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(a) To order h we cannot distinguish between counterterms of the form omli/ift and Z2omli/ift since (Z2om)[1] = Om[1]. (b) In the Landau gauge.le .... CIJ the ultraviolet divergences cancel in Z2 1 to order one. Unfortunately,
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no unique choice of gauge eliminates all ultraviolet divergences perturbatively.
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(c) All counterterms introduced up to now have a structure similar to the terms in the original
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lagrangian, pointing toward the success of the renormalization program. The original observation that the electron self-mass is only logarithmically divergent is due to Weisskopf (1939), and the first complete calculation to Karplus and Kroll (1950).
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7-1-3 Vertex Function
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After studying the two-point functions we now face the three-point vertex function
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(eey). The unique one-particle irreducible diagram pertaining to this process is
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shown in Fig. 7-10 and is given by the expression
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-lel/L (p, p)
d k 1 [gp"
k 2 _ fl2
+ is + (k 2 i
(1 - A)kpk"
+ is)(Ak2
- fl2
+ is)
x ( y"
P' - ~ -
m + is Y/L
P- ~ -
m + is Y
Using the same notation, the zeroth-order value is y/L. Accordingly we shall write the complete one-particle irreducible three-point function
A/L(p', p) = Y/L
+ 1/L(p', p)
with Eq. (7-45) giving the first nontrivial contribution to 11'"
q =p'_p
Figure 7-10 Vertex diagram to the one-loop order.
Figure 7-11 Insertion of an external photon line in an electron selfenergy diagram leading to the Ward identity.
If we recall the introduction of the electromagnetic coupling through the minimal substitution p-+ p - eJ we may expect a close relationship between the electron propagator and the vertex function. This is indeed the case and is embodied in an identity due to Ward. Consider the insertion of an external photon line of zero momentum in an electron self-energy diagram, in all possible manners, on the internal charged particle propagators (Fig. 7-11). The term of zeroth order yields i(p - m)-+ - ieJ, while the one-loop contribution as depicted on Fig. 7-11. leads to - ieI[l](p) -+ - ieAl'rJl](p, p), and so on. The graphical correspondence may be interpreted as follows. In each internal charged particle propagator of a selfenergy diagram we substitute p - eJ for the running momentum p with AI' a fixed constant four-vector. We then expand in powers of eA and extract the coefficient of the linear term. This is r I'(p, p).
For instance, if we apply the procedure to the one-loop contribution to L(P) we find
"(P) ---> L..
- - ( -Ie
0 oeA"
. )2' I
d k 1[ g p. (2n)4 i k 2 - J12
+ iE
(1 - J,,)k pk. + ---co--~--'-~---=---] (e - J12 + iE)(J"e - J12 + iE)
x yp
p- ~ -
ej - m + iE 1
P- ~ -
ej - m + iE
p - ~ - m + iE + P- ~ - m + iE ej p - ~ - ej - m + iE
after differentiation, and setting A equal to zero, we recover an expression for P(p, p) consistent with Eq. (7-45).
We could ask whether the above operation yields all vertex diagrams. The answer is obviously "no." Consider in a self-energy diagram an internal closed fermion loop. According to Furry's theorem an even number of photon lines are attached to this loop (if we agree to cut through each internal photon line starting and ending on this loop). Again according to Furry's theorem, it is therefore impossible from the above procedure to find a vertex contribution where an internal fermion loop would be attached to the rest of the diagram by an odd number of photon lines, such as depicted on the example of Fig. 7-12. This
Figure 7-12 Example of a diagram that cannot be obtained from by the procedure described in the text.
means that we obtain only those diagrams where the external photon is attached to the fermionic line that carries the charge flow originating in the external line. This may at first seem troublesome if we are to find a relation between Land r. Fortunately, the sum of all diagrams where the external line is attached in all possible ways to an internal electron loop vanishes when evaluated at zero momentum transfer q. This we can see after a suitable Pauli-Villars regularization of the loop by isolating a factor of the form
( Lf
d4 p tr
1 . p _ m + IS I'fl.
1 . 1'2 m + IS
P+ r1i2
. + IS
P+ 42 + 43 - m + is P where the sum of all momenta entering the loop q2 + q3 + ... + q2p vanishes.
As before,
X 1'3
1'2 )
1 m + is I'fl.
1 m + is
1 m + is
Summing over all possible insertions of the external photon for fixed values of the internal photon momenta we get a total derivative as an integrand. The integral vanishes provided it is regularized in a gauge invariant way. We conclude that the prescription of attaching an external photon may be restricted to the distinguished set of electron propagators carrying the external charge flow. The same set may be seen to carry the electron momentum flow and the prescription is then equivalent to a derivative with respect to this momentum. This yields the Ward identity as
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