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T'(k) ~ in .NET
T'(k) ~ PDF417 2d Barcode Recognizer In Visual Studio .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in VS .NET applications. PDF 417 Creator In .NET Using Barcode drawer for .NET Control to generate, create PDF417 image in .NET applications. ,~o(4n
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g2C [5 1 + 16n2 "3 + "2"(1  1] 2 A ) ~
(12114) To this order, the term 2/10 may be regarded as corresponding to In (A 2/J12) in a conventional regularization. The wavefunction renormalization Z3 is gauge dependent. This may be traced to the fact that nonabelian gauge fields also play the role of charged fields. The fundamental consequence of the transversity of the function (12110) is that no counterterm in A2 nor (a A)2 has been required. 590 QUANTUM FIELD THEORY
Figure 122 Threepoint function.
The divergent terms in (12110) come from the poles of the r function. However, when the computation is carried out in a different way, they may arise as singularities of the B function. This is a reflection of the fact that in a massless theory, the distinction between ultraviolet and infrared divergences becomes loose. For instance, the term of the integrand of P' proportional to (1  r 1 )b~' has the form (k2 _ q2)2 q2(q
+ k)4 and leads, within the dimensional regularization, to an ultraviolet finite, but infrared divergent, integral However, after a change of variable, q ..... q' infrared finite: q + k, this integral looks ultraviolet divergent but
ddq'Ck2  (q'  k)2]2 ex 4B (d  1  r ( d   2B (d   1) d) 3 )  d (q'  k)2q'4 2'2 2 2' 2 (2  d) 2 Of course, the two expressions coincide, but it is important to pick all singular terms when expanding around d = 4. 1233 Other Functions
For other superficially divergent proper functions we only give the structure of the divergent term. Three diagrams contribute to the threepoint function (Fig. 122). The required counterterm reads (12115) Figure 123 Fourpoint function.
NON ABELIAN GAUGE FIELDS
_~ _ Figure 124 Ghost selfenergy to the oneloop order.
Similarly, the diagrams of Fig. 123 for the fourpoint function necessitate a counterterm
(12116) The counterterms have the same form as the initial terms of the lagrangian.
It is of no wonder that this is the case for [;!l'A3, as the expression (12115) is the
only one which is Lorentz invariant, cubic in the field, of dimension four, and invariant under (global) group transformations. This is no longer true for the quartic term and the form of the counterterm (12116) is therefore a gift. The counterterms pertaining to functions with external ghosts must also be computed. Thanks to the structure of the operator Q defined in (12102) and (12103), or equivalently to the Feynman rules (1292), the momentum of an outgoing ghost line may always be factored out. This reduces the effective superficial degree of divergence of functions involving ghost fields, and leaves us with only two divergent functions: the ghost selfenergy (Fig. 124) and the ghostvector vertex (Fig. 125). The former does not need any mass counter term, owing to the above property, and we find [;!l''i~
(23  1)( ijao2Yja) (12117) [;!l''iA~
= (21  1)(CabcA~o!lijbYjc) _ Z1
g2C A12 1  16n 2 2 ~
(12118) In fact 21 reduces to one in the Landau gauge (A  1 = 0) to all o~ders, as a consequence of the transversity of the vector propagator and of the factorization of the incoming ghost momentum. In practical cases, matter fields are also coupled to the gauge field. We list _ .t
\0 _  Figure 125 Oneloop contributions to the ghostvector vertex.
QUANTUM FIELD THEORY
(i2A 2, (i2A 3, Figure 126 Fermion selfenergy.
the counterterms involving spinor fields, as well as the extra contributions to and (i2A 4 that they induce. The coupling has been written in Eq. (1293). We use the following notations for the quadratic Casimir operators in the representation of the fields: tr (TaT b) =  Tj(iab (P)2 = CjI
(12119) If r stands for the order of the group [number of generators of the Lie algebra, for example, r = N 2  1 for SU(N)] and nj is the dimension of the fermion representation [nj = N for the fundamental representation of SU(N)], we have the relation (12120) For the adjoint representation, n = r and hence T = C [ = N for SU(N)], while Tj = i and C j = (N 2  1)j2N for the fundamental representation of SU(N). The counterterms generated by the diagrams of Figs. 126 and 127 read

