The EMC PVDIS Experiment

A Constraint on Isovector-Dependent Nuclear Modification EffectsUsing Parity-Violating Deep Inelastic Scattering

A Proposal to PAC 44

Seamus Riordan, Rakitha Beminiwattha, and John Arrington

June 6, 2016

A. Deshpande, N. Hirlinger Saylor, K. S. Kumar, T. Kutz, S. Riordan∗†, and Y.X. ZhaoStony Brook University

W. R. Armstrong, J. Arrington∗, I. C. Cloet, K. Hafidi, M. Hattawy, P. E. Reimer, B. P. Waidyawansa and Z. YeArgonne National Laboratory

R. Beminiwattha∗, R. Holmes, and P. SouderSyracuse University

S. BarkanovaAcadia University

K. AniolCalifornia State University, Los Angeles

H. Gao, X. Li, T. Liu, C. Peng, W. Xiong, X. Yan, and Z. ZhaoDuke University

P. Markowitz and M. SargsianFlorida International University

S. P. Wells and N. SimicevicLouisiana Tech University

A. AleksejevsGrenfell Campus of Memorial University

N. KalantariansHampton University

D. McNultyIdaho State University

V. Bellini, C. SuteraINFN - Sezione di Catania

J. Bericic, S. Sirca, and S. StajnerJozef Stefan Institute and University of Ljubljana, Slovenia

O. HenMassachusetts Institute of Technology

J. Dunne, D. Dutta and L. El FassiMississippi State University

P. M. King and J. RocheOhio University, Athens, Ohio

R. Gilman, K. E. MesickRutgers University

J. Benesch, A. Camsonne, J. P. Chen, S. Covrig, D. Gaskell, J.-O. Hansen, C. E. KeppelThomas Jefferson National Accelerator Facility

A. J. PuckettUniversity of Connecticut

P. BlundenUniversity of Manitoba

R. MiskimenUniversity of Massachusetts, Amherst

ii

X. Bai, D. Di, C. Gal, K Gnanvo, C. Gu, N. Liyanage, H. Nguyen, K. D. Paschke, V. Sulkosky, and X. ZhengUniversity of Virginia

F. R. WesselmannXavier University of Louisiana

A. W. ThomasUniversity of Adelaide, Australia

and the SoLID Collaboration

∗Spokesperson†Contact, riordan@jlab.org

Abstract

We propose to directly measure flavor-dependent nuclear medium modification effects on quarks usingparity-violating deep inelastic scattering on a 48Ca target. This measurement could provide evidence ofnuclear parton distribution function modification that is dependent on the proton or neutron excess of anucleus. Such an effect would represent new and important information on our understanding of nucleonmodification at the quark level, the EMC effect, which has been known for over 30 years but is still not fullyunderstood theoretically, with essentially no experimental constraint on the flavor dependence. In addition,such an effect has great importance in the extraction of nuclear parton distribution functions using a varietyof techniques and processes, such as using neutrino scattering, Drell-Yan processes, or flavor-dependentobservables at a future EIC.

Parity-violating deep inelastic scattering at Jefferson Lab kinematics measures an asymmetry betweenhelicity states of longitudinally polarized electrons scattered from an unpolarized target. This asymmetryarises from the interference between the virtual photon and Z0 exchange and is effectively the ratio of weakto electromagnetic interactions between the target and electrons. Within the quark-parton model it is directlysensitive to the ratios of quark flavors and isovector degrees of freedom. Such a measurement is also cleanlyinterpretable with minimal model dependence and offers the best direct access with available experimentaltechniques. With 60 days of 11 GeV beam at 80µA using the PVDIS SoLID configuration with a 48Catarget we will obtain 0.7-1.3% statistical precision on the parity violating asymmetry APV over a range of0.2 < x < 0.7 with about a 0.7% systematic error. This precision would test a prominent model of mediummodification to better than the 5σ level.

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Contents

List of Figures iii

List of Tables iv

1 Introduction 21.1 Deep Inelastic Scattering, Nuclear Modification, and the EMC Effect . . . . . . . . . . . . . 31.2 Nuclear Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Possible Signals of Flavor Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 The NuTeV Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 PDF Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Neutron PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Short Range Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Predictions for Light Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Measurement with PVDIS 132.1 Size of the Isovector EMC Effect and Necessary Precision . . . . . . . . . . . . . . . . . . 13

2.1.1 Nambu-Jona-Lasinio Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Nuclear Parton Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Short-Range Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Choice of Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Relation to Other Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Experimental Design 193.1 Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 SoLID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 GEMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.4 Light Gas Cherenkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.5 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Tracking, Optics, and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Radiation Dose in the Hall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Projections, Uncertainties, and Beam Time Request 344.1 Statistical Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Pion Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Radiative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.3 Hadronic and Nuclear Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.4 Uncertainties from Free Parton Distributions . . . . . . . . . . . . . . . . . . . . . 374.2.5 Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Beam Time Request . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A Quark Parton Model 40

ii

References 41

iii

List of Figures

1 A comparison between the two measurements for this proposal (left) and E12-10-008 (right). 22 Data demonstrating the EMC effect over a range of A and x from E139 at SLAC [4]. . . . . 63 Dependence on A at x = 0.22 and x = 0.6 from [4]. . . . . . . . . . . . . . . . . . . . . . 64 Constraining world data on the running of sin2 θW including the published NuTeV result [19]. 85 Isovector parton distribution modification in Fe from Cloet et al. [2]. . . . . . . . . . . . . 96 Differences in nuclear correction factors R = FA2 /F

N2 for charged current neutrinos (left)

and neutral current leptons and Drell-Yan (right) at Q2 = 5 GeV2 from [23]. . . . . . . . . 107 A fitted EMC slope parameter against the SRC data “plateau” parameter a2 shows a close

relationship [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Isospin dependence of the EMC effect vs. fractional neutron excess of the nucleus for the

four scaling models. Fraction of momentum distribution above 300 MeV/c (black open cir-cles), average kinetic energy (red triangles), average density (green circles), and probabilityto be within 1 fm of another nucleon (blue diamonds). The lines are simple unweightedlinear fits. The short-dashed line shows the N/Z value corresponding to 48Ca. [32] . . . . . . 12

9 a1 predictions from Cloet [33] for 48Ca (note the reference uses the nomenclature a2 for theequivalent quantity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

10 a1 predictions from the nCTEQ data set assuming various weighting between pure DIS andDrell-Yan (w = 0) and pure neutrino data w =∞ from Refs. [23] and [24] for 48Ca. Inclu-sion of any fraction of neutrino data dramatically shifts the fit demonstrating the weaknessof DIS data to this observable. The change in slope is about 5% over our x range and isconsistant with the CBT calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

11 Projections of the isoscalar EMC F2 ratios for 40Ca and 48Ca from the SLAC E139 pa-rameterization [4], a prediction from the CBT model [1], and data point projections forE12-10-008 [37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

12 Ratios for isoscalar-corrected F2 for 48Ca and 40Ca using SLAC E139 [4] and the CBTmodel [1] with data point projections for E12-10-008 [37]. . . . . . . . . . . . . . . . . . . 17

13 Existing pionic Drell-Yan data for ratios of heavy target to deuterium data compared to theCBT model [38]. A slight preference is shown for the CBT model, but is not conclusive. . . 17

14 Our statistical precision for APV for x and Q2 bins in %. . . . . . . . . . . . . . . . . . . . 1915 Our measurement projections in the context of the CBT model. . . . . . . . . . . . . . . . 1916 SoLID side-view of the Geant4 configuration with full detector setup. . . . . . . . . . . . . 2017 A CAD design of the existing 48Ca target for E08-014. Two identical 40Ca and 48Ca targets

were mounted in the ladder design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118 Drawings of the existing 48Ca target for E08-014. . . . . . . . . . . . . . . . . . . . . . . . 2219 A schematic a conceptual frame for the target. About a 1.5 mm lip is allowed on the outer

radius downstream to allow for 35 acceptance when the raster is included. . . . . . . . . . 2220 Projection of the baffle lead “spokes” to block low momentum particles. . . . . . . . . . . . 2321 Electron and π− acceptances from the baffles. Differences between these are due to varying

angular distributions and the fact that π− have longer interaction lengths. . . . . . . . . . . 2422 CAD drawing of a GEM plane for the PVDIS configuration. . . . . . . . . . . . . . . . . . 2423 Geant4 calculation results for photon interaction probabilities with GEM chambers from

Ref. [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424 The design of the baffle structure minimizes the EM background rates at the GEM detectors.

The solid lines shows background rates with no baffles and the dashed lines show the rateswith the baffles. The baffle structure reduce the background rates by almost a factor of 10. . 26

25 Cross section of an electromagnetic calorimeter module and absorber sheets. . . . . . . . . . 27

iv

26 π− rejection and electron efficiency for calorimeter. Red points and curves are for calorime-ter in PVDIS configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

27 Electromagnetic calorimeter trigger performance for the low rate azimuthal region for e−

(left) and π− (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2828 Electromagnetic calorimeter trigger performance for high rate azimuthal region for e− (left)

and π− (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2929 Geant4 cross section of the light gas Cherenkov detector for the SIDIS (left) and PVDIS

(right) configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3130 A comparison of π− to e− for LD2 and 48Ca targets. The ratio for 48Ca is about 50% larger. 3231 Simulation results for collected photoelectrons for the PVDIS LD2 experiment for the mid-

dle of the target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3232 Simulation results for the pion rejection factor and electron detection efficiency for a “nomi-

nal” case where π− rejection is maximized and and minimizing the loss of electrons as wellas the ∼ 10% and ∼ 20% e− loss cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

33 Layout of the FADC crate for the shower and preshower systems. Each crate contains acrate trigger processor (CTP), signal distribution module (SD), and trigger interface (TI). . . 34

34 a1 predictions from the CJ12 PDF fit [54] assuming no modification for 40Ca, 48Ca withdifferent nuclear correction sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

35 Anticipated data for measurements on d/u, see text for references. The constraints providedby these data will allow for accurate tests of an isovector EMC effect at larger x. . . . . . . . 38

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List of Tables

1 Summary of uncertainties and possible size of the effect in a common model. “x-dependence”is given as the variation in R or a1 between x=0.2 and x=0.7 (DIS region) from Cloet, etal. [1, 2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 The standard model axial, vector, and effective couplings to the quarks and electron. . . . . . 53 GEM design parameters for the SoLID PVDIS configuration. . . . . . . . . . . . . . . . . . 254 The low energy EM background radiation at GEM detectors compared for 48Ca and LD2

targets with and without baffles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Breakdown of rates based on the particle types for 48Ca target at 80 µA. . . . . . . . . . . . 256 Breakdown of DIS rates for 48Ca target at 80 µA. . . . . . . . . . . . . . . . . . . . . . . . 267 Calorimeter trigger rates based on 48Ca target. DIS and background rates that enter full

coverage of the EC are considered for the resulting trigger rates. Trigger is broken down top < 1 GeV and p > 1 GeV particles and for the “low” and the “high” background regions.The total rate for the sum of 30 sectors are shown here. The simulated pion rejection andelectron efficiency values are shown in Figs. 27 and 28. . . . . . . . . . . . . . . . . . . . . 29

8 Cherenkov trigger rates for 48Ca target at 80 µA is estimated using simulated pion rejectionand electron efficiency values from Fig. 32. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Detector channel counts for each sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010 Breakdown of coincidence trigger rates (Cherenkov+EM calorimeter) for momentuma> 1 GeV

from 48Ca target at 80 µA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011 Breakdown of radiation power seen by a cylindrical detector in the hall compared for 48Ca

and LD2 targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512 Breakdown of relative radiation power and dose increase in 48Ca with respect to LD2 seen

by a cylindrical detector in the hall. Notice same relative increase for dose since both LD2

and 48Ca dosages are estimated for 60 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 3513 Summary of the systematic error contributions to our measurement. . . . . . . . . . . . . . 3614 Beam time request for this experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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Updates from PAC42 submission

The overall goals and general methods of this proposal are much the same as the previous version whichwas submitted to PAC42. The previous proposal was deferred, with the PAC noting that an effect matchingthe prediction of the Cloet, Bentz, and Thomas (CBT) model [1, 2] might be observable by E12-10-008,which includes a measurement of the cross section ratios of 48Ca to 40Ca. We summarize below the newinformation included in this proposal, mainly in Secs. 1 and 2. In addition, there are updated projectionsfor the experiment, background and radiation estimates, based on feedback from PAC 42 and the continuingdevelopment of the SoLID apparatus.

While the previous proposal focused on the CBT model to estimate how large the flavor dependenceof the EMC effect in non-isoscalar nuclei might be, this update includes discussion of a variety of effectswhich would be expected to yield a flavor-dependent EMC effect. We argue that this flavor dependence isa critical observable needed to understand nuclear modification of PDFs, and that the EMC effect cannotbe fully understood, or even quantified, without this information. Such an understanding, or at least aquantification of the effect, is required to have a reliable description of nuclear PDFs for studies involvingthe weak interaction (e.g. the NuTeV anomaly), and for nuclei with N/Z ratios that are away from thetypical nuclei measured in the EMC effect (e.g. in the comparison of DIS from 3H and 3He). Further,examination of models using ab initio calculations of nuclear structure in light nuclei suggest that the flavordependence is also an important observable in understanding the microscopic origin of the EMC effect.

Therefore, while the 48Ca/40Ca cross section ratio has sensitivity to the presence of large flavor-dependenteffects (< 3σ ability to differentiate between no flavor dependence and the CBT prediction), this is notsufficient to quantify the effect. The quark flavors are only observed through the electromagnetic charge-weighted combination and which leave open degrees of freedom. Further, the approved cross section mea-surement does not have the sensitivity to exclude significant flavor dependence. As such, we argue that themeasurement utilizing parity-violating electron scattering will be important no matter what is observed inthe 48Ca/40Ca ratios. A direct comparison between the two is given in Table 1 and Fig. 1.

PVEMC EMCStatistics 0.7-1.3% 0.8-1.1%Systematics 0.5% 0.7%Normalization 0.4% 1.4%CBT x-dependence 5% 3%CBT sensitivity 5.6σ < 3σ

Table 1: Summary of uncertainties and possible size of the effect in a common model. “x-dependence” isgiven as the variation in R or a1 between x=0.2 and x=0.7 (DIS region) from Cloet, et al. [1, 2].

2

Aµ=12%, 60 days, 800Ca x/X48 from CBT, 1a

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1a

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1 CBT1a

naive1a

Our Projections (stat, stat+ pt to pt sys)

Shared sys. uncert

Aµ=12%, 60 days, 800Ca x/X48 from CBT, 1a

bjx0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ca

402,

/F

Ca

482,

is

Nor

m. F

0.85

0.9

0.95

1

1.05

1.1

1.15

E139 Param

CBT

Ca Ratio (stat + pt to pt sys)40Ca/ 48

Shared systematic

Ca Ratios40Ca/ 48

Figure 1: A comparison between the two measurements for this proposal (left) and E12-10-008 (right).

1 Introduction

Within QCD our descriptions of protons and neutrons are with the underlying quark and gluon degrees offreedom. In addition, protons and neutrons also arrange themselves into the more complex objects of nucleiand this transition between QCD and nuclear physics is still out of reach for modern theory. The effectivetheories we have for the description of inter-nucleon interactions have been widely successful in producingdetailed descriptions of systems such as nuclear structure and scattering processes. However, they are basedaround the concept that nucleons in the nuclear environment strongly maintain their identities and have few,if any, provisions for how they change.

An open and important question for hadronic physics today is how protons and neutrons are modifiedwhen they are bound in a nucleus and how one makes the transition between traditional nuclear physics toQCD. This is enormously important not only as a theoretical question, but as a practical one, as measure-ments often rely on data from bound neutrons to produce “effectively free” neutron data to complement ouraccessible free protons. For example, measurements for elastic neutron electromagnetic form factors andform factor flavor separation, and deep inelastic scattering flavor separation all rely on some level of nu-clear modeling. Even highly exotic systems such neutron stars, where protons and neutrons are compressedto volumes several times smaller than their charge radii suggest, frequently use models that maintain theirconstituent’s identities.

At what level all of these assumptions hold have broad reaching effects on complementary processesthat rely on this extracted flavor data. Neutrino deep inelastic scattering is highly quark flavor dependentas it probes different couplings than the standard electromagnetic probes in the neutral current process,or changes their flavor in the charged current processes. Parity-violating elastic and neutrino quasi-elasticscattering require reliable nucleon form factor data for both protons and neutrons. Drell-Yan processes andelectron scattering at a future EIC rely on knowledge of the quark and antiquark flavor distributions.

While we have evidence that modification occurs, the exact mechanism for it is not well understood. Itis clear from simple models that the Fermi motion of bound nucleons is insufficient to describe observedeffects in deep inelastic scattering or nucleon quasielastic scattering. The concept of “nuclear shadowing,”where a reduction in scattering cross section is observed relative to the naive sum is also important but stilldoesn’t offer a complete picture. The “EMC effect” was some of the first data observed that there is asignificant change on the quark level and that the identities of the bound nucleons are different. It showed adepletion of quarks in the valence region and despite sophisticated modeling, cannot be described by simplebinding effects. Despite decades of theoretical efforts, a rigorous explanation has been elusive.

3

One of the primary goals of Jefferson Lab is to study nuclear modification and to study how one can re-late the basic QCD degrees of freedom, quarks and gluons, to the objects that nature most readily presents tous, nucleons and nuclei, through the use of the well-understood electron probes. With the 12 GeV upgrade,we will have unprecedented access to the valence quark kinematic region allowing for new constraints onmodification. By far, most of the data available on parton distributions is through electromagnetic scatter-ing, which is heavily weighted to the the u-quark distributions and is only sensitive to one particular linearcombination of quarks. Parity-violating deep inelastic scattering with leptonic probes provides a power-ful method to access flavor ratios of quark distributions that have not been as well explored and offersopportunities to study difficult-to-obtain flavor dependent effects within nuclear modification.

The scattering cross section for weak neutral currents is dependent on both the amplitudes for the ex-changed virtual photon and neutral Z boson, which interfere

σ ∝ |Aγ +AZ |2 . (1)

For Q2 MZ , the dominant term for the scattering rates is |Aγ |2 and for the parity-violating component,the interference term |A∗γAZ |. One can then form a parity-violating quantity which is the ratio of these twoterms, and is measured by the differences between left and right-handed polarized lepton cross sections

APV =σR − σLσR + σL

∼|A∗γAZ ||Aγ |2

. (2)

This asymmetry provides a particularly sensitive method to obtain flavor-dependent effects in nuclearmodification as it is a ratio of the weak-to-electromagnetic interactions, which gives access to ratios of quarkdistributions. Systematic contributions from effects such as quark-parton evolution, higher twist, and photonradiation are also suppressed compared to a cross section measurement. It also has a diminished need foradditional measurements involving super-ratios to flavor symmetric (isoscalar) systems if charge symmetryviolating effects are subdominant.

1.1 Deep Inelastic Scattering, Nuclear Modification, and the EMC Effect

Deep inelastic scattering has provided one of the most important tools in understanding modern hadronicstructure. From studying this scattering process we have some of our best evidence for the concepts ofquarks as strongly interacting, point-like spin-1/2 objects, the running of the strong coupling constant αs,and the validity of perturbative QCD, and confinement. It has been used for decades as a tool to mapnucleon structure through parton distribution functions (PDFs) for which we have no predictions from firstprinciples. The universality of these parton distribution functions is absolutely critical in our modern studiesof deep inelastic neutrino scattering and of high-energy physics at facilities like the RHIC and the LargeHadron Collider.

At sufficiently large momentum and energy transfer from an electromagnetic probe to a hadronic target,a transition takes place where the underlying QCD degrees of freedom are exposed and the target appears asan incoherent sum of weakly interacting partons which we identify as quarks. The differential cross sectionfor the electromagnetic scattering interaction can be written in the lab frame as

d2σ

dΩdE′=

4αE′2

Q4cos2

θ

2

(F2(x,Q

2)

ν+

2F1(x,Q2)

Mtan2 θ

2

)(3)

where α is the fine structure constant, Q2 is the negative of the four-momentum transfer, E and E′ are theinitial and final probe energies, ν = E − E′, M is the nucleon mass, and F1 and F2 are the quark-partonstructure functions with x, the Bjorken scaling variable,

x =Q2

2Mν, (4)

4

F2 is expressed in terms of the quark and anti-quark parton distribution functions

F2(x,Q2) = x

∑q

e2q(q(x,Q2) + q(x,Q2)

), (5)

where eq are the quark charges. This scaling variable has the interpretation of the fraction of momentumcarried by that quark when the nucleon is boosted to the speed of light. The parton distribution functionsq(x,Q2) carry the soft, non-perturbative nucleon structure and represent the probability that the quark carriesthat fraction of momentum x. The Q2 dependence is predominantly logarithmic which is successfullypredicted within the framework of perturbative QCD where it is order-by-order identified with phenomenasuch as soft gluon emission. Related to F2 is F1 through the longitudinal structure function FL = F2−2xF1.Equating FL = 0 is the Callan-Gross relation and represents treating the partons as free, point-like spin-1/2objects.

One challenge in this framework is accessing the quark flavors since the interaction is only sensitive tothe charge-squared-weighted sum of the parton distributions, and therefore most heavily to the u-quarks.Exploiting charge symmetry between protons and neutrons, the idea that the u and d-quark distributionsare symmetric between the two, and suppressing the sea quark contributions through studies at high x, adeconvolution can be performed. However, since there are no sufficiently high luminosity free neutrontargets, neutrons are typically studied bound in a nucleus such as deuterium. How representative such atarget is to the free neutron beyond simple binding effects is an open question, but deuterium measurementsremain a standard in extracting neutron physics.

A method of accessing quark flavor information without having to consider such binding effects isthrough parity-violating processes which measure weak force couplings to the quarks. For now we assumethe Callan-Gross relation and Q2 M2x2; the full framework is presented in Appendix A. In the quark-parton model, the left-right polarized lepton scattering asymmetry given in Eq. 2 can be expressed in termsof the parton distribution functions by

σR − σLσR + σL

= APV = − GFQ2

4√

2πα[Y1a1(x) + Y3(y)a3(x)] , (6)

where GF is the Fermi constant and

Y1 ≈ 1;Y3(y) ≈ 1− (1− y)2

1 + (1− y)2(7)

with

a1(x) = geAF γZ1F γ1

= 2

∑iC1iei(qi + qi)∑i e

2i (qi + qi)

(8)

a3(x) = geVF γZ32F γ1

= 2

∑iC2iei(qi − qi)∑i e

2i (qi + qi)

. (9)

with y = ν/E, geA and geV the axial and vector couplings to the electron respectively, and C1i and C2i

the effective quark couplings dependent on the weak-mixing angle sin2 θW , given in Table 2. In practicebecause the C2i couplings are suppressed by an order of magnitude to the C1i the a1 term dominates theasymmetry.

The power of this method is elucidated when one considers nuclear quark distributions for the lightflavors uA and dA and expand a1 about the isoscalar uA = dA limit and at high enough x where the seaquarks do not contribute significantly

a1 '9

5− 4 sin2 θW −

12

25

u+A − d+A

u+A + d+A(10)

5

geA = 1guA = 1

2gdA = −1

2geV = −1 + 4 sin2 θW ≈ -0.0478guV = −1

2 + 43 sin2 θW ≈ -0.19

gdV = 12 −

23 sin2 θW ≈ 0.34

C1u = geAguV = −1

2 + 43 sin2 θW ≈ -0.19

C1d = geAgdV = 1

2 −23 sin2 θW ≈ 0.34

C2u = geV guA = −1

2 + 2 sin2 θW ≈ -0.030C2d = geV g

dA = 1

2 − 2 sin2 θW ≈ 0.025

Table 2: The standard model axial, vector, and effective couplings to the quarks and electron.

with the convention that q± = q(x) ± q(x). Parity-violating deep inelastic asymmetry measurements aretherefore directly sensitive to differences in the quark flavors. In turn, for isoscalar targets (and neglecting seaquarks) a1 roughly becomes a constant and the measurement becomes a test for charge symmetry violation.

1.2 Nuclear Modification

The EMC effect, first reported by the European Muon Collaboration collaboration [3] over 30 years ago,provided the first direct evidence that the quark distributions in nucleons bound in a nucleus are significantlydifferent from those of free nucleons. This was demonstrated by observing a difference in the ratios of deep-inelastic muon scattering cross sections between a heavy nucleus (in this case iron) and deuterium. Theyshowed that this ratio deviated from a simple constant expectation across a range of x.

When compared to more advanced models, it was clear that Fermi motion alone is insufficient to explainthe behavior. Extensive studies have been performed mapping out this effect over a broad range of nucleisince its discovery and there is a large amount of precise data on the structure functions for many nucleiranging from helium to gold, e.g. Ref [4] shown in Fig. 2, and Ref. [5]. To compare between nuclei whereN 6= Z, a model-dependent “isoscalar” correction is made to the interaction cross section to account for anexcess of neutrons.

In addition, there has also been significant theoretical efforts but despite decades of work there are stillno satisfactory models that fully encompass the mechanism from first principles. We refer the reader toreviews Refs. [6, 7, 8].

The data generally show the following features in the ratios to deuterium,

• a slowly increasing effect growing with A in the ratio

2σAAσD

(11)

after the cross section is corrected for neutron excess Fig. 3,

• a suppression in the range 0.3 < x < 0.8 with a minimum around x = 0.65 dubbed the “EMC effect”,

• an enhancement attributed to Fermi motion for x > 0.9,

• a strong suppression for x < 0.1 from nuclear “shadowing”,

• a small enhancement from 0.1 < x < 0.3 or “anti-shadowing”.

6

Figure 2: Data demonstrating the EMC effect over a range of A and x from E139 at SLAC [4].

Figure 3: Dependence on A at x = 0.22 and x = 0.6 from [4].

7

Fermi motion gives a strong enhancement to the high x region due to the relative availability of effec-tively higher momentum quarks from the moving nucleons and their rarity in free nucleons. The very smallx range x < 0.1 is typically explained by nuclear shadowing where nucleons overlap or “shadow” eachother, leading to a smaller total cross section (a detailed review of this can be found in [9]).

The EMC effect region is less understood and a variety of models attempt an explanation. For example,the presence of multi-quark clusters, e.g. [10, 11, 12], or, through a similar concept, short-range nucleon cor-relations [13] cause a depletion in the mid-x range. Another model suggests that if the scale of confinementchanges due to medium modification, i.e. the radius of the bound nucleus becomes larger, it effectivelychanges the QCD evolution scale parameter [14, 15]. Qualitatively, it becomes easier to have soft gluonemission or generate quark-anti-quark pairs (which carry color) from the struck quark as the volume it canproduce these in is larger.

It is typically assumed that the nuclear modification of the quark distribution is flavor-independent,yielding an identical modification of the up- and down-quark distributions. There is no reason to expectthis to be the case, and in fact many of the calculations attempting to understand the EMC effect includeeffects which would be expected to have at least some flavor dependence. Even simple Fermi motion in anisoscalar nucleus will yield slightly different modification of the up- and down-quark PDFs, as the down-quark PDF falls more rapidly at large x, increasing the impact of Fermi smearing on down quarks. Moresignificantly, binding, Fermi motion, and separation energy can be different for protons and neutrons innon-isoscalar nuclei, yielding differences in the modification of up- and down-quark distributions. Modelsincluding more exotic effects can also generate a flavor-dependent EMC effect, for example the contributionof something like a 6-quark bags or hidden color states could yield a flavor-dependent EMC effect if thesestates are different for pp, np, and nn configurations. Studies of short-range correlations note a dramaticdifference between the prevalence of n−p and p−p pairs [16], and the presence of SRCs shows an empiricalcorrelation with the size of the EMC effect in nuclei [13, 17]).

Despite the fact that there are many ways to generate flavor dependent nuclear effects, this possibilityis generally not discussed, as the existing data have very little sensitivity to any possible flavor dependence.As noted above, the data for heavier nuclei show a slow increase in the nuclear modification as A increases.While heavier nuclei have N > Z, and are in principle sensitive to both the overall A dependence andany flavor-dependent effects, there is a strong correlation between A and N/Z for heavier nuclei, makingit difficult to disentangle A-dependent effects from any flavor dependence. In light nuclei, the sensitivity todetailed nuclear structure observed in recent measurements [5] is not yet understood, making it even moredifficult to separate A-dependent and flavor-dependent effects.

While a flavor-dependent effect is challenging to study in conventional measurements of the EMC effect,the different flavor dependence in parity-violating electron scattering provides an ideal probe of such effects.The parity-violating asymmetry in deep inelastic scattering provides direct access to flavor dependence inthe EMC effect, cleanly separated from flavor-independent nuclear modification of the PDFs.

Historically, explanations of the EMC effect have neglected the potential impact of flavor-dependenteffects, and many have neglected the impact of the nuclear structure, focusing on effects which have simplescaling behavior with nuclear mass or density. Recent measurements of the EMC effect in light nucleihave made it clear that a complete understanding of nuclear PDFs requires a detailed understanding ofthe connection between the nuclear effects and details of the nuclear structure. It is also clear that ourunderstanding of the EMC effect will not be complete without an understanding of the flavor-dependenceof the nuclear modification to PDFs. The Particle Data Group states in their 2013 review of the electroweakmodel that “it would be important to verify and quantify this kind of effect experimentally, e.g., in polarizedelectron scattering.” [18]

8

Figure 4: Constraining world data on the running of sin2 θW including the published NuTeV result [19].

1.3 Possible Signals of Flavor Dependence

Current models of nuclear PDFs do not include flavor-dependent effects, and so any true flavor dependenceis absorbed into the parameterization of the A dependence. This can have an important impact on anymeasurement where the flavor sensitivity differs from direct measurements of the nuclear PDFs. Any mea-surement involving weak coupling to the nuclear quark distributions will have a different contribution fromup and down quarks, and a flavor-independent modification of the nucleon PDFs will not yield the correctresult. Similarly, nuclei with unusual N/Z ratios will not be well represented by conventional parameter-izations of the EMC effect. For most nuclei, the A-dependent parameterizations of the EMC effect willbe fairly reliable, as nuclei in a given mass region tend to have a relatively narrow range on N/Z values.However, a difference between the EMC effect for up-quark and down-quark distributions could change thenuclear effects in 3H or 48Ca from those observed in 3He or 40Ca.

1.3.1 The NuTeV Experiment

The NuTeV experiment [19] at Fermilab was designed as a measurement of the electroweak mixing anglesin2 θW through neutrino deep inelastic scattering measuring together charged and neutral current neutrinoand anti-neutrino scattering. With those cross sections, one can measure the weak mixing angle using thePachos-Wolfenstein relation [20],

RPW ≡σ(νµN → νµX)− σ(νµN → νµX)

σ(νµN → µ−X)− σ(νµN → µ+X). (12)

which reduces to 12−sin2 θW in the case of an isoscalar target and the absence of charge symmetry violation.

This quantity is particularly attractive to study as a large number of systematic uncertainties cancel, includinga great deal of nuclear structure. However, important corrections must be made in the case where there is anexcess of neutrons which is the case for heavy nuclei typically used as targets in neutrino experiments.

For NuTeV, high purity ν and ν beams from the decay of charged pions or kaons were produced bythe Tevatron and the neutrino interactions were detected 1.5 km downstream in a large detector array. Thisarray consisted of steel-scintillator target calorimeter followed by an iron-toroid spectrometer. The pub-lished result on sin2 θW was approximately 3σ from the standard model prediction, Fig. 4 and has caused asignificant amount of discussion in the community regarding the discrepancy [21].

A large class of possible explanations has been generated involving unconsidered nuclear effects in theNuTeV analysis. In particular we focus on an isovector EMC effect by Cloet et al. [1, 2], which we discuss

9

Q2 = 5GeV2

Z/N = 26/30 (iron)

0.6

0.7

0.8

0.9

1

1.1

EMC

ratios

0 0.2 0.4 0.6 0.8 1x

RγZFe

RγFe

dA/df

uA/uf

Figure 5: Isovector parton distribution modification in Fe from Cloet et al. [2].

in Sec. 2.1.1, that could account for the remaining discrepancy. In this model, a flavor dependence wherethe nuclear u and d-quarks within the iron nucleus are modified irrespective of the nucleon they are boundin by the mean isovector field from the surrounding nucleus, Fig. 5.

Various other models include higher order QCD evolution, a strange sea asymmetry, charge symmetryviolation, or nuclear shadowing. There is significant literature showing many potential effects that impactthe NuTeV anomaly. Without quantification of the flavor dependence of the EMC effect, which must be thereat some level, we do not have a quantitative understanding of what additional effects would be required toresolve the NuTeV anomaly. A recent calculation [1, 2] suggests that 2/3 of the effect is related to flavordependence, essentially removing the anomaly, and limiting the size of additional corrections.

1.3.2 PDF Fits

An analysis was done by Schienbein et al. [22, 23] noted a striking difference between the nuclear correctionfactors FA2 /F

D2 by fitting charged lepton with Drell-Yan data as well as charged current neutrino scattering,

Fig. 6. In this method, comparisons between “free” nucleon PDFs to the nuclear PDFs are made. LaterKovarik et al. [24] performed a global analysis using partitions of the neutrino-nucleus DIS, charged lepton-nucleus DIS, and Drell-Yan data to test the compatibility between these data sets.

In the Kovarik “nCTEQ” analysis, a goodness of fit test is used while varying the contribution weightsbetween the lepton with Drell-Yan data and the neutrino data. In those two partitions, they find no compro-mise fit that has acceptable χ2/NDoF simultaneously for both sets at the 90% confidence level. Further-more, individual data sets from the NuTeV neutrino iron results and from lepton-Fe exceed the 99% limit inthe compromise fits.

This result contrasts one by de Florian et al. [25] which was a global fit where the nuclear effects areparameterized and included in the fit. In that analysis they claim compatible fits within their errors betweenall data and cites possible differences in the overall deuterium normalization (Kovarik calculates the deu-terium from free PDFs, rather than use the sparse neutrino data) or possibly “disregarded uncertainties” and“theoretical ambiguities”. An analysis by Paukkunen et al. [26] notes that only the NuTeV neutrino data anddata from CHORUS (lead), CDSHW (iron), and the NuTeV antineutrino data show no controversy. Laterthey argue [27] that unnoticed fluctuations in the overall normalization to the NuTeV data are sufficientlylarge to cause tension.

A review of these issues is given in Ref. [9], Section 8. There they suggest that when the full uncertaintiesfor all the measurements are taken in quadrature, as in the de Florian fit, the discriminating power for tension

10

Figure 6: Differences in nuclear correction factors R = FA2 /FN2 for charged current neutrinos (left) and

neutral current leptons and Drell-Yan (right) at Q2 = 5 GeV2 from [23].

may be reduced. The amount of controversy and uncertainty demands measurements such as this proposalwhich could unambiguously address the situation.

1.4 Neutron PDFs

The neutron structure function is often extracted from measurements on the deuteron, with significant cor-rections to account for the proton contribution, especially at large x values. The neutron can also be extractedfrom comparisons of 3H and 3He structure functions. In this case, only the difference of the nuclear cor-rections for these two nuclei enters into the extraction of F2n, and this is directly sensitive to the differencein the nuclear effects for the proton and neutron. Even with the assumption that the proton distributionsin 3He are identical to the neutron distributions in 3H, there is a difference in the relative nuclear effectswhen including only conventional smearing corrections [28]. While this portion of the isospin-dependence(and its estimated uncertainty) is accounted for in calculations aimed at extracting F2n/F2p from the ratio of3He to 3H, additional flavor dependence associated with nuclear effects beyond simple binding and Fermimotion could yield an additional correction.

1.4.1 Short Range Correlations

Another important connection is the relationship between short-range correlations between nucleons in thenucleus and the slope of the EMC effect. This connection represents a very important crossroad in the studyof bound-nucleonic degrees of freedom and QCD degrees of freedom. It’s been shown that the cross section“plateau” observed in DIS at x > 1 that is generated by very high momentum nucleons is empiricallyrelated the slope of the EMC effect [13, 29], Fig. 7. In addition, short-range correlations measured throughcoincident detection of high momentum pairs note a dramatic difference between counting n− p and p− ppairs, the former being about 20 times more likely [16].

If there is a true relation between these these two phenomena, then the isovector nature preferring n− ppairs over p − p pairs has consequences for the EMC slope and introduces flavor dependence naturally.Figure 7 shows a strong correlation between the SRC data and the EMC effect slope, but also shows non-linear deviations attributed to the fact that heavy nuclei tend to have a neutron excess.

11

Figure 7: A fitted EMC slope parameter against the SRC data “plateau” parameter a2 shows a close rela-tionship [29].

1.4.2 Predictions for Light Nuclei

The observed correlation between the EMC effect and the contribution of SRCs has generated much interestand several ideas as to what underlying physics might connect these phenomena [30]. Unfortunately, we donot have calculations that provide quantitative predictions for the EMC effect or its flavor dependence. Wecan, however, take various ideas that have been proposed to explain the correlation and use this to predict thescaling of the EMC in various nuclei. Older parameterizations assumed that the EMC effect depended on thedensity, so density distributions for protons and neutrons in various nuclei can be used to make predictionsof the relative size of the EMC effect in these nuclei, as well as predicting the flavor dependence based onthe difference of the densities observed by protons and neutrons in the nucleus.

More recent proposals include the idea that it is the local environment of the struck nucleon drives theEMC effect, and so the EMC effect would scale based on the local density for protons and neutrons in thenucleus. It has also be proposed that the EMC effect might be associated with high-virtuality nucleons,which would suggest that the EMC effect might scale with the fraction of the nucleon distribution at veryhigh momenta, the average virtuality of the nucleons, or their average kinetic energy.

Several of these pictures were evaluated in light nuclei, up to A = 12, using the one- and two-bodymomentum and density distributions from ab initio Quantum Monte Carlo Calculations [31]. The predictedA dependence of the EMC effect in light nuclei showed a clear difference between the assumption of scalingwith average density and the other pictures [32]. However, the predictions based on the idea of scaling withlocal density or high virtuality show little difference, which is not surprising since a nucleus with a largeprobability of having 2 nucleons very close together will also tend to have very high-momentum nucleonscaused by their strong, short-range interactions.

However, the short-range interactions have a significant isospin dependence [16] and the predictions forthe flavor-dependence of the EMC effect are more sensitive to the assumed origin of the EMC effect. So inaddition to providing a direct measurement of the flavor dependence, needed to characterize the EMC effect,these data provide greater sensitivity to the underlying physics. While the same one- and two-body distri-butions used in this analysis are not yet available for 48Ca, significant progress is being made in calculatingnuclear structure in this mass range.

12

Figure 8: Isospin dependence of the EMC effect vs. fractional neutron excess of the nucleus for the fourscaling models. Fraction of momentum distribution above 300 MeV/c (black open circles), average kineticenergy (red triangles), average density (green circles), and probability to be within 1 fm of another nucleon(blue diamonds). The lines are simple unweighted linear fits. The short-dashed line shows the N/Z valuecorresponding to 48Ca. [32]

1.5 Summary

The EMC effect is one of the best cases we have for the medium modification of bound nucleons and hasbeen known for over 30 years, but the mechanism is still not well understood. Experimentally, there isconsiderable room for additional investigation, in particular in the realms of asymmetric nuclei and smallflavor differences in the modified quark distributions. There are several hints that the underlying assump-tion that quark modification is entirely isoscalar in nature may be inconsistent in measurements that wouldbe sensitive to such effects. Parity-violating measurements offer an excellent window in exploring theseassumptions.

13

Ca48 from Cloet-Bentz-Thomas for 1a

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1a

0.8

0.85

0.9

0.95

1

1.05

CBT1anaive1a

Wθ29/5 - 4 sin

Ca48 from Cloet-Bentz-Thomas for 1a

Figure 9: a1 predictions from Cloet [33] for 48Ca (note the reference uses the nomenclature a2 for theequivalent quantity).

2 Measurement with PVDIS

2.1 Size of the Isovector EMC Effect and Necessary Precision

Quantitative predictions for the possible size of this effect are sparse, so we must rely on the few calculationsavailable. However, the authors would like to stress that this proposal is to provide data in a sector wherethe data available must be improved rather than test specific models. Such models however, provide usefulguidance in the required precision for a significant experiment.

2.1.1 Nambu-Jona-Lasinio Model

It was proposed by Cloet et al. [1] that one possible resolution to the NuTeV anomaly was through the exis-tence of an isovector EMC effect. Calculations, which we will call the Cloet-Bentz-Thomas (CBT) model,were carried out in the Nambu-Jona-Lasinio Model, which is a chiral effective theory treating the quark in-teractions as four-point contact interactions and contains important QCD concepts, such as confinement. Toproduce a nucleon model, the Faddeev equations are solved for a quark-diquark configuration. An isovectormean field is introduced and the free parameters are constrained to reproduce nucleon and nuclear propertiessuch as nucleon masses and the empirical symmetry energy from the Bethe-Weizsaker formula. With theseingredients, quark distributions can be obtained for symmetric and antisymmetric matter.

This type of model has been very successful in reproducing the quark distributions for the EMC effectand the measured structure functions. In Ref. [1], the impact of the flavor-dependent nuclear PDF modifica-tion on the NuTeV anomaly was evaluated in the CBT model. The calculation was able to explain two-thirdsof the anomaly, suggesting that the size of the flavor-dependent EMC effect predicted in this model is of thecorrect scale to resolve this long-standing issue.

Later predictions were made within this model specifically for the PVDIS a1 term for iron and lead [2].Calculations for 48Ca were also provided for this proposal [33] and are shown in Fig. 9. The effect isqualitatively similar to lead, but slightly smaller due to the smaller neutron excess.

The size of the effect clearly shows an enhancement in a1 over the isoscalar-corrected “naive” case andgrows with x. There is essentially no difference at x = 0.2, and a 5% difference at x = 0.7. We will be ableto measure a1 across this x range with a statistical precision of ∼1% and systematic uncertainties of 0.7%.

14

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-a

1

x

48Ca from nCTEQ

w=0w=1/7w = inf

Figure 10: a1 predictions from the nCTEQ data set assuming various weighting between pure DIS and Drell-Yan (w = 0) and pure neutrino data w = ∞ from Refs. [23] and [24] for 48Ca. Inclusion of any fractionof neutrino data dramatically shifts the fit demonstrating the weakness of DIS data to this observable. Thechange in slope is about 5% over our x range and is consistant with the CBT calculation.

2.1.2 Nuclear Parton Distributions

We consider data from PDF fits to understand present constraints in this sector and to consider the precisionfor new data required to become a significant test. First, we look at the nCTEQ nuclear PDF fits done byRefs. [23] and [24], discussed in Sec. 1.3.2. There the authors varied the contributions of neutrino chargedcurrent data with the standard DIS and Drell-Yan data. If there are flavor-dependent variations in the nucleardistributions, the discrepancy between the two in those fits provides an idea on their size.

The results of the a1 calculations from this fit are shown in Fig. 10. The change in slope of a1 of about5% is remarkably consistant CBT calculation in the range of 0.2 < x < 0.7. The fits including any neutrinodata have a very different behavior than the neutrino-free fit in terms of the modification. This suggests thatthere is a lack of constraint on the order of a few percent in the DIS/Drell-Yan data in this observable, whichis perhaps not surprising considering the unique flavor sensitivity of neutrino scattering.

2.1.3 Short-Range Correlations

The tie between short-range correlations and the EMC effect in principle can also make predictions for anisovector EMC effect. Though the SRC model presented in [32], Fig. 8 only presents results for the EMCslope dR/dx, the black line fit (p > 300 MeV) is near the CBT model predictions. This suggests the twomodels perhaps capturing the same physics in this context and reinforces the prediction of the CBT model.While this work is still under development, more detailed calculations for heavier nuclei are possible.

2.2 Choice of Target

It is clear that a large proton-neutron asymmetry is favorable for such a measurement through PVDIS. Anisoscalar target would test for charge symmetry violating terms which are likely subdominant to an isovectoreffect. A measurement has already been approved for deuterium which can test charge symmetry violationthrough the same technique and will provide precision on the order of 1% up to x = 0.7 [34]. If those effectsin deuterium or this effect turn out to be surprisingly large, an isoscalar measurement on a medium-to-heavysymmetric nucleus such as 40Ca would likely be well motivated.

There are many choices of possible target, though very high Z targets present additional complicationswhich must be carefully considered. Our nucleus criteria include

15

• providing a high (N − Z)/A ratio and relatively large EMC effect;

• minimizing beam radiative corrections which scale as Z2 while scattering rates only scale with A;

• the size of Coulomb corrections with the additional consideration of how they affect a parity-violatingasymmetry.

A wide range of nuclei are possible candidates with reasonable values of (N − Z)/A as well as a rela-tively large EMC effect. 9Be, 48Ca, and 208Pb have (N − Z)/A values of 0.11, 0.17, and 0.21 respectively.So 9Be would be expected to have a slightly smaller flavor asymmetry, as well as a slightly smaller overallEMC effect, although a thicker target could be used. While the light nuclei may be of interest in testing mi-croscopic calculations, such calculations do not yet exist and we focus on heavier nuclei, where the expectedeffect is larger.

For this experiment, we choose a target of 48Ca due to the clear advantages over a heavier target, suchas 208Pb. N/Z is comparable between these two targets (and other targets such as depleted uranium andgold), as is the projected impact on a1 in the calculations by Cloet (i.e. Fig. 9). Sufficient rates can beachieved with 48Ca using only a 12% radiator, whereas an equivalent DIS rate in lead would require morethan a 60% radiator. These radiative effects contribute non-linearly to deconvolution scheme due to therising cross section with lower-energy electrons having undergone radiation. In addition the photon spectraincrease linearly with radiation length which are a dominant component of induced rates in detectors andpion production.

Coulomb corrections for high-Z targets have often been calculated in the so-called effective momentumapproximation [35] and have been shown to be quite successful at lower energies even for targets as heavyas lead in quasielastic scattering [36]. In this approximation, the relative size of the correction is smalleras one goes to higher energy. For a heavy target like lead, a correction factor VC ≈ 18 MeV is applied tothe incoming electron. VC for 48Ca is ∼ 5 MeV and would have a sub-0.1% effect in this framework. Theauthors are unaware of calculations that have been carried through the full DWBA including the nuclearweak potentials for DIS.

2.3 Relation to Other Experiments

In light of the importance of such an effect, several avenues should be explored and by using data frommultiple techniques, a more complete picture can emerge. While we discuss several possibilities, we stressthat the parity-violating technique presented here offers one of the most direct and theoretically clean accessto these observables and would serve as the strong underlying foundation for all of these studies.

Deep Inelastic Scattering Cross section measurements are planned on 40Ca and 48Ca (amongst manyother targets) in JLab experiment E12-10-008 [37], which allows for a test of flavor dependence through theratio of the two. The small difference A does not modify the naive effect dramatically allowing for a goodcomparison. Of note is that the 48Ca cross section measurement is the denominator in the asymmetry forthis proposal and would be for a similar range in x. Having both greatly improves determining the absoluteflavor contributions. Comparisons to isoscalar EMC parameterizations and the CBT model are shown inFigs. 11 and 12. The ratio shows a similar trend as the PVDIS measurement of larger deviations at higherx. The overall sensitivity is less than 3σ with dominant scale uncertainties from the target thickness.

Such comparisons are more difficult in light nuclei, as contributions related to clustering and/or short-range correlations are strongly A dependent in light nuclei. So a similar measurement comparing 10B to9Be or 11B would have a large model-dependent uncertainty. As noted above, parity-violating scattering isinsensitive to the EMC effect, as long as it is identical for up and down quarks, so even a nucleus like 9Be,

16

Calcium Isotope EMC Effect

bjx0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2D/F

is 2AF

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2Ca40

Ca, No Isovector48

Ca, CBT48

Ca Ratio (stat + pt to pt sys)40Ca/ 48

Shared systematic

Calcium Isotope EMC Effect

Figure 11: Projections of the isoscalar EMC F2 ratios for 40Ca and 48Ca from the SLAC E139 parameteri-zation [4], a prediction from the CBT model [1], and data point projections for E12-10-008 [37].

which has significant cluster structure, can be used to examine the flavor dependence without worry aboutthe detailed structure of the nucleus.

A comparison of the EMC effect in 3H and 3He would be sensitive to the flavor dependence of the EMCeffect, but this comparison is much more sensitive to the neutron structure function. While the MARATHONexperiment will perform this comparison, it will use estimated upper limits on the difference of nuclear ef-fects in 3H and 3He to provide an improved extraction of F2n/F2p. Other experiments can also providemodel-independent extractions of the neutron structure function, but this will not allow a comparison ofthe nuclear effects unless the precision is significantly better than the extraction from MARATHON. Morespecifically, the total uncertainty in the neutron extraction must be smaller than the model-dependent con-tribution to MARATHON’s extraction before limits can be set beyond the assumed upper limits taken inMARATHON’s extraction of F2n/F2p.

Drell-Yan Drell-Yan measurements offer access to complementary data from DIS probes as they are sen-sitive to the flavors through the annihilation of sea quarks with the valence quarks of a heavy target. Inparticular, for pionic Drell-Yan the production ratios of π−A/π−D would have cancellation in the pionquark distributions, which are not well known. It was presented by Dutta. et al. [38] that within the CBTmodel there is a slight preference in existing pionic Drell-Yan measurements to support such an effect overthe N = Z predictions, but is not statistically strong, Fig. 13.

The COMPASS-II experiment has the possibility of providing more data on this with the inclusion of aheavy target. This data would be an excellent complementary measurement, in particular if done on a highA target providing a stronger constraint for extrapolation across the periodic table.

Parity Violating DIS on Light Targets The existing approved SoLID PVDIS proposal [34] on LD2 andLH2 does not constrain modification effects, but does provide very important information which necessaryto obtain a measurement with adequate sensitivity. For LD2, critical data on hadronic effects such as possibledifferences betweenRγZ andRγ and charge symmetry violation will be obtained. In addition, as this is alsoa test for physics beyond the standard model, important radiative correction calculations which are neededwith a higher sensitivity at our kinematics will be completed and are applicable to this proposal. LH2 PVDISwill provide data on the free quark distribution ratio d/u (as well as two other independent experiments at

17

bjx0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ca

402,

/F

Ca

482,

is

Nor

m. F

0.85

0.9

0.95

1

1.05

1.1

1.15

E139 Param

CBT

Ca Ratio (stat + pt to pt sys)40Ca/ 48

Shared systematic

Ca Ratios40Ca/ 48

Figure 12: Ratios for isoscalar-corrected F2 for 48Ca and 40Ca using SLAC E139 [4] and the CBT model [1]with data point projections for E12-10-008 [37].

Figure 13: Existing pionic Drell-Yan data for ratios of heavy target to deuterium data compared to the CBTmodel [38]. A slight preference is shown for the CBT model, but is not conclusive.

18

JLab) in our kinematic range. This is necessary to constrain the free parton distributions which underly thetests for modification to a much better precision than presently exist. A more quantitative discussion of theseeffects is presented in Secs. 4.2.3 and 4.2.4.

As noted above, measurements on light nuclei with significant fractional neutron excess (7Li, 9Be) couldalso be studied using parity-violating electron scattering. The upcoming Hall C measurement of the EMCeffect in light nuclei will provide information on the overall magnitude of the EMC effect in these nuclei,while models of the EMC effect can provide predictions of the difference in the EMC effect for up and downquarks. Thus, as improved models of the EMC effect including the connection with the detailed nuclearstructure become available, additional measurements could be demonstrated to have additional sensitivity tothe underlying physics of nuclear PDF modification.

19

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

)2 (

GeV

2Q

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4

6

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14

1.33

1.01

0.760.70

0.700.78

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1.21

Ca Target48A Electron Beam on 12% µAsymmetry Uncertainty (%) with 60 Days of 85% Polarized 80

Figure 14: Our statistical precision for APV for x and Q2 bins in %.

3 Experimental Design

We choose a 2.4 g/cm2 48Ca target (x/X0 = 12%) at 80 µA with maximum longitudinal beam polarizationas the general production conditions. The experimental layout we propose is identical to the existing SoLIDPVDIS proposal [34], with the replacement of the LD2 target ladder with our 48Ca target ladder. For60 days of production, a 0.7-1.3% statistical uncertainty for 0.2 < x < 0.7 can be obtained, Figs. 14. A∼0.5-0.7% systematic uncertainty per-bin is anticipated, discussed in Sec. 4.2. Our sensitivity to the CBTmodel is shown in Fig. 15 and is more than 5σ including all shared systematics.

This is a true “counting mode” parity-violating experiment in contrast to parity-violating experiments inthe past that have used integrating detectors gated over the beam-helicity window. The full SoLID programis still under a detailed design and review process, which is more fully described in the SoLID pre-CDRdocument [39]. Here we cover the relevant aspects of the design and the anticipated performance.

Aµ=12%, 60 days, 800Ca x/X48 from CBT, 1a

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1a

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1 CBT1a

naive1a

Our Projections (stat, stat+ pt to pt sys)

Shared sys. uncert

Aµ=12%, 60 days, 800Ca x/X48 from CBT, 1a

Figure 15: Our measurement projections in the context of the CBT model.

20

Figure 16: SoLID side-view of the Geant4 configuration with full detector setup.

Most aspects of this experiment are less demanding than the approved PVDIS LD2 experiment for thefollowing reasons:

• this proposal utilizes a solid target with good thermal properties and is not subject to effects such asboiling,

• less total mass of the material, generally providing lower rates,

• the physically short target has better controlled acceptance and collimation.

This target has a 12% radiation length, which is a factor of two larger than the LD2 target, and presentsa few challenges. First, the overall radiative corrections due to external bremsstrahlung is approximatelydoubled. Direct photons from the target are increased as well as the pion photoproduction rates. Combinedwith the fact the target has a neutron excess, the relative number of π− to DIS events is increased.

A suite of simulations has been developed in the Geant4 framework which includes a complete anddetailed representation of the target and detector geometries, Fig. 16, and includes particle showering andoptical photon production. In addition, event generators for deep inelastic scattering based on the CTEQ6MPDFs [40] and pion production based on a SLAC photoproduction data parameterization [41] with the equiv-alent photon approximation for the electroproduciton data have been developed. We utilize this simulationto generate the following rate and response estimates.

3.1 Targets

The target proposed is a 2.4 g/cm2 isotopically pure 48Ca target at least 1 inch in diameter. A 0.8 g/cm2

90% pure target was employed by the x > 2 experiment [42] at 50 µA, Figs. 17 and 18 and is also plannedfor the parity violating experiment, CREX [43] with beam current of 150 µA at 2.2 GeV. The existing designwill need to be modified to allow for full acceptance of all scattering angles when the raster is employed.

It would be advantageous to have the target separated into disks separated by an amount discernibleto tracking to aid in radiative corrections, but to allow for a solid frame to hold the target, the materialmust be in a contiguous block as the 48Ca disk is 2.5 cm in diameter. A 1.5 mm lip can provide support

21

Figure 17: A CAD design of the existing 48Ca target for E08-014. Two identical 40Ca and 48Ca targets weremounted in the ladder design.

and not interfere with the maximum accepted scattering angle of 35 when a 4× 4 mm2 raster is employed,Fig. 19. Additionally, 5 cm clearance should be provided from the entrance and exit windows so that windowbackground will not contribute to the main signal.

Calcium has relatively robust thermal properties which are advantageous for this experiment: a meltingpoint of 840 C and a high thermal conductivity of 200 W/m/K. While the melting point is an absolute upperlimit, calcium undergoes crystalline structure changes at lower temperatures. A 4 × 4 mm2 raster will beused to distribute the heat load. This experimental configuration will have a power deposited from the beamof about 600 W, but thermal calculations showed that with sufficient cooling on the support frame held atroom temperature, operate at only 100 C, within heating limits and below the CREX ∆T .

Auxiliary targets will be required in the same ladder to provide calibrations and tests, described in Sec 3.4.In particular, a set of several carbon foils spaced ±20 cm (with one specifically at the center position of the48Ca target). Aluminum targets with known radiation thicknesses of 1%, 5%, and 10% will help providevalidation of unfolding external radiative effects and a LH2 target will be used for momentum calibration.

3.2 SoLID

The SoLID project is a large acceptance, high luminosity spectrometer and detector system designed for ex-periments that require a broad kinematic acceptance at high rates. It presently has five approved experimentscovering physics topics such as PVDIS on LD2 and LH2, semi-inclusive DIS on polarized targets, and J/ψproduction at threshold. We will focus on the PVDIS configuration, which consists of

• A 3 m diameter solenoidal magnet that provides a central field of ∼ 1.5 T and field integral of about1.8 T ·m.

• A set of collimators (“baffles”) which block low momentum particles and line-of-sight photons.

• A set of five GEM layers which provide high resolution, hit-based tracking in a high luminosityenvironment.

• A light gas Cherenkov detector for pion identification.

• An electromagnetic calorimeter in a shower-preshower configuration which also provides some pionrejection capabilities and acts as the primary trigger and an additional point in tracking.

This configuration with baffles nominally has 2 rad azimuthal acceptance, polar angle acceptance of 22-35, and momentum acceptance of 1− 7 GeV. Azimuthally it is divided into 30 predominantly independent

22

SECTION B-B

SCALE 1 : 1

PARTS LIST

DESCRIPTIONPART NUMBERQTYITEM

CONFLAT SEAL CELL BLOCK ASMTGT-101-1001-010011

Entrance window assyTGT-101-1001-041622

20 cm He Exit WindowTGT-101-1001-040123

1/4-28 x 1.00 12 pt Cap Screw SST Certified MaterialSBX 1/4-28-1 12 pt124

1/4-28 x 1.25 12 pt Cap Screw SST Certified MaterialSBX 1/4-28-125 12 pt125

BB

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

A A

B B

C C

D D

E E

F F

G G

H H

SHEET 1 OF 1

DWG NO

TGT-101-1001-0420

ASSY. NO.SCALE

REV

FINISH

MATERIAL

Var

N/A

DEBUR AND BREAK ALL SHARP EDGES

ACCEPT AS NOTED

±1/64

±0.5'

UNLESS OTHERWISE SPECIFIED

DIMENSIONS ARE IN INCHES

TOLERANCES ARE:

DIM & TOL PER ASME Y14.5 .

UNLESS

OTHERWISE

NOTED

FRACTIONS DECIMAL

ANGLE

.X±0.1

.XX±0.01

.XXX±0.005

Hall A Cryogenic Target

CF Style Cell Calcium Target

Assembly

JEFFERSON LAB TARGET GROUP

THOMAS JEFFERSON NATIONAL ACCELERATOR FACILITY

NEWPORT NEWS VIRGINIA

U.S. DEPARTMENT OF ENERGY

-

SIZE

DRAWN

CHECKED

APPROVED

APPROVED

APPROVED

E -

THIRD ANGLE PROJECTION

DOCUMENT CONTROL STAMP DATE

APPROVALS

D. Meekins

2/27/2011

1

Assemble CF joints with aluminum gaskets only.

Use no more than 11 ft-lb torque.

NOTE:

Do not pressure test final assembly with Ca target installed.

Assembly to be made inside inert environment (glove box).

2

54

7.387

.197

3.777

Figure 18: Drawings of the existing 48Ca target for E08-014.

Raster Area

o35

Beam

Figure 19: A schematic a conceptual frame for the target. About a 1.5 mm lip is allowed on the outer radiusdownstream to allow for 35 acceptance when the raster is included.

23

Figure 20: Projection of the baffle lead “spokes” to block low momentum particles.

sectors which can operate at a total of ∼ 600 kHz in inclusive running. A representation of this setup fromour Geant4 simulation is shown in Fig. 16.

3.2.1 Baffles

The baffles provide a reduction in the large low-momentum flux and block line-of-sight photons from thedownstream detectors. They consist of 11 lead “wheels” which divide the acceptance into 30 sectors, Fig. 20.The curvature of the arms are designed in such a way that particles within a specific momentum windowwill pass in between the arms to the detectors.

The coverage of the baffles defines the azimuthal and momentum acceptance for the spectrometer. Nom-inally, the first baffle reduces the flux by a factor of two and particles less than 1 GeV are blocked by suc-cessive baffles, leading to an overall charged rate reduction of about an order of magnitude. The momentumacceptance for the accepted particles follows from several geometric and design effects and is shown for the48Ca configuration in Fig. 21.

3.2.2 GEMs

The GEM (gas electron multiplier) trackers originally developed at CERN provide high resolution trackingin high rate environments. They have been demonstrated to work at rates up to 100 MHz/cm2 and providea hit resolution up to 70 µm with a 200 µm readout pitch. We employ five planes of GEM chambers, threeinterleaved with the rear baffle planes and two after the light-gas Cherenkov detector, detailed in Table 3.Each plane consists of 30 individual GEM modules and are aligned such that the gaps of the first threechambers lie over a baffle spoke, Fig. 22. The pitch will be 0.4 mm for the first three GEMs and 0.6 mm inthe rear GEMs as the rates are lower.

Significant contributions to the GEM rates come from not only DIS electrons, but also π− and photons.For the latter, the response is highly dependent on the photon momentum and the radiation thickness of thedetector. Figure 23 shows the results from Geant4 simulation for the photon response with a 0.5% radiationlength GEM, which drops for photons < 1 MeV. A comparison between hit rates for our proposal andsimulations for the LD2 measurement are shown in Table 4.

24

Figure 21: Electron and π− acceptances from the baffles. Differences between these are due to varyingangular distributions and the fact that π− have longer interaction lengths.

Figure 22: CAD drawing of a GEM plane for the PVDIS configuration.

Figure 23: Geant4 calculation results for photon interaction probabilities with GEM chambers fromRef. [44].

25

Location Z (cm) Rmin (cm) Rmax (cm) Surface (m2) # chan1 157.5 51 118 3.6 24 k2 185.5 62 136 4.6 30 k3 190 65 140 4.8 36 k4 306 111 221 11.5 35 k5 315 115 228 12.2 38 k

Total ≈ 36.6 ≈ 164 k

Table 3: GEM design parameters for the SoLID PVDIS configuration.

GEM plane LD2 background 48Ca EM background 48Ca EM background (no baffles)(kHz/mm2/µA) (kHz/mm2/µA) (kHz/mm2/µA)

1 6.8 4.8 49.4

2 3.0 2.1 32.3

3 1.1 0.8 9.9

4 0.7 0.5 6.4

Table 4: The low energy EM background radiation at GEM detectors compared for 48Ca and LD2 targetswith and without baffles.

The radial GEM rates are presented in Fig. 24. The particle rates at the last GEM, which will beincident on the EM calorimeter, are broken down by particle type and shown in the Table 5. The initial π+

background is heavily suppressed by the combination of baffle design and the solenoidal magnetic field butwill also be produced in interactions within the baffle material. A combination of triggering and off-lineanalysis is required to suppress the π− background to desired level. The DIS electron rates at the last GEMfor various x-cuts is shown in the Table 6.

The background rates are much greater than the DIS rates at the entrance to the EM calorimeter. Therates below p < 1.0 GeV are predominantly electromagnetic backgrounds. The high energy p > 1.0 GeVbackgrounds are dominated by contributions from pions and protons.

3.2.3 Calorimeter

The electromagnetic calorimeter serves as the primary trigger as well as an independent means for rejectingπ− backgrounds. It it configured in a hexagonal preshower-shower configuration and consists of “shashlyk”-style blocks with 50 cm of interleaved sampling lead and scintillator plates with a fiber readout, Fig. 25,

Momentum π− π+ π0(γ) Proton EM (γ, e±)range (GeV) (MHz) (MHz) (MHz) (MHz) (GHz)p > 0.0 GeV 618 283 70123 483 844p > 0.3 GeV 439 153 438 417 n/ap > 1.0 GeV 123 18 37 51 0.0p > 3.0 GeV 2 0.01 0.04 0.004 0.0

Table 5: Breakdown of rates based on the particle types for 48Ca target at 80 µA.

26

Figure 24: The design of the baffle structure minimizes the EM background rates at the GEM detectors. Thesolid lines shows background rates with no baffles and the dashed lines show the rates with the baffles. Thebaffle structure reduce the background rates by almost a factor of 10.

Momentum 48Carange (GeV) (kHz)

DIS Total 228

W > 2.0 GeV, xBjk > 0.20 207

W > 2.0 GeV, xBjk > 0.55 15

W > 2.0 GeV, xBjk > 0.65 3

Table 6: Breakdown of DIS rates for 48Ca target at 80 µA.

with a radial coverage of 110-265 cm or ∼18 m2. Each module has a lateral coverage of about 100 cm2

providing adequate position resolution, background sensitivity, and cost for a total of about 1800 modules.The shower and preshower are read out through 100 1-mm-diameter waveshifting fibers that are threadeddown the module and run to the rear of the solenoid. They are coupled to clear fibers and then to multianodePMTs (1 PMT per module).

The momentum resolution requirements for the shower are relatively modest, as it primarily must pri-marily provide a trigger above the low energy background flux, and reconstruct a reasonably good positionand energy. For our modules, a 4%/

√E resolution has been simulated. A good position resolution is

important as in the high luminosity environment, the reconstructed point will serve as a base for track re-construction. Accounting for the energy distributions of tracks that are not normal to the face of the detectorleads to a RMS of <1 cm is achieved in the radial and azimuthal directions.

As a method of pion rejection (in combination with the gas Cherenkov), the preshower and showerenergy deposition information can be used. For our configuration, at a 100:1 rejection factor is anticipated(improving with particle momentum) for E > 2 GeV, Fig. 26 while maintaining a 95% electron efficiency.

Our simulations to determine the performance of the EM calorimeter are based on DIS events withrealistic backgrounds incident on the EM calorimeter. These simulations provide trigger efficiencies forDIS electrons and all the background types. The trigger efficiencies and rates on particles incident on thefull coverage of the EM calorimeter are used to extract the total trigger rate for the EM calorimeter. In orderto further optimize the calorimeter performance, each 12 azimuthal sector in the calorimeter is divided into

27

WLS fibersShower

Preshower WLS fiber connectors fibersclearfiber

11.2mm 20mm434.5mm (194 layers) 100mm

0Shower, 18 X , Shashlyk

Preshower, 2X lead + scintillator0

Figure 25: Cross section of an electromagnetic calorimeter module and absorber sheets.

Momentum (GeV)1 2 3 4 5 6 7 8

1/R

eje

ction

0

0.005

0.01

0.015

0.02

0.025

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> = 28.5θPVDIS <

> = 12.0θSIDIS­Forward <

> = 20.5θSIDIS­Large <

rejection­π

Momentum (GeV)1 2 3 4 5 6 7 8

Eff.

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

efficiency­e

Figure 26: π− rejection and electron efficiency for calorimeter. Red points and curves are for calorimeter inPVDIS configurations

two 6 segments based on the background rate which varies due to the baffle structure. The low rate sectionis shown in Fig. 27 and high rate in Fig 28.

It is observed by dedicated EM calorimeter simulations that the pile-up effects are not significant forparticles with momentum p > 1 GeV but is an important effect for particles with p < 1 GeV. Due to thiseffect, the trigger rates for the lowest energy particles cannot be broken down to particle type in a straightforward manner and we quote only a total trigger rate. Table 7 summarizes this for 48Ca target.

The luminosity for this experiment is∼ 2×1037 Hz/cm2 (calculated per nucleus) or a factor of 3 smallerper-nucleon than the LD2 measurement. The additional radiation dose is about 36 kRad compared to thedesign specification of 400 kRad and present program of less than 200 kRad of running.

3.2.4 Light Gas Cherenkov

The light gas Cherenkov detector provides rejection of π− background, which is difficult to otherwise sup-press from the e− DIS signal. In the PVDIS configuration, it is proposed to consist of a ∼ 1 m gas radiatorof 65% C4F8O and 35% N2 (refractive index 1.001 or π± threshold of 3.2 GeV) and is divided up into30 sectors matched to the baffle segregation. For each sector there are two spherical mirror sections to pro-vide light collection over a broad radial range which focus into a Winston cone/PMT set. The PMTs are8×8 pixel multi-anode bialkali PMTs arranged in a 3×3 array and are shielded from the residual field witha mu-metal cone. To help reject pion triggers, the Cherenkov is placed in coincidence with the calorimeterthrough a sum of all 9 PMTs.

The photoelectron distributions generated as a function of angle for DIS electrons is shown in Fig. 31.

28

Momentum (GeV/c)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

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R=113 cm, Trig = 3.8 GeV

Electron

Momentum (GeV/c)

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1R=230 cm, Trig = 1.5 GeV

R=180 cm, Trig = 2.0 GeV

R=150 cm, Trig = 2.5 GeV

R=120 cm, Trig = 3.5 GeV

R=113 cm, Trig = 3.8 GeV

Pion

Figure 27: Electromagnetic calorimeter trigger performance for the low rate azimuthal region for e− (left)and π− (right).

A threshold was chosen dependent on the momentum where the pion rejection efficiency was maximizedwhile losing minimum number of electrons and is shown in Fig. 32. The rejection factor for 2-3 GeV pionsis 1000:1 - 400:1, worse for the higher energy π. In combination with the 100:1 independent rejection factorfrom the preshower/shower providing an overall rejection of 105 - 4 × 104 up to 3 GeV and at least 100:1above that. The total rates seen by the Cherenkov and the estimated trigger rates are summarized in theTable 8. The π−/e− ratio we anticipate as a function of momentum is shown in Fig. 30 for both the LD2

measurement and 48Ca.

3.2.5 Data Acquisition

Due to the large number of channels and the necessity to keep the readout size small, the data acquisitionsystem for SoLID is complex, even for the inclusive, independent-sector running for a PVDIS measurement.To approach this, SoLID utilizes pipelined electronics similar to the Hall D GlueX design. The readout forthe calorimeter and Cherenkov is a VME JLAB FADC250 16-channel 12-bit FAC sampling at 250 MHz anda schematic of the FADC crate layout is shown in Fig. 33. The GEM readout is based on the APV25 chipdeveloped at CERN which is a pipelined readout system that includes a shaper/amplifier and the scalablereadout system (SRS). A summary of channel and module counts per sector is shown Table 9.

For our experiment, we anticipate about a total 155 kHz coincidence trigger, Table 10, for all sectorsusing both the calorimeter and gas Cherenkov signal compared to the 500 kHz trigger for the LD2 measure-ment. We will run the 30 sectors independently requiring about 5 kHz/sector for the primary measurement.

A level 1 trigger can be formed by summing all modules for all sectors simultaneously every 4 ns andsending the signal to the crate trigger processor (CTP). To account for overlapping sectors on the calorimeterplane, neighboring CTPs are connected through optical links to share the overlapping 16 channels. Sup-pression will be invoked on hits outside of a cluster. The gas Cherenkov configuration is similar but lesscomplicated as each PMT array is unique to a sector and there is no overlap.

The APV25 system for the GEMs was developed at CERN and provides a pipelined readout and ashaper/amplifier which allows for background suppression through multisampling. It has 128 channels perchip and can hold 192 consecutive time samples at 40 MHz continuously. This allows for a deadtime-lessreadout of the GEM signals. The scalable readout system (SRS) developed by the RD51 collaboration atCERN interfaces with the APV25 chips to handle triggering of the readout, data transfer, and event building.

29

Momentum (GeV/c)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Trig

ge

r E

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0.2

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R=230 cm, Trig = 1.5 GeV

R=180 cm, Trig = 2.0 GeV

R=150 cm, Trig = 2.5 GeV

R=120 cm, Trig = 3.5 GeV

R=113 cm, Trig = 3.8 GeV

Electron

Momentum (GeV/c)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Trig

ge

r E

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1R=230 cm, Trig = 1.5 GeV

R=180 cm, Trig = 2.0 GeV

R=150 cm, Trig = 2.5 GeV

R=120 cm, Trig = 3.5 GeV

R=113 cm, Trig = 3.8 GeV

Pion

Figure 28: Electromagnetic calorimeter trigger performance for high rate azimuthal region for e− (left) andπ− (right).

region full high lowrate entering the EC (kHz)

e− 240 129 111π− 5.9× 105 3.0× 105 3.0× 105

π+ 2.7× 105 1.5× 105 1.2× 105

γ(π0) 7.0× 107 3.5× 107 3.5× 107

p+ 4.8× 105 2.1× 105 2.7× 105

sum 7.1× 107 3.6× 107 3.6× 107

Rate for p < 1 GeV (kHz)sum 8.4× 108 4.2× 108 4.2× 107

trigger rate for p > 1 GeV (kHz)e− 152 82 70π− 4.0× 103 2.2× 103 1.8× 103

π+ 0.2× 103 0.1× 103 0.1× 103

γ(π0) 3 3 0p 1.6× 103 0.9× 103 0.7× 103

sum 5.9× 103 3.3× 103 2.6× 103

trigger rate for p < 1 GeV (kHz)sum 2.8× 103 1.4× 103 1.4× 103

Total trigger rate (kHz)total 8.7× 103 4.7× 103 4.0× 103

Table 7: Calorimeter trigger rates based on 48Ca target. DIS and background rates that enter full coverageof the EC are considered for the resulting trigger rates. Trigger is broken down to p < 1 GeV and p > 1 GeVparticles and for the “low” and the “high” background regions. The total rate for the sum of 30 sectors areshown here. The simulated pion rejection and electron efficiency values are shown in Figs. 27 and 28.

30

Total Rate for p > 0.0 GeV Rate for p > 3.0 GeV(kHz) (kHz)

DIS 240 73π− 5.9× 105 1.6× 103

π+ 2.7× 105 40γ(π0) 7.0× 107 40p 4.8× 105 4

Sum 7.1× 107 1.7× 103

Trigger Rate from Cherenkov (kHz)Trigger Rate for p > 1.0 GeV Trigger Rate for p > 3.0 GeV

(kHz) (kHz)DIS 223 66π− 193 49π+ 22 1.6γ(π0) 0 0p 0 0

Sum 438 116

Table 8: Cherenkov trigger rates for 48Ca target at 80 µA is estimated using simulated pion rejection andelectron efficiency values from Fig. 32.

Detector Module TypeNumber of Number ofChannels Modules

Electromagnetic Calorimeter (EC) FADC 122 8Light Gas Cherenkov (GC) FADC 9 1

GEM VME 4700 3

Table 9: Detector channel counts for each sector.

Particle DAQ Coincidence Trigger Rate (kHz)P > 1 GeV P > 3 GeV

DIS e− 144 61π− 11 7π+ 0.4 0.2

Total 155 68

Table 10: Breakdown of coincidence trigger rates (Cherenkov+EM calorimeter) for momentuma > 1 GeVfrom 48Ca target at 80 µA.

31

Figure 29: Geant4 cross section of the light gas Cherenkov detector for the SIDIS (left) and PVDIS (right)configurations.

The data is then fed to the level 3 farm for data reduction. For this experiment the level 1 event data sizeis expected to be less than 50 kB with a 200 kHz level 1 rate. Further data reduction will be done correlatingdetectors such as the GEMs and calorimeter clusters together in space and time. Afterwards, crude trackreconstruction must be performed for the GEMs. In particular, this is important for the GEM data rates asthe occupancy will be on the order of 10-20%, or ∼ 3000 hits, making full hit recording untenable.

3.3 Polarimetry

A precise determination of the beam polarization is required to relate the asymmetry to the underlyingphysics measurement. As our statistical precision is about 1%, we require an uncertainty from the polar-ization better than that. We will utilize two independent techniques to measure this, Compton and Møllerpolarimetry. The upgrade to the existing Hall A Compton polarimeter is expected to provide a 0.4% preci-sion. The existing Møller polarimeter can achieve a level better than 1%.

3.4 Tracking, Optics, and Calibration

To precisely determine the kinematics of individual scattering events, tracking must first be performed usingthe GEM chambers and calorimeter, and then a sufficient optics model must be in place to reconstruct theevent. In particular, the momentum p, scattering angle θ, and the scattering vertex along them beamlinez must be known to sufficient precision to determine x and Q2 as well as eliminate background windowscattering events or events that originate outside of the target. Due to the relatively high luminosity andlarge acceptance for the experiment, efficient and fast tracking is important.

The overall background rates for this experiment are generally a factor of 2 smaller than the LD2 exper-iment and therefore less demanding. As this is an inclusive measurement, tracking only needs to be doneacross a single sector. Presently, simulations are underway testing several tracking algorithms under theSoLID experimental conditions. These include a detailed model of SoLID and the individual GEM planesas well as a model detailing the ionization and GEM front-end electronics response based on real data [45]implementing an existing framework used for the Super Bigbite project, [46]. Those studies have shown90% track reconstruction efficiency in occupancies that exceed the worst-case estimates of SoLID.

One additional challenge of the SoLID experiments is to reduce the data rate so at least crude trackingmust be done “on the fly”. This requires extensive simulation and testing of reconstruction algorithms beforethe experiment can run. The SoLID collaboration is actively working at realizing this.

32

Figure 30: A comparison of π− to e− for LD2 and 48Ca targets. The ratio for 48Ca is about 50% larger.

electron angle (deg)25 30 35

Mea

sure

d p

.e.

0

20

40

60

80

100upstream-Z 5cm

electron angle (deg)25 30 35

Mea

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80

100mid-Z 30cm

electron angle (deg)25 30 35

Mea

sure

d p

.e.

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60

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100downstream-Z 5cm

Figure 31: Simulation results for collected photoelectrons for the PVDIS LD2 experiment for the middle ofthe target.

33

Momentum (GeV/c)2.0 2.5 3.0 3.5 4.0

pi r

ejec

tio

n f

acto

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Momentum (GeV/c)2.0 2.5 3.0 3.5 4.0

elec

tro

n e

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1.0

Nominal Efficiency

0.9 of Nominal

0.8 of Nominal

Figure 32: Simulation results for the pion rejection factor and electron detection efficiency for a “nominal”case where π− rejection is maximized and and minimizing the loss of electrons as well as the ∼ 10% and∼ 20% e− loss cases.

Optics models were implemented based on ray-traced tracks in the Geant4 simulation using a field mapgenerated by the Poisson/Superfish package with a realistic coil and yoke geometry. The 0th order termswere based on the trajectories of particles in a uniform field and then deviations were fit using first orderpolynomials of generic track parameters. It was determined for SoLID in the PVDIS configuration that themomentum resolution is multiple scattering limited and about 1%, the angular resolution is GEM resolutionlimited and 0.5%, and the beamline vertex resolution of 7 mm. The derived quantities Q2 and x were 1.5%and 1% respectively.

The calibration of the system requires several steps. First, GEM alignment must be done using “straightthrough” tracks with the magnetic field off and a combination of a set of thin carbon foils to ensure accurateinteraction vertex reconstruction and a sieve to ensure angle reconstruction. Second, we utilize elasticscattering from a liquid hydrogen target and lower the beam energy to 4.4 GeV. The position of the elasticpeak provides a point of calibration and the magnetic field can be scanned to provide additional points.

To aid with the determination of radiative effects, independent aluminum targets with x/X0 = 1%, 5%,and 10% will be included. These will aid in the verification of scattering rate distributions under differentradiative conditions and the overall unfolding procedure, which will be limited by the determination ofquantity of event bin-migration.

3.5 Radiation Dose in the Hall

Radiation dose is generated from the 48Ca target by direct electron beam interactions as well as scatteredelectrons making secondary interactions in the hall. For reference, we compare radiation budget without theadditional shielding presented in Ref. [39] from the 12% 48Ca target to the approved LD2 measurement. Fornormalization, we note that we are requesting 60 days at 80 µA and the approved 11 GeV LD2 measurement,60 days at 50 µA. Due to 80 µA on 12% 48Ca target, the total exposure in the hall is larger due to the higherZ and increased radiation length, but within a factor of 2 (see Table 12). The iron core of the solenoidalmagnet provides self shielding for high energy neutrons and will help to reduce the site boundary radiationbudget. There are more extensive radiation and shielding studies for all SoLID experiments underway bythe collaboration to minimize the radiation to the hall and to site boundary. Based on preliminary studies theneutron radiation on superconducting coils are expected be an order of magnitude lower then the radiationdose limit for the coils.

34

VME C

PU

FADC250

FADC250

FADC250

FADC250

FADC250

CTP

SD TI

VME CPU

CTP

SDTI

ECSH

ECSH

ECSH

ECSH

CC

1) FADC ADC channels to CTP

VXS 2 x 4 Gbps

1000 MB/s

Optical link 10 Gbps

1250 MB/s

2) Optical link 10 Gbps

1250 MB/s

ADC neigboring channels sent to next CTP

3) CTP computessums and generates 64 bit pattern

corresponding to clustersto be read out

2) Optical link 10 Gbps

1250 MB/s

ADC neigboring channels sent to next CTP

4) TI sends pattern to SD

5 ) Each FADC receivesa 16 bit trigger word and puts the relevantchannels in the event buffer

FADC250

FADC250

FADC250

FADC250

ECPS

ECPS

ECPS

ECPS

Figure 33: Layout of the FADC crate for the shower and preshower systems. Each crate contains a cratetrigger processor (CTP), signal distribution module (SD), and trigger interface (TI).

4 Projections, Uncertainties, and Beam Time Request

4.1 Statistical Uncertainty

Our statistical uncertainty for 60 days at 80 µA on a 2.4 g/cm2 48Ca target of APV for x and Q2 bins isshown in Fig. 14. This translates to a sensitivity in a1 shown in Fig. 15 assuming the standard model for C1i

andC2i. The Y a3/2 term in Eq. 6 is small for our kinematics only contributes to about 5% to the asymmetryand approximately proportional to the a1 term due to the small contributions from sea quarks.

4.2 Systematics

Total systematic uncertainties are shown in Table 13.

Polarimetry Two independent polarimeters will be deployed for this experiment. A continuous monitor-ing of the polarization will be done by the upgraded Compton polarimeter, which is anticipated to give 0.4%systematic uncertainty using both the photon and electron detectors. The iron-foil Møller polarimeter willprovide an additional measurement periodically, as it is invasive, but with a projected uncertainty of about0.8%. We assume a 0.4% shared systematic for our measurement.

4.2.1 Pion Contamination

Our anticipated pion contamination to the electron signal based on the combined rejection factors in theCherenkov and the preshower/shower is expected to be no worse than 4% in a given bin, but still making it animportant effect. The contamination is worst at the highest x due to the increased difficulty of separating highenergy pions from electrons, but better for the higherQ2 bins at a fixed x due to the relative number of fewer

35

Radiation Power in the HallRadiation E-Range 48Ca LD2

Type (MeV) (W/µA) (W/µA)e± E < 10 0.11 0.11

E > 10 0.18 0.16n E < 10 0.0002 0.0003

E > 10 0.005 0.010γ E < 10 0.02 0.02

E > 10 0.04 0.04

Table 11: Breakdown of radiation power seen by a cylindrical detector in the hall compared for 48Ca andLD2 targets.

Relative Power Increase Relative Dose IncreaseRadiation E-Range From LD2 From LD2

Type (MeV) (%) (%)e± E < 10 37 37

E > 10 44 44n E < 10 6 6

E > 10 -30 -30γ E < 10 42 42

E > 10 34 34

Table 12: Breakdown of relative radiation power and dose increase in 48Ca with respect to LD2 seen by acylindrical detector in the hall. Notice same relative increase for dose since both LD2 and 48Ca dosages areestimated for 60 days.

pions at large angle. To make a correction this requires good characterization of the pion contamination andthe pion asymmetry.

Pion contamination level can be determined to good accuracy by using the gas Cherenkov and preshower/showerto cross-check one another. There is a large phase space in which to characterize, including momentum andposition, but also drifts over time. A fraction of triggers without the gas Cherenkov will be dedicated tocharacterize this.

The pion asymmetry in the 6 GeV Hall A PVDIS experiment was measured to be a few times smallerthan and the same sign as the LD2 asymmetry [47]. For our estimations, we will conservatively assumethat the π− asymmetry is zero. For this scenario, to control the total pion systematic to less than 0.5%, thefractional contribution must be known to to a relative 8% and the π− asymmetry must be known to 10 ppm.

For the asymmetry across all our bins, this would require at least an additional dedicated 10 kHz overall sectors of dedicated pion triggers over the course of the run, which is relatively small. This rate wouldalso be sufficient to generate statistics to verify the contamination rates with a drift on the order of hours.We assign a systematic of 0.1-0.5% bin-to-bin, larger at larger x.

4.2.2 Radiative Corrections

Several factors need to be applied to extract the PDF-dependent quantities from our measured asymme-try. First, an unfolding procedure will need to be included to account for the hard radiative events. This

36

Effect Uncertainty [%]Polarimetry 0.4RγZ/Rγ 0.2Pions (bin-to-bin) 0.1-0.5Radiative Corrections (bin-to-bin) 0.5-0.1Total for any given bin ∼0.5-0.7

Table 13: Summary of the systematic error contributions to our measurement.

causes the average Q2 to be reduced, causes events from lower energy transfer (including resonant) eventsto become convoluted into the asymmetry. Fortunately, there is good momentum acceptance up to andeven beyond elastic events, so these contributions will all be measured to sufficient accuracy automatically.It was also shown that resonance event asymmetries are in general agreement with quark-hadron dualityarguments [48].

The theory for radiative corrections is well understood e.g. [49], though for our kinematics calculationsare ongoing by a dedicated working group within the collaboration. Up to 40% of the reconstructed DISsignal will come from resonance or migrated DIS and are worst for the lowest x bins. However, theseevents have asymmetries that are only a few percent different from the primary asymmetry which makes thecorrections relatively small. We claim we can understand the size of the tails to at least 10% relative in theunfolding procedure, and we assign a 0.5%-0.1% bin-to-bin systematic, worse for small x.

The electroweak couplings for theCij in Table 2 are valid for all energy scales in the absence of radiativeloop corrections. With these corrections, in one parameterization they become [50]

C1u = ρ′e

(−1

2+

4

3κ′es

2Z

)+ λ′ (13)

C1d = ρ′e

(1

2− 2

3κ′es

2Z

)+ 2λ′ (14)

C2u = ρe

(−1

2+ 2κes

2Z

)+ λu (15)

C2d = ρe

(1

2− 2κes

2Z

)+ λd (16)

with, for Q2 lim 0, ρ′e = 0.9887, ρe = 1.0007, κ′e = 1.0038, κe = 1.0297, λ′ = −1.8 × 10−5, λu =−0.0118, λd = 0.0029, and s2Z = sin2 θW = 0.2312. These are being calculated for our kinematics and arenot likely to change in a way that is sensitive to this experiment.

4.2.3 Hadronic and Nuclear Uncertainties

There are potential contributions that can arise from higher order hadronic effects. Higher twist, chargesymmetry violation (which mimics our isovector EMC effect in our signal), PDF uncertainties, and free PDFnuclear-model uncertainties can all potentially interfere with the extraction of a signal. Fortunately, theseeffects will be greatly constrained by approved measurements, within and outside of the SoLID program.

Charge symmetry violation will be measured in LD2 to better precision than ours across the same kine-matic range using the same apparatus, as described in Ref. [34] and are likely to be smaller than the proposedisovector EMC effect. If charge symmetry violation is found to be large in LD2 or if the measurement hereis found to be unexpectedly large, that may motivate a proposal similar to this one, but on 40Ca. At presentwe assign no systematic uncertainty to charge symmetry violating effects.

37

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1-a

0.8

0.85

0.9

0.95

1

1.05

Ca - Minimum Correction48

Ca - Maximum Correction48

Ca - Middle Correction40

Our Statistical Uncertainty

- No Modification, CJ12 pdf1a

Figure 34: a1 predictions from the CJ12 PDF fit [54] assuming no modification for 40Ca, 48Ca with differentnuclear correction sizes.

R = σL/σT has been determined for proton and deuterium DIS over a broad kinematic range, e.g.Ref. [51], and is about 0.2 for our kinematics. For our measurement of an asymmetry, the effects of this willmostly cancel, though one potential concern for this proposal is the nuclear dependence of of R. It has beensuggested that these may be on the order of a few percent for our kinematics [52], which is negligibly smallfor our experiment due to cancellations and we ignore it.

For a symmetric target, specific higher twist effects can contribute in APV even though other Q2 de-pendent effects such as the DGLAP evolution are highly suppressed. If they do turn out to be significant,the LD2 measurement will provide constraints which then can be used as corrections to our measurement.For a 10% variation variation on a symmetric target at Q2 = 5 GeV2, these appear as a 0.5% correction toAPV [53]. The LD2 experiment uses a combination of 11 GeV and 6.6 GeV beam with the kinematic reachfrom 3 - 8 GeV2 and as these effects to first order scale with 1/Q2, this provides a useful lever arm. We willbe at high Q2 kinematics and assign a shared systematic of 0.2% to our measurement.

4.2.4 Uncertainties from Free Parton Distributions

If the free parton distributions are not well constrained for our kinematics, either due to insufficient dataor model-dependent nuclear corrections, it presents an effective systematic when testing for modification.While individual flavors are often shown to be well constrained by themselves, we are pursuing an unusualcombination which requires careful consideration. We choose the recent CJ12 set [54] and make calculationsfor a1 assuming that there is no modification, but appropriately weighting for Z and N . The results withour projected statistical uncertainties are shown in Fig. 34.

The fit uncertainty in a1 assuming no modification is on the order of or less than ∼1% from 0.2 < x <0.7 and is presently smaller than our statistical uncertainties. Of even greater importance at x > 0.5 isthe uncertainty in the model dependence on extracting quark flavor distributions from nuclear targets. Atx ∼ 0.7, this becomes large enough that it is close to the uncertainty on our largest error bar. Precision datafrom the SoLID proton program [34], BoNuS [55], and the 3H/3He ratio [56] are sensitive to the ratio d/u,Fig. 35 and will constrain the uncertainties on a1 for 48Ca from the free PDFs to better than 0.2%.

While the presence of these would reduce the sensitivity to the discussed isovector EMC effect, ulti-mately, we are performing a measurement to be included into the global nuclear PDF fits. If this measure-ment is statistically incompatible with the models within those fits, then they must be revised. To this end,

38

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

d/u

0

0.2

0.4

0.6

0.8

1 CJ12 - PDF + nucl uncert.

He DIS3H/ 3BigBite CLAS12 BoNuS

CLAS12 BoNuS, relaxed cutsSoLID PVDIS

SU(6)

pQCD

DSE

Broken SU(6)BoNuS sys. uncert.

Projected 12 GeV d/u Extractions

Figure 35: Anticipated data for measurements on d/u, see text for references. The constraints provided bythese data will allow for accurate tests of an isovector EMC effect at larger x.

we assign no systematic uncertainty from this to our measurement, but make the comment that the uncer-tainties from the CJ12 fits when combined with upcoming d/u data will propagate to a1 in 48Ca in our xrange to better than 0.2%.

4.2.5 Beam Parameters

Corrections to the measured asymmetry from helicity-correlated beam parameters are likely to be small dueto the existing excellent beam quality at Jefferson Lab, the level of control at systematically reducing theseerrors, such as the double Wien filter, and the relative size of our asymmetry, which is on the order of 10−4

compared to the 10−6 level for experiments such as Qweak and PREX. Corrections are likely to be on the0.1% level.

39

4.3 Beam Time Request

We request 60 days of production data at 11 GeV at 80 µA with full beam polarization. We also requesttime for commissioning, calibration runs, and polarimetry, summarized in Table 14.

Table 14: Beam time request for this experiment.

Time (days) E (GeV) Current (µA)48Ca Production 60 11 80Optics 2 4.4 Up to 80Moller Polarimetry 4 11 2Commissioning 5 11 Up to 80Total 71

40

A Quark Parton Model

In Eq. 6 higher order corrections for Y1 and Y3 were neglected and the Callan-Gross relation F2 = 2xF1

was invoked. Here we follow the convention of Ref. [53]. We define

Rγ(γZ) ≡σγ(γZ)L

σγ(γZ)T

= r2Fγ(γZ)2

Fγ(γZ)1

− 1 (17)

r2 = 1 +Q2

ν= 1 +

4M2x2

Q2(18)

The full parity-violating asymmetry is in terms of the structure functions F γ1 (γZ) and F γ2 (γZ)

APV = −(GFQ

2

4√

2πα

) geA

(2xyF γZ1 − 2 [1− 1/y + xM/E]F γZ2

)+ geV x(2− y)F γZ3

2xyF γ1 − 2 [1− 1/y + xM/E]F γ2(19)

We can then write it in the reduced from by

APV = −(GFQ

2

4√

2πα

)[geAY1

F γZ1F γ1

+geV2Y3F γZ3F γ1

](20)

with

Y1 =1 + (1− y)2 − y2

(1− r2/(1 +RγZ)

)− 2xyM/E

1 + (1− y)2 − y2 (1− r2/(1 +Rγ))− 2xyM/E

(1 +RγZ

1 +Rγ

)(21)

Y3 =1− (1− y)2

1 + (1− y)2 − y2 (1− r2/(1 +Rγ))− 2xyM/E

(r2

1 +Rγ

)(22)

and

F γ1 =1

2

∑i

e2i (qi(x) + qi(x)) ;F γ2 = 2xF γ1 , (23)

F γZ1 =∑i

eigiV (qi(x) + qi(x)) ;F γZ2 = 2xF γZ1 , (24)

F γZ3 = 2∑i

eigiA (qi(x)− qi(x)) . (25)

41

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