February 13, 2004 10:42 WSPC/Trim Size: 9in x 6in for Proceedings p

POWER CORRECTIONS IN THE DEEP INELASTIC SCATTERING

S. ALEKHIN Institute for High Energy Physics (IHEP), Pobeda, 1, Protvino 142281 Russia E-mail: alekhin@sirius.ihep.su

We outline the impact of the power corrections on the study of deep inelastic scattering and describe the determination of the power corrections from the existing deep inelastic scattering data.

One of the main ingredients of the high energy phenomenology of the hadrons interaction is factorization. It allows to greatly simplify theoretical calculations using the connection of different processes through convolution of the coefficient functions (i.e. parton cross sections) with the parton distributions functions (PDFs). However factorization is valid only for the hard interactions, i.e. in the limit of large masses, transverse momenta, etc. For the soft interactions it is broken due to parton correlations being non-negligible. The formalism for the description of the factorization breakdown is given by the operator-product expansion (OPE)^{1}. In terms of this expansion the factorized part of the cross section corresponds to the contribution of the leading-twist (LT) operators and the factorization-breaking part – to the higher-twist (HT) ones. The coefficient functions for the LT terms and the dependence of the PDFs on the transverse momentum Q can be calculated in the pQCD-improved parton model. In accordance to these calculations the LT (twist-2) structure functions of deep inelastic scattering (DIS) demonstrate logarithmic Q-dependence. The Q-dependence of the HT terms is mainly defined by the factor of 1/Q^{τ−2}, where τ is the corresponding value of twist. Due to the 1/Q factors the HT terms rapidly fall with Q and in the limit of large Q the LT terms give the dominant contribution. However the coefficients at the power-like factors cannot be calculated using existing approaches and therefore the on-set scale of this asymptotic regime is a priori unclear. The importance of this problem was recognized just after first attempts to confront the pQCD predictions to

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the earliest available data since it was understood that the measured Qdependence of the DIS structure functions can be equally well described both by the log-like and the power-like behavior^{2}. With more precise measurements in the wide region of Q, which appeared later, we can better separate the LT and the HT terms, however the impact of the HT terms on different aspects of the DIS phenomenology is not understood completely. In this report we describe the modern status of this problem and outline possible ways to resolve it.

Figure 1. The x-dependence of twist-4 term in the proton structure function F_{2} determined from the fit of Ref.^{20} to the combined SLAC-BCDMS-NMC-H1-ZEUS data. The value of α_{s} given in the panel was obtained from the same fit.

For DIS the OPE up to twist-4 terms reads

F_{i}(x, Q^{2}) = F_{i}^{LT,TMC}(x, Q^{2}) + \frac{H_{i}(x, Q^{2})}{Q^{2}} + O (1/Q^{4}) ,  (1)

where i marks different structure functions and F^{LT,TMC} are the LT terms corrected for the target mass effects^{3}. These target mass effects give contribution to the power-like terms of the OPE, but are regularly considered separately since they are just of kinematical origin and can be expressed as

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a convolution of the LT terms with the coefficient functions proportional to negative powers of Q (see^{4}). The functions H give the dynamical HT contributions and correspond to the operators, which include correlations of the quark and gluon fields^{5}. Calculation of matrix elements of these operators in field theory is very difficult^{6} and only very limited results have been obtained so far^{7,8}. The predictions of most of the models for the HT terms contain essential free parameters and cannot be used without a fit to the data. For this reason the main source of information about the HT comes from phenomenological studies. These studies are regularly based on the QCD extrapolation from the high-Q region to the low-Q one, where a discrepancy between the extrapolation and data is considered as a manifestation of the HT terms. This approach requires accurate treatment of all sources of theoretical uncertainties, which can have an impact on the extrapolation, and therefore to bias the results or even generate fake HT terms. Besides, since the the existing DIS data from any one single experiment do not cover the whole kinematic region necessary for the study of such kind, the combination of all available data is desirable. In this case the fake HT terms can be also generated due to incomparability of the separate data sets, especially if the kinematics of different experiments do not overlap.

The first firm evidence of the HT dynamical terms was found in^{9}. In this analysis the twist-4 terms in the structure functions F_{2} for proton and deuteron were determined from the next-to-leading order (NLO) QCD fit to the combined SLAC-BCDMS data with account of the HT terms parameterized in the factorized form:

F_{2}(x, Q^{2}) = F_{2}^{LT,T MC}(x, Q^{2}) [1 + \frac{h_{2}(x)}{Q^{2}}] .  (2)

This form is equivalent to (1) modulo the impact of the anomalous dimensions of LT and HT operators. The latter must exist as follows from the theoretical expectations^{10} and were indeed found in the phenomenological analysis of^{11}. However at x ≤ 0.75, which are relevant for the analysis of^{9}, the results of fits based on (1) and (2) are comparable within the experimental errors^{12}. This indicates that the logarithmic corrections to the Q-dependence of the HT terms, which are generated by the anomalous dimensions, are unimportant in this region of x. The interpretation of the HT terms observed in the NLO analysis of^{9} for a long time was questionable due to the concern that they can be simulated by the missing higher-order QCD corrections. This concern is based on the observation that the Q-

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Figure 2. Comparison of the isospin asymmetry in the HT terms of the structure functions F_{2} and F_{T} = F_{2} + F_{L} extracted from the global fit of Ref.^{27} (solid curves) and calculated from the IRR model (dashes).

dependence of the terms proportional to the large powers of the strong coupling constant α_{s} can simulate power-like behavior in the limited range of Q (see^{13}). However such a simulation is possible only for the specific choice of powers of α_{s} and Q and/or insufficiently precise data. In DIS the next-to-next-to-leading (NNLO) QCD corrections do not simulate the power-like behavior within the precision of existing data and as a result the change of the dynamical twist-4 terms from the NLO to the NNLO fit does not exceed the experimental errors^{14} (similar stability was observed earlier in^{15}, although conclusion about the non-vanishing of the dynamical HTs was not drawn due to the fact that they were not separated from the target-mass effects). The same stability should be valid also for the higherorder corrections since the N^{3}LO correction is important at very high and very low x only as follows from the analyses based on the resummation

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technique^{16,17,18,19}.

The x-dependence of the twist-4 term in the proton structure function F_{2} extracted from the data with the assumption, that functions H(x, Q) do not depend on Q, is given in Fig.1. This term is maximal at x ∼ 0.6 and is determined in this region with the accuracy of O(10%) that allows to put a constraint on some matrix elements of the twist-4 operators^{21}.

Figure 3. The HT terms in the nucleon structure function F_{L} determined from the global fit of Ref.^{20}.

The impact of the HT terms is important up to relatively large values of Q. E.g. at x = 0.6 the experimental error in F_{2} averaged over global data set is less than the HT terms at Q^{2} ≳ 20 GeV^{2} only. In practice this means that inaccount of the HT terms can lead to bias of different quantities extracted from the DIS data. It is of particular importance for the extraction of α_{s}. Indeed, in the global fit with constraints H = 0 the value of α_{s}(M_{Z}) = 0.1215 + −0.0003 (c.f. α_{s}(M_{Z}) = 0.1143 + −0.0014 with the HT terms fitted)^{22}. For the fit without account of the HT terms we observe the shift of the central value and an essential decrease of the error. Both effects are due to the value of α_{s} and are strongly anti-correlated with H_{2}. Essentially this means that for the global fit of existing data the error

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in α_{s} is determined by the uncertainty in the HT terms and if these terms could be reliably estimated (e.g. from lattice calculations) the error in α_{s} would be suppressed several times. The cut on data with Q^{2} ≲ 20 GeV^{2} aimed to decrease the uncertainty in α_{s} due to the HT terms is ineffective since in this case the error in α_{s} rises due to loss of data corresponding to the fastest rate of the QCD evolution. The promising way in this direction is to collect new statistically significant samples of data at HERA.

Figure 4. The 1σ band of the global fit (curves) compared to the data on F_{2} used in the fit.

A good precision in the determination of H_{2} allows a conclusive comparison to the theoretical models. One of the most predictive model of the HTs is based on the infrared renormalon (IRR) approach^{23}. This model does not fix the general normalization of the HT terms, but puts a stringent constraint on their x-dependence. For the non-singlet terms the IRR model gives an especially simple predictions^{24,25}

H_{i}^{NS} = A · F_{2}^{LT,NS} ⊗ C_{i}^{IRR},  (3)

where ⊗ denotes the Mellin convolution, C_{i}^{IRR} are the corresponding coefficient functions and A is the normalization factor. These predictions

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are in qualitative agreement to the data, however the subtle details are not reproduced. For example a study of the isospin asymmetry, which can be used for a test of the basic features of the IRR model, since C_{i}^{IRR} are flavor-independent^{26}, demonstrates significant disagreement between the HT terms extracted from the data and the IRR predictions (see Fig.2).

Figure 5. The JLAB and SLAC measurements of R = σ_{L}/σ_{T} compared to 1σ band for R^{LT,TMC} extracted from the NNLO global fit of Ref.^{20}.

The HT term in F_{L} extracted from the global fit to the DIS data is given in Fig.3. The observed x-dependence of H_{L} is less regular than the one of H_{2}. It might be that due to F_{L} extracted from the global fit it is rather sensitive to disagreements in the data. This point is illustrated in Fig.4, which contains the comparison of the results of the global fit to the data on F_{2} at x ≈ 0.012. It is evident that due to the large gap between separate data sets any disagreement between the NMC data at low Q and/or the HERA ones at high Q can be compensated by the corresponding HT term in F_{L}, if one performs the fit to the data of the cross section in order to extract H_{L}. Therefore the spike at small x might be driven by such a disagreement and more data at small x and Q^{2} = 2 ÷ 30 GeV^{2} will be extremely useful for clarification of this point. An alternative explanation of this spike is the

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effect of missing higher-order QCD corrections that is partially supported by the relative perturbative instability of H_{L}, at least at x ∼ 0.2. Indeed, the higher-order corrections at small x obtained using the k_{T} resummation technique are very large^{28} and with the use of the approximate 3-loop coefficient function for F_{L}^{LT} from^{15} based on the calculations of^{28,29} one probably should observe a decrease of H_{L} extracted from the fit. However, one has to take into account that the uncertainty in the 3-loop coefficient function of^{15} due to missing terms in the 3-loop corrections is very large and the impact of the exact 3-loop correction might turn insufficient to explain the observed spike. The spike might be generated by a new phenomena such as the parton saturation^{30} and, eventually, the OPE up to twist-4 might be insufficient^{31}. In agreement with the latter the non-vanishing twist-6 contribution to F_{L} was found in the analysis of^{11}. This analysis was based on the data from the resonance region and therefore in order to obtain this result the duality hypothesis^{32} was heavily used. However, the recent JLAB-E94-110 data^{33} beyond the resonance region confirm this observation. In Fig.5 the JLAB measurements of R = σ_{L}/σ_{T} are compared to the earlier SLAC data^{34} and the LT term extracted from the global fit. Excess of the SLAC data over the LT term with account of the target mass correction at Q^{2} ∼ 2 GeV^{2} is interpreted as a manifestation of the HT term in R and, correspondingly, in F_{L}^{35}. At small Q the relation between data and the LT term is opposite and the most natural interpretation within OPE is that the effect of the positive twist-4 contribution is overwhelmed by the negative twist-6 contribution. If so, the structure of the HT terms is very non-trivial, at least for F_{L}. In summary, despite the predictions based on the HT terms given in Fig.3 are in good agreement with the independent measurements of F_{L} at small x and Q by the H1 collaboration (see^{36} for the details), the status of the determination of H_{L} is rather unclear and for a better understanding we need efforts both from the theoretical and the experimental side.

The dynamical HT terms in the polarized structure functions are even less known than for the unpolarized ones, although the potential impact of the HT terms is very important in this case due to the typical relatively low Q of the data used in the analysis. The recent indication on the the HT terms in g_{1}^{37} was not entirely confirmed in the latter study^{38} and, evidently, a more careful examination is necessary.

The HT terms in the DIS of neutrinos are also poorly known because of the low precision of the available data (see recent review^{39} for more details). The essential progress in this area can be achieved only with the very

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intense neutrino beams (neutrino factories)^{40}. These facilities would allow to constraint the dynamical HT terms in the neutrino-nucleon structure function at the level of precision comparable to the case of the charged leptons’ DIS. However, since the neutrino experiments provide a much wider set of constraints, this additional input would allow to significantly improve disentangling the matrix element of different operators, which describe the partons’ correlation.

In conclusion, our understanding of the power corrections in the DIS structure functions is rather limited. The kinematical component of the power correction (target mass effects) can be calculated with a reasonable accuracy, but the dynamical HT terms are essentially unknown for the most of structure functions. Meanwhile further study of HT terms is necessary in order to outline the kinematical region, where the QCD factorization can be applied and the universality of PDFs can be proven. From a practical point of view it is necessary to provide unbiased extrapolations of PDFs to the high-energy region for the searches of the effects of new physics. Eventually, study of the dynamical HT terms give information about the regime intermediate between asymptotic freedom and confinement. All this makes the precise determination of the HT terms an important and promising topic.

The research was supported in part by the RFBR grant 03-02-17177 and by the National Science Foundation under Grant No. PHY99-07949.

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