First, let us emphasize that IR renormalon ambiguities are not a problem of QCD, but a problem of doing perturbative calculations in QCD, which implicitly or explicitly require some kind of factorization, and an expansion in a ratio like Λ/Q. If we could do non-perturbative calculations, IR renormalons would just be artifacts that appear in the expansion of the exact (and well-defined) result.
Let us imagine that an observable R, which depends on at least the two scales Q and Λ, can be written as an expansion33
R(Q, Λ) =X
n
Cn(Q, µ)
Qdn ⊗ hhOnii(µ, Λ). (4.31) The product may be a normal product or a convolution and the operator Onof dimension dn may be local or non-local. The matrix element may be a vacuum matrix element or a matrix element between hadron states. (We use the double bracket to indicate that the external state may be complicated.) We assume that the Cn(Q/µ) can be calculated as series in αs. It is not obvious that such an expansion in powers and logarithms of Λ/Q always exists. Or one may know only the form of the first term, but not the form of power corrections to it. This is the most interesting situation from the point of view of IR renormalons.
We assume that factorization is done in dimensional regularization. If one uses an-other factorization scheme, the wording of the following changes but the conclusions do not. In dimensional regularization the coefficient functions Cn(Q, µ) have IR renor-malons from integrating Feynman integrals over loop momenta much smaller than Q.
With regard to power corrections we note:
(i) Renormalon ambiguities in Cn(Q, µ) are power-suppressed. Non-perturbatively they are cancelled by ambiguities in defining the (renormalized) matrix elements hhOmii with dm > dn. Contrary to the σ model one cannot trace this cancellation non-perturbatively in QCD. However, if QCD is a consistent theory and if R is physical, this cancellation must occur. In this way, IR renormalons in Cn lead one to introduce parameters for power corrections with a dependence on Q (given by Cm/QdmhhOmii) that matches the scaling behaviour of the renormalon ambiguity. This is the minimalistic, but also most rigorous and most universally applicable use of IR renormalons.
(ii) The analysis of Feynman diagrams gives some information on the form of the operator Om. But IR renormalons provide no information on the magnitude of hhOmii.
33We assume that R has been made dimensionless.
It is natural to think of hhOmii as at least as large as the renormalon ambiguity. The σ model is an example where the matrix elements are parametrically larger than their ambiguities, both at large N and at g ≪ 1.
(iii) The IR renormalon approach to power corrections does not provide a ‘non-perturbative method’. Viewed from the low-energy side IR renormalons are related to ultraviolet properties of operators and not to matrix elements. The analysis of the σ model shows that the IR renormalon in Cn is related to a power divergence of degree dm−dnof Om. In a cutoff factorization scheme with factorization scale µ divergent series appear in the expansion of matrix elements for µ/Λ ≫ 1 and the same statement holds.
In section 4.2.2 we demonstrate this for deep inelastic scattering in QCD by evaluating the ultraviolet contributions to twist-4 operator matrix elements perturbatively. As a consequence of being ultraviolet with respect to the scale Λ, IR renormalons do not distinguish matrix elements of the same operator but taken between different states.
(iv) Renormalon factorial divergence is closely connected with logarithms of loop momentum which in turn are related to the running coupling. This leads to the universal appearance of β0, β1 etc. in the large-order behaviour. On the other hand, power corrections inferred from IR renormalons and power corrections in general have nothing to do with the low-energy properties of the running coupling. They are process-dependent and, generally speaking, non-universal.
IR renormalons can be universal for a restricted set of observables, if the same opera-tor appears in their short-distance expansion.34 However, universality of IR renormalons does not imply universality of non-perturbative effects. This is true only if the operator is not only the same, but is also taken between the same external states.
(v) If this strong form of universality holds for a set of observables, one can relate power corrections to them on the basis of knowing only the IR renormalon behaviour of coefficient functions. In particular, one can relate the leading power correction on the basis of the perturbative expansion at leading power. For simplicity, consider two observables
R = C0+ C1
QdhhOii, (4.32)
R = C0+C1
QdhhOii, (4.33)
and denote by δC0|t=−d/(2β0) the renormalon ambiguity in C0 of order 1/Qd, which is related directly to large-order behaviour. Then it follows that
δC0
δC0|t=−d/(2β0) = C1
C1
, (4.34)
34If the operator is non-local and is multiplied with the coefficient function in the sense of a convolu-tion, the situation is more complicated, because one also has to unfold the convolution.
and this ratio can be expanded in αs(Q). In particular, the ratio of the uncalculable normalizations of IR renormalon behaviour is given by the ratio of the coefficient func-tions C1, C1 evaluated to lowest order in αs. Conversely, knowing the left-hand side of (4.34) one can predict the relative magnitude of 1/Qd power corrections of the two observables systematically as an expansion in αs(Q). One observable has to be used to fix the absolute normalization, i.e. to determine C1hhOii from data. The procedure described parallels the phenomenological use of the OPE in standard situations such as QCD sum rules or deep inelastic scattering.
Note that in practice, in connection with renormalons, universality often takes the status of an assumption. This is so because to establish universality one needs to know enough of the operator structure of power corrections that it may be possible to compute C1 and C1 directly, thus by-passing (4.34) and the IR renormalon argument.
(vi) There is the problem of consistently combining (divergent) perturbative expan-sions in dimensional renormalization with phenomenological parametrizations of power corrections. For the purpose of discussion let us consider the simplified structure of (4.32) with only one parameter and a single, corresponding IR renormalon singularity in C0 at t = −d/(2β0). If we knew the singularity exactly, we could subtract it from the series. Recalling that the µ-dependence of the IR renormalon singularity is an overall factor (µ/Q)d (up to logarithms), we write (4.32) as
R =
C0− C0as
µ Q
!d
+ 1 Qd
hC0asµd+ C1hhOiii, (4.35)
where Λ < µ < Q and C0as denotes the exact asymptotic behaviour. Both square brackets are now separately well-defined. Note that this rewriting results in exactly the same representation that one would obtain with cutoff factorization. In reality the subtraction can be carried out at best approximately. Moreover, C0 is known only in first few orders.
Suppose we choose µ as close to Λ as possible for αs(µ) to be perturbative. In this case the subtraction is effective only as one gets close to the minimal term of the series expansion of C0. It may turn out that the phenomenological determination of the power suppressed term is large compared to the last term kept in the expansion of C0. In this case C0asµd is small and IR renormalons are not an issue of concern. It is sometimes argued (Novikov et al. 1985) that such a numerical fact is at the basis for the success of QCD sum rules.
It may also turn out that the phenomenological determination of the power sup-pressed term is not large, but of the order of the last known term in the truncated series. In this case the phenomenological power correction may parametrize the effect of higher order perturbative corrections rather than a truly non-perturbative effect. It is still reasonable to use such an effective parametrization, because, as illustrated by (4.35)
the dominant contribution to perturbative coefficients in sufficiently large orders can be combined with hhOii. Moreover, if the minimal term in C0 is reached not at very high orders, the sum of higher order corrections parametrized in this way, may indeed scale approximately like a power correction.
The important conclusion is that combining power corrections with truncated per-turbative series is meaningful in the sense that the error incurred is never larger and most likely smaller than the error one would obtain without using information on power corrections. The improvement comes from the fact that the error is now determined by the degree to which the perturbative correction is non-universal in intermediate orders rather than by the size of the perturbative correction itself. For a related discussion see (David 1984; Martinelli & Sachrajda 1996).