---
abstract: 'A qualitative (and selective) discussion of current activities and problems in the field is given.'
---

CERN-TH/97-134\
June 1997\
hep-ph/9706457

[**Renormalon Phenomenology: Questions and Directions**]{} [^1]

[M. Beneke]{}\
[*Theory Division, CERN, CH-1211 Geneva 23, Switzerland*]{}\

The basic idea of ‘renormalon phenomenology’ is simply this: to parametrise, or at least unravel, power-like infrared (IR) sensitive contributions to hard processes. Just as perturbative IR logarithms in QCD lead to the introduction of non-perturbative parameters, such as parton densities in DIS processes, so do power-like dependences on an IR factorisation scale indicate power-suppressed non-perturbative contributions. In DIS they are known as higher-twist corrections. Now one aims at a more general understanding of power corrections, including truly Minkowskian processes. The fact that this is connected with renormalons, that is, large-order behaviour in perturbation theory, can be considered as an accident of history. \[The connection between large orders and small momentum is by itself quite interesting and has led to a better understanding of the systematics of exact multi-loop results. This, however, is not the subject of the present talk.\] In a wider sense the problem is the generalisation of perturbative factorisation beyond leading power accuracy.

DIS structure functions provide a useful example to start with. The twist expansion of the longitudinal structure function can be written as $$\begin{aligned}
\label{fl}
F_L(x,Q)/(2 x) 
&=& \sum_i\int\limits_x^1\frac{d\xi}{\xi}\,C_i^{[2]}(x,\xi,Q,\mu)
\,f_i(\xi,\mu)
\nonumber\\
&&\,+\frac{1}{Q^2}\sum_j\int \!d\xi_1 d\xi_2\,C_j^{[4]}(x,\xi_1,\xi_2,Q,\mu)
\,T_j(\xi_1,\xi_2,\mu) + \ldots.\end{aligned}$$ It is well-known that due to logarithmic operator mixing, the factorisation scale dependence in the leading twist term cancels only over different contributions (quarks and gluons) in the sum over $i$. It is less known that the separation of twist-2 and twist-4 is also not unique. It would be obvious, if the factorisation in transverse momentum were introduced explicitly, in which case powers of $\mu^2/Q^2$ would arise. In dimensional regularisation, the ambiguity in separating twists appears as renormalons: the series expansion of the coefficient function $C^{[2]}_i$ diverges in large orders in $\alpha_s$ and can not be unambiguously defined. This IR renormalon divergence comes from small momenta in the loops. The ambiguity is compensated by corresponding ultraviolet contributions to the matrix elements $T_j$ of twist-4 (non-local) operators. This point is crucial: although IR renormalons are IR compared to the scale $Q$, they correspond to ultraviolet effects viewed from the scale $\Lambda$, the scale of QCD. As a consequence, we can learn very little on the specifics of non-perturbative effects. What one does learn is the scaling of power corrections with the scale $Q$, just from the consistency requirement that a physical quantity must be unambiguous.

Calculations rely on approximations and to date these correspond to an analysis of IR sensitive regions in one-loop virtual corrections or one gluon emission. The set of bubble diagrams is one way to trace these regions through the large-order behaviour of these diagrams [@Z]. The same set of graphs can also be evaluated through a dispersion relation for the running coupling [@BB1; @DMW]. The IR contributions can then be found as non-analytic terms at small values of the dispersion variable. For sufficiently inclusive observables the dispersion variable can be identified with a gluon mass [@BBZ]. For other interesting quantities like event shape variables and fragmentation functions calculations with a gluons mass correspond to an alternative scheme of IR regularisation, not related to renormalons.

The past two years have seen many phenomenological applications of renormalons. Usually they involve the type of calculations just described, together with additional assumptions that can be judged only by their empirical success. The second line of interest pursues ‘operator interpretations’ of renormalons in Minkowski space, in analogy to higher-twist operators in DIS. If successful, one could then dispense of particular sets of diagrams and approach an understanding of power corrections comparable to applications of operator product expansions in Minkowski space. The following gives a telegram overview of some of the problems that have been addressed recently.\
[*The Drell-Yan process.*]{} This process has been the first process without operator product expansion, where power corrections have been analysed with the help of renormalons. Since collinear factorisation can be extended to $1/Q^2$ corrections [@QIU91], the main question is whether soft gluons could invalidate this result and introduce $1/Q$ corrections. \[$Q$ is the mass of the Drell-Yan pair.\] The Drell-Yan process seems to be well-suited to address this question, as soft gluon radiation has been extensively studied and the resummation of corresponding large logarithms is well understood.

The first investigations [@DY1] of renormalons in Drell-Yan production accordingly started from the soft gluon resummation formula and reported the presence of $1/Q$ corrections. It was then realized [@BBDY] that the approximations legitimate for a systematic resummation of logarithms lose $1/Q$ power corrections and that $1/Q$ corrections cancel in the full result, when these approximations are abandoned. While the leading logarithms originate from the region $k_\perp\ll k_0\ll Q$, the region $k_\perp\sim k_0$, that is, large angle gluon radiation, is equally important for power corrections.

The result of [@BBDY] is based on the analysis of one-loop diagrams. Although the absence of $1/Q$ corrections to all orders may be plausible, this has been shown so far only in an abelian theory [@ASZ], where it follows from the fact that one gluon emission is already the only building block for multiple soft gluon emission amplitudes. For QCD, a proof is still missing. Since the non-abelian vertices enter only at two loops, it might be useful to extend the analysis of IR sensitive regions to two-loop diagrams. This would also provide a check on possible interpretations of power corrections to Drell-Yan production in terms of some operators. If $1/Q$ corrections are indeed absent, as we believe, the same twist-4 multi-parton correlations that enter DIS would seem the best bet for these operators at order $1/Q^2$ [@QIU91].\
[*Hadronic event shape variables*]{}[@E1; @E2; @E3]. These are the simplest observables for which $1/Q$ corrections have been found. They come only from the soft region and it is natural to associate them with hadronization corrections. In addition, experimental information exists for various center-of-mass energies $Q$ in $e^+ e^-$ collisions and a $1/Q$ term fits the difference between data and NLO perturbation very well [@delphi].

One can take a step further towards the absolute magnitude of power corrections by assuming that $1/Q$ corrections are universal [@E2], i.e. that a single non-perturbative number controls $1/Q$ corrections to all event shape variables. To be precise, one assumes that for any (averaged) event shape $S$ (such as the average $1-T$ etc.), the $1/Q$ power correction is given by $$\label{s} 
S_{|1/Q} = K_S\cdot\frac{\langle \mu_{had}\rangle}{Q}$$ with a unique parameter $\langle \mu_{had}\rangle$ and a calculable coefficient $K_S$ that depends on $S$. Note that if one identifies $K_S$ with the residue of the IR renormalon pole that leads to the $1/Q$ ambiguity, $K_S$ can not be calculated, because all higher order diagrams contribute to the residue. However, the universality assumption implies that they contribute equally to different event shapes so that the ratio $K_{S_1}/K_{S_2}$ can be found by a one-loop calculation up to corrections of order $\alpha_s(Q)$. Since one event shape measurement is required to fix $\langle \mu_{had}\rangle$, knowledge of the ratio is sufficient to predict $1/Q$ corrections to other event shapes.

The universality assumption seems to work well phenomenologically, to an accuracy of about 20%. The calculation has now been done also for event shape variables in DIS [@DWDIS] and a good fit to the data is obtained with the same value of $\langle \mu_{had}\rangle$ as in $e^+ e^-$ collisions [@EDIS]. This could not have been expected theoretically.

It is fair to say that even for event shape variables in $e^+ e^-$ collisions alone a good theoretical argument in favour of universality still remains to be found. Diagrammatically it is evident that higher order contributions do not contribute equally to the renormalon residue [@E3], mainly because event shapes resolve large angle soft gluon emission at the level of $1/Q$ power corrections even in the two-jet limit [@BBDY]. As a consequence every event shape corresponds to a different weight on the distribution of soft gluons [@sterman] (see also [@BBM]), which can not be described by a single number $\langle \mu_{had}\rangle$. Another problem is the ‘uniqueness problem’: even if universality held (or held approximately), we do not know how to calculate ratios $K_{S_1}/K_{S_2}$ unambiguously. As mentioned above, calculations based on bubble diagrams or on a finite gluon mass as IR regulator lead to different results [@BBDY; @E3; @BBM]. Since there is no obvious reason to prefer one or the other, this difference must be considered as an uncertainty in $K_{S_1}/K_{S_2}$. For the longitudinal cross section, this difference is small [@BBM], about 20%. Given these reservations, the fact that universality appears to work approximately is by itself quite interesting and remains to be understood.\
[*$x$-dependence of twist-4 corrections in DIS*]{} [@DMW; @x]. Since the moments of DIS structure functions have an operator product expansion, one may not expect renormalons to give any further insight on their power behaviour. However, if one assumes that the $x$-dependence of the twist-4 correction follows the $x$-dependence of the corresponding IR sensitive contribution in perturbation theory, the ‘unknowns’ at twist-4 are reduced from functions to numbers. This is a very strong assumption and one may consider the result as a model for twist-4 multi-parton correlation functions. For example, eq. (\[fl\]) reduces to $$\label{fl2}
F_L(x,Q)/(2 x) 
= \sum_i\int\limits_x^1\frac{d\xi}{\xi}\,f_i(x/\xi,\mu)\,
\left[C_i^{[2]}(\xi,Q,\mu)+A_i(\xi)\frac{\Lambda^2}{Q^2}\right]
+ \ldots$$ with calculable functions $A_i(\xi)$. Comparison with twist-4 corrections extracted from DIS data shows surprising agreement of the model with the $x$-dependence of the data.

Theoretically it seems rather obscure that the model should work so well and again it is the data itself that teaches us an interesting lesson. Two suggestions have been put forward: the first [@DMW] is based on the idea of universality, which postulates that all non-perturbative effects are generated through integrals over an IR finite coupling, at least to first approximation. This postulate itself appears hard to digest; the second [@BBM] is an a posteriori explanation based on the correspondence of IR renormalons and cut-off dependence of higher-twist operators: the above model could be justified, if the matrix elements of twist-four operators were dominated by their cut-off piece rather than by ‘genuine’ non-perturbative effects. Again, there is no dynamical understanding why this should be so, unless we consider both $A_i(\xi)$ and the data on twist-4 corrections as effective parametrisation of higher order perturbative corrections beyond NLO. Another possibility is that one is mainly seeing generic $x$-dependences that follow from kinematics such as $A_i(x)/C_i^{[2]}(x)\sim 1/(1-x)$ as $x\to 1$.

Whatever the agreement with the data, the model by construction gives no insight into hadron structure. In other words, in terms of moments $$\label{moment}
\frac{M_n^{twist-4}}{M_n^{twist-2}}|_{\mbox{hadron 1}} - 
\frac{M_n^{twist-4}}{M_n^{twist-2}}|_{\mbox{hadron 2}} \equiv 0.$$ The model can only work for those aspects of twist-4 corrections which are not hadron-specific. Let me also note that in present applications, the added twist-4 correction does not have the correct scale-dependence. This constitutes an additional uncertainty, although the problem is not fundamental and could be solved by higher-loop calculations of $A^i(\xi)$.\
[*Fragmentation*]{} [@BBM; @DW]. The same model has been suggested for fragmentation processes in $e^+ e^-$ annihilation, although there is as yet no data to compare with. The uniqueness problem mentioned for event shape variables affects the calculation of the functions $A^i(x)$ in fragmentation even more severely. On the other hand fragmentation processes are particularly interesting, because although the leading-twist formalism is analogous to DIS, there is no operator product expansion to provide information beyond leading power. It has been found that for non-zero value of the scaling variable $x$, the leading power corrections are of order $1/Q^2$, in agreement with the collinear expansion of fragmentation functions in [@BBOPE]. But this statement ceases to be true for moments in $x$, which include the soft region at small $x$. The power expansion of fragmentation processes has strong soft gluon singularities at small $x$, which are non-integrable. After integration over $x$, including the small-$x$ region, every term $(\Lambda^2/(Q^2 x^2))^n$ in the expansion at non-zero $x$ can contribute to the leading power correction to the moments. The leading power correction depends on the order of the moment, a situation that is excluded for moments of DIS structure functions. In particular, the total longitudinal cross section receives a $1/Q$ power corrections similar to other event shapes. Only for high enough moments is the soft region suppressed and power corrections scale universally as $1/Q^2$ due to collinear regions. As a consequence, a light-cone expansion for moments of fragmentation expansions does not exist. Note that in DIS and fragmentation, the twist expansion also breaks down as $x\to 1$. However, the singularities as $x\to 1$ are integrable and do not alter the power behaviour of the power expansion of moments.

It is known that at small $x$ multiple gluons emission and coherence effects lead to a suppression of soft hadron production, which formally follows from resummation of logarithms in $x$ in perturbation theory. One may wonder how multiple soft gluon emission would affect the $x$-behaviour of power corrections in the small-$x$ region, which has been crucial in understanding the emergence of a $1/Q$ correction to the longitudinal cross section. It may well be that the two problems are in fact disconnected because they refer to different momentum regions in Feynman integrals. On the other hand, the question of how to reconcile the two different sets of higher order diagrams that correspond to multiple gluon emission and to renormalons has not yet been addressed.\
[*Soft gluon cancellations beyond logarithms.*]{} A more general understanding of under which circumstances $1/Q$ corrections exist would be highly desirable. From what we have learnt, such corrections, if present, would arise only from soft, but not from collinear partons. The approach to the problem taken in [@AZ] makes essential use of the KLN and Low theorems, but still has to be completed for the non-abelian theory. As shown in [@AZ] the KLN theorem alone already guarantees the absence of $1/Q$ corrections to a process inclusive over degenerate final and initial states, basically because the KLN transition amplitudes have no $1/k^0$ contributions (where $k^0$ is the emitted gluon’s energy). As a consequence, the amplitude squared integrated over phase space is proportional to $dk^0 k^0$ for small $k^0$ which implies at most $1/Q^2$ corrections. The main obstacle in this approach appears to be the generalisation of the Low theorem required to dispense of the sum over degenerate initial states implied by the KLN theorem.

Another possibility would be to investigate in what situations the Ward identities that guarantee the cancellation of soft gluon divergences in the factorisation proofs generalise to the cancellation of $1/Q$ corrections as well.\
More theoretical work is needed particularly in this direction. Meanwhile, the phenomenology of renormalons is encouraging, but some magic seems to be at work: things work that needn’t have to.

[99]{} Zakharov, V.I., [*Nucl. Phys.*]{} [**B385**]{}, 452 (1992); Beneke, M., [*Nucl. Phys.*]{} [**B405**]{}, 424 (1993) Beneke, M. and Braun, V.M., [*Phys. Lett.*]{} [**B348**]{}, 513 (1995); Ball, P., Beneke, M. and Braun, V.M., [*Nucl. Phys.*]{} [**B452**]{}, 563 (1995). Dokshitser, Yu.L, Marchesini, G. and Webber, B.R., [*Nucl. Phys.*]{} [**B469**]{}, 93 (1996). Beneke, M., Braun, V.M. and Zakharov, V.I., [*Phys. Rev. Lett.*]{} [**73**]{}, 3058 (1994). Qiu, J. and Sterman, G., [*Nucl. Phys.*]{} [**B353**]{}, 105 (1991), [*ibid.*]{} [**B353**]{}, 137 (1991). Contopanagos, H. and Sterman, G., [*Nucl. Phys.*]{} [**B419**]{}, 77, (1994); Korchemsky, G.P. and Sterman, G., [*Nucl. Phys.*]{} [**B437**]{}, 415, (1995). Beneke, M. and Braun, V.M., [*Nucl. Phys.*]{} [**B454**]{}, 253, (1995). Akhoury, R., Sotiropoulos, M.G. and Zakharov, V.I., [*Phys. Rev.*]{} [**D56**]{}, 377 (1997) Manohar, A.V. and Wise, M.B., [*Phys. Lett.*]{} [**B344**]{}, 407 (1995); Webber, B.R., [*Phys. Lett.*]{} [**B339**]{}, 148 (1994). Dokshitser, Yu.L. and Webber, B.R., [*Phys. Lett.*]{} [**B352**]{}, 451 (1995); Akhoury, R. and Zakharov, V.I., [*Phys. Lett.*]{} [**B357**]{}, 646 (1995). Nason, P. and Seymour, M.H., [*Nucl. Phys.*]{} [**B454**]{}, 291 (1995). See, e.g., DELPHI collaboration, DELPHI 96-107 CONF 34, paper pa02-003 submitted to ICHEP’96, Warsaw 1996. Dasgupta, M. and Webber, B.R., Cavendish-HEP-96/5 \[hep-ph/9704297\] Martyn, H.-U. and Rabbertz, K. (for the H1 Collaboration), [*these proceedings*]{}. Sterman, G., [*these proceedings*]{}. Beneke, M., Braun, V.M., Magnea, L., [*Nucl. Phys. B (Proc. Suppl.)*]{} [**54A**]{}, 183 (1997) \[hep-ph/9609266\]; CERN-TH/96-362 \[hep-ph/9701309\], to appear in [*Nucl. Phys.*]{} [**B**]{}. Stein, E. et al., [*Phys. Lett.*]{} [**B376**]{}, 177 (1996); Dasgupta, M. and Webber, B.R., [*Phys. Lett.*]{} [**B382**]{}, 273 (1996); Maul, M. et al., [*Phys. Lett.*]{} [**B401**]{}, 100 (1997). Dasgupta, M. and Webber, B.R., [*Nucl. Phys.*]{} [**B484**]{}, 247 (1997) Balitsky, I.I. and Braun, V.M., [*Nucl. Phys.*]{} [**B361**]{}, 93 (1991). Akhoury, R. and Zakharov, V.I., [*Phys. Rev. Lett.*]{} [**76**]{}, 2238 (1996); Akhoury, R., Stodolsky, L. and Zakharov, V.I., MPI-PH-96-88 \[hep-ph/9609368\]

[^1]: Talk presented at the Fifth International Workshop on ‘Deep Inelastic Scattering and QCD’ (DIS’97), Chicago, IL, April 1997.
