Model-dependence of the \gamma Z dispersion correction to the parity-violating asymmetry in elastic ep scattering

Model-dependence of the dispersion correction to the parity-violating asymmetry in elastic scattering

Mikhail Gorchtein Center for Exploration of Energy and Matter, Physics Department,
Indiana University, Bloomington, IN 47403
mgorshte@indiana.edu
   C. J. Horowitz Center for Exploration of Energy and Matter, Physics Department,
Indiana University, Bloomington, IN 47403
   Michael J. Ramsey-Musolf Department of Physics , University of Wisconsin-Madison, Madison, WI 53706 USA
and
Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125 USA
mjrm@physics.wisc.edu
August 4, 2019
Abstract

We analyze the dispersion correction to elastic parity violating electron-proton scattering due to exchange. In particular, we explore the theoretical uncertainties associated with modeling contributions of hadronic intermediate states. Taking into account constraints from low- and high-energy, parity-conserving electroproduction measurements, choosing different models for contributions from the non-resonant processes, and performing the corresponding flavor rotations to obtain the electroweak amplitude, we arrive at an estimate of the uncertainty in the total contribution to the parity-violating asymmetry. At the kinematics of the Q-Weak experiment, we obtain a correction to the asymmetry equivalent to a shift in the proton weak charge of . This should be compared to the value of the proton’s weak charge of that includes Standard Model contributions at tree level and one-loop radiative corrections. Therefore, we obtain a new Standard Model prediction for the parity-violating asymmetry in the kinematics of the Q-Weak experiment of . The latter error leads to a relative uncertainty of 2.8% in the determination of the proton’s weak charge, and is dominated by the uncertainty in the isospin structure of the inclusive cross section. We argue that future parity-violating inelastic asymmetry measurements at low-to-moderate and could be exploited to reduce the uncertainty associated with the dispersion correction. Because the corresponding shift and error bar decrease monotonically with decreasing beam energy, a determination of the proton’s weak charge with a lower-energy experiment or measurements of “isotope ratios” in atomic parity-violation could provide a useful cross check on any implications for physics beyond the Standard Model derived from the Q-Weak measurement.

pacs:
preprint: NPAC-11-02

I Introduction

Precise measurements of low-energy observables can provide powerful probes of physics beyond the Standard Model that complement high energy collider studies Erler:2004cx (); RamseyMusolf:2006vr (). In particular, measurements of parity-violating (PV) observables in atomic physics and electron scattering have provided key tests of the neutral weak current sector of the Standard Model and constrained possible new physics in this sectorerler (); RamseyMusolf:2006vr (); RamseyMusolf:1999qk (); Young:2007zs (). In this work, we consider parity-violating (PV) elastic scattering of longitudinally polarized electrons from hydrogen, which is the subject of the Q-Weak experiment at the Jefferson Lab (JLab) qweak (). This experiment draws on a rich history of parity-violating electron scattering (PVES) at various facilities and aims to provide the most precise determination of , the weak charge of the proton, ever made.

In PVES, the weak charge is operationally defined through the forward scattering limit of the PV asymmetry:

(1)

where the ratio of response functions is defined below. Here and in the rest of the article, denotes the Fermi constant, as taken from the muon lifetime (often denoted by ). The weak charge – defined as a static property of the proton – is then the leading term the expansion of the ratio in powers of :

(2)

where the reason for specifying zero beam energy will become apparent below. In the one-boson exchange (OBE) approximation, the weak charge is just given by

(3)

where the characterize the effective four-fermion parity-violating electron-quark interaction

(4)

In the Standard Model, it is possible to make precise predictions for the , including the effects of electroweak radiative corrections erler (); erler2 (); sirlin (). These corrections include the effects of one-loop contributions to the gauge boson and fermion propagators and gauge boson-fermion vertices. Ultraviolet (UV)divergences are removed through renormalization, and in what follows we will use the modified minimal subtraction ( ) scheme for doing so.

Additional, UV-finite corrections arise from the two-boson exchanges (“box graphs”): , , and . Those involving two heavy vector bosons are dominated by loop momenta of order and are properly included in the radiatively-corrected coefficients. On the other hand, the box graph corrections involving one or more photons are sensitive to low-momentum scales where target-dependent hadronic structure effects may be significant. In what follows, we focus on the box correction. For a review of recent work on the corrections, see Ref. TPEreview ().

Recently, the box graph contribution has been the subject of renewed scrutiny. In Refs. erler (); erler2 (); Musolf:1990ts (), the short-distance part of this correction was computed, confirming the earlier computation of Ref. sirlin (). It carries a logarithmic dependence on the hadronic scale, , with the latter requiring the presence of a “low energy constant” to yield a result independent of the hadronic matching scale. The authors of Ref. erler (); erler2 () assigned a generous error to associated with the difficult-to-compute long-distance hadronic effects.

The authors of Ref. ja_chuck () subsequently observed that there exists an additional contribution from the box graph that grows with the electron beam energy and that is independent of the hadronic cutoff parameter111For related work considering the effects of the box graph away from the forward limit – relevant to the strange quark form factor determinations – see Refs. Zhou:2007hr (); Nagata:2008uv (); Chen:2009mza ().. Given the energy-dependence of this “dispersion correction”, it is more appropriate to consider it as a new term in the PV asymmetry than as a contribution to the weak charge that is nominally a static property of the proton. Nevertheless, in the forward limit of Eq. (2), its effect is to shift the apparent value of . Moreover, unlike the short-distance and terms that are suppressed by , the energy-dependent correction is not accidentally suppressed. For the energy of the Q-Weak experiment, the authors of Ref. ja_chuck () estimated that the correction was several percent, raising the possibility that the estimated theoretical uncertainty in the PV asymmetry could be larger than given in Refs. erler (); erler2 ().

A follow-up study sibirtsev () repeated the computation of Refs. ja_chuck (); deltagZ_conf () using a somewhat different hadronic model framework and drawing upon recent structure function measurements carried out at the Jefferson Laboratory. These authors argued that the expressions used in Ref. ja_chuck () contained numerical errors but nonetheless obtained a quantitatively similar result for the size of the correction. An estimate of the uncertainty in the correction was also provided, suggesting that the theoretical uncertainty associated with the energy-dependent term is well below the uncertainty quoted in Refs. erler (); erler2 (). Recently, another study of this correction was reported in Ref. carlson (). The latter work employed yet another parametrization of virtual photoabsorption data from Jefferson Lab, and a different treatment of the isospin structure and of the uncertainty was applied. The results is consistent with that of Ref. sibirtsev () with an error bar that is also smaller than that of Refs. erler (); erler2 (). We will review these works in greater detail below. For the moment, we display in Table 1 the results of the previously mentioned studies along with the results of this work. While all of the recent results (ours and Refs. sibirtsev (); carlson ()) are consistent within quoted error bars, we obtain a larger uncertainty by roughly a factor of two. As we discuss below, this larger theory uncertainty results from taking into account hadronic model-dependence in computing the dispersion correction.

Ref. ja_chuck () Ref. sibirtsev () Ref. carlson () This work
Table 1: Estimates for the dispersion correction Re obtained in various works, as indicated in the Table. Originally, Gorchtein and Horowitz in ja_chuck () only quoted the value of Re %, as calculated relative to . This corresponds to the number given in Table.

Obtaining a robust theoretical prediction for in the Standard Model is essential for the proper interpretation of the asymmetry in terms of possible contributions from physics beyond the Standard Model. In light of the recent history and disagreements in the literature on the question of the box correction, we revisit here the computations of Refs. ja_chuck (); deltagZ_conf (); sibirtsev (); carlson (). Our goal is three-fold. First, we seek to clarify the apparent disagreements about the numerical factors in the analytic expressions for the energy-dependent part of the correction. Second, we attempt to provide an estimate of the theoretical uncertainty associated with hadronic modeling required for its computation. While the study of Ref. sibirtsev () included an uncertainty associated with the experimental data used as input for the calculation, no estimate of the theoretical error related to the choice of model framework was given. Finally, we discuss additional experimental input that would be useful to improve the reliability of the calculated correction.

The remainder of our treatment of these points is organized as follows. Section II outlines the elastic electron-nucleon scattering kinematics and observables that are analyzed to one-loop order. In Section III, we derive a forward dispersion relation for the dispersion corrections. In Section IV, we discuss the input in these sum rules, perform an isospin decomposition of the inclusive electroproduction data and isospin-rotate these data in order to obtain the inclusive parity violating data. We combine different data sets to obtain an estimate of the uncertainty associated with such rotation in the flavor space. Detailed discussion of the isospin rotation of the resonant contributions is reported in Appendix A. In Section V, we present our results for the dispersion correction and the respective theory uncertainty at the kinematics of the QWEAK experiment. Section VI is dedicated to the study of the -dependence of the dispersion correction that is important for translating the value obtained from dispersion relation in the exact forward direction to the experimental kinematics. In Section VII, we compare the existing calculations of the energy-dependent dispersion correction to the weak charge of the proton in detail. We close the article with a short summary in Section VIII.

Ii PVES in the Forward Scattering Regime

We consider elastic scattering of massless electrons off a nucleon, , in presence of parity violation (and in absence of -violation). The scattering amplitude can be cast in the following form involving six scalar amplitudes , ,

(5)

where only electromagnetic and weak neutral currents are considered. stands for the Fermi constant, as taken from the muon lifetime, according to the scheme. The amplitudes are parity conserving (PC), and are explicitly parity violating (PV). Above, stands for the initial (final) electron momenta, and for the initial (final) nucleon momenta, respectively, and denotes the mass of the nucleon (we take ). All six amplitudes are functions of energy (with and ) and the elastic momentum transfer is , with . At tree level (one boson exchange, OBE) and to leading order in and , the amplitudes reduce to the electromagnetic and weak form factors of the nucleon (the index takes values denoting proton and neutron, respectively),

(6)

Above, and . Radiative corrections induce terms , leading generically to . We denote the usual Dirac (Pauli) form factors by , respectively, and the nucleon axial form factor at tree level by . Similarly, stand for the form factors describing the vector coupling of the to the nucleon. One introduces the conventional combinations,

(7)

with . In absence of radiative corrections, these amplitudes reduce to the electroweak Sachs form factors . In terms of these generalized form factors, the unpolarized cross section on a nucleon target can be written as

(8)

with the electron Lab scattering angle, the incoming (outgoing) electron Lab energy, and the virtual photon longitudinal polarization parameter. The reduced cross section , up to and including terms of order , reads

In what follows, we will concentrate on the case of electron-proton scattering. Therefore, we will understand everywhere and suppress the index in all expressions, unless explicitly stated otherwise.

The parity violating asymmetry is defined in Eq. (1) with the ratio of the response functions is given by

(10)

Here, are the cross sections for positive and negative helicity electrons, and .

Since we are interested in very forward scattering angles corresponding to the Q-Weak kinematics qweak (), thus , the expressions for the cross section and PV asymmetry can be fruther simplified.

For the reduced cross section the leading contribution in Eq. (LABEL:eq:sigmaR) comes from the term, and we obtain

(11)

The three distinct corrections quoted above are defined as follows: is a kinematic correction that arises at tree level due to the magnetic part and other subleading kinematic effects of order , that do not contain effects; stands for order corrections that are energy-independent (such as vacuum polarization, self energy and vertex corrections); finally, denotes the two-photon exchange correction that is an energy-dependent correction.

Similarly, for the PV asymmetry the leading order contribution in Eq. (10) originates from the term.

As discussed in Ref.  erler (), the Standard Model prediction for the PV asymmetry in the forward regime can be expressed as

(12)

where is the running weak mixing angle in the scheme at zero momentum transfererler2 (). The correction is a universal radiative correction to the relative normalization of the neutral and charged current amplitudes; the and give, respectively, non-universal corrections to the axial vector and couplings; the for give the non-universal box graph corrections; and the “” indicate terms that vanish with higher powers of in the forward limit, such as those arising from the magnetic and strange quark form factors and the two-photon dispersion correction, . The weak charge of the proton, considered as a static property, is given by the quantity in the squark brackets in the zero-energy limit.

Within the radiative corrections, the TBE effects are separated explicitly. This is done because the TBE corrections, unlike other corrections in the above equation, are in general and -dependent. In particular, the (or ) dependence of the -box is believed to be responsible for the discrepancy between the Rosenbluth and polarization transfer data for gegm_exp (). It should be noted that in the exact forward direction vanishes as a consequence of electromagnetic gauge invariance.

The and -box diagrams were first considered in sirlin (), and subsequently investigated in Refs. mjrm (); erler (). The contribution from in particular is relatively large. Both corrections are -independent at any hadronic energy scale since they are dominated by exchange of hard momenta in the loop . Higher-order perturbative QCD corrections to and were computed in Ref.erler (), and the overall theoretical uncertainty associated with these contributions is well below the expected uncertainty of the QWEAK experiment.

In contrast to and , receives substantial contributions from loop momenta at all scales. For the electron energy-independent contribution, this situation leads to the presence of a large logarithm where is a typical hadronic scalesirlin (); mjrm (); erler (). Since the asymmetry must be independent of the latter, includes also a “low-energy constant” whose hadronic scale dependence compensates for that appearing in the logarithm. An analogous box correction enters the vector current contribution to neutron and nuclear -decay. Importantly for the PV asymmetry, these energy-independent box contributions are suppressed by , thereby suppressing the associated theoretical uncertainty.

In Ref. ja_chuck (), the -box contribution was re-examined in the framework of dispersion relations and it was found that it possesses a considerable energy dependence, so that at energies in the GeV range its value can differ significantly from that found at zero energy. Moreover, the energy-dependent correction contains a term that is not suppressed, so the theoretical uncertainty associated with hadronic-scale contributions is potentially more significant. This energy dependence comes through contributions from hadronic energy range inside the loop that cannot be calculated reliably using perturbative techniques.

At present, a complete first principles computation is not feasible, forcing one to rely on hadronic modeling. For a proper interpretation of the PV asymmetry, it is thus important to investigate the theoretical hadronic model uncertainty. The remainder of the paper is devoted to this task. In so doing, we will attempt to reduce this model uncertainty by relating – wherever possible – contributions from hadronic intermediate states to experimental parity-conserving electroproduction data through the use of a dispersion relation and isospin rotation. As a corollary, we will also identify future experimental measurements, such as those of the parity-violating inelastic asymmetry in the regime of moderate and , that could be helpful in reducing the theoretical uncertainty.

Iii Dispersion corrections

To calculate the real part of the direct and crossed box diagrams showed in Fig. 1, we follow ja_chuck () and adopt a dispersion relation formalism. We start with the calculation of the imaginary part of the direct box (the crossed box contribution to the real part will be calculated using crossing),

(13)

where denotes the virtuality of the exchanged photon and (in the forward direction they carry exactly the same ), and we explicitly set the intermediate electron on-shell. In the center of mass of the (initial) electron and proton, one has , with the full c.m. energy squared and the invariant mass of the intermediate hadronic state. Note that for on-shell intermediate states, the exchanged bosons are always spacelike.

Figure 1: Direct and crossed diagrams for -exchcange. Dashed lines correspond to an exchange of a -boson, and wavy lines to an exchange of a photon. The blob stands for an inclusive sum over intermediate hadronic states.

The leptonic tensor is given by

(14)

We next turn to the lower part of the diagrams in Fig. 1. The blobs stand for an inclusive sum over all possible hadronic intermediate states, starting from the ground state (i.e., the nucleon itself) and on to a sum over the whole nucleon photoabsorption spectrum. The case of the elastic hadronic intermediate state was considered in blunden (). Here, we concentrate on the inelastic contribution. Such contributions arise from the absorption of a photon (weak boson). In electrodynamics, for a given material, the relation between its refraction coefficient and the dependence of the latter on the photon frequency (i.e., dispersion) on one hand, and the photoabsorption spectrum of that material on the other hand, is historically called a dispersion relation. It is exactly this dependence of the forward scattering amplitude (see Eq. (5)) on the energy that arises from its relation to the electroweak -absorption spectrum that is the scope of an investigation in this work. This explains the origin of the term “dispersion correction” used for the inelastic contributions to the -box correction.

In the forward direction, the imaginary part of the doubly virtual “Compton scattering” () amplitude is given in terms of the interference structure functions , with the Bjorken variable. Making use of gauge invariance of the leptonic tensor, we have

Contracting the two tensors, one obtains after a little algebra two contributions that are due respectively to the axial and vector couplings of the to the electron,

(16)

where the imaginary parts will appear in a dispersion relation for the real parts in Eq. (20) below. The full correction is the sum of the two,

(17)

In Eqs. (16), stands for the pion production threshold, and the -integration is constrained below a maximum value

(18)

as a condition of on-shell intermediate states for an imaginary part calculation. Eq. (16) is in agreement with Refs. sibirtsev (); carlson (). In particular, we confirm the correctness of the claim made in Ref. sibirtsev () that in Ref. ja_chuck () a factor of 2 was missing.

In order to write down the dispersion relation for the function , one should consider its behavior under crossing. We distinguish two contributions, and that have different crossing behavior ja_chuck ():

(19)

Correspondingly, the two contributions obey dispersion relations of two different forms,

(20)

where the presence or absence of the factor of in the integrands follows from the behavior of the under crossing symmetry.

The result in Eq. (20) gives a model-independent relation between the dispersion correction to the weak charge of the proton and the parity violating structure functions appearing in Eq. (16). This relation does not rely on any assumption, other than the neglect of higher order radiative corrections and the number of subtractions needed for convergence of the dispersion relation. The advantage for this formulation is that the are in principle measurable. However, in absence of any detailed parity violating inclusive electron scattering data, the input in the dispersion integral will depend on a model. In the following, we will investigate the extent to which this model dependence can be constrained by existing or future experimental data.

Iv Input to the dispersion integral

In the previous section, the contribution of the forward hadronic tensor to the box diagram was considered. In this section, we will address the possibility of relating the interference hadronic tensor of Eq. (LABEL:eq:hadron_tensor)

(21)

to the pure electromagnetic one,

(22)

Using unitarity, we rewrite these matrix elements as an inclusive sum over intermediate hadronic states,

(23)

and

respectively. We now proceed to investigate the possible relationships between the products of transition matrix elements appearing in each inclusive sum (23) and (LABEL:eq:Wgg_def).

Theoretically, calculating the full set of contributions to the inclusive sum represents a fundamental difficulty since in QCD, the basis for intermediate states is infinite, and the matrix elements are non-perturbative. Under certain kinematic conditions, one can organize this basis into leading and subleading (kinematically suppressed) sub-sets. We depict this situation schematically in Fig. 2, where we show in the plane the approximate kinematic areas where various mechanisms dominate.

Figure 2: The plane vs. and kinematic regions corresponding roughly to various physical contributions.

At high energy and , and finite Bjorken , the leading set of states is ( denotes a quark), where to leading order in , is a spectator. Thus, in this regime the electromagnetic (weak) current directly probes a single quark within the nucleon, and gives access to the parton distribution functions (deep inelastic scattering, DIS in Fig. 2). At high energy and , and small , however, the picture changes, as the leading set is . In this regime, the photon polarizes the QCD vacuum at the periphery of the hadron, and the resulting -pair forms a color dipole that interacts with the nucleon (diffractive DIS in Fig. 2). This picture was first realized in the Vector Meson Dominance model (VDM) that capitalized on the fact that since vector mesons and the photon have the same quantum numbers, the latter can fluctuate into former vdm (); gvd (). This simple model works quite well at low (VDM area in Fig. 2). Such “hadron-like” behavior of a photon in scattering processes also results in the e.-m. data following the Regge behavior, as a function of (respective Regge area in Fig. 2). At higher values of , rescattering effects in vector meson-nucleon scattering become increasingly important but can still be accounted for in what is called the “generalized VDM” (GVDM region in Fig. 2). At low energies, the relevant degrees of freedom are hadronic (that is, highly non-perturbative), etc. In this regime, the inelastic cross section is tyically dominated by resonances on top of a non-resonant background (Resonance area in Fig. 2). The boundaries of each kinematic region are, of course, approximate. Their meaning is that the farther one departs from a kinematical region, to the lesser extent the respective mechanism works. Consequently, a large area on the plane, that overlaps with all the depicted regions but not covering them completely is the so-called shadow region where none of the mechanisms can be considered as fully dominant.

If data for the interference cross section existed throughout all these distinct regimes, we would not need to know details of any of the aforementioned models. In principle, such data could be obtained with measurements of the PV inelastic asymmetries in the various kinematic regimes shown in Fig. 2. At present, however, either no or very poor data on PV inelastic scattering exist. Consequently, we will instead pursue an alternate strategy, endeavoring to make use of extensive data sets for real and virtual photoabsorption that exist through vast kinematic region in energy and . To that end, we will rely on models that adequately describe the photoabsorption cross section in different regimes and for each attempt to establish relationships between the matrix elements and for each intermediate hadronic state of definite isospin. We will approach this problem by extracting the electromagnetic matrix elements from inclusive e.-m. data, and then isospin-rotate every such matrix element. We begin with a brief review of the experimental situation and discuss various model descriptions.

iv.1 Real and virtual photoabsorption data

We find that the dispersion integral for Re is dominated by moderate values of GeV and GeV (see Fig. 15 in Section V). Consequently, we need to analyze in detail contributions from the resonance regime and portions of what we have called the VDM, GVDM, and regge regimes. Our goal will be to draw upon existing experimental data for inclusive and semi-inclusive electromagnetic data to infer the interference structure functions that appear in the dispersion integrals. To that end, we first summarize the experimental situation.

While it is equally possible to use structure functions to describe resonance data, in the following we opt to use total photoabsorption cross sections with transverse or longitudinal (for virtual photons only) photon polarization. These cross sections are unambiguously related to the electromagnetic structure functions,

(25)

with the usual Bjorken scaling variable . This choice is convenient because in what follows, we will address transitions between helicity states of the nucleon and resonances, and it is preferrable to work with matrix elements of the electromagnetic current with definite helicities. As is evident from Eq. (25), the two helicity states are mixed in . Similar relations hold between the interference cross sections and interference structure functions . Note that the definition of the transverse and longitudinal polarizations of the photon and the -boson are identical since in both cases they are fixed by the lepton kinematics of the reaction .

Real photoabsorption data exhibit the following general features: i) a resonance structure on top of ii) a smooth non-resonant background between the threshold of pion production and GeV, and iii) Regge behavior at high values of with the cross section that grows slowly with energy, , with the parameter of the pomeron.

(26)

where , etc. stand for pomeron and Regge trajectories. In this work, the most recent fit in terms of two trajectories (pomeron plus ) is used regge_fit ()

(27)

with parameter of the pomeron . The threshold factor is necessary to make the continuation of the Regge fit into the resonance region meaningful. In this work, we take it in the same form as in bianchi ()

(28)

For virtual photons in the range of of interest here, the picture remains the same, with the -dependence of the resonance contributions described by the form factors measured for a number of resonances, at least in certain channels.

Figure 3: (Color online) World data on total photoabsorption bloom69 (); armstrong72 (); caldwell73 (); caldwell78 (); chekanov02 (); vereshkov03 () (see PDG () for the complete list) compared to the two models described in the text. The experimental errors are not shown.

We will next specify two models that provide a smooth extrapolation between the real photoabsorption data and the virtual photoabsorption data and that can to certain extent be used to describe data all the way up into the diffractive DIS region. The two models differ in the form of the -dependence of the background contribution:

Parameter Ref. bosted () Model I Model II
2.124 2.24 2.2
2.569 2.73 2.73
0.064 0.155 0.155
0.549 0.549 0.7
1.914 1.914 2.5
1.0 1.0 1.5
3.419 5. 5.
11. 15. 15.
Table 2: The list of the resonance parameters and their values for Model I and Model II, as compared to the original fit of Ref. bosted () (see Table III of that reference). The notation of bosted () was kept.
  • Model I: The model used in ja_chuck () utilized the resonance parameters obtained in bianchi () and the non-resonant Regge contribution from regge_fit () that was fitted to the real photoabsorption data at high energies. The -dependence of the high-energy part was taken from the hybrid GVD/color dipole (CDP) approach of Ref. cvetic (). For the estimates of ja_chuck (), a simple dipole model with the dipole mass GeV for all the transition resonance form factors was employed. Because it was found that this simple dipole form fails dramatically throughout the resonance region, we adopt the resonance part from bosted () with a few parameters minimally adjusted in order to fit the data with the background of a different form, rather the one used in bosted () originally. We list those parameters and the respective changes in Table 2.

  • Model II: To test the sensitivity of our calculations to the specific model, we use another form of the background from the “naïve” GVD model of Ref. alwall () (cf. Eqs. (3,4) of that Ref.), and we add the resonance contributions from bosted () on top of that. Again, some resonance parameters are slightly adjusted to the background, and all changes are quoted in Table 2.

In Fig. 3 we confront the two models with the total photoabsorption cross section. The Model I is shown by solid red lines, Model II by the dashed blue line.

Figs. 4-6 display the comparison of the two models with the data for the differential cross section for inclusive electroproduction in the resonance region. Both models in general provide a good description of the data in the resonance region. The areas between the lower and upper thin curves in each plot correspond to the range of values of the helicity amplitudes for the photoexcitation of each resonance included in Models I and II, as given by the PDG PDG (). It can be seen that the experimental data are always contained within these areas for GeV, even without including the experimental errors. At the same time, we note that just above the resonance region, in the limited range 4 GeV GeV, and at moderate values of , the background systematically lacks strength. However, we stress that this lack of strength is observed only in very limited range of energies, and the deficit is less than 20% which makes the impact of this effect on the dispersion correction small.

Figure 4: (Color online) Differential cross section data in the resonance region from E00 () are shown in comparison with the two models. The experimental errors are not shown. Thick solid line is the result of Model I, and thick dashed line is the result of Model II. Thin solid lines show the error bar due to the uncertanties in helicity amplitudes for the photoexcitation of the resonances on the proton, according to PDG ().
Figure 5: (Color online) Differential cross section data in the resonance region from E00 () are shown in comparison with Model I and II. Notation as in Fig. 4
Figure 6: (Color online) Differential cross section data in the resonance region from E94 () are shown in comparison with Model I and II. Notation as in Fig. 4
Figure 7: (Color online) Differential cross section data in the resonance region from E94 () are shown in comparison with Model I and II. Notation as in Fig. 4
Figure 8: (Color online) Differential cross section data in the resonance region from E94 () are shown in comparison with Model I and II. Notation as in Fig. 4
Figure 9: (Color online) Differential cross section data in the resonance region from E94 () are shown in comparison with Model I and II. Notation as in Fig. 4

We next turn to the deep inelastic (DIS) data. For DIS, a natural choice would be to use the PDF parametrizations from MRST or CTEQ, DGLAP-evolved to the necessary value of . However, this is only applicable at large enough , and extrapolating them below GeV introduces additional systematic error. In Figs. 10, 11, the naïve GVD model of Ref. alwall () (Model II) is shown along with the GVD/CDP model of cvetic () (Model I). One can see that while the GVD/CDP model reproduces the data in a wide range of , the naive GVD model overshoots the data at large starting at moderate , and underestimates the low- behavior for all . One needs to keep in mind, however, that both models work reasonably well at moderate and large which give the main contributions to the dispersion correction.

Figure 10: Comparison of the -dependence of the DIS structure function at fixed and as a function of , in GVD/CDP model of cvetic () (solid lines) and the naive GVD model of alwall () (dashed lines) to the low- DIS data of H1 Collaboration F2_H1 (). The experimental errors are not shown.
Figure 11: Comparison of the -dependence of the DIS structure function at fixed and as a function of , in GVD/CDP model of cvetic () (solid lines) and the naive GVD model of alwall () (dashed lines) to the DIS data of NMC Collaboration F2_NMC () and E665 Collaboration E665 () where the -binning corresponds to that of NMC. The experimental errors are not shown.

The following comment is in order here. The authors of Ref. sibirtsev () argued that our description of the data is unsatisfactory not only in the resonance region but also beyond (cf. Fig. 1 of sibirtsev ()). While the model of the resonance form factors of Ref. ja_chuck () was definitely not accurate (one of the instances on which we improve that calculation in the present work), the model for the background in ja_chuck () is exactly the same as that of Model I here. We believe that Figs. 4-11 presented in this section provide abundant evidence of a satisfactory description of the experimental data by our phenomenological model. In view of this, we find it puzzling that Ref. sibirtsev () quotes a discrepancy of 40-50% at as low as 0.6 GeV just above the resonance region (cf. the upper left panel of Fig. 1 of that reference).

iv.2 Isospin rotation of the resonance contributions

In Standard Model, the and hadronic currents are related by means of a simple isospin rotation,

(29)

with

(30)

The e.m. charges given by

(31)

whereas the weak charges are

with being a shorthand for (for purposes of this argument). This isospin decomposition is used to relate weak proton form factors to the proton and neutron electromagnetic form factors,

(32)

where we neglected strangeness contributions that are generally small strangequarks ().

-1.0 -0.51 -0.77 -0.28 -0.27 -0.62 -1
Table 3: Ratios with respective uncertainties for seven resonances.

The above relation is valid for transitions to resonances, as well:

(33)

It is then straightforward to relate the contribution of a resonance with isospin to the interference cross section entering Eq. (16) to its contribution to the electromagnetic cross section:

(34)

Consequently, for each resonance, we define two ratios describing the relative strength of its contribution to the -interference cross sections with respect to the purely electromagnetic ones as

(35)

In the Appendix A we discuss in detail the -dependence of these ratios, as well as the ratios of the longitudinal cross sections . Basing on the discussion in Appendix A, we will use the value

(36)

to rescale the contribution of a resonance to both transverse and longitudinal cross section. Possible discrepancies (which, if known, are model-dependent) from this rule are accounted for by assigning a conservative uncertainty to the ratios . This is done by using the PDG values and respective errors for the transition helicity amplitudes. These PDG values represent an average over different data sets and different extraction procedures adopted in the various experiments. Consequently, they automatically include an enhanced error due to model dependence of this extraction.

The first term in Eq. (36) is a constant that is model-independent, arising from Eq. (34). This model independence reflects the cancelation of the proton-to-resonance transition matrix elements involving the e.m. currents. The second term in Eq. (36) , , is given by the ratio of combinations of neutron and proton transverse helicity amplitudes (we refer the reader to the Appendix for details). We summarize the values of obtained using the PDG values for the helicity amplitudes with the respective errors in Table 3. The lower and upper limits correspond to taking extreme values of the transition helicity amplitudes for the proton and neutron from PDG ().

For the resonance, we assign a conservative 10% error on its isospin structure. According to the PDG, this error should be precisely zero. However, the analyzes of Refs. bosted (); bosted2 () return slightly different results for the excitation on the proton and the neutron, both for real and virtual photons. The discrepancy stays below relative 10% for GeV, although this conclusion is definitely model-dependent. This observation provides motivation for assigning a conservative 10% error to for the .

Similarly, for the resonance, the uncertainty is driven by the analyses of Refs. bosted (); bosted2 (). The fit of bosted () for the proton returns a very mild monopole form factor, whereas the neutron data require a dipole form factor for the same resonance bosted2 (). Also the strength strongly depends on the form of the background, as found in our work (see Table 2). This motivated us to assign a conservative 100% uncertainty due to this resonance.

We note that for both resonances listed in the Table 3 the error bar exceeds 100%. This is mostly due to the quality of the extracted values for the neutron. It is also worth noting that the quark model expectations (see Table I of Ref. deltagZ_conf () for the isospin scaling factors within the quark model of Ref. koniuk ()) are not too far from the central values quoted in Table 3.

iv.2.1 Uncertainty in isospin rotating the resonances

To summarize the results of the previous subsection, we propose to obtain the contribution of a resonance to the -interference cross sections by multiplying the purely electromagnetic cross sections with a scaling factor that is independent of and . Furthermore, to the precision required here, we rescale the trasverse and longitudinal cross sections with the same factor. Each such factor contains two parts, as per Eq. (36): the first one is model-independent, whereas the second one is obtained from the analysis of the proton and neutron electromagnetic data, and involves model dependence and experimental uncertainties. The values of are listed in Table 3 with the respective uncertainties. Correspondingly, for each resonance we simply obtain its contribution to the interference structure functions from that to the electromagnetic structure functions as

(37)

To compute Re, we use Eqs. (20) and (16) with the input from Eqs. (37) and (25). Finally, we use the parametrizations of the transverse and longitudinal electromagnetic cross sections from Model I and Model II, and values of factors from Table 3. The uncertainty on the contribution of each resonance is obtained according to the definition

(38)

where are the uncertainties quoted in Table 3. Using the steps described above for the individual contributions of resonances to Re, we can also compute the uncertainties Re associated with each such contribution. Because most resonances do not overlap, we treat all these uncertainties as independent, thus we define

(39)

iv.3 Isospin rotation of the high energy contribution

We need to employ a well-motivated model to describe the isospin dependence of the background contribution. One option is to employ the the VDM picture, incorporating the simple observation that the photon has the same quantum numbers as vector mesons (VM). Therefore, it can fluctuate into or that then scatter off the nucleon. This approach underlies the background in both Models I and II, so we proceed generally at first.

According to the VDM, the photon can be represented as a superposition of a few vector mesons,

(40)

with the VM decay constant. Assuming this basis to be complete and orthogonal (no VM mixing), one can express the total photoabsorption cross section through a combination of total cross sections for vector meson-proton scattering,

(41)

At high energies, the total cross section should be independent of the VM flavor and the above equation becomes simply a flavor decomposition of the electromagnetic total cross section, although this representation is of limited use because