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DOE OSTI2022

Measurement of the inclusive and differential $t\overline{t}γ$ cross sections in the dilepton channel and effective field theory interpretation in proton-proton collisions at $\sqrt{s}$ = 13 TeV

The production cross section of a top quark pair in association with a photon is measured in proton-proton collisions in the decay channel with two oppositely charged leptons (e ± μ ∓ , e + e - , or μ + μ - ). The measurement is performed using 138 fb -1 of proton-proton collision data recorded by the CMS experiment at √s = 13 TeV during the 2016–2018 data-taking period of the CERN LHC. A fiducial phase space is defined such that photons radiated by initial-state particles, top quarks, or any of their decay products are included. An inclusive cross section of 175.2 ± 2.5(stat) ± 6.3(syst) fb is measured in a signal region with at least one jet coming from the hadronization of a bottom quark and exactly one photon with transverse momentum above 20 GeV. Differential cross sections are measured as functions of several kinematic observables of the photon, leptons, and jets, and compared to standard model predictions. The measurements are also interpreted in the standard model effective field theory framework, and limits are found on the relevant Wilson coefficients from these results alone and in combination with a previous CMS measurement of the tt¯γ production process using the lepton+jets final state.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Measuring Electron Diffusion and Constraining the Neutral Current $\pi^0$ Background for Single-Photon Events in MicroBooNE

Liquid Argon Time Projection Chambers (LArTPCs) are a rising technology in the field of experimental neutrino physics. LArTPCs use ionization electrons and scintillation light to reconstruct neutrino interactions with exceptional calorimetric and position resolution capabilities. Here, I present two analyses conducted in the MicroBooNE LArTPC at Fermilab: a measurement of the longitudinal electron diffusion coefficient, $D_L$, in the MicroBooNE detector and a constraint of the systematic uncertainty on MicroBooNE's single-photon analysis due to the dominant neutral current (NC) \piz{} background. Longitudinal electron diffusion modifies the spatial and timing resolution of the detector, and measuring it will help correct for these effects. Furthermore, current measurements of $D_L$ in liquid argon are sparse and in tension with one another, making the MicroBooNE measurement especially valuable. We report a measurement of $3.74^{+0.28}_{-0.29}$ cm$^2$/s. MicroBooNE is searchin g for si ngle-photon events as a potential explanation for the MiniBooNE low-energy excess (LEE) of electron neutrino-like events, which has been interpreted as evidence for low-mass sterile neutrinos. However, this search is overwhelmed by a large NC \piz{} background. By performing a sideband selection of NC \piz{} events, we apply a data-driven rate constraint to the single-photon analysis to reduce the systematic uncertainties. At present, this constraint improves the single-photon analysis' median sensitivity to the LEE-like signal from 0.9$\sigma$ to $1.5\sigma$. This sensitivity is expected to improve significantly as more data become available. Both of these measurements will not only benefit MicroBooNE, but also inform future LArTPC experiments.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Nucleon-Nucleon Scattering Up to N5LO in Chiral Effective Field Theory

During the past few decades a large effort has been made toward describing the NN interaction in the framework of chiral Effective Field Theory (EFT). The main idea is to exploit the symmetries of QCD to obtain an effective theory for low energy nuclear systems. In 2003, the first accurate charge-dependent NN potential in this scheme was developed and it has been applied to many ab-initio calculations, opening the possibility to study nuclear systems in a systematic and accurate way. It was shown that the fourth order (N3LO) was necessary and sufficient to describe the NN scattering data with a χ 2 /d.o.f on the order of so-called high precision potentials. However the systematics of chiral EFT also allow to relate two- and many-body interactions in a well-defined way. Since many-body forces make their first appearance at higher order, they are substantially smaller than their two-body counterparts, but may never-the-less be crucial for some processes. Thus, there are observables where they can have a big impact and, for example, there are indications that they solve the long standing Ay puzzle of N-d scattering. The last few years, have also seen substantial progress toward higher orders of chiral EFT which was motivated by the fact that only three-body forces of rather high order may solve some outstanding issues in microscopic nuclear structure and reactions. In this chapter we will review the latest contributions of the authors to development of chiral EFT based potentials up to N4LO as well as first calculations conducted for NN scattering at N5LO.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Trigger-DAQ and Slow Controls Systems in the Mu2e Experiment

The muon campus program at Fermilab includes the Mu2e experiment that will search for a charged-lepton flavor violating processes where a negative muon converts into an electron in the field of an aluminum nucleus, improving by four orders of magnitude the search sensitivity reached so far. Mu2e's Trigger and Data Acquisition System (TDAQ) uses otsdaq as its solution. Developed at Fermilab, otsdaq uses the artdaq DAQ framework and art analysis framework, under the-hood, for event transfer, filtering, and processing. otsdaq is an online DAQ software suite with a focus on flexibility and scalability, while providing a multi-user, web-based, interface accessible through the Chrome or Firefox web browser. The detector Read Out Controller (ROC), from the tracker and calorimeter, stream out zero-suppressed data continuously to the Data Transfer Controller (DTC). Data is then read over the PCIe bus to a software filter algorithm that selects events which are finally combined with the data flux that comes froma Cosmic Ray Ve to System (CRV). A Detector Control System (DCS) for monitoring, controlling, alarming, and archiving has been developed using the Experimental Physics and Industrial Control System (EPICS) Open Source Platform. The DCS System has also been itegrated into otsdaq. The installation of the TDAQ and the DCS systems in the Mu2e building is planned for 2021-2022, and a prototype has been built at Fermilab's Feynman Computing Center. We report here on the developments and achievements of the integration of Mu2e's DCS system into the online otsdaq software.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Splitting regions and shrinking islands from higher point constraints

We study constraints from higher-point amplitudes on 2 → 2 scattering in the context of effective field theory (EFT) using the perturbative numerical S-matrix bootstrap. Specifically, we investigate the class of weakly coupled EFTs with amplitudes that obey the hidden zero and split conditions that are known to hold both for Tr(Φ 3 ) theory and for certain string tree amplitudes, including at 4-point the beta function. Requiring the splitting condition for the 5-point amplitude not only fixes nearly all its contact terms, but it also imposes non-linear constraints among the 4-point EFT Wilson coefficients. When included in the bootstrap, the resulting allowed region consistent with positivity is no longer convex but is restricted to a smaller non-convex region — which has a sharp corner near the string beta function! Assuming the absence of an infinite spin tower at the mass gap, the allowed region bifurcates into a trivial region (with states only above a chosen cutoff) and an island that continues to shrink around the string as more constraints are included in the bootstrap. The numerics indicate that in the absence of single-mass infinite spin towers the string beta function is the unique 4-point amplitude compatible with hidden zero and the 5-point splitting constraints. The analysis provides a prototype example for how features of higher-point amplitudes constrain the bootstrap of 4-point amplitudes.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Background studies and normalization of signal events in the Mu2e experiment

The Mu2e experiment is currently being constructed at Fermilab to search for the neutrino-less conversion of negative muons into electrons in the field of an aluminum nucleus. The experiment aims at a sensitivity of four orders of magnitude higher than previous related experiments, which implies highly demanding accuracy requirements both in the design and during the operation. To achieve such a goal, two important tasks should be accomplished. First, it is essential to estimate precisely the particle yields and all the backgrounds that could mimic the monoenergetic conversion electron signal. Second, it is necessary to normalize the signal events accurately. The normalization of the signal events is planned to be done using a detector system made of an HPGe detector and a Lanthanum Bromide detector, which will measure the rate of muons stopped on the aluminum target by looking at the emitted characteristic X-and γ-rays of energies up to 1809 keV. Therefore, it is essential to evaluate the detector system's performance before the start of the actual experiment. In this study, the first task was addressed by an extensive campaign of Monte Carlo simulations to investigate the relevant parameters and their impact on the experiment's sensitivity. The second task was handled by taking advantage of the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) pulsed Bremsstrahlung photon beam at the ELBE facility. The detector system was tested at the ELBE facility under timing and background conditions similar to the ones expected at the Mu2e experiment. The study presents and discusses the simulation results and the detector system testing campaign.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

The straw tracking detector for the Fermilab Muon g-2 Experiment

The Muon g-2 Experiment at Fermilab uses a gaseous straw tracking detector to make detailed measurements of the stored muon beam profile, which are essential for the experiment to achieve its uncertainty goals. Positrons from muon decays spiral inward and pass through the tracking detector before striking an electromagnetic calorimeter. The tracking detector is therefore located inside the vacuum chamber in a region where the magnetic field is large and non-uniform. As such, the tracking detector must have a low leak rate to maintain a high-quality vacuum, must be non-magnetic so as not to perturb the magnetic field and, to minimize energy loss, must have a low radiation length. The performance of the tracking detector has met or surpassed the design requirements, with adequate electronic noise levels, an average straw hit resolution of (110 ± 20) μm, a detection efficiency of 97% or higher, and no performance degradation or signs of aging. The tracking detector's measurements result in an otherwise unachievable understanding of the muon's beam motion, particularly at early times in the experiment's measurement period when there are a significantly greater number of muons decaying. This is vital to the statistical power of the experiment, as well as facilitating the precise extraction of several systematic corrections and uncertainties. This paper describes the design, construction, testing, commissioning, and performance of the tracking detector.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

Dark photon dark matter in the presence of inhomogeneous structure

Dark photon dark matter will resonantly convert into visible photons when the dark photon mass is equal to the plasma frequency of the ambient medium. In cosmological contexts, this transition leads to an extremely efficient, albeit short-lived, heating of the surrounding gas. Existing work in this field has been predominantly focused on understanding the implications of these resonant transitions in the limit that the plasma frequency of the Universe can be treated as being perfectly homogeneous, \ie neglecting inhomogeneities in the electron number density. In this work we focus on the implications of heating from dark photon dark matter in the presence of inhomogeneous structure (which is particularly relevant for dark photons with masses in the range $10^{-15} \, {\rm eV} \, \lesssim m_{A^\prime} \lesssim 10^{-12}$ eV), emphasizing both the importance of inhomogeneous energy injection, as well as the sensitivity of cosmological observations to the inhomogeneities themselves. More specifically, we derive modified constraints on dark photon dark matter from the Ly-$\alpha$ forest, and show that the presence of inhomogeneities allows one to extend constraints to masses outside of the range that would be obtainable in the homogeneous limit, while only slightly relaxing their strength. We then project sensitivity for near-future cosmological surveys that are hoping to measure the 21cm transition in neutral hydrogen prior to reionization, and demonstrate that these experiments will be extremely useful in improving sensitivity to masses near $\sim 10^{-14}$ eV, potentially by several orders of magnitude. Finally, we discuss implications for reionization, early star formation, and late-time $y$-type spectral distortions, and show that probes which are inherently sensitive to the inhomogeneous state of the Universe could resolve signatures unique to the light dark photon dark matter scenario, and thus offer a fantastic potential for a positive detection.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Measurement of the polarisation of single top quarks and antiquarks produced in the $t$-channel at $ \sqrt{s} $ = 13 TeV and bounds on the $tWb$ dipole operator from the ATLAS experiment

A simultaneous measurement of the three components of the top-quark and top-antiquark polarisation vectors in t-channel single-top-quark production is presented. This analysis is based on data from proton–proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 139 fb –1 , collected with the ATLAS detector at the LHC. Selected events contain exactly one isolated electron or muon, large missing transverse momentum and exactly two jets, one being b-tagged. Stringent selection requirements are applied to discriminate t-channel single-top-quark events from the background contributions. The top-quark and top-antiquark polarisation vectors are measured from the distributions of the direction cosines of the charged-lepton momentum in the top-quark rest frame. The three components of the polarisation vector for the selected top-quark event sample are P x' = 0.01 ± 0.18, P y' = –0.029 ± 0.027, P z' = 0.91 ± 0.10 and for the top-antiquark event sample they are P x' = –0.02 ± 0.20, P y' = –0.007 ± 0.051, P z' = 0.79 ± 0.16. Normalised differential cross-sections corrected to a fiducial region at the stable-particle level are presented as a function of the charged-lepton angles for top-quark and top-antiquark events inclusively and separately. These measurements are in agreement with Standard Model predictions. The angular differential cross-sections are used to derive bounds on the complex Wilson coefficient of the dimension-six O tW operator in the framework of an effective field theory. The obtained bounds are C tW ϵ [–0.9, 1.4] and C itW ϵ [–0.8, 0.2], both at 95% confidence level.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

The very forward CASTOR calorimeter of the CMS experiment

The physics motivation, detector design, triggers, calibration, alignment, simulation, and overall performance of the very forward CASTOR calorimeter of the CMS experiment are reviewed. The CASTOR Cherenkov sampling calorimeter is located very close to the LHC beam line, at a radial distance of about 1 cm from the beam pipe, and at 14.4 m from the CMS interaction point, covering the pseudorapidity range of $-$6.6 $\lt\eta\lt$ $-$5.2. It was designed to withstand high ambient radiation and strong magnetic fields. The performance of the detector in measurements of forward energy density, jets, and processes characterized by rapidity gaps, is reviewed using data collected in proton and nuclear collisions at the LHC.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

Theory, phenomenology, and experimental avenues for dark showers: a Snowmass 2021 report

In this work, we consider the case of a strongly coupled dark/hidden sector, which extends the Standard Model (SM) by adding an additional non-Abelian gauge group. These extensions generally contain matter fields, much like the SM quarks, and gauge fields similar to the SM gluons. We focus on the exploration of such sectors where the dark particles are produced at the LHC through a portal and undergo rapid hadronization within the dark sector before decaying back, at least in part and potentially with sizeable lifetimes, to SM particles, giving a range of possibly spectacular signatures such as emerging or semi-visible jets. Other, non-QCD-like scenarios leading to soft unclustered energy patterns or glueballs are also discussed. After a review of the theory, existing benchmarks and constraints, this work addresses how to build consistent benchmarks from the underlying physical parameters and present new developments for the PYTHIA Hidden Valley module, along with jet substructure studies. Finally, a series of improved search strategies is presented in order to pave the way for a better exploration of the dark showers at the LHC.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

New Horizons: Scalar and Vector Ultralight Dark Matter

The last decade has seen unprecedented effort in dark matter model building at all mass scales coupled with the design of numerous new detection strategies. Transformative advances in quantum technologies have led to a plethora of new high-precision quantum sensors and dark matter detection strategies for ultralight ($<10\,$eV) bosonic dark matter that can be described by an oscillating classical, largely coherent field. This white paper focuses on searches for wavelike scalar and vector dark matter candidates.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2023

Search for new physics using effective field theory in 13 TeV pp collision events that contain a top quark pair and a boosted Z or Higgs boson

A data sample containing top quark pairs ($\mathrm{t\bar{t}}$) produced in association with a Lorentz-boosted Z or Higgs boson is used to search for signs of new physics using effective field theory. The data correspond to an integrated luminosity of 138 fb$^{-1}$ of proton-proton collisions produced at a center-of-mass energy of 13 TeV at the LHC and collected by the CMS experiment. Selected events contain a single lepton and hadronic jets, including two identified with the decay of bottom quarks, plus an additional large-radius jet with high transverse momentum identified as a Z or Higgs boson decaying to a bottom quark pair. Machine learning techniques are employed to discriminate between $\mathrm{t\bar{t}}$Z or $\mathrm{t\bar{t}}$H events and events from background processes, which are dominated by $\mathrm{t\bar{t}}$ + jets production. No indications of new physics are observed. The signal strengths of boosted $\mathrm{t\bar{t}}$Z and $\mathrm{t\bar{t}}$H production are measured, and upper limits are placed on the $\mathrm{t\bar{t}}$Z and $\mathrm{t\bar{t}}$H differential cross sections as functions of the Z or Higgs boson transverse momentum. The effects of new physics are probed using a framework in which the standard model is considered to be the low-energy effective field theory of a higher energy scale theory. Eight possible dimension-six operators are added to the standard model Lagrangian and their corresponding coefficients are constrained via fits to the data.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Quantum Computing for High-Energy Physics: State of the Art and Challenges

Quantum computers offer an intriguing path for a paradigmatic change of computing in the natural sciences and beyond, with the potential for achieving a so-called quantum advantage—namely, a significant (in some cases exponential) speedup of numerical simulations. The rapid development of hardware devices with various realizations of qubits enables the execution of small-scale but representative applications on quantum computers. In particular, the high-energy physics community plays a pivotal role in accessing the power of quantum computing, since the field is a driving source for challenging computational problems. This concerns, on the theoretical side, the exploration of models that are very hard or even impossible to address with classical techniques and, on the experimental side, the enormous data challenge of newly emerging experiments, such as the upgrade of the Large Hadron Collider. In this Roadmap paper, led by CERN, DESY, and IBM, we provide the status of high-energy physics quantum computations and give examples of theoretical and experimental target benchmark applications, which can be addressed in the near future. Having in mind hardware with about 100 qubits capable of executing several thousand two-qubit gates, where possible, we also provide resource estimates for the examples given using error-mitigated quantum computing. The ultimate declared goal of this task force is therefore to trigger further research in the high-energy physics community to develop interesting use cases for demonstrations on near-term quantum computers.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

$|\Delta \mathcal{B}| =2$: A State of the Field, and Looking Forward--A brief status report of theoretical and experimental physics opportunities

The origin of the matter-antimatter asymmetry apparently obligates the laws of physics to include some mechanism of baryon number ($\mathcal{B}$) violation. Searches for interactions violating $\mathcal{B}$ and baryon-minus-lepton number $\mathcal{(B-L)}$ represent a rich and underutilized opportunity. These are complementary to the existing, broad program of searches for $\mathcal{L}$-violating modes such as neutrinoless double $\beta$-decay which could provide deeper understandings of the plausibility of leptogenesis, or $\mathcal{B}$-violating, $\mathcal{(B-L)}$-conserving processes such as proton decay. In particular, a low-scale, post-sphaleron violation mechanism of $\mathcal{(B-L)}$ could provide a \textit{testable} form of baryogenesis. Though theoretically compelling, searches for such $\mathcal{(B-L)}$-violating processes like $\Delta\mathcal{B}=2$ dinucleon decay and $n\rightarrow\bar{n}$ remain relatively underexplored experimentally compared to other rare processes. By taking advantage of upcoming facilities such as the Deep Underground Neutrino Experiment and the European Spallation Source, this gap can be addressed with new intranuclear and free searches for neutron transformations with very high sensitivity, perhaps greater than three orders of magnitude higher than previous experimental searches. This proceedings reports on recent theoretical and experimental advances and sensitivities of next-generation searches for neutron transformations were detailed as part of the Amherst Center for Fundamental Interactions Workshop, "Theoretical Innovations for Future Experiments Regarding Baryon Number Violation," directly coordinated with the Rare Processes and Precision Measurements Frontier.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Measurement of the properties of Higgs boson production at $\sqrt{s}$ = 13 TeV in the H → γγ channel using 139 fb -1 of pp collision data with the ATLAS experiment

Measurements of Higgs boson production cross-sections are carried out in the diphoton decay channel using 139 fb -1 of pp collision data at $\sqrt{s}$ = 13 TeV collected by the ATLAS experiment at the LHC. The analysis is based on the definition of 101 distinct signal regions using machine-learning techniques. The inclusive Higgs boson signal strength in the diphoton channel is measured to be $1.04^{+0.10}_{-0.09}$. Cross-sections for gluon-gluon fusion, vector-boson fusion, associated production with a W or Z boson, and top associated production processes are reported. An upper limit of 10 times the Standard Model prediction is set for the associated production process of a Higgs boson with a single top quark, which has a unique sensitivity to the sign of the top quark Yukawa coupling. Higgs boson production is further characterized through measurements of Simplified Template Cross-Sections (STXS). In total, cross-sections of 28 STXS regions are measured. The measured STXS cross-sections are compatible with their Standard Model predictions, with a p-value of 93%. The measurements are also used to set constraints on Higgs boson coupling strengths, as well as on new interactions beyond the Standard Model in an effective field theory approach. No significant deviations from the Standard Model predictions are observed in these measurements, which provide significant sensitivity improvements compared to the previous ATLAS results.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Collective dynamics of polarized spin-half fermions in relativistic heavy-ion collisions

Standard relativistic hydrodynamics has been successful in describing the properties of the strongly interacting matter produced in the heavy-ion collision experiments. Recently, there has been a significant theoretical advancement in this field to explain spin polarization of hadrons emitted in these processes. Although current models have successfully explained some of the experimental data based on the coupling between spin polarization and vorticity of the medium, they still lack a clear understanding of the differential measurements. This is commonly interpreted as an indication that the spin needs to be treated as an independent degree of freedom whose dynamics is not entirely bound to flow circulation. In particular, if the spin is a macroscopic property of the system, in equilibrium its dynamics should follow hydrodynamic laws. Here, we develop a framework of relativistic hydrodynamics which includes spin degrees of freedom from the quantum kinetic theory for Dirac fermions and use it for modeling the dynamics of matter. Following experimental observations, we assume that the polarization effects are small and derive conservation laws for the net baryon current, the energy–momentum tensor and the spin tensor based on the de Groot–van Leeuwen–van Weert definitions of these currents. We present various properties of the spin polarization tensor and its components, analyze the propagation properties of the spin polarization components, and derive the spin-wave velocity for arbitrary statistics. We find that only the transverse spin components propagate, analogously to the electromagnetic waves. Finally, using our framework, we study the space–time evolution of the spin polarization for the systems respecting certain space–time symmetries and calculate the mean spin polarization per particle, which can be compared to the experimental data. We find that, for some observables, our spin polarization results agree qualitatively with the experimental findings and other model calculations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗