Off-shell processes

In QCD off-shell, euclidian Green functions of external (electromagnetic or weak) cur-rents are of interest. They are related to physical processes such as the total cross section in e⁺e⁻ → hadrons or moments of deep inelastic scattering structure functions through dispersion relations.

In the flavour expansion the Borel transform of a diagram with chains is represented by the integral (3.8). We suppose that there are no power-like infrared divergences.

Then for off-shell Green functions at euclidian momenta it follows from properties of analytic regularization that the Borel transform has IR renormalon singularities at non-negative integer u.²³ However, the structure of the singularity in terms of subgraphs is different from (3.10) as different notions of irreducibility apply to ultraviolet and infrared properties. The methods used in (Beneke & Smirnov 1996) could be extended to this situation.

Consider the two-point function of two quark currents, defined in (2.14), with external momentum q. IR renormalons arise from regions of small loop momentum k ≪ q, where the integrand becomes infrared sensitive. For massless, off-shell Green functions the IR sensitive points are those where a collection of internal lines has zero momentum. There has to be a connected path of large external momentum from one external vertex to

23Recall that in the flavour expansion u = −β0ft, where t is the Borel parameter. In QCD u = −β⁰t, so that in both cases, QED and QCD, IR renormalons are located at positive u.

H

q q

(a)

q H

J

J S

(b)

Figure 8: Infrared regions that give rise to infrared renormalons. (a) For a current-current two-point function at euclidian momentum. The external currents are shown as dashed lines. (b) For an event-shape variable in e⁺e⁻ annihilation near the two-jet limit. Wavy lines represent collections of soft lines.

the other. Hence, a general graph can be divided into a sum of contributions of the form shown in Figure 8a: A ‘hard’ subgraph to which both external vertices connect and a ‘soft’ subgraph of small momentum lines which connects to the hard subgraph through an arbitrary number of soft lines. In terms of the operator expansion (OPE), the soft subgraph corresponds to the matrix element of an operator and the hard part to the coefficient function. An analysis of the leading IR renormalon contribution (t =

−2/β⁰) to the current-current correlation function based on factorization of hard and soft subgraphs can be found in (Mueller 1985).

In section 2.2 we considered the leading-order diagrams of Figure 1 in the loop mo-mentum region, where the soft part consisted of a single gluon line (or chain). The general classification would also allow a quark line or more than one line in the soft part.

These parts are associated with condensates in the OPE containing quark fields. For the analysis of IR renormalons soft quark lines alone play no role, because they cannot be

‘dressed’ with bubbles, which is necessary in order to turn IR sensitivity in a skeleton diagram into a factorially divergent series expansion.

An immediate consequence of the factorization expressed by Figure 8a is that in order that the diagram contributes to an IR renormalon at t = −m/β⁰, the soft part must connect to the hard part by not more than 2m gluon lines. This follows from the fact that each additional such line adds one hard propagator to the hard part which counts as 1/q. On dimensional grounds this factor must be compensated by a power of one of the small momenta ki. Such factors result in suppression of large-order behaviour which

is related to integrals that generalize

q

Z

0

dk²k^m−2hβ0ln(k²/q²)ⁱⁿ∼ −2β0

m

!n

n!. (3.49)

In general, the location of IR renormalons and the possible contributions to a singularity at a particular point follow from such IR power counting arguments.

The leading-order diagrams in the flavour expansion, Figure 1, result in d⁴k/k² for small k. This leads to a singularity at t = −1/β⁰ for each diagram, which can be as-sociated with the operator A^A_µA^µ,A. Gauge invariance of the current-current two-point function requires that these leading contributions cancel in the sum of diagrams. After this cancellation the leading term is d⁴k, associated with a singularity at t = −2/β0

and the operator G^A_µνG^µν,A as discussed in section 2. Consider now the diagram with two chains shown in Figure 9a. If both gluon momenta are small, power counting gives d⁴k1/k₁²d⁴k2/k₂² which can contribute to the singularity at t = −2/β⁰. This contribution must be associated with the (Aµ)⁴ term in the operator G^A_µνG^µν,A and hence it is re-lated to the leading order in the flavour expansion by gauge invariance. Except for this trivial contribution, the region when both gluon momenta are small contributes only to subleading renormalon singularities at t > −2/β⁰. When one of the gluon lines is hard and only one is soft, one obtains a contribution to the order αs correction of the coefficient function of G^A_µνG^µν,A. Because one looses one power of αs, this contribution is 1/n-suppressed in large orders relative to the leading order in the flavour expansion.

We conclude that the leading IR renormalon at u = 2 is determined by diagrams with only a single soft chain, up to contributions constrained by gauge invariance and up to a calculable multiplicative factor that follows from the coefficient function of G^A_µνG^µν,A. These diagrams are shown in Figure 9b, where the shaded circle denotes an arbitrary collection of soft lines. Note the difference to the corresponding analysis for UV renor-malon singularities in which case diagram 9a was found to be enhanced relative to the leading order in the flavour expansion rather than suppressed. The diagrams of type 9b have been considered further in (Zakharov 1992; Grunberg 1993; Beneke & Zakharov 1993). It was found that the residue of the IR renormalon singularity receives contribu-tions from arbitrarily complicated graphs in the shaded circle and remains uncalculable (Grunberg 1993; Beneke & Zakharov 1993) despite the simpler overall diagram structure compared to the UV renormalon case. A graph-wise comparison of some contributions to the first IR and first UV renormalon is summarized in Table 2.

A complete characterization of IR renormalon singularities must account not only for powers of small momenta but also for logarithms of k/q. The soft subgraphs contains renormalization parts, when some soft momenta are larger than others: k₁ ≪ k2. These renormalization parts lead to logarithms whose coefficients are given by renormalization group functions and introduce the effect of higher-order coefficients in the β-function and

(a) (b)

Figure 9: Some diagrams at higher order in the flavour expansion.

operator anomalous dimensions into the large-order behaviour. Technically, in the flavour expansion, this occurs in a way similar to the UV renormalon case. In particular, there is no difference between UV and IR renormalons as far as the mechanism is concerned that restores the non-abelian β-function coefficient β0 (see section 3.2.2).

Once factorization is established, the most elegant characterization of IR renormalon singularities follows from first identifying the ‘operator content’ of the soft subgraph and then from deriving an evolution (renormalization group) equation for it. Consider a physical quantity such as the Adler function (2.15) or its discontinuity and its series ex-pansion^Prnαⁿ⁺¹_s (Q) in αsnormalized at Q. IR renormalon behaviour of the coefficients rnleads to an ambiguity in the Borel integral with a certain scaling behaviour in Q. This scaling behaviour must be matched exactly by higher-dimension terms in the OPE. For simplicity, we assume that there is only one operator O of dimension d with anomalous dimension γ as defined in (3.35) and coefficient function C(1, αs(Q)) = c0+c1αs(Q)+. . ..

The scaling behaviour is given by 1

Q^dC^Q²/µ², αs

 h0|O|0i(µ) = const · e^{d/(2β0αs(Q))}(−β⁰αs(Q))^dβ1/(2β20)

Diagram Fig. 1 Fig. 9a Fig. 9b

UV n n² n ln n

IR 1 1/n^† ln n

†Ignoring O(1) fixed by gauge invariance.

Table 2: Comparison of contributions of various diagrams to the leading UV and IR renormalon behaviour. For the UV renormalon the displayed factor multiplies βⁿ₀n!, for the IR renormalon (−β0/2)ⁿn!. In case of Figure 9b we refer to the diagram with a chain inserted into a chain analogous to Figure 4b.

· F (α^s(Q))^d/2 exp where F is defined in (3.37). Using (2.7) and (2.10), the large-order behaviour

rn n→∞

follows. Note the different signs of the anomalous dimension terms compared to (3.48).

(Otherwise the first UV renormalon can formally be obtained from setting d = −2.) The global normalization K is not determined. This equation is valid provided the renormalization counterterms do not absorb factorial divergence into the definition of renormalized parameters (Mueller 1985; Beneke 1993b), see also section 3.4.

For current-current correlation functions the leading IR renormalon corresponds to d = 4 and O = α^sG^A_µνG^µν,A. Taking into account that for this operator γ0 = 0 and γ1 = 2β1 one reproduces the leading asymptotic behaviour and the 1/n correction, obtained in (Mueller 1985) and (Beneke 1993b), respectively. The 1/n² correction could be computed also, if the two-loop correction to the coefficient function of the gluon condensate were known.

An important point is that the unknown constant K is a universal property of the soft part in Figure 8a, that is a property of the operator O. Hence for correlation functions with different currents, which differ only in their hard part, the difference in the leading IR renormalon behaviour is calculable. We refer to this property as universality of the leading IR renormalon or 1/Q⁴ power correction. Note, however, that universality is more restricted for IR renormalons than for UV renormalons, because it refers to a specific class of processes, in the present case given by various current-current correlation functions. Let us also note that for certain operators K can be exactly zero. These are operators like ¯qq which are protected from perturbative contributions to all orders in perturbation theory.

Our discussion has focused on the current-current correlation functions. The gener-alization to other off-shell quantities is straightforward.