IvrA j7[ffVflV 1 f7gv s

Doing the /c integral and replacing the factor by unity give

As advertised, this term cancels (16.109), giving zero.

The final term, and the only non-zero contribution to A*"1*, comes from the A3 terms, which are already linear in q and therefore quite easy to reduce. To reduce them, recall that Y is unchanged if q —q and 77 —> -tj, and hence A3(fc, -q) can be written in a form similar to A%{k,q) provided 77 —> -77 in the integral which defines it [recall Eq. (16.102)]. Hence, retaining terms linear in q only, the numerator of the A£"a contributions can be combined as follows:

N3 = tr {7" h 7A h V rf (1 - 2^77)- é 7" (« (1 + 2Çti)] K }

-, (1 - 2 fr)k2 [tr (7" ft 7 V it) + tr (rf 7V i 7M Ji)]

- (1 4- 2^)k2 [tr (7" |Î7AN/) + tr (7" rf jf 7A ff 7M)] •

Again, as above, we may use (C.9) and 7a7M7Q = -2^ immediately to get jV3 - —4A;4(1 - 2£tj) [9 V + <zV" - sY1 + 4fe4(l + 2^)<?Y". Hence the A3 contributions are

AfA = - — / de r dV Ug^q» + g"xq") (1 - 2^) - 2<?V

Combining denominators and doing the A: integral gives a2 1 / 11.2 \e Z-1 /■*

47T2 e

This is the final result for the fourth order self-energy, and combined with our previous results gives

The result (16.114) shows how the renormalization constant Z3 must be redefined in fourth order. The fact that the singular term is constant is another example of the requirements of renormalizability. In addition, we have obtained the finite, high q2 part of the vacuum polarization correction to fourth order. Note that it adds to the second order corrections, further enhancing the effects already discussed in Chapter 11. In the next chapter we will show how this result may be used to calculate the QCD corrections to R, which were discussed in Sec. 10.4.

The calculation of (16.114) is long, but note that none of the singular contributions arises from singularities in the integrals over the Feynman parameters, as they did in the <p3 example discussed in Sec. 16.2 above. This is because we calculated (16.114) from diagrams with no overlapping divergences.


We conclude this chapter with a brief discussion of the renormalization of QCD. First, consider the four types of interactions which can occur in QCD (with quarks). There are three-gluon (3g) and four-gluon (4g) vertices, a quark-gluon vertex (gqq), and the ghost-gluon vertex (gcc). Two of these vertices have derivative operators, which introduce an extra power of momentum into the vertex and add a single unit to the formula for the index of divergence (see Prob. 16.1). The new formula for the index is

where no is the number of derivatives at the vertex. Evaluating this formula in four dimensions for each vertex gives the following computations:

4 g: 4 +§(0)+ 0-4 = 0 gqq: 1 + |(2)+0-4 = 0 gcc: 3+§(0) + l-4 = 0

In every case the index of divergence is zero. Note that ghosts are treated like bosons for this estimate, since their propagator goes like jr2. Hence, by the definition given in Sec. 16.1, QCD is superficially renormalizable.

In QCD, the divergent subgraphs are seven in number: the gluon, ghost and quark self-energies, renormalized by the constants Z3, Z3, and Z[, and the three-gluon coupling, with renormalization constant Z\, the gcc coupling with constant Zi, the gqq coupling with constant Z[, and the four-gluon vertex with constant Zi. In addition, there is a mass shift for the quark, which we shall ignore.

A central issue in QCD is to demonstrate that the SU(3) local gauge invariance is preserved by renormalization. Since the freedom to choose a, the gauge parameter in the gluon propagator, is a consequence of this freedom, it should be true that this parameter is also unaffected by renormalization. We will discuss the implication of these remarks now.

Recall that the QCD Lagrangian, including ghost fields and gauge fixing terms, can be written (assuming, for simplicity, that the quarks are massless)

Cf ~ Cg + Cf + Cg + CC + £3 g + £4 g + Cggg + CgCC

where the "free" Lagrangians and interaction terms are

¿9 =

-i {d»Aav - dvAl) (d»Aav - d"Aa»)

Cf =


- Ip Qip

Cc =

C3g =

yjabc Ab" A™ {dpAl - d„Al)

Cig =

~"\92RfabeicdeAallA\,AC>1 Adu

r . — ^gqq

-gRin»\\a1> Al

Cgcc ~

-9Rfabcd^ccAb» .

These were discussed in Chapters 13 and 15. As in our discussion of QED, the fields and coupling constant gR are assumed to be finite, renormalized quantities.

Counterterms must be added to the Lagrangian to cancel infinities which arise from the seven types of diverging graphs or subgraphs. Since the gauge fixing trrm, Cf, is associated with the gluon propagator, and the gauge fixing parameter a is unchanged, counterterms are introduced as follows:

+ (Z! - 1) C3g + (Z4 - 1) C4g + (Zf - 1) Cm + (Zi -1) c

Hence the bare Lagrangian, which includes the counterterms, is gcc ■ (16.118)

Absorbing the infinite renormalization constants into the fields gives the following bare fields, denoted by a superscript (0):

and we see that the bare coupling constant, go, depends on four different combinations of renormalization constants:

Clearly, each of these quantities must be identical if the SU (3) gauge symmetry is to survive the renormalization. Hence, we must have generalizations of the Ward-Takahashi identity, referred to as the Slavnov-Taylor identities [Ta 71, SI 72]:

For a further discussion of these identities, the reader is referred to the literature [La 81, MP 78]. We will not pursue the discussion of the renormalizability of QCD further.

In the next chapter we will assume that QCD is renormalizable and the identities (16.122) can be proved and show that QCD is an asymptotically free neory.


16.1 Consider a theory with an interaction Lagrangian with no space-time derivatives. For example, pseudovector nN coupling has the form

with no = 1 space-time derivative. Using the methods worked out in Sec. 16.1, show that Eq. (16.5) is unchanged but that the index of divergence of the theory becomes

Discuss the significance of this result.

16.2 Compute the counterterm Ao which arises from the four-point function shown in Fig. 16.3A when calculated in d = 8 dimensions.

16.3 Using the BS equation [in particular, Eq. (12.50) may be helpful], prove that the Dyson equation illustrated in Fig. 16.13A is equivalent to the equation illustrated in Fig. 16.14. (You will also need to use the equation in Fig. 16.13B.)

16.4 Show that the counterterms needed to remove the subdivergences from the sixth order self-energy, graph 16.11, are as shown in Fig, 16.16. You may use Figs. 16.14 and 16.15 to construct your argument. (Hint: Don't forget the contribution from r(3) and its counterterms.)

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH


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