# Weak Polarized Electron Scattering

###### Abstract

Scattering polarized electrons provides an important probe of the weak interactions. Precisely measuring the parity-violating left-right cross section asymmetry is the goal of a number of experiments recently completed or in progress. The experiments are challenging, since is small, typically between 10 and 10. By carefully choosing appropriate targets and kinematics, various pieces of the weak Lagrangian can be isolated, providing a search for physics beyond the Standard Model. For other choices, unique features of the strong interaction are studied, including the radius of the neutron density in heavy nuclei, charge symmetry violation, and higher twist terms. This article reviews the theory behind the experiments, as well as the general techniques used in the experimental program.

Annu. Rev. Nucl. Part. Sci. \jyear2013 \jvol1 \ARinfo1056-8700/97/0610-00

eak Neutral Currents, Parton Distributions, Neutron Stars, Physics Beyond the Standard Model

## 1 Introduction

### 1.1 Overview

In 1978, SLAC experiment E122 [1, 2] published the observation of parity violation in the deep inelastic scattering of polarized electrons from deuterium. It settled the issue as to whether or not the then recently discovered weak neutral currents were parity-violating and led to the universal acceptance of the Standard Model (SM). The experiment also demonstrated that parity violation in electron scattering (PVES) is a viable tool for particle and nuclear physics.

In the following 35 years, many new PVES experiments were performed or are planned at various laboratories around the world, including SLAC, Mainz, MIT–Bates, and JLab. The goals of the new experiments included searching for non-zero strange elastic form factors [3], measuring the weak mixing angle, , at low energies [4], measuring a set of electroweak (EW) couplings, measuring the radius of the distribution of neutrons in heavy nuclei, searching for charge symmetry violation (CSV) at the quark level, and measuring higher-twist effects in deep inelastic scattering (DIS). The first two topics have been the subject of previous reviews [3, 4]; here we will focus on the other topics.

The basic idea of PVES is to measure the parity-violating EW asymmetry,

(1) |

where is the cross section for the scattering of electrons with left (right) helicity. The leading order effect arises from an interference between photon and exchange, resulting in small asymmetries proportional to the four momentum transfer and also to the EW couplings. Values of in the range from to can be measured with good accuracy. By selecting optimal kinematics and targets, the wide variety of physics mentioned above can be accessed.

A useful feature of is that in the ratio a number of experimental uncertainties cancel, such as those due to target thickness and the solid angle. Some possible theoretical uncertainties, such as elastic form factors, also cancel. The kinematics of the various experiments are chosen either so that unknown hadronic effects cancel, probing EW physics, or so that the hadronic effects remaining are of interest and cannot be measured by other techniques.

### 1.2 Electroweak Physics

An efficient way to represent the sensitivities of different experiments and observables to the underlying EW physics, including possible new physics beyond the SM, is to use the language of a low-energy effective field theory in a way described in Section 3. The parity-violating (PV) part of the neutral-current (NC) interactions of charged fermions with electrons is

(2) |

where GeV and is the Fermi constant. The EW coefficients are real-valued and at the SM tree level given by

(3) |

(4) |

We now briefly review how these couplings are extracted experimentally.

Currently, the most precise determination of any combination derives from observations [5] of atomic parity violation (APV), most notably in Cs where one also obtained [6, 7] the best understanding of atomic structure [8] which is crucial for the interpretation in terms of EW physics. The result,

(5) |

is lower than the SM prediction of 36.66. In the future, one may constrain different linear combinations by studying neutron rich nuclei like Fr or Ra, or by considering isotope ratios in which atomic physics uncertainties cancel.

Note that we are using the notation introduced in Reference [9]. More familiar are the so-called weak charges, , which at the tree level are also given by the coherent sum of the corresponding quark couplings (as in Equation 5), but multiplied by a factor of and with a different set of radiative corrections applied. This is indicated by the use of and in place of and . Our notation serves as a reminder that corrections have been applied (see Section 3.1) allowing direct comparison and combination of the couplings when extracted from different observables, experiments, and kinematic conditions.

PVES offers an alternative to APV and the possibility to cross-examine Equation 5 with entirely different experimental and theoretical challenges and uncertainties. E.g., the Qweak Collaboration [10] has measured the left-right asymmetry [11] in elastic polarized scattering, ,

(6) |

where is the center-of-mass energy, the EM fine structure constant, and is a QED correction factor. Here and in the following, is the fractional energy transfer from the electrons to the hadrons which is perturbatively small for Qweak kinematics, , and will be smaller yet for the experiment at the MESA facility in Mainz. The analysis of the Qweak commissioning data (about 4% of the total) translates into the constraint,

(7) |

to be compared to SM prediction of . The final result is expected to be four times as precise, and even greater precision will be possible in Mainz. If additionally elastic scattering off isoscalar nuclei like C was used to extract , one could disentangle and from PVES alone.

The analogous asymmetry in Møller scattering [12],

(8) |

has been obtained by the SLAC–E158 Collaboration [13] from which one can extract [9],

(9) |

while the SM predicts the value 0.0225. The MOLLER Collaboration [14] at Jefferson Lab aims to reduce the uncertainty in by a factor of five by taking advantage of the energy upgraded CEBAF.

Deep inelastic PVES (PVDIS) experiments [2, 15] are sensitive to the interference of the quark-level amplitudes corresponding to with photon exchange. Scattering from an isoscalar target provides information on the charge weighted combinations, and . In the quark model and in the limit of zero nucleon mass one can write [9] in a simple valence quark approximation,

(10) |

One has to correct for higher twist effects, CSV, sea quarks, target mass effects, longitudinal structure functions and nuclear effects. Some of these issues are of considerable interest in their own right, and their discussion is deferred to Sections 4 and 5.

The proposed Electron Ion Collider (EIC) [16], currently under study as the next step in exploring the QCD frontier, is a high luminosity (cm s) machine that will use highly polarized () electron and nucleon beams with a variable center of mass energy and a wide variety of nuclear targets. Such a facility will allow further improved precision studies of PVDIS asymmetries. Its wide kinematic range, along with other experiments, will allow one to disentangle various hadronic effects such as CSV or higher twist effects. In turn, it can also allow for more precise extractions of the contact interaction couplings that are sensitive to new physics and the weak mixing angle at different values of .

We will return to the EW couplings, their radiative corrections and the associated uncertainties in Section 3.

### 1.3 Hadronic and Nuclear Physics

As discussed earlier, sound theoretical control over various hadronic effects that contribute to the asymmetries is essential for a reliable interpretation of the EW physics. Alternatively, one can view PVDIS as a probe of the hadronic effects themselves as a means to further our understanding of QCD and nuclear dynamics. Within this context, we discuss some of the important hadronic effects that affect PVDIS asymmetries in Section 4.

We describe elastic PVES from heavy nuclei in Section 5. These experiments can precisely locate neutrons in the nucleus because the weak charge of a neutron is much larger than that of a proton. These experiments target neutron radii that have important implications for nuclear structure, astrophysics and APV.

## 2 Experimental Issues

### 2.1 Introduction

A large number of PVES experiments have been completed or are in progress. A list is given in Table 1 which gives a flavor of the variety of both apparatus and physics goals. We will first give a brief description of selected experiments. Then we will discuss general design criteria that apply to all of the experiments.

Experiment | Magnets | Detector | Count | Angles | Physics |
---|---|---|---|---|---|

SLAC–E122 | Dipoles | Pb Glass | No | 12 | , |

Mainz | None | Air C | No | 146 | |

MIT–Bates | Q | Lucite C | No | 20 | |

SAMPLE | None | Air C | No | 146 | Strange FF |

HAPPEX | QQDDQ | Pb-Lucite | No | 15 | Strange FF |

GØ | Toroid | Scintillator | Yes | 20-50 | Strange FF |

Mainz–A4 | None | PbF | Yes | 30 | Strange FF |

SLAC–E158 | QQQQ | Cu-Quartz | No | 2 | |

HAPPEX–He | QQDDQ | Cu-Quartz | No | 5 | Strange FF |

PREX | QQDDQ | Quartz | No | 5 | |

PREX-II | QQDDQ | Quartz | No | 5 | |

CREX | QQDDQ | Quartz | No | 4 | |

JLab–Hall A | QQDDQ | Pb Glass | Yes | 19 | |

Qweak | Toroid | Pb-Quartz | No | 5 | |

MOLLER | Toroid | Quartz | No | 1 | |

SoLID | Solenoid | Package | Yes | 22-35 | , CSV, HT |

Mainz–P2 | Solenoid | Quartz | No | 20 | |

Mainz–C | Solenoid | Quartz | No | 40 |

### 2.2 Brief Descriptions of Selected Experiments

In the Qweak experiment [10] electrons scattered from a 35 cm long LH target with angles between 5 and 11 and passed through a collimator. They were deflected by a toroid and focussed onto quartz bars that served as Cherenkov detectors. Virtually all of the apparatus for Qweak was custom fabricated.

By contrast, the PVDIS [15] and PREX [17] experiments both used the same apparatus, the JLab–Hall A high-resolution spectrometers (HRS) [18]. The HRS was designed not for PVES but rather to measure cross sections with excellent energy resolution. For that purpose, the HRS spectrometers were instrumented with a detector package with drift chambers for tracking, Cherenkov counters and an electron calorimeter made of Pb glass for particle identification.

For PREX, the spectrometers needed sufficient energy resolution to reject inelastically scattered electrons that had lost more than a few MeV. The HRS detector package was not used, but rather elastic events were focussed onto a small quartz bar and the signal was integrated. The drift chambers in the spectrometers were used in special calibration runs to measure the average of the events giving signals in the quartz.

For PVDIS, by contrast, energy resolution was irrelevant. However, Cherenkov and Pb Glass detectors were used to identify the DIS electrons, separating them from the more copious pions. Since the electrons were identified by a coincidence between two detectors, the events must be counted.

Many PVES experiments are built around magnets originally designed for a different purpose. E122 [1], MOLLER [14], Bates [19], SoLID [20], and P2 [21] are examples. Another variation are experiments like A4 [22] and SAMPLE [23], where there is no magnet but special detectors are designed that can detect the elastic events amidst a large background of lower energy particles.

#### 2.2.1 Spectrometers:

The optimization of PVES experiments is as follows. The differential cross sections and PV asymmetries are approximately,

(11) |

where is typically 10 to GeV, depending on the reaction. At first glance, it may appear that higher energies are better. However, for small angles, where , we can show that the statistics are independent of . The reason is that since depends strongly on , the angular acceptance cannot be too large; typically . At fixed , . Then, the solid angle , and is approximately constant. The statistical error varies as the ratio , which is independent of . Thus, the main criteria for designing an experiment is to maximize the acceptance .

Toroids, such as the one used for Qweak, typically have a large . For MOLLER, where there are two electrons in the final state, full coverage can be obtained with a toroid. With a solenoid, the full acceptance in can be achieved [24], as is planned for the P2 experiment. The JLab HRS spectrometers have a fixed solid angle acceptance. As a consequence, they are used at the most forward angles possible where they achieve a respectable of about .

Another important requirement on the spectrometer, especially for experiments with elastic scattering, is energy resolution, which must be good enough to reject inelastic events. For proton scattering, pion production is the first background, so the resolution must be better than 100 MeV. For nuclear targets, inelastic levels at the 5 MeV levels must be rejected. Often a low beam energy is optimal because the rate is independent of energy but the absolute resolution improves with lower energies. By contrast, for the JLab HRS spectrometers with their excellent energy resolution, backgrounds can be rejected even with high energies and rates go up with energy due to the fixed solid angles of the HRS system.

#### 2.2.2 Detectors:

One important decision that influences both the design of the spectrometer and the detectors is whether to count events or to integrate the signal. Integration has the advantage that there are no pile-up or dead-time corrections. E.g., with the high resolution spectrometers in Hall A at JLab, elastically scattered events are physically separated from all inelastic events, and very clean data samples are obtained even though the signal is integrated.

For a counting experiment, the statistical error in is simply , where is the number of events detected. For an integrating experiment with average signal and detector resolution , the statistical error is given by,

(12) |

Thus, if the detector has reasonable resolution, little statistics are lost by integrating. For example, for a moderate resolution , the statistical error is increased by only 2%. However, care must be taken. Half of the statistics would be lost if 1% of the events has 10 times the average signal. With a thin scintillator, the Landau tail would causes to be large, and integrating the signal would cause unacceptable loss in statistics. For high energy electrons, a crude shower counter with coarse granularity will suffice. For high rates, radiation damage becomes important, and fused silica (henceforth called quartz) is the material of choice. For energies of 1 GeV and below, the resolution of a granular detector becomes poor, and thin quartz has better performance. The design is critical. If the quartz is too thin, there are too few photoelectrons detected and increases. If the quartz is too thick, the electrons start to shower and create a high-signal tail, which increases . With great care in optimizing the collection of light, acceptable performance can be achieved.

For experiments with larger asymmetries and thus lower event rates, counting techniques are practical, taking advantage of the continuos wave nature of modern electron facilities. For SoLID, a traditional electron spectrometer with tracking, Cherenkov counter, and a calorimeter is proposed.

#### 2.2.3 Data Acquisition and Electronics:

Perhaps the most specialized aspect of PVES is the electronics. Since integration is only used for PVES, the integrating electronics is custom made. For the high rate experiments, such as SLAC–E158, PREX, and Qweak, the problem is that the statistical noise in a given helicity window is only on the order of 100 ppm, and keeping the noise of the electronics below that level is a special challenge.

For the counting experiments, the large rate of accepted events separates PVES experiments from other experiments. Custom counting electronics were a main feature of the A4 program, the JLab PVDIS experiment, and SoLID.

#### 2.2.4 Backgrounds:

Backgrounds are an important source of error. The easiest case is elastic scattering where the highest energy particles possible are the desired events, and backgrounds can be eliminated simply by achieving good resolution, as is done with the JLab–Hall A HRS spectrometers. For DIS, pions are the main background, and can be separated by the usual methods.

General electromagnetic background can be a problem, especially for toroidal spectrometers. Helicity-dependence of the beam width, or other higher-order parameters which are very difficult to monitor, can propagate to the background if it arises from beam spraying from a small collimator. There can be physics asymmetries in some backgrounds, such as decay products from hyperons or pions produced by DIS events. Detailed studies are required to untangle these effects

### 2.3 Precision of PVES Experiments

As the field of PVES has developed, the precision achieved or proposed, both in terms of the absolute size of the error and of the fractional error in the asymmetry, has improved dramatically, as shown in Table 2. Most of the techniques that have been perfected over the years are common to all PVES experiments.

Experiment | Reference | Year | (stat) | (syst) | /(%) | |

SLAC–E122 | [2] | 1978 | 7 | 6 | 8 | |

Mainz | [25] | 1989 | 1.8 | 0.5 | 20 | |

MIT–Bates | [19] | 1990 | 1.62 | 0.37 | 0.11 | 24 |

SAMPLE | [23] | 1990 | 0.67 | 0.88 | 20 | |

HAPPEX | [26] | 2001 | 0.98 | 0.56 | 7.5 | |

GØ | [27] | 2005 | 0.15 | 0.2 | 13 | |

Mainz–A4 | [22] | 2009 | 0.8 | 0.9 | 5 | |

SLAC–E158 | [13] | 2005 | 0.014 | 0.010 | 13 | |

HAPPEX–He | [28] | 2007 | 6.40 | 0.23 | 0.12 | 4.1 |

PREX | [17] | 2012 | 0.656 | 0.060 | 0.014 | 9.4 |

PREX-II | 2016 | (0.51) | 0.015 | 0.005 | 3 | |

CREX | 2017 | 2.0 | 0.04 | 0.02 | 2.4 | |

JLab–Hall A | [15] | 2014 | 6.4 | 3.1 | 4.4 | |

Qweak | [10] | 2013 | 0.035 | 0.031 | 17 | |

MOLLER | 2020 | 0.035 | 0.0007 | 0.0004 | 2.3 | |

SoLID | [20] | 2022 | 2 | 4 | 0.6 | |

Mainz–P2 | [21] | 2018 | 0.00025 | 0.00019 | 1.7 | |

Mainz–C | 2020 | 0.8 | 0.0025 | 0.00017 | 0.3 |

#### 2.3.1 Measuring Small Asymmetries:

The basic technique used to observe the small asymmetries in PVES is the ability to rapidly reverse the helicity of the beam without changing any of its other properties, including energy, intensity, position, and angle. Many systematic errors that are important for measurements of cross sections, such as target thickness, spectrometer solid angle, etc., which are hard to measure and tend to drift with time, cancel in the asymmetry.

The source of the polarized electrons in PVES is photo-emission from a crystal, and the helicity of the beam is reversed by reversing that of the laser light producing the photo-electrons. Today, polarizations of more that 85% and beam currents of more than 100 A are now routinely achieved by using a strained GaAsP crystal [29, 30].

The helicity of the electron beam is determined by that of the laser light producing the photo-emission. The helicity of the light is reversed by using a device called a Pockels cell, which is a crystal whose birefringence is controlled by high voltage applied across the cell. To first order, the helicity is the only beam property changed by the voltage on the Pockels cell.

The improvement in the achievable precision in PVES is in part due to great progress in understanding and correcting the imperfections in the helicity reversals. The basic problem is that the laser light is partially linearly polarized, and the linear polarization also reverses with helicity. Flipping the linear component of the light causes systematic differences in the intensity, position and width of the photo-emitted electrons, the latter effects arising from a spatial dependence of the linear polarization. Extensive studies of these effects resulted in specialized techniques that have greatly improved the ability to provide clean helicity reversals.

Systematic errors can further be reduced by using independent means to reverse the beam helicity. These include insertion of a half wave plate in the laser beam, using a Wien filter in the electron beam before it is accelerated, and sometimes by running the experiment at a nearby energy where the helicity is reversed by an extra flip in the accelerator or beam transport. The half-wave plate reversals can be done every few hours; the other reversal methods are done less frequently because the accelerator needs to be retuned after the reversal.

The spectrometers in PVES experiments are designed so that they are as insensitive as possible to the beam parameters. By making the apparatus symmetrical, the signal is very insensitive to differences in beam position and angle. Careful design of apertures can reduce the sensitivity to beam size. In addition, careful tuning of the accelerator reduces the size of the helicity-correlated beam differences at the target.

Systematic differences in the beam parameters are controlled by a set of position-sensitive monitors that together determine the beam position and angle on target. A position monitor in a point of high dispersion determines energy differences in the beam. The sensitivity of the spectrometer to these beam parameters can be measured by dithering the beam with coils and an RF cavity so corrections can be made for any residual beam differences [31]. Feedback is another tool, especially useful for eliminating the helicity-dependence of the beam intensity.

#### 2.3.2 Scale errors:

Although the asymmetries measured in PVES experiments are small, in many cases the fractional error is on the order of 5% and the SoLID experiment proposes to measure to 0.5%, which is a higher precision than most cross section experiments. Systematic errors in scale factors, such as the beam polarization and , become important.

There are two methods for measuring , each with variations. One is Compton scattering from a polarized laser beam. Either the scattered electron or photon can be detected; indeed both can be detected to provide independent measurements. Theoretical errors for Compton scattering are small, and the beam polarization can be measured simultaneously with . Precisions as good as 1% have been achieved and an additional factor of two is possible.

The other method is Møller scattering from a target with polarized electrons. For a ferromagnetic material in a low magnetic field, errors in the target polarization are on the order of 3%. By saturating the target in a 4 T field, this error can be reduced to below 1%. Unfortunately, the targets are thick and can tolerate currents on the order of a few A and the polarimetry cannot be done while is measured. A proposed variation [32], which uses cold H atoms in a magnetic trap, features a thin target that can be run during the measurement, and promises to provide precision below 0.5%

## 3 Theoretical Issues

### 3.1 Effective Couplings

In an effective field theory one absorbs the effects of the heavy degrees of freedom, such as the , the , and the Higgs boson mediating the weak interaction, into effective couplings. The degrees of freedom of the effective theory relevant to PVES are then electrons, first generation quarks, and nucleons. At the level of radiative corrections, discussed in Section 3.2, it is important to define the couplings in a process-independent manner and for an arbitrary gauge theory, so as to allow for direct comparisons and global analyses of the different types of experimental information. We use the conventions proposed in Reference [9] which aim to strike a balance between formal, practical and historical considerations.

After EW gauge symmetry breaking, the SM fermion fields (with mass ) interact with the Higgs field , the intermediate vector bosons and , and the photon field , according to the Lagrangian

(13) |

where is the vacuum expectation value of . We have chosen a common normalization in which the charged (CC), electromagnetic (EM) and weak neutral currents (NC) are given by,

(14) |

(15) |

(16) |

(17) |

Here, denote chiral projections, contains the quark mixing matrix , is the weak mixing angle, and fermion generation indices have been ignored. Furthermore,

(18) |

are vector and axial-vector couplings with .

At low energies, , one finds the effective four-fermion interaction Lagrangians,

(19) |

### 3.2 Radiative Corrections

The low-energy couplings defined in Equation 2 are modified by radiative corrections, which in general depend on energies, experimental cuts, etc. To render them universal, adjustments have to be applied to the underlying processes. One defines [9] the one-loop radiatively corrected couplings to include the purely EW diagrams and certain photonic loops and -exchange graphs, while the remaining corrections, as e.g., the interference of two photon exchange diagrams with single or exchanges, are assumed to be applied individually for each experiment. Moreover, since the genuine EW radiative corrections will in general depend on the specific kinematical points or ranges at which the low-energy experiments are performed, one needs to introduce idealized EW coupling parameters [9] defined at the common reference scale , and expect the experimental collaborations to adjust to their conditions. According to the calculations from References [33, 34, 35, 36], the SM expressions for the NC couplings defined in this way [9] are given by,

(20) |

(21) |

where using the abbreviations and , one has,

(22) |

(23) |

(24) |

(25) |

(the QCD correction needs to be dropped if refers to a lepton) and is given by with . Furthermore,

(26) |

In these relations, the parameter [37] renormalizes the NC interaction strength at low energies, and and denote EW box diagrams. The logarithms entering the quark charge radii, and , are regulated at the strong interaction scale, , introducing a hadronic theory uncertainty into the unless they are extracted from DIS (in these expressions by definition).

Numerically most important are vacuum polarization diagrams of - mixing type giving rise to a scale-dependence [38] of the weak mixing angle. As long as one stays in the perturbative QCD domain these effects can be re-summed [39], while passing introduces a hadronic uncertainty (see Section 3.3). At the tree level we employ the weak mixing angle at the scale and abbreviate, , while in the EW box graphs the use of is more suitable (the caret indicates the definition in the -renormalization scheme). In and in the box diagrams discussed in Section 3.3 we use [9]. The resulting SM values of the NC couplings are given in Table 3.

C | |||||||
---|---|---|---|---|---|---|---|

0.0225 | 0.3419 | 0.4951 | |||||

0.0225 | 0.0248 | 0.0144 |

Unlike the quantities and defined [33] in the context of APV we exclude the small axial current QED renormalization factors from the definitions of and , because QED and QCD corrections to external lines are not considered part of the EW couplings. They can be included together with experiment specific initial and final state radiation effects and box graphs in explicit QED radiative correction factors [40] multiplying the asymmetries. In addition, and the box contributions need to be adjusted for , which in the case of Møller scattering amounts to shifts [9]

(27) |

for the SLAC [13] and Jefferson Lab [14] experiments, respectively.

In the DIS regime, the box graphs need to be adjusted for the relevant -values and beam energies, but this is feasible [41] and the extra -dependences are expected to change the by at most a few . Ignoring this issue, the adjustments for the SLAC [2] and Jefferson Lab [15, 42] experiments, all with -values around the charm quark threshold, are

(28) |

(29) |

(30) |

The most precise current constraints on the corrected and as functions of are shown in Figure 1.

### 3.3 Theoretical Uncertainties

The corrections that need to be applied in order to extract the coefficients and introduce additional uncertainties. These are relatively enhanced whenever the tree-level expression is suppressed by a factor , as is the case for all the , as well as the for the electron and the proton. The most important associated theoretical uncertainties are from the renormalization group evolution (running) of the weak mixing angle [38, 39] and -box diagrams (both involving hadronic effects), as well as from unknown higher order EW corrections.

#### 3.3.1 :

The running of from (where it has been measured precisely in -pole experiments) to arises from loop-induced - mixing diagrams and amounts to a 3.2% effect. There is currently a relative 0.9% theoretical uncertainty in its calculation [39] which translates to a 0.6% uncertainty in the Møller asymmetry and a 0.4% uncertainty for the asymmetry (it would be a negligible 0.03% for measurements in carbon).

Most of the scale evolution (%) can be computed reliably within perturbation theory, but the contribution below requires other considerations. In this regime one can estimate the effect by relating it to that of the EM coupling, [43], which in turn is determined by computing a dispersion integral over cross-section data and (up to isospin breaking effects) to decay spectral functions [44]. The corresponding error contribution is about in and fully correlated with the one in .

This strategy is limited by the necessary separation of the contributions from the various quark flavors and from QCD annihilation (or singlet) diagrams which enter and differently. The largest uncertainty () is induced by the need to quantify the strange quark contribution relative to the first generation quarks. In the future it may be possible to decrease this error by including information about the strange meson production fraction, but this is complicated due to the occurrence of secondary strange pair production. Isospin breaking due to contributes an additional but much smaller () uncertainty. Currently an error of is assigned to the singlet separation [39], but since QCD annihilation diagrams are related to Okubo-Zweig-Iizuka (OZI) rule violations which are phenomenologically known to be strongly suppressed, this is quite likely an overestimate.

In addition, there are parametric uncertainties from the imperfect knowledge of , and . They could increase the above error estimates by up to 30%, but it can be expected that they will be much better determined in the future, e.g., by means of lattice simulations. In any case, these are fundamental fit parameters allowed to vary in EW fits, and should not be added to the purely theoretical uncertainties reviewed here.

#### 3.3.2 Hadronic box:

From Equation 26 one can see that the contributions from box diagrams are logarithmically enhanced. Moreover, and more importantly, these logarithms are sometimes regulated at the scale (we chose the reference value while the full effect depends on experimental and kinematical details). This signals a hadronic uncertainty and is indicated when the parton model expression introduces the logarithm of the mass of a quark. Fortunately, the sum of uncrossed and crossed box diagrams entering APV gives rise to the chiral structure which is itself suppressed by . This is the reason why this kind of uncertainty is small in APV [34, 45].

The finite beam energy, , in PVES, on the other hand, upsets the cancellation between uncrossed and crossed box graphs, effectively introducing the wrong-chirality and unsuppressed structure [46]. Several groups estimated the effect and there is consensus regarding the central value. The most recent evaluation [47] of the sum of the two chiral structures applicable to the Qweak condition with GeV implies the correction,

(31) |

The uncertainty is dominated by the structure and, respectively, smaller and larger by more than a factor of two compared to the ones quoted in [48] and [49]. A reduction and robust estimate of the error will be important for a solid interpretation of the Qweak experiment [50]. Further hadronic effects of relative order and their uncertainties () are treated by extrapolating the Qweak and other PVES data points to , as e.g., in Reference [51]. The box corrections, uncertainties, and correlations need to be applied to each data point in the extrapolation.

The beam energy of MESA for the P2 project is still subject to optimizations, but it will be low enough that the uncertainty dominates. For MeV, the analysis of Reference [47] implies the correction,

(32) |

Effects due to in the shifts 31 and 32 are negligible [48], and the -dependence of the weak mixing angle can be ignored if the asymmetry is normalized using the fine structure constant in the Thomson limit. But the electron charge radius induces the additional shift [9],

(33) |

In the DIS regime, the box can be calculated perturbatively, but the appropriate event generators [52] need to be examined for consistency with more recent conventions and refinements. This will introduce an additional -extrapolation.

#### 3.3.3 Unknown higher orders:

Several EW one-loop corrections are large, most notably the box contribution to the asymmetry which exceeds the anticipated experimental precision by an order of magnitude. Thus, the terms [36] in Equations 24 and 25, as well as other higher order effects need to be included or induce additional uncertainties. For example, the uncertainties displayed in Equation 27 arise from scale uncertainties in one-loop terms. Some reducible contributions can be determined by renormalization group and similar techniques, but a full two-loop calculation (ideally including enhanced three-loop effects) appears feasible and should be vigorously pursued [53].

## 4 Hadronic Structure

The idealized form of the DIS asymmetry for a deuteron target in Equation (10), is modified by various hadronic corrections to

(34) |

where the parameters have the form

(35) |

and where and . The quantities and denote corrections arising from sea-quarks, charge symmetry violation, target mass effects, and higher twist effects, respectively. Below we discuss some of these hadronic effects and their potential impact on the theoretical interpretation of precision PVDIS measurements.

Within the SM, it is often convenient to write Equation 34 in terms of the five EW structure functions and as

(36) |

Note, that deviating from common conventions, we have absorbed and , in the first and second term, respectively, into the structure function. The functions and functions have the form

(37) |

where and denote the ratios of the longitudinal to transverse virtual photon cross-sections, for the EM () and the interference () contributions, respectively. They are given in terms of the structure functions as

(38) |

In the (Bjorken) limit, , in which the Bjorken variable is kept fixed, the structure functions satisfy the Callan-Gross relations, , and

(39) |

In this limit and ignoring sea quarks and CSV, for the isoscalar deuteron target takes the simple form in Equation (10). All structure function effects cancel, allowing for a clean extraction of the and coefficients.

### 4.1 Higher Twist Effects

The PVDIS asymmetry for electron-deuteron scattering is particularly interesting as a probe of long range quark and gluon correlations, which go beyond the leading twist (twist-2) parton model. They give rise to higher twist -dependent power corrections encoded in and . It was first shown in References [54, 55] that the twist-4 contribution to is due to the deuteron () matrix element of a single four-quark operator,

(40) |

, which receives contributions from several twist-4 operators, is relatively suppressed by and can be isolated from through its -dependence. It was also shown in References [54, 55] that is related to the higher twist effects appearing in neutrino-deuteron charged-current DIS. These properties allow for a relatively clean theoretical interpretation of the quark-quark correlation in Equation 40, provided that is large enough to be observed.

Several earlier works [56, 57, 58, 59, 60, 61, 62] have explored various aspects of HT effects in PVDIS. More recently [63, 64], the HT phenomenology was revisited in the context of the current generation of high precision experiments. In the form of given in Equation 36, the term can receive contributions from HT effects if and from the ratio . In the absence of empirical data on , the impact of was explored in Reference [63] and it was found that the variation resulted in a shift in . In subsequent work [64], the Bjorken-Wolfenstein [54, 55] argument was applied to the term and it was shown that the equality was true even at twist-4 up to perturbative corrections — a consequence of the twist-4 structure functions associated with satisfying the tree-level Callan-Gross relation [65, 66, 67]. Instead, the dominant HT effects arises through the ratio , giving rise to the correction

(41) |

where () is the () quark parton distribution function (PDF) in the proton.

The first estimates of the twist-4 contribution to were obtained [56, 57] within the MIT Bag Model [68] by rescaling the twist-2 contribution by the ratio of their leading moments. This computation was recently [64] extended to include the effects of higher spin operators. A model for the nucleon wave functions in the light-cone formalism [69, 70, 71] was used to obtain an independent estimate [72] for . Most recently [73], quark orbital angular momentum dynamics was added and its effect on was studied. It was found that was largely insensitive to quark angular momentum due to cancellations between its different components. All of these recent studies yield similar results in the range to 0.005 in the valence region of . This is likely too small to be measured by SoLID which is expected to measure to precision, and indicates that can probe CSV or new physics without HT contamination. On the other hand, if a large -dependent effect is observed in the -term, it would be a clear indication of interesting long range quark-quark correlations that are not adequately described by current theoretical models.

### 4.2 Charge Symmetry Violation

Most early phenomenological work on PDF assumed charge symmetry, meaning that the () quark PDF in the proton was set equal to the quark PDF in the neutron

(42) |

CSV is expected to arise from quark mass differences and the effects of QED splitting functions [74, 75, 76] on the DGLAP evolution of PDF. Several non-perturbative models [77, 78, 79] of the nucleon have been studied to estimate the size of CSV effects. While no conclusive evidence of CSV has been observed, it is constrained from an analysis of high energy data [74]. The strongest limit on CSV arises from comparison of the structure functions between charged lepton DIS and neutrino charged current DIS on isoscalar nuclear targets. For a recent detailed review on CSV and experimental constraints, we refer the reader to Reference [80].

The CSV effects can be parameterized in terms of the quantities and ,

(43) |

Then, the CSV corrections to in Equations 34 and 35 take the form,

(44) |

A global fit to high energy data was performed by the MRST group [74] using the phenomenological form,

(45) |

that yielded a best fit value of and a confidence interval given by . In Figure 2, we show the CSV correction for the end points of this interval. For comparison we also show the MIT Bag model estimate [64] of the HT correction . This estimate for , similar in size to other estimates [56, 57, 72, 73], indicates that the -term of can probe percent level CSV effects, without much contamination from HT effects. The CSV effects on the total asymmetry, including both and , are also at the percent level [63] of the confidence interval, which should be contrasted with the expected precision of SOLID at the level.

Flavor-dependent effects can also arise for PDF in heavier nuclei through isospin-dependent nuclear forces affecting the - and quark distributions differently. Several works [81, 82, 83, 84, 85, 86, 87] have been devoted to understanding the underlying mechanisms responsible for differences in the cross sections per nucleon among various nuclei. Recently, nuclear-dependent effects were also studied [88] using a newly introduced [89, 88, 90, 91] DIS event shape called 1-jettiness [92]. Differences in the PDF in the valence region of , due to nuclear effects, are referred to as the EMC [93] effect. An unambiguous picture of the mechanisms responsible for the EMC effect is still lacking after more than two decades of investigation.

Recently, PVDIS on nuclei has been proposed [94] as a novel probe of the flavor dependence of the nuclear EMC effect, which appears dominantly as an isovector correction to the term in Equation 34,

(46) |

where is the parton momentum fraction in the nucleus multiplied by the atomic weight and . For isoscalar targets, ignoring heavy quark flavors, quark mass differences, and EW corrections, this correction vanishes since the and quark nuclear distributions and are identical. However, for other nuclear targets with , this isovector EMC correction could be large. Such a PVDIS analysis on an iron or lead target could provide insight into the impact of this EMC effect on the extraction of the weak mixing angle at NuTeV [93]. In general, combining from PVDIS analyses with data on EM DIS, may allow to extract flavor-dependent nuclear PDF.

### 4.3 Extracting the Ratio of Parton Distribution Functions

A measurement of on a proton target is sensitive [95] to the ratio of the to quark PDF. The standard determination of the ratio relies on fully inclusive DIS on a proton target compared to a deuteron target. In the large region, nuclear corrections in the deuteron target lead to large uncertainties in the ratio. Several methods [96, 97, 98, 99, 100, 101] to control these nuclear uncertainties have been investigated. However, they can be completely eliminated if the ratio is obtained from the proton target alone. For this reason, precision measurements of on a proton target can be a powerful probe of the ratio.

in Equation 36 for a proton target at leading twist takes the form,

(47) |

where the coefficients and depend on the ratio,