofthe internal lines is irrelevant. 2. To the jth external line, assign the factor i/(pJ - m 2 + is). 3. To the lth internal line, assign [d 4k//(2n)4] [i/(kf - m 2 + is)]. 4. To each vertex, assign (- iA)(2n)4 J4(qj) where qj is the sum of all incoming momenta at vertex j. 5. Integrate over the k variables the product of all these contributions and divide by the symmetry factor of internal lines and vertices of the diagram. 6. Sum the contributions of all topologically distinct diagrams. Generally, if V is the number of vertices (called p previously) and I the number of internal lines, there are V-I conservation rules among the k after separation of the global conservation (6-21), and thus at most I - V + 1 nontrivial integrations. The Feynman rules must be slightly modified when some external lines are connected to no vertex at all. For instance, the single propagator of Fig. 6-4 gives in configuration space the contribution
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