Higher Twist, ξ_{w} Scaling, and Effective LO P DF s for Lepton Scattering in the Few GeV Region

A Bodek†and U K Yang‡§

† Department of Physics and Astronomy, University of Rochester, Rochester, New York 14618, USA ‡ Enrico Fermi Institute, University of Chicago,Chicago, Illinois 60637, USA

Abstract. We use a new scaling variable ξ_{w}, and add low Q^{2} modifications to GRV98 leading order parton distribution functions such that they can be used to model electron, muon and neutrino inelastic scattering cross sections (and also photoproduction) at both very low and high energies.

1. Higher Twists and Previous Results with GRV94 PDFs and x_{w}

The quark distributions in the proton and neutron are parametrized as Parton Distribution Functions (PDFs) obtained from global fits to various sets of data at very high energies. These fits are done within the theory of Quantum Chromodynamics (QCD) in either leading order (LO) or next to leading order (NLO). The most important data come from deep-inelastic e/μ scattering experiments on hydrogen and deuterium, and ν_{μ} and \overline{ν}_{μ} experiments on nuclear targets. In previous publications [1, 2, 3] we have compared the predictions of the NLO MRSR2 PDFs to deep-inelastic e/μ scattering data [4] on hydrogen and deuterium from SLAC, BCDMS and NMC. In order to get agreement with the lower energy SLAC data for F_{2} and R down to Q^{2}=1 GeV^{2}, and at the highest values of x (x = 0.9), we found that the following modifications to the NLO MRSR2 PDFs must be included.

(i) The relative normalizations between the various data sets and the BCDMS systematic error shift must be included [1, 2].

(ii) Deuteron binding corrections need to be applied and the ratio of d/u at high x

must be increased as discussed in ref. [1]. (iii) Kinematic higher-twist originating from target mass effects [5] are very large and

must be included. (iv) Dynamical higher-twist corrections are smaller but also need to be included [1, 2].

(v) In addition, our analysis including QCD Next to NLO (NNLO) terms shows [2] that most of the dynamical higher-twist corrections needed to fit the data within a NLO QCD analysis originate from the missing NNLO higher order terms.

§ Emails: bodek@fnal.gov; ukyang@fnal.gov. This contribution to J. Phys. G. is based on two talks by Arie Bodek at the NuFact′02 conference, Imperial College, London, England, July 2002.

Figure 1. Electron and muon data (SLAC, BCDMS and NMC) for F_{2p} [a] and R [b] compared to the predictions with MRSR2 NLO PDFs including both NNLO and target mass corrections with (solid line) and without (dashed line) higher twist corrections (From Yang and Bodek Ref. [2]). These studies indicate that in QCD LO or NLO fits, the extracted higher twist corrections originate from target mass effects and the missing QCD NNLO higher order terms (for Q^{2} > 1 GeV^{2}).

Figure 1 shows that the NLO MRSR2 PDFs with target mass and NNLO higher order terms describe electron and muon scattering F_{2} and R data with a very small contribution from higher twists. Studies by other authors [6] also show that in NNLO analyses the dynamic higher twist corrections are very small. If (for Q^{2} > 1 GeV^{2}) most of the higher-twist terms needed to obtain agreement with the low energy data actually originate from target mass effects and missing NNLO terms (i.e. not from interactions with spectator quarks) then these terms should be the same in ν_{μ} and e/μ scattering. Therefore, low energy ν_{μ} data should be described by the PDFs which are fit to high energy data and are modified to include target mass and highertwist corrections that fit low energy e/μ scattering data. However, for Q^{2} < 1 GeV^{2} additional non-perturbative effects from spectator quarks must also be included [7].

In a previous communication [7] we used a modified scaling variable x_{w} and fit for modifications to the GRV94 leading order PDFs such that the PDFs describe both high energy low energy e/μ data. In order to describe low energy data down to the photoproduction limit (Q^{2} = 0), and account for both target mass and higher twist effects, the following modifications of the GRV94 LO PDFs are need:

(i) We increased the d/u ratio at high x as described in our previous analysis [1]. (ii) Instead of the scaling variable x we used the scaling variable x_{w} = (Q^{2} +

B)/(2Mν + A) (or =x(Q^{2} + B)/(Q^{2} + Ax)). This modification was used in early fits to SLAC data [8]. The parameter A provides for an approximate way to include both target mass and higher twist effects at high x, and the parameter B allows the fit to be used all the way down to the photoproduction limit (Q^{2}=0).

Figure 2. Electron and muon F_{2} data (SLAC, BCDMS, NMC, H1 94) used in our fits compared to the predictions of the unmodified GRV98 PDFs (LO, dashed line) and the modified GRV98 PDFs fits (LO+HT, solid line); [a] for F_{2} proton,

[b] for F_{2} deuteron, and [c] for the H1 and NMC proton data at low x.

(iii) In addition as was done in earlier non-QCD based fits [9] to low energy data, we multiplied all PDFs by a factor K=Q^{2} / (Q^{2} +C). This was done in order for the fits to describe low Q^{2} data in the photoproduction limit, where F_{2} is related to the photoproduction cross section according to

σ(γp) = \frac{4π^{2}α_{EM}}{Q^{2}} F_{2} = \frac{0.112mb GeV^{2}}{Q^{2}} F_{2}

(iv) Finally, we froze the evolution of the GRV94 PDFs at a value of Q^{2} = 0.24 (for Q^{2} < 0.24), because GRV94 PDFs are only valid down to Q^{2} = 0.23 GeV^{2}.

In our analyses, the measured structure functions were corrected for the BCDMS systematic error shift and for the relative normalizations between the SLAC, BCDMS

Figure 3. Comparisons to data not included in the fit. (a) Comparison of SLAC and JLab (electron) F_{2p} data in the resonance region (or fits to these data) and the predictions of the GRV98 PDFs with (LO+HT, solid) and without (LO, dashed) our modifications. (b) Comparison of photoproduction data on protons to predictions using our modified GRV98 PDFs. (c) Comparison of representative CCFR ν_{μ} and \overline{ν}_{μ} on iron at 55 GeV and the predictions of the GRV98 PDFs with (LO+HT, solid) and without (LO, dashed) our modifications.

and NMC data [1, 2]. The deuterium data were corrected for nuclear binding effects [1, 2]. A simultaneous fit to both proton and deuteron SLAC, NMC and BCDMS data (with x > 0.07 only) yields A=1.735, B=0.624 and C=0.188 (GeV^{2}) with GRV94 LO PDFs (χ^{2} = 1351/958 DOF). Note that for x_{w} the parameter A accounts for both target mass and higher twist effects.

2. New Analysis with ξ_{w}, G_{D} and GRV98 PDFs

In this publication we use a new improved scaling variable ξ_{w} and fit for modifications to more modern GRV98 LO PDFs such that the PDFs describe both high energy and low energy electron/muon data. We now also include NMC and H1 94 data at lower x. Here we freeze the evolution of the GRV98 PDFs at a value of Q^{2} = 0.8 (for Q^{2} < 0.8), because GRV98 PDFs are only valid down to Q^{2} = 0.8 GeV^{2}. In addition, we use different photoproduction limit multiplicative factors for valence and sea. Our proposed new scaling variable is based on the following derivation. Using energy momentum conservation, it can be shown that the factional momentum ξ = (p_{z} + p_{0})/(P_{z} + P_{0}) carried by a quark of 4-mometum p in a proton target of mass M and 4-momentum P is given by ξ = xQ^{′2}/[0.5Q^{2}(1 + [1 + (2Mx)^{2}/Q^{2}]^{1/2})], where

2Q^{′2} = [Q^{2} + M_{f}^{2} − M_{i}^{2}] + [(Q^{2} + M_{f}^{2} − M_{i}^{2})^{2} + 4Q^{2}(M_{i}^{2} + P_{T}^{2})]^{1/2}.

Here M_{i} is the initial quark mass with average initial transverse momentum P_{T} and M_{f} is the mass of the quark in the final state. The above expression for ξ was previously derived [5] for the case of P_{T} = 0. Assuming M_{i} = 0 we use instead:

ξ_{w} = x(Q^{2} + B + M_{f}^{2})/(0.5Q^{2}(1 + [1 + (2Mx)^{2}/Q^{2}]^{1/2}) + Ax)

Here M_{f}=0, except for charm-production processes in neutrino scattering for which M_{f}=1.5 GeV. For ξ_{w} the parameter A is expected to be much smaller than for x_{w} since now it only accounts for the higher order (dynamic higher twist) QCD terms in the form of an enhanced target mass term (the effects of the proton target mass are already taken into account using the exact form in the denominator of ξ_{w} ). The parameter B accounts for the initial state quark transverse momentum and final state quark effective ΔM_{f}^{2} (originating from multi-gluon emission by quarks).

Using closure considerations [10] (e.g.the Gottfried sum rule) it can be shown that, at low Q^{2}, the scaling prediction for the valence quark part of F_{2} should be multiplied by the factor K=[1-G_{D}^{2}(Q^{2})][1+M(Q^{2})] where G_{D} = 1/(1+Q^{2}/0.71)^{2} is the electric proton elastic form factor, and M(Q^{2}) is related to the magnetic elastic form factors of the proton and neutron. At low Q^{2}, [1-G_{D}^{2}(Q^{2})] is approximately Q^{2}/(Q^{2} +C) with C = 0.71/4 = 0.178 (versus our fit value C=0.18 with GRV94). In order to satisfy the Adler Sum rule [11] we add the function M(Q^{2}) to account for terms from the magnetic and axial elastic form factors of the nucleon). Therefore, we try a more general form K_{valence}=[1-G_{D}^{2}(Q^{2})][Q^{2}+C_{2}v]/[Q^{2} +C_{1}v], and K_{sea}=Q^{2}/(Q^{2}+C_{sea}). Using this form with the GRV98 PDFs (and now also including the very low x NMC and H1 94 data in the fit) we find A=0.418, B=0.222, and C_{1v}=0.604, C_{2v}=0.485, and C_{sea}=0.381 (all in GeV^{2}, χ^{2} = 1268/1200 DOF). As expected, A and B are now smaller with respect to our previous fits with GRV94 and x_{w}. With these modifications, the GRV98 PDFs must also be multiplied by N=1.011 to normalize to the SLAC F_{2p} data. The fit (Figure 2) yields the following normalizations relative to the SLAC F_{2p} data (SLAC_{D}=0.9863, BCDMS_{P} =0.9640, BCDMS_{D}=0.9772, NMC_{P} =1.003, NMC_{D}=0.9881, H1_{P} =0.9776, and BCDMS systematic error shift of 1.7).

Comparisons of predictions using these modified GRV98 PDFs to other data which were not included in the fit is shown in Figure 3. From duality [13] considerations, with the ξ_{w} scaling variable, the modified GRV98 PDFs should also provide a reasonable description of the average value of F_{2} in the resonance region. Figure 3(a) shows a comparison between resonance data (from SLAC and Jefferson Lab, or parametrizations of these data [14]) and the predictions with the standard GRV98 PDFs (LO) and with our modified GRV98 PDFs (LO+HT). The modified GRVB98 PDFs are in good agreement with SLAC and JLab resonance data down to Q^{2} = 0.07 (although resonance data were not included in our fits). There is also very good agreement of the predictions of our modified GRV98 in the Q^{2} = 0 limit with photoproduction data as shown in Figure 3(b). We also compare the predictions with our modified GRV98 PDFs (LO+HT) to a few representative high energy CCFR ν_{μ} and \overline{ν}_{μ} charged-current differential cross sections [3, 12] on iron (neutrino data were not included in our fit). In this comparison we use the PDFs to obtain F_{2} and xF_{3} and correct for nuclear effects in iron [7]. The structure function 2xF_{1} is obtained by using the R_{world} fit from reference [4]. There is very good agreement of our predictions with these neutrino data on iron.

In order to have a full description of all charged current ν_{μ} and \overline{ν}_{μ} processes, the contribution from quasielastic scattering [15] must be added separately at x = 1. The best prescription is to use our model in the region above the first resonance (above W =1.35 GeV) and add the contributions from quasielastic and first resonance [16] (W =1.23 GeV) separately. This is because the W = M and W =1.23 GeV regions are dominated by one and two isospin states, and the amplitudes for neutrino versus electron scattering are related via Clebsch-Gordon rules [16] instead of quark charges (also the V and A couplings are not equal at low W and Q^{2}). In the region of higher mass resonances (e.g. W =1.7 GeV) there is a significant contribution from the deepinelastic continuum which is not well modeled by the existing fits [16] to neutrino resonance data (and using our modified PDFs should be better). For nuclear targets, nuclear corrections [7] must also be applied. Recent results from Jlab indicate that the Fe/D ratio in the resonance region is the same as the Fe/D ratio from DIS data for the same value of ξ (or ξ_{w}). The effects of terms proportional to the muon mass and F_{4} and F_{5} structure functions in neutrino scattering are discussed in Ref. [15, 17].

In the future, we plan to investigate the effects of including the initial state quark P_{T} in ξ_{w}, and institute further improvements such as allowing for different higher twist parameters for u, d, s, c, b quarks in the sea, and the small difference (expected in the Adler sum rule) in the K factors for axial and vector terms in neutrino scattering. In addition, we can multiply the PDFs by a modulating function [8, 10] A(W,Q^{2}) to improve modeling in the resonance region (for hydrogen) by including (instead of predicting) the resonance data [14] in the fit. We can also include resonance data on deuterium [14] and heavier nuclear targets in the fit, and low energy neutrino data. Note that because of the effects of experimental resolution and Fermi motion [18] (for nuclear targets), a description of the average cross section in the resonance region is sufficient for most neutrino experiments.

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