By F. J. Ynduráin (auth.)

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**Extra info for Quantum Chromodynamics: An Introduction to the Theory of Quarks and Gluons**

**Example text**

Produces finite answers and, in some sense, as e ~ 0, J . ~ f . Due to Bohr and Rosenfeld's (1933, 1950) classical work, we know that field operators are intrinsically singular ; therefore, any regularization must destroy some physical feature of the theory . Thus , Pauli-Villars regularization destroys hermiticity and gauge invariance for non-Abelian theories; lattice regularization destroys Poincare invariance, etc. ). Because gauge and relativistic invariance are essential for QCD , we will use dimensional regularization that only destroys scale invariance.

4c) o Here we have defined N. = l_YE + log4'17. 10 In fact, all poles in dimensional regularization appear in this combination. 3) and rearrange it to read: 2:,t/itf} = CF~ij SD~(P) = i{jI-m + g2CFL(2)} _ . 4) into 2 jI-m{I - CFg BD ~(P)} + higher orders . 5a) takes into account exactly the contribution of all the diagrams of Figure 4 and that, replacing L(2) ~ Lexa ct, is the most general form for S . As we see, there are two divergences: g2 1- CF 16'17 2 (I - ~)N. 6) which multiplies the entire S, and I +3CF - g2 - 16'17 2 N.

QeD as a Field Theory q q q ... - .. ~ ( a) k, i q ,v,b q ,~ . O (b) +-@-@--+... 1'>. -@-- + .... ~ + ~ + (c ) Figure 6. The gluon propagator. 27 The part arising from a quark loop (Figure 6b) is, for each flavor f of quark rr;:;uark ;ab = • 2 -lg a bJ dDk 4-D Tr(le"+ mf)y IL(le"+ q+ mf)y" ~ij tij~i 2 D "o ( 7T) (k 2 2) [ - mf ( k + q) 2 - mf2 ] The calculation may be carried out using standard techniques. The result is exactly the same as that for the photon vacuum polarization, apart from the factor Tr tat b • The result is, with nf the total number of quark flavors : II:" quark s;ab = l)ab X - 2T g2 2 F (- gIL q + qILq l67T 2 P 2 { "3 N,nf - 4 (' Jo{) dx· x(l - P ) nf x) ~ log /=\ mJ- x(1 2 ~ - X)q2 ) .