Nuclear shadowing in neutrino-nucleus deeply inelastic scattering

In the framework of the collinear factorized pQCD approach we calculate the small-$x_B$ process-dependent nuclear modification to the structure functions measured in neutrino-nucleus deeply inelastic scattering. We include both heavy quark mass corrections $(M^2/Q^2)$ and resummed nuclear-enhanced dynamical power corrections in the quantity $(\xi^2/Q^2)(A^{1/3}-1)$ with $\xi^2$ evaluated to leading order in $\alpha_s$. Our formalism predicts a measurable difference in the shadowing pattern of the structure functions $F_2^A(x_B,Q^2)$ and $F_3^A(x_B,Q^2)$ and a significant low- and moderate-$Q^2$ modification of the QCD sum rules. We also comment on the relevance of our results to the NuTeV extraction of $\sin^2\theta_W$.


I. INTRODUCTION
Recent surprising results on sin 2 θ W , reported by the NuTeV collaboration and based on a comparison of charged and neutral current neutrino interactions with an iron rich target [1], renewed our quest for understanding the nuclear dependence in neutrino-nucleus deeply inelastic scattering (DIS). A possibility that processdependent nuclear shadowing might affect the NuTeV extraction of the Weinberg angle θ W was raised by Miller and Thomas [2]. Although such scenario was considered unlikely by the collaboration [3], a systematic study and a clear understanding of the process-dependent nuclear effects in neutrino-nucleus scattering will strengthen the importance of the NuTeV result.
Like all nuclear dependences in the physical cross sections [4], the small-x B shadowing in lepton-nucleus DIS has both process-dependent and process-independent contributions. While its universal part can be factorized in the leading twist nuclear parton distribution functions (nPDFs), the DIS-specific modifications arise from the higher twist (or power) corrections to the structure functions [2,5,6]. In this letter, we present a calculation of the process-dependent shadowing in neutrino-nucleus deeply inelastic scattering by resumming heavy quark mass corrections, M 2 /(2p · q) = x B M 2 /Q 2 , and nuclear size enhanced dynamical power corrections, (ξ 2 /Q 2 )(A 1/3 − 1) with ξ 2 ∝ F +⊥ F + ⊥ , the gluon density in a large nucleus. The numerical value for the characteristic scale of higher twist ξ 2 [6], extracted from DIS data on µ-A interactions [7], is much less than Q 2 in the region which is perturbatively accessible. Therefore, we only evaluate ξ 2 to the leading order in α s , while resumming the power corrections to all orders in (ξ 2 /Q 2 )(A 1/3 − 1). Using ξ 2 = 0.09 − 0.12 GeV 2 [6], our results provide a good description of the deviation between the Gross-Llewellyn Smith QCD sum rule [8] adjusted for O(α s ) scaling violations [9] and the existing data [10]. At small Bjorken x B , the high twist components to the calculated structure functions F A 2 (x B , Q 2 ) and F A 3 (x B , Q 2 ) in neutrino-iron DIS qualitatively describe the low-x B and low-Q 2 suppression trend in the preliminary data, recently reported by the NuTeV collaboration at DIS 2003 [11].
In the next Section we briefly review the DIS kinematics and coherence at small Bjorken x B . In Section III we demonstrate that at the tree level mass corrections and dynamical power corrections "commute" and their resummation can be carried out in a closed form. We derive analytic expressions for the process-dependent nuclear modification to the transverse and longitudinal structure functions in neutrino-nucleus DIS. In Section IV we predict the difference in the shadowing pattern of , and give quantitative results for the x B -, A-and Q 2 -dependence of the nuclear modification to the charged current ν(ν)-A DIS structure functions. We find sizable small-and moderate-Q 2 corrections to the Gross-Llewellyn Smith QCD sum rule. In Section V we comment on the relevance of our results to the NuTeV extraction of sin 2 θ W . Finally, we give our conclusions in Section VI.

II. DIS KINEMATICS AND COHERENCE AT SMALL xB
The charged current DIS cross section of a neutrino (or antineutrino) beam (k) off a nuclear target (P A ), as illustrated in Fig. 1(a), probes three independent structure functions, where the Bjorken variable x B = Q 2 /(2p · q) with p = P A /A, the exchanged W -boson momentum q and its virtuality Q 2 = −q 2 , and y = p · q/(p · k). In Eq. (1), the "(−)" represents the sign for an antineutrino beam, m N = M A /A with nuclear mass M A , M W is the W -boson mass, and E is the beam energy. The often-referred longitudinal structure function, F [9]. Here, we are predominantly interest in the small-x B region and neglect the target mass (rescaling) corrections [12].
The DIS cross section with an exchange of a W -or Z-boson of virtuality Q 2 and energy ν = Q 2 /(2m N x B ) has an effective resolution in transverse area A ⊥ = 1/Q 2 , which is much less than the nucleon size, and an uncertainty in longitudinal direction ∆z (−) = 1/(x B p + ) with boosted nucleon momentum p + . If ∆z (−) = 1/(x B p + ) ≥ 2r 0 (m N /p + ) or x B ≤ x N = 1/(2m N r 0 ) ∼ 0.1, the neutrino will coherently interact with more than one nucleon inside the nucleus, and probe the nuclear dependence at a perturbative scale Q 2 [5,6,13].

III. CALCULATING MASS AND DYNAMICAL POWER CORRECTIONS
Electroweak charged and neutral current processes necessitate a discussion of final state charm mass effects in neutrino-nucleus DIS even if the leading twist charm quark parton distribution is neglected, φ c (x, Q 2 ) = φc(x, Q 2 ) = 0 [14]. It is, therefore, critical to develop a systematic approach to the interplay of a heavy quark final state and the resummed nuclear enhanced power corrections discussed in [6]. We define the boost invariant mass fraction and choose a frame such that p µ = p +nµ and q µ = −x B p +nµ + Q 2 /(2x B p + )n µ , wheren µ = [1, 0, 0 ⊥ ] and n µ = [0, 1, 0 ⊥ ] specify the "+" and "−" lightcone directions, respectively. With a non-vanishing quark mass M , the Feynman rule for the final state cut line of quark momentum x i p + q in Fig. 2(a) is For M → 0 we recover the known massless case, in which the scattered quark is moving along the "−" lightcone direction. A direct consequence of this Feynman rule is a tree-level coupling for longitudinally polarized vector mesons ∝ M 2 /Q 2 . Contracting ǫ µν L [15] with the charged current hadronic tensor W µν [9] yields: where the CKM matrix elements V ij parametrize the electroweak and mass eigenstate mixing with up-type quark U = (u, c, t) and down-type quark D = (d, s, b) [9], and φ A i represent the flavor-i universal twist-2 parton distribution functions (PDFs) of a nucleon (A = 1) or a nucleus [5]. Eqs. (5) and (6) give a novel leading order (α 0 s ) power suppressed (M 2 /Q 2 ) quark mass contribution to the ratio of longitudinal and transverse structure functions for both nucleons and nuclei.
We calculate the nuclear enhanced dynamical power corrections in the lightcone A + = 0 gauge. In this gauge, other than the initial-state contact-term contributions, all leading order nuclear enhanced power corrections are from final-state multiple gluon interactions of the scattered quark in a large nucleus shown in Fig. 2(b) [6]. To resum all order nuclear enhanced power corrections with a non-vanishing (anti)quark mass, we examine its propagator structure [16]. For a quark momentum x i p + q where ±i, ±iǫ correspond to propagators to the left or right of the t = ∞ cut. In the Fourier space conjugate to x i p + the first term, free of x i pole, is ∝ δ(y − i ). The operators in the hadronic matrix element that this contact term (−→ | ) separates can be evaluated in the same nucleon state [6]. In contrast, the Fourier transform of the second term is ∝ θ(y − i ). Therefore, this pole term (−→ × ) is the source of the A 1/3 nuclear size enhancement to the power corrections. The operators that it connects in the multi-field multi-local hadronic matrix element can be long-distance separated and thus approximately evaluated in different nucleon states [6]. Alternative operator decompositions as well as other terms that arise from a formal operator product expansion (OPE) [17] are suppressed by powers of the nuclear size.
The case of massive final state quarks could be much more involved than the M → 0 limit. The complexity of the calculation stems from the potentially dangerous exponential growth of the number of terms coming from products of propagators, see Fig. 2(b). In our calculation we first observe that γ ·p+ M in the cut line, Eq. (3), and the numerator of the pole-term, Eq. (7), arise from an on-shell momentump. Since the exchange gluons at the vertices connected by quark propagators are transversely polarized in the For the diagrams in Fig. 2(b) only one alternating sequence of short and long distance parts of the propagators Eq. (7), initiated by the t = ∞ cut, survives. Therefore, there must be an even number of gluon interactions between the cut line and any surviving pole term of a propagator in Fig. 2(b). We also note that leaves no mass dependence in the spinor trace of the diagrams. The basic unit for two-gluon exchange with a net momentum fraction flow x i − x i−1 and two-quarkpropagators (one contact plus one pole) [6], see Fig. 2(b), now reads: In Eq. (8) the boost invariant λ i = p + y − i , the two cases correspond to a vertex to the left or right of the final state cut andF 2 (λ i ) is given by the intra-nucleon two-gluon field strength correlator defined in [6]: We conclude that the dynamical nuclear-enhanced all twist contributions from the leading order in α s Feynman diagrams with a massive quark final state are identical to the massless case up to the substitution x B → x B + x M (rescaling) in the δ-function in the cut, Eq. (3), and the propagator poles, Eq. (8). Effectively, we have shown that the mass and nuclear enhanced power corrections "commute" and x i = x B + x M for all "i". This allows us to take all possible final state interaction diagrams and all possible cuts [18] to explicitly carry out the resummation of coherent high-twist contributions to neutrino-nucleus DIS structure functions, In Eqs. (10) and (11) the "±" signs refer to F 1 (parity conserving) and F 3 (parity violating) transverse struc-ture functions, respectively. The factor "{2}" gives the standard normalization for F 3 only [9] and the isospin average in the PDFs over the protons and neutrons in the nucleus is implicit. In Eqs. (10) and (11) x HT is the momentum fraction shift (rescaling) induced by nuclear enhanced dynamical power corrections and derived in Ref. [6]: where ξ 2 represents the effective scale for the dynamical power corrections. To the leading order in α s it is given by where p |F 2 | p depends on the small-x limit of the gluon distribution in the nucleon/nucleus [6]. While x N ≈ 0.1 is the limiting value for the onset of coherence, at x A = 1/(2m N r 0 A 1/3 ) < x N the exchange vector meson already probes the full nuclear size, see Fig. 1. To first approximation, the function in Eq. (12) represents the interpolation between the two regimes based on the uncertainty principle [20]. The nuclear enhancement factor (A 1/3 − 1) in Eq. (12) comes from the integration dλ i in Eq. (8), the lower limit of which was chosen such that the effect vanishes for the proton (A = 1) case. Including the dynamical power corrections, the longitudinal structure functions in Eqs. (5) and (6) become: In Eqs. (15) and (16) we include the O(α s ) leading twist longitudinal structure functions F (LT) L (x, Q 2 ) [9] since they are of the same order as the leading ξ 2 power. For numerical evaluation in the next Section we consider two quark generations, U = (u, c) and D = (d, s), and use |V ud | 2 = |V cs | 2 = cos 2 θ c = 0.95, |V us | 2 = |V cd | 2 = sin 2 θ c = 0.05 with Cabibbo angle θ c [19]. The u, d and s quarks are treated as massless and the charm quark mass is set to M c = 1.35 GeV [1].

IV. HIGH TWISTS, SHADOWING AND QCD SUM RULE
We first quantify analytically the differences in the "shadowing" pattern induced by valance and sea quarks, neglecting the charm mass effects that are shown to be small below. For isoscalar-corrected (Z = N = A/2) target nuclei we average over neutrino-and antineutrino-initiated charged current interactions, In the leading-order and leading twist parton model , a singlet distribution, is proportional to the momentum density of all interacting quark constituents and for x B ≪ 0.1 is dominated by the sea contribution, φ sea (x) ∝ x −αsea . Therefore, the x B -dependent shift from dynamical nuclear enhanced power corrections, x HT in Eq. (12), generates different modification to The predicted nuclear modification for isoscalarcorrected 12 C, 56 F e and 208 P b to the neutrino-nucleus DIS stricture functions F A 2 (xB, Q 2 ) (top) and xBF A 3 (xB, Q 2 ) (bottom) versus Bjorken xB (left) and Q 2 (right). The bands correspond to ξ 2 = 0.09 − 0.12 GeV 2 [6]. in x HT to obtain: Since α val. ≈ 0.5 and α sea ≈ 1 vary slowly with Q 2 , Eq. (17) predicts a measurable difference in the nuclear shadowing for the structure function F 2 (F 1 ) in comparison to F 3 . Figure 3 shows the modification to the DIS structure functions from Eqs. (10)-(16) for large nuclei ( 12 C, 56 F e and 208 P b) relative to the deuteron, calculated with the CTEQ6L parton distribution functions [21]. The bands correspond to a scale for power corrections ξ 2 = 0.09 − 0.12 GeV 2 , extracted from the analysis [6] of the NMC and E665 data [7]. The transition 0.01 ≤ x B ≤ 0.1 region represents the onset of coherent interactions, Eq. (14), and the modest x B -dependence for x B ≤ 0.01 is driven by the change in the local slope of the PDFs. The right panels show the Q 2 dependence of the modification from the resummed nuclear-enhanced power corrections, which is noticeably stronger than the DGLAP evolution of leading twist shadowing in the nPDFs [22]. The difference in the suppression pattern of F A 2 and x B F A 3 in Fig. 3 is qualitatively described by Eq. (17). In contrast, it has been suggested in the framework of a Gluber-Gribov approach [23] that the suppression of the non-singlet distribution may be significantly larger than of the singlet one (R A/A ′ sea/val. > 1 for A > A ′ ). Such distinctly different predictions should be testable in the future ν-Factory exper-iments, for example at the Fermilab NuMI facility [24].
Although the current ν(ν)-A DIS measurements are mostly on nuclear targets [25], these data lack the necessary atomic weight systematics to identify small-x B nuclear shadowing. We do, however, note that our results provide a consistent explanation of the observed small-and moderate-Q 2 power law deviation at smallx B of the preliminary NuTeV data [11] on F A 2 (x B , Q 2 ) and x B F A 3 (x B , Q 2 ) from the next-to-leading order leading twist QCD predictions using MRST parton distribution functions [26].
The latest global QCD fits include ν(ν)-A DIS data without nuclear correction other than isospin [21]. Such analysis would tend to artificially eliminate most of the higher twist contributions discussed here due to a trade off between the power corrections in a limited range of Q 2 and the shape of the fitted input distributions at Q 2 0 , especially within the error bars of current data. An effective way to verify the importance of the nuclear enhanced power corrections for neutrino-nucleus deeply inelastic scattering is via the QCD sum rules, in particular, the Gross-Llewellyn Smith (GLS) sum rule [8] At tree level Eq. (18) counts the number of valance quarks in a nucleon, S GLS = 3. Since valence quark number conservation is enforced in the extraction of twist-2 nucleon/nucleus PDFs, the adjustments of input parton distributions can alter their shape but not the numerical contributions to the GLS sum rule. The effect of scaling violations can modify S GLS , and at O(α s ) [9] Loop contributions to the GLS sum rule are known to O(α 3 s ) [27]. Although power corrections can also modify the shape of nucleon structure functions, recent precision DIS data on both hydrogen and deuterium targets from JLab [28] indicate that effects from higher twist to the lower moments of structure functions are very small at Q 2 as low as 0.5 GeV 2 , which confirms the Bloom-Gilman duality [29]. A recent phenomenological study [30] also suggests that power corrections to the proton F 2 (x B , Q 2 ) have different sign in the small-and large-x B regions and largely cancel in the QCD sum rules.
On the other hand, the coherence between different nucleons inside a large nucleus is only relevant for x B ≤ x N . The suppression of structure functions at small Bjorken x B in Fig. 3, caused by the nuclear enhanced dynamical power corrections, cannot be canceled in the moments and further reduces the numerical value of S GLS . Figure 4 shows a calculation of ∆ GLS from Eqs. (10) and (11) for 56 F e. While the effect of charm mass is seen to be small relative to α s /π, for Q 2 ∼ 1 GeV 2 nuclear enhanced higher twists may contribute as much as ∼ 10% to ∆ GLS . Their Q 2 behavior is consistent with the trend in the current data [10]. For comparison, we found no deviations for this sum rule induced by the EKS98 scale dependent parameterization of nuclear effects [22], which is again a consequence of the valance quark number conservation in leading twist shadowing.

V. IMPLICATIONS FOR EXTRACTION OF sin 2 θW
Based on a comparison of charged and neutral current neutrino interactions (separated on the basis of event topology) with an iron-rich heavy target, the NuTeV collaboration reported a measurement of sin 2 θ (on−shell) W = 0.2277 ± 0.0013(stat.) ± 0.0009(syst.) , neglecting the very small top quark and Higgs mass corrections. This result is approximately 3 standard deviations [1] above the Standard Model (SM) expectation value sin 2 θ W = 0.2227 ± 0.0004. The NuTeV's result was derived from a quantity that is a close approximation to the Paschos-Wolfenstein relationship Corrections to Eq. (20) include higher order and nonperturbative QCD effects, higher order electroweak effects and nuclear effects. Based on a QCD global analysis of parton structure, Kretzer et al. argue [31] that the uncertainties in the theory which relates R − to sin 2 θ W are substantial on the scale of the precision NuTeV data and suggest that the sin 2 θ W measurement, NuTeV dimuon data and other global data sets used in QCD parton structure analysis can all be consistent within the SM. Because a heavy target was used, several nuclear effects can enter the cross sections to influence the extraction of sin 2 θ W [32]. Since nuclear enhanced power corrections were not included in NuTeV's analysis, Miller and Thomas pointed out that nuclear shadowing from a vector meson dominance (VMD) model could affect the charged and neutral current neutrino scattering differently, and therefore change the predictions for the ratios of neutral current (NC) over charged current (CC) cross sections, R ν(ν) = σ NC (ν(ν))/σ CC (ν(ν)), and the extraction of sin 2 θ W [2]. The NuTeV collaboration argued [3,32] that such possibility was considered unlikely because R − has little sensitivity to process-dependent nuclear effects.
In this letter we calculated the process-dependent nuclear effects in the neutrino-nucleus differential cross sections, Eq. (1), in the perturbatively accessible DIS region. Our predictions on the nuclear modification to the ν(ν)-A structure functions in Fig. 3 should be relevant for Q 2 between 1 and 10 GeV 2 . While for the mean Q 2 ν = 25.6 GeV 2 and Q 2 ν = 15.4 GeV 2 the effect of dynamical power corrections is small, a large fraction of the final data sample cover the x B < 0.1, Q 2 < 10 GeV 2 range where shadowing can be as large as ∼ 20%. We note that the NuTeV measurement constitutes a ∼ 2% increase in the value of sin 2 θ W relative to the SM, or equivalently ∼ 4% reduction of the expected total neutrino-nucleus cross section. Including shadowing into the expected total cross sections will certainly reduce the discrepancy of sin 2 θ W . However, without knowing the nuclear enhanced power corrections to the structure functions at Q 2 < 1 GeV 2 , and the detailed Monte Carlo simulation of event distributions, it is difficult to estimate the precise corrections to the extraction of sin 2 θ W . We, nevertheless, note that at small Bjorken x B , the calculated nuclear structure functions F A 2 (x B , Q 2 ) and F A 3 (x B , Q 2 ) in neutrino-iron DIS qualitatively describe the low-x B and low-Q 2 suppression trend in the preliminary data, presented by the NuTeV collaboration at DIS 2003 [11].

VI. CONCLUSIONS
In the framework of the perturbative QCD collinear factorization approach [33,34], we computed and resumed the tree level perturbative expansion of nuclear enhanced power corrections to the structure functions measured in inclusive (anti)neutrino-nucleus deeply inelastic scattering. We demonstrated that these corrections commute with the final state heavy quark effects and identified the new contributions to the longitudinal structure function F A L (x B , Q 2 ). Our calculated Q 2 -dependent modification to the Gross-Llewellyn Smith sum rule agrees well with the existing measurements on an iron target [10]. Our approach predicts a non-negligible difference in the small-x B shadowing of the structure functions F A 2 (x B , Q 2 ) (F A 1 (x B , Q 2 )) and F A 3 (x B , Q 2 ), which is consistent with the trend in the preliminary NuTeV data [11]. Although our results, valid in the perturbative region, are unlikely to have an immediate impact on the NuTeV's extraction of sin 2 θ W , the predicted x B -, Q 2 -, and A-dependence of the structure functions in the shadowing region can be tested at the future Fermilab NuMI facility [24].