large logarithms
15.2 The renormalization group and related ideas in QED .1 Where do the large logs come from?
We have taken the title of this section from that of Section 18.1 in Weinberg (1996), which we have found very illuminating, and to which we refer for a more detailed discussion.
As we have just mentioned, the phenomenon of ‘large logarithms’ arises also in the simpler case of QED. There, however, the factor corresponding to αsβ0 ∼ 14 is α/3π ∼10−3, so that it is only at quite unrealistically enor-mous|q2|values that the corresponding factor (α/3π) ln(|q2|/m2e) (whereme
is the electron mass) becomes of order unity. Nevertheless, the origin of the logarithmic term is essentially the same in both cases, and the technicalities are much simpler for QED (no photon self-interactions, no ghosts). We shall therefore forget about QCD for a while, and concentrate on QED. Indeed, the discussion of renormalization of QED given in chapter 11 will be sufficient to answer the question in the title of this subsection.
For the answer does, in fact, fundamentally have to do with renormaliza-tion. Let us go back to the renormalization of the charge in QED. We learned in chapter 11 that the renormalized chargeewas given in terms of the ‘bare’
charge e0 by the relation e = e0(Z2/Z1)Z312 (see (11.6)), where in fact due to the Ward identityZ1 andZ2 are equal (section 11.6), so that onlyZ312 is needed. To ordere2 in renormalized perturbation theory, including only the e+e− loop of figure 15.2,Z3 is given by (cf (11.31))
Z3[2]= 1 + Π[2]γ (0) (15.8)
15.2. The renormalization group and related ideas in QED 117 where, from (11.23) and (11.24),
Π[2] 2) 2i
{ 1 { d4k' x(1 − x)
(q = 8e dx (15.9)
γ (2π)4 (k'2 − Δγ + iE)2
2 2 2
0
and Δγ = me − x(1 − x)q with q < 0. We regularize the k' integral by a cut-off Λ as explained in sections 10.3.1 and 10.3.2, obtaining (problem 15.1)
e2 { 1 { (
Λ + /
Λ2 + Δγ )
Λ }
dx x(1 − x) ln −
Π[2]γ (q 2) = − .
π2 0 Δ γ 12 (Λ2 + Δγ )1/2 (15.10) Setting q2= 0 and retaining the dominant ln Λ term, we find that
( [2]
Z )12
= 1 −( α )
ln(Λ/me). (15.11)
3 3π
It is not a coincidence that the coefficient α/3π of the ultraviolet divergence is also the coefficient of the ln(|q2|/m2e ) term in (11.55)–(11.57); we need to understand why.
We first recall how (11.55) was arrived at. It refers to the renormalized self-energy part, which is defined by the ‘subtracted’ form
Π¯[2] γ (q 2) =Π[2] γ (q 2)− Π[2]γ (0). (15.12) In the process of subtraction, the dependence on the cut-off Λ disappears and we are left with
2
Π¯[2](q 2) = −2α { 1
dx x(1 − x) ln| me |
(15.13)
γ π 0 m2 e − q2x(1 − x)
as in equation (11.34). For large values of |q2| this leads to the ‘large log’
term (α/3π) ln(|q2|/m2e ). Now, in order to form such a term, it is obviously not possible to have just ‘ln |q2|’ appearing: the argument of the logarithm must be dimensionless, so that some mass scale must be present, to which
|q2| can be compared. In the present case, that mass scale is evidently me, which entered via the quantity Π[2]γ (0), or equivalently via the renormalization constant Z3 [2](cf (15.11)). This is the beginning of the answer to our questions.
Why is it me that enters into Π[2]γ (0) or Z3? Part of the answer – once again – is of course that a ‘ln Λ’ cannot appear in that form, but must be
‘ln(Λ/some mass)’. So we must enquire: what determines the ‘some mass’ ? With this question we have reached the heart of the problem (for the moment).
The answer is, in fact, not immediately obvious: it lies in the prescription used to define the renormalized coupling constant; this prescription, whatever it is, determines Z3.
The value (15.8) (or (11.31)) was determined from the requirement that the O(e2) corrected photon propagator (in ξ = 1 gauge) had the simple form
15.2. The renormalization group and related ideas in QED 119 working always to one-loop order with an e+e− loop. The relation between eμ andeis then
to leading order in α. Equation (15.19) indeed represents, as anticipated, a finite shift from ‘e’ to ‘eμ’, but the problem with it is that a ‘large log’
has resurfaced in the form of ln(μ/me) (remember that our idea was to take μ2>>m2e). Although the numerical coefficient of the log in (15.19) is certainly small, a similar procedure applied to QCD will involve the larger coefficient β0αs as in (15.5), and the correction analogous to (15.19) will be of order 1, invalidating the approach.
We have to be more subtle. Instead of making one jump from m2e to a large valueμ2, we need to proceed in stages. We can calculate eμ from e as long as μ is not too different from me. Then we can proceed to eμ' for μ' not too different from μ, and so on. Rather than thinking of such a process in discrete stages me → μ → μ' → . . ., it is more convenient to consider infinitesimal steps – that is, we regardeμ'at the scaleμ'as being a continuous function ofeμ at scaleμ, and of whatever other dimensionless variables exist in the problem (since the e’s are themselves dimensionless). In the present case, these other variables are μ'/μ and me/μ, so that eμ' must have the form
eμ' =E(eμ, μ'/μ, me/μ). (15.20) Differentiating (15.20) with respect to μ' and lettingμ' =μwe obtain
μdeμ
Forμ>>me equation (15.21) reduces to μdeμ
dμ =βem(eμ,0)≡βem(eμ), (15.23) which is a form of Callan–Symanzik equation(Callan 1970, Symanzik 1970);
it governs the change of the coupling constanteμas the renormalization scale μchanges.
118 15. QCD II: Asymptotic Freedom, the Renormalization Group
−igμν /q2 as q2 → 0; that is, as the photon goes on-shell. Now, this is a perfectly ‘natural’ definition of the renormalized charge – but it is by no means forced upon us. In fact the appearance of a singularity in Z3 [2] as me → 0 We have to concede, however, that if the point of renormalization is to render amplitudes finite by taking certain constants from experiment, then any choice of such constants should be as good as any other – for example, the ‘charge’ come back to this important point shortly.
15.2.2 Changing the renormalization scale
The recognition that the renormalization scale (−μ2 in this case) is arbitrary renor-malization prescription we have ‘tamed’ the large logarithms.
However, we have forgotten that, for consistency, the ‘e’ we should now be
118 15. QCD II: Asymptotic Freedom, the Renormalization Group
−igμν/q2 as q2 → 0; that is, as the photon goes on-shell. Now, this is a perfectly ‘natural’ definition of the renormalized charge – but it is by no means forced upon us. In fact the appearance of a singularity inZ3[2] as me → 0 suggests that it is inappropriate to the case in which fermion masses are neglected. We could in principle choose a different value ofq2, sayq2=−μ2, at which to ‘subtract’. Certainly the difference between Π[2]γ (q2 = 0) and Π[2]γ (q2=−μ2) is finite as Λ→ ∞, so such a redefinition of ‘the’ renormalized charge would only amount to a finite shift. Nevertheless, even a finite shift is alarming, to those accustomed to a certain ‘sanctity’ in the valueα= 1371 ! We have to concede, however, that if the point of renormalization is to render amplitudes finite by taking certain constants from experiment, then any choice of such constants should be as good as any other – for example, the ‘charge’
defined atq2=−μ2 rather than atq2= 0.
Thus there is, actually, a considerablearbitrariness in the way renormal-ization can be done – a fact to which we did not draw attention in our earlier discussions in chapters 10 and 11. Nevertheless, it must somehow be the case that, despite this arbitrariness, physical results remain the same. We shall come back to this important point shortly.
15.2.2 Changing the renormalization scale
The recognition that therenormalization scale(−μ2 in this case) is arbitrary suggests a way in which we might exploit the situation, so as to avoid large
‘ln(|q2|/m2e)’ terms: we renormalize at a large value of μ2! Consider what renor-malization prescription we have ‘tamed’ the large logarithms.
However, we have forgotten that, for consistency, the ‘e’ we should now be using is the one defined, in terms ofe0, via
15.2. The renormalization group and related ideas in QED 119 rather than small, a similar procedure applied to QCD will involve the larger coefficient β0αs as in (15.5), and the correction analogous to (15.19) will be of order 1,
15.2. The renormalization group and related ideas in QED 121 is to understand how this can help us with the large −q2 behaviour of our cross section, the problem we originally started from.
15.2.3 The RGE and large –q2 behaviour in QED
To see the connection we need to implement the fundamental requirement, stated at the end of section 15.2.2, that predictions for physically measurable quantities must not depend on the renormalization scale μ. Consider, for example, our annihilation cross section σ for e+e− → hadrons, pretending that the one-loop corrections we are interested in are those due to QED rather than QCD. We need to work in the spacelike region, so as to be consistent with all the foregoing discussion. To make this clear, we shall now denote the 4-momentum of the virtual photon by q rather than Q, and takeq2 <
0 as in sections 15.2.1 and 15.2.2. Bearing in mind the way we used the
‘dimensionless-ness’ of the e’s in (15.20), let us focus on the dimensionless ratio σ/σpt ≡ S. Neglecting all masses, S can only be a function of the dimensionless ratio|q2|/μ2 and ofeμ:
S =S(|q2|/μ2, eμ). (15.31) But S must ultimately have no μdependence. It follows thatthe μ2 depen-dence arising via the |q2|/μ2 argument must cancel that associated with eμ. This is why the μ2-dependence of eμ controls the |q2|dependence of S, and hence ofσ. In symbols, this condition is represented by the equation
( ∂
Equation (15.33) is referred to as ‘the renormalization group equation (RGE) for S’. The terminology goes back to Stueckelberg and Peterman (1953), who were the first to discuss the freedom associated with the choice of renormalization scale. The ‘group’ connotation is a trifle obscure – but all it really amounts to is the idea that if we do one infinitesimal shift inμ2, and then another, the result will be a third such shift; in other words, it is a kind of
‘translation group’. It was, however, Gell-Mann and Low (1954) who realized how equation (15.33) could be used to calculate the large|q2|behaviour ofS, as we now explain.
It is convenient to work in terms ofμ2andαrather thanμande. Equation (15.33) is then
120 15. QCD II: Asymptotic Freedom, the Renormalization Group To this one-loop order, it is easy to calculate the crucial quantity βem(eμ). would be a higher-order correction to (15.24). Now the unrenormalized cou-pling is certainly independent ofμ. Hence, differentiating (15.24) with respect
120 15. QCD II: Asymptotic Freedom, the Renormalization Group To this one-loop order, it is easy to calculate the crucial quantityβem(eμ).
Returning to (15.17), we may write the bare couplinge0 as e0 = eμ where the last step follows from the fact thateandeμ differ byO(e3), which would be a higher-order correction to (15.24). Now the unrenormalized cou-pling is certainly independent ofμ. Hence, differentiating (15.24) with respect toμat fixede0, we find
Working to ordere3μ we can drop the last term in (15.25), obtaining finally (to one-loop order)
We can now integrate equation (15.26) to obtaineμat an arbitrary scaleμ, in terms of its value at some scaleμ=M, chosen in practice large enough so that for variable scalesμgreater thanM we can neglectme compared withμ, but small enough so that ln(M/me) terms do not invalidate the perturbation theory calculation of eM from e. The solution of (15.26) is then (problem 15.2) denominator(and has coefficientαM/3π!). We note that the general solution of (15.23) may be written as
We have made progress in understanding how the coupling changes as the renormalization scale changes, and how ‘large logarithmic’ change as in (15.19) can be brought under control via (15.29). The final piece in the puzzle
15.2. The renormalization group and related ideas in QED 121
is to understand how this can help us with the large −q2 behaviour of our cross section, the problem we originally started from.
15.2.3 The RGE and large –q2 behaviour in QED
To see the connection we need to implement the fundamental requirement, stated at the end of section 15.2.2, that predictions for physically measurable quantities must not depend on the renormalization scale μ. Consider, for example, our annihilation cross section σ for e+e− → hadrons, pretending that the one-loop corrections we are interested in are those due to QED rather than QCD. We need to work in the spacelike region, so as to be consistent then another, the result will be a third such shift; in other words, it is a kind of
‘translation group’. It was, however, Gell-Mann and Low (1954) who realized how equation (15.33) could be used to calculate the large |q2| behaviour of S,
15.2. The renormalization group and related ideas in QED 123 This is a remarkable result. It shows that all the dependence of S on the (momentum)2 variable |q2| enters through that of the running coupling α(|q2|). Of course, this result is only valid in a regime of−q2 which is much greater than all quantities with dimension (mass)2– for example the squares of all particle masses, which do not appear in (15.31). This is why the technique applies only at ‘high’−q2. The result implies that if we can calculateS(1, αμ) This is almost exactly the formula we proposed in (11.57), on plausibility grounds.1
Suppose now that the leading QED perturbative contribution toS(1, αμ) isS1αμ. Then the terms contained inS(1, α(|q2|)) in this approximation can
Comparing (15.45) and (15.46) we see that each power of the large log factor appearing in (15.46) comes with one more power ofαμ than in (15.45). Pro-videdαμis small, then, theleadingterms int, t2, . . . are contained in (15.45).
It is in this sense that replacingS(1, αμ) byS(1, α(|q2|)) sums all ‘leading log terms’.
In fact, of course, the one-loop (and higher) corrections toS in which we are really interested are those due to QCD, rather than QED, corrections. But the logic is exactly the same. The leading (O(αs)) perturbative contribution to S = σ/σpt at q2 = −μ2 is given in (15.1) as αs(μ2)/π. It follows that the ‘leading log corrections’ at high −q2 are summed up by replacing this expression byαs(|q2|)/π, where the runningαs(|q2|) is determined by solving (15.39) with the QCD analogue of (15.36) – to which we now turn.
1The difference has to do, of course, with the different renormalization prescriptions. Eq (11.57) is written in terms of an ‘α’ defined atq2= 0, and without neglect ofme.
122 15. QCD II: Asymptotic Freedom, the Renormalization Group where βem(αμ) is defined by
Now introduce the important variable
t = ln(|q 2|/μ2). (15.37)
122 15. QCD II: Asymptotic Freedom, the Renormalization Group whereβem(αμ) is defined by
βem(αμ)≡ μ2∂αμ
∂μ2 e0. (15.35)
From (15.35) and (15.26) we deduce that, to the one-loop order to which we are working, This is a first-order differential equation which can be solved by implicitly defining a new function – therunning coupling α(|q2|) – as follows (compare (15.30):
To see how this helps, we have to recall how to differentiate an integral with respect to one of its limits – or, more generally, the formulae
∂
∂a J f(a)
g(x)dx=g(f(a))∂f
∂a. (15.40)
First, let us differentiate (15.39) with respect tot at fixedαμ; we obtain
1 = 1 the minus sign coming from the fact that αμ is the lower limit in (15.39).
From (15.41) and (15.42) we find
15.2. The renormalization group and related ideas in QED 123
This is a remarkable result. It shows that all the dependence of S on the (momentum)2 variable |q2| enters through that of the running coupling α(|q2|). Of course, this result is only valid in a regime of −q2 which is much greater than all quantities with dimension (mass)2 – for example the squares of all particle masses, which do not appear in (15.31). This is why the technique applies only at ‘high’ −q2. The result implies that if we can calculate S(1, αμ)
Comparing (15.45) and (15.46) we see that each power of the large log factor appearing in (15.46) comes with one more power of αμ than in (15.45). Pro-vided αμ is small, then, the leading terms in t, t2, . . . are contained in (15.45).
It is in this sense that replacing S(1, αμ) byS(1, α(|q2|)) sums all ‘leading log terms’.
In fact, of course, the one-loop (and higher) corrections to S in which we are really interested are those due to QCD, rather than QED, corrections. But the logic is exactly the same. The leading (O(αs)) perturbative contribution
15.3. Back to QCD: asymptotic freedom 125
FIGURE 15.3
q¯q vacuum polarization correction to the gluon propagator.
the motivations for a colour SU(3) group discussed in the previous chapter, led rapidly to the general acceptance of QCD as the theory of strong interactions, a conclusion reinforced by the demonstration by Coleman and Gross (1973) that no theory without Yang-Mills fields possessed the property of asymptotic freedom.
In section 11.5.3 we gave the conventional physical interpretation of the way in which the running of the QED coupling tends to increaseits value at distances short enough to probe inside the screening provided by e+e− pairs (|q|−1 <<m−e1). This vacuum polarization screening effect is also present in (15.49) via the term−2N12πf, the value of which can be quite easily understood.
It arises from the ‘q¯q’ vacuum polarization diagram of figure 15.3, which is precisely analogous to the e+e− diagram used to calculate ¯Π[2]γ (q2) in QED.
The only new feature in figure 15.3 is the presence of the λ2-matrices at each vertex. If ‘a’ and ‘b’ are the colour labels of the ingoing and outgoing gluons, the λ2-matrix factors must be
E3 α,β=1
(λa
2 )
αβ
(λb
2 )
βα
(15.51)
since there are no free quark indices (of typeα, β) on the external legs of the diagram. It is simple to check that (15.51) has the value 12δab(this is, in fact, the way the λ’s are conventionally normalized). Hence for one quark flavour we expect ‘α/3π’ to be replaced by ‘αs/6π’, in agreement with the second term in (15.49).
The all-important, positive, first term must therefore be due to the gluons.
The one-loop graphs contributing to the calculation ofβ0 are shown in figure 15.4. They include figure 15.3, of course, but there are also, characteristically, graphs involving the gluon self-coupling which is absent in QED, and also (in covariant gauges) ghost loops. We do not want to enter into the details of the calculation ofβ(αs) here (they are given in Peskin and Schroeder 1995, chapter 16, for example), but it would be nice to have a simple intuitive picture of the
‘antiscreening’ result in terms of the gluon interactions, say. Unfortunately no fully satisfactory simple explanation exists, though the reader may be inter-124 15. QCD II: Asymptotic Freedom, the Renormalization Group