CALCULATION OF ANOMALOUS DIMENSIONS 161

and (as in §11.1)

^i,a— 1 a

— 1 + A-Z^.

(12^.101⁾

(12.102) Calculating the non-trivial diagrams in Feynman gauge (for this gauge invariant8 object) gives

g2M eXtC3Sij 4 1 16n2s /(/ + 1) /!

—(trace terms)

— ... p^1 -f- permutations)

(12.103) where g is the dimensionless coupling constant and M is the renormalisation scale, as in Chapter 11, and the group theory factor C3 is defined in (11.53).

g2M el*C38u ( ' 4 \ 1

= ---— - 2^— - (y^P**2 • • • P* + permutations)

—(trace terms). (12.104)

Also

= (diagram 2). (12.105)

Returning to (12.98), and using (12.99)-(12.105), we obtain for the renormalisation constant in the m s scheme

(mo6>

Taking the fermion wave function renormalisation counter term AZ^, from (11.59), we find

Thus, using (12.107), and the renormalisation mass dependence of § given in (11.23), we find

with b as in (12.30), and the appropriate flavour non-singlet combination of structure functions understood (e.g. the difference of vW2 off protons and vW2 off neutrons). chapter may also be extended to deep inelastic neutrino production9, where moments of structure functions which involve a single power of ln( — q2/M 2) may be found, with the aid of charge conjugation invariance, without considering different targets.

12.7 COMPARISON WITH EXPERIMENT, AND AqcD 163 12.7 Comparison with experiment, and AQCD

To compare with experiment the prediction (12.112) for the variation with q2 of the moments of flavour non-singlet structure functions, it is convenient to take logarithms. Thus Since the dl a are as in (12.111), there is a clean prediction, which is in quite good agreement with experiment.

The alert reader will have noticed that, following on from (12.91), we have written the prediction for the structure function moment, (12.112), in terms of a reference mass M which is entirely arbitrary. Moreover, we have not yet fixed the value of the ^{q c d} coupling constant g in (12.29), defined as the value of the renormalised coupling constant for renormalisation scale M. The reason for this is that (12.29) is only valid when 2bg2 In s is very much greater than one (otherwise g2(s) is not necessarily small, and expansion in powers of g(s) in Pg(g(s)) is not valid), and then, to leading order, g2 divides out.

Correspondingly, (12.112) is only valid when ln( — q2/M 2) is very much greater than one, at which stage ln( — q2) > ln M 2 (so to speak) and the scale M cannot be determined reliably. However, by going to next-to-leading order in the coupling strength g2, in performing the ^{q c d} calculations and comparing with experiment, it is possible to determine the scale on which g2, and consequently the structure function moments, vary. We may introduce this scale in the following way. In the leading order expression (12.29) take

5 = M /M (12.116)

where M is some new renormalisation mass. Then,

g2(M/M) = g2(M) = g2/[l+ 2 b g 2 In (M/M)] (12.117) where we have used (12.25), and

g= g(M ). (12.118)

We may rewrite (12.117) in the form

g2(M) = [2b IiHM/Aqcd)] “ 1 (12.119)

valid for M §> Aqcd where AqcD is defined by

ln(AQCD/M )= ~(2 bg2)~ x. (12.120) The value of the coupling constant g(M) at the new renormalisation scale M cannot depend on the original renormalisation scale M. Thus (12.119) must be independent of M, and A.qcD must be independent of M (the M dependence cancelling between In M and g = g{M)). The q c d coupling constant at the Z mass g(mz ), as determined from comparison with experiment of next-to-leading-order ^{q c d} calculations is given by

a5(mz) = 0.113 mz = 91.18 GeV (12.121) where

as(A?) = g2(M)/4n. (12.122)

With b as in (12.30) with 5 flavours of quark operative in the range of energy up to the Z mass, the corresponding value of the q c d scale parameter AQCD is

A q c d = 0 065 GeV. (12.123)

We may determine g2(M) for any renormalisation scale M from this value

A qC D .

The coupling constant for q e d may be treated in a similar way by writing e \ M ) = - [2b ln(AQED/M )] ' 1 (12.124) valid for M « AQED. In this case b is negative, which accounts for the slight difference in form (12.124) and (12.119), and, for A? in the range of energy up to the Z mass, we take b to be given by (12.31) with two complete generations and the top quark contribution of 4/9 omitted for the third generation. Then AQED may be determined from the known value of epHYS/4n where ^cPHy s t h e coupling constant for on-mass-shell electrons.

From §11.3, we see that e2(M) for M = me, the electron mass, differs from CpHYs by less than 1%, so we write

a ( M ) ^ 1/137 - 7.3 x 10"3 M = me (12.125) where

oc(M) = e2(M )/4n. (12.126)

e+e" ANNIHILATION 165

A q e d — 2.5 x 1066me (12.127)

This is such an enormous number that e2(M) grows exceedingly slowly with M. Thus, for example,

e2{mz )/e2{me)= 1.09 (12.128)

so that

a - 1(wz) = 126.1. (12.129)

12.8 e +e~ annihilation

The total cross section into hadrons for the inclusive process

e +e ~ - X (12.130)

where X is an arbitrary unobserved hadronic final state, provides another test10 of q c d. Treating the process to lowest order in the electromagnetic interaction, what we have to study is the total cross section for a photon to produce hadrons (see figure 12.2). One way to approach this problem is to use the optical theorem to relate the required cross section to the ^{o p i} Green function for two photon fields, and then to use the renormalisation group equation with two running coupling constants, e and g, because both the electromagnetic coupling constant e, and the ^{q c d}coupling constant g enter the diagrams. The variation of the electromagnetic coupling constant with renormalisation scale is extremely slow (see §12.7) and to a good approximation only the q c d

coupling constant need be allowed to run. This approach is described in detail in the review of Politzer11.

Using (12.124) we then find

Figure 12.2 Electron-positron annihilation into hadrons.

There is an alternative less rigorous approach, which has the virtue that it can be extended to study the differential cross section for quark or gluon jets, as well as the total cross section for e +e “ annihilation. It can also be extended to other situations where the operator product expansion is not applicable. In

this approach, one calculates (zero- and) one-loop diagrams for e +e"

annihilation into hadrons of figure 12.2. We have to include not only diagrams for e +e~ -> qq (where q is a quark) but also diagrams for e +e~ qqg (where g is a gluon), because, as a matter of principle, any apparatus used for observations has a finite energy resolution, and there is no way of excluding the possibility that a sufficiently low energy gluon has been emitted. Because of the zero mass of the gluon, there are infrared divergences in the diagrams of figure 12.2(b), (c) and (d) which appear in dimensional regularisation as poles in e which remain after the counter terms have been subtracted using the diagrams of figure 12.4. However, after integrations over phase space have been made to obtain observable cross sections, these infrared divergences are cancelled by the infrared divergences in the diagrams of figures 12.3(c) and 12.3(h). This is true in particular of the total cross section for e +e"

annihilation. Thus, when all diagrams of figures 12.3 and 12.4 are included, a finite result is obtained for the total cross section into hadrons for virtual photon four-momentum, namely,

* . . - . X , - £ ( 3 l < 8 ) ( . + ? * £ U ) ,12.131, where the sum over / is a sum over the charges squared of all (active) quark flavours, the preceding factor of 3 is from the three quark colours, and the group theory factor c3 (as defined in (11.53)) is for the 3 of colour SU(3), and so has the value f . As observed earlier, the electromagnetic coupling constant varies only very slowly with renormalisation scale, and the fine structure

Figure 12.3 One-loop diagrams for e+e —>qqand e+e —>qqg.

PROBLEMS 167

ib)

( c )

Figure 12.4 Counter term diagrams for e+e —>qq.

constant ^a in (12.131) may be taken to be 1/137. The ^{q c d} fine structure constant as must be allowed to run, and is given by (12.119). More detail of this calculation, together with extensions of the method to eN eX and other processes may be found in the reviews of Pennington12 and Sachrajda13.

Problems

12.1 Calculate the renormalisation group coefficients of (12.10)-( 12.14) from