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DOE OSTI2020

Monte Carlo Global QCD Analyses of the Pion Parton Distribution Functions

As the lightest hadron, the pion presents itself as a dichotomy. While being the pseudo Goldstone boson associated with chiral symmetry breaking, it is simultaneously regarded as the lightest pseudoscalar meson typically composed of constituent quark and antiquark. As the chiral symmetry shares the same global symmetry as QCD, studying the pion and its internal structure is key in understanding one of the four fundamental forces of nature, namely the quantum chromodynamic strong interaction. We study the pion's parton distribution functions (PDFs), which are universal quantities that describe the structure of the pion in terms of its constituent quarks, antiquarks, and gluons. Our goal is to discover what the available data reveal for these universal quantities, namely the PDFs. Through the use of factorization theorems and perturbative QCD, we use Monte Carlo (MC) methods to extract the PDFs from the available Drell-Yan (DY) and leading neutron (LN) data. While the DY process involves two hadrons colliding, the LN electroproduction involves an electron beam incident on a target nucleon. We can constrain well the pion PDFs at large momentum fraction using DY data, and at low momentum fraction using LN data. We also use threshold resummation in the DY process to gather predictable higher order terms associated with the soft gluon radiation because their contributions to the cross section are nontrivial. In this dissertation, we parametrize the pion PDFs and fit those parameters to the available data. We make use of Bayesian inference and use MC techniques to sample the parameter space. Three sets of results are presented. We first extract the pion PDFs by fitting the DY and LN data. Then, we include transverse momentum dependent DY data in conjunction with the DY and LN data to extract the pion PDFs. Finally, we apply various methods of threshold resummation to the DY cross section and extract pion PDFs. The pion PDFs presented here are the first global QCD analyses performed as well as the first MC extracted pion PDFs, which are at the forefront of the pion PDF community.

Barry, Patrick↗

DOE OSTI2017

Dispersion relation for hadronic light-by-light scattering: two-pion contributions

In our third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon (g - 2) μ , including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of γ*γ* → ππ. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, a$π-box\atop{μ}$ =-15.9(2) × 10 -11 . As an application of the partial-wave formalism, we present a first calculation of ππ-rescattering effects in HLbL scattering, with γ*γ* → ππ helicity partial waves constructed dispersively using ππ phase shifts derived from the inverse-amplitude method. In this way, the isospin-0 part of our calculation can be interpreted as the contribution of the f0(500) to HLbL scattering in (g - 2) μ . We also argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its S-wave rescattering corrections reads a$π-box\atop{μ}$ + a$ππ, π-pole LHC\atop{μ, J=0}$ = -24(1) × 10 -11 .

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Measurement of the Pion Exclusive Electro-Production Cross-Section in the E12-19-006 Experiment in Hall-C at Jefferson Lab

One of the most effective methods for exploring the transition from hadronic degrees of freedom to quark-gluon degrees of freedom in Quantum Chromodynamics (QCD) is through the investigation of \exclusive" pion and kaon electro-production reactions at various Q2 and ?t values. The E12-19-006 experiment is conducted within the confi?nes of experimental Hall C at the Thomas Jefferson National Accelerator Facility, USA, for such studies. The primary aim of the experiment is to ?first enhance our comprehension of the pion electro-production cross-section and its form factor at Q2 = 0.38 and 0.42 GeV2. This is the fi?rst run period of the E12-19-006 experiment which ran in summer 2019. A more profound understanding of the pion electro-production reaction, 1H(e,e'?+)n, at low Q2 is deemed essential to employ this electro-production reaction (an indirect technique) for the high Q2 studies, thereby delving deeper into the realm of QCD. Consequently, this dissertation presents a thorough analysis of the experimental data acquired in the ?first run period of the E12-19-006 experiment. In pursuit of precision, a series of systematic studies (target boiling correction study, the elastic reaction cross-section measurements, study for determining vari- ous kinematics o?sets, etc.) are conducted to discern the accuracy of the analyzed data, a prerequisite for the use of Rosenbluth separation technique to separate the pion electro-production cross-section terms in t bins. The separated pion electro-production cross-section through the Rosenbluth separation technique is then used to extract the pion electromagnetic form factor. In this dissertation, the pion electro-production cross-section is carefully dissected into its four constituent components: longitudinal (?L), transverse (?T ), longitudinal-transverse (?LT ), and transverse-transverse (?TT ), using the full version of Rosenbluth separation technique for the Q2 = 0.38 GeV2. The technique is simultaneously fi?tted to the unseparated pion electro-production cross-sections at the three values of polarization of the virtual photon (?), i.e., ? = 0.286, 0.629 and 0.781. An iterative process is applied to re?ne the parameters of the model cross-sections until the yield ratio of experimental and Monte Carlo simulation converges. In this study, 21 iterations are conducted to re?ne the model cross-section parameters. The fi?nal pion electro-production cross-section terms are then determined for 7 t bins using the optimized parameters of the model cross-sections.

Kumar, Vijay↗

DOE OSTI2024

Measurement of the differential cross section for neutral pion production in charged-current muon neutrino interactions on argon with the MicroBooNE detector

We present a measurement of neutral pion production in charged-current interactions using data recorded with the MicroBooNE detector exposed to Fermilab’s booster neutrino beam. The signal comprises one muon, one neutral pion, any number of nucleons, and no charged pions. Studying neutral pion production in the MicroBooNE detector provides an opportunity to better understand neutrino-argon interactions, and is crucial for future accelerator-based neutrino oscillation experiments. Using a dataset corresponding to 6.86 ×10 20 protons on target, we present single-differential cross sections in muon and neutral pion momenta, scattering angles with respect to the beam for the outgoing muon and neutral pion, as well as the opening angle between the muon and neutral pion. Data extracted cross sections are compared to generator predictions. We report good agreement between the data and the models for scattering angles, except for an over-prediction by generators at muon forward angles. Similarly, the agreement between data and the models as a function of momentum is good, except for an underprediction by generators in the medium momentum ranges, 200–400 MeV for muons and 100–200 MeV for pions.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2017

Measurement of Charged Current Coherent Pion Production by Neutrinos on Carbon at MINER$\nu$A

Neutrino-nucleus coherent pion production is a rare neutrino scattering process where the squared four-momentum transferred to the nucleus is small, a lepton and pion are produced in the forward direction, and the nucleus remains in its initial state. This process is an important background in neutrino oscillation experiments. Measurements of coherent pion production are needed to constrain models which are used to predict coherent pion production in oscillation experiments. This thesis reports measurements of νµ and νµ charged current coherent pion production on carbon for neutrino energies in the range 2 < Eν < 20 GeV. The measurements were made using data from MINERνA, which is a dedicated neutrino-nucleus scattering experiment that uses a fi scintillator tracking detector in the high-intensity NuMI neutrino beam at Fermilab. Coherent interactions were isolated from the data using only model-independent signatures of the reaction, which are a forward muon and pion, no evidence of nuclear breakup, and small four-momentum transfer to the nucleus. The measurements were compared to the coherent pion production model used by oscillation experiments. The data and model agree in the total interaction rate and are similar in the dependence of the interaction rate on the squared four- momentum transferred from the neutrino. The data and model disagree significantly in the pion kinematics. The measured νµ and νµ interaction rates are consistent, which supports model predictions that the neutrino and antineutrino interaction rates are equal.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2023

DDFRG2: Double-Differential FRaGmentation Model with Pion Production

A set of Double-Differential FRaGmentation (DDFRG) models for production of fragments from nucleon-nucleus and nucleus-nucleus collisions, relevant to space radiation, is currently being developed. DDFRG1 was a previous model developed for production of protons and light ions. A new model for pion production is developed in the present work and is called DDFRG2, which is based upon thermal production of pions from a central fireball source. The Lorentzinvariant double-differential cross section is calculated in the central fireball frame, and is Lorentz transformed to the laboratory frame, which results in a closed-form analytic formula involving no numerical integration, and which can be run very efficiently in radiation transport codes. The Lorentz-invariant double-differential cross section is then integrated over angle to give the singledifferential energy spectral distribution. Two versions of the DDFRG2 pion production are developed, namely DDFRG2-SIMPLE and DDFRG2-FULL. Both versions produce pion production Lorentz-invariant double-differential cross sections and also pion production single-differential spectral (energy) distributions. The SIMPLE version contains only one adjustable parameter. The FULL version contains three additional adjustable parameters. Both DDFRG2 versions are compared to over 1,000 data points. The SIMPLE version describes the p + 12 C → π − much better than the current HZETRN pion model. For other reactions, the SIMPLE version is of comparable quality to the current HZETRN pion model compared to data. For both the p + 12 C → π − reaction and all other reactions, DDFRG2-FULL describes the data much more accurately than the current HZETRN pion model.

John W. Norbury↗

DOE OSTI2017

Gravitational wave from dark sector with dark pion

In this work, we investigate the spectra of gravitational waves produced by chiral symmetry breaking in dark quantum chromodynamics (dQCD) sector. The dark pion (π) can be a dark matter candidate as weakly interacting massive particle (WIMP) or strongly interacting massive particle (SIMP). For a WIMP scenario, we introduce the dQCD sector coupled to the standard model (SM) sector with classical scale invariance and investigate the annihilation process of the dark pion via the 2π → 2 SM process. For a SIMP scenario, we investigate the 3π → 2π annihilation process of the dark pion as a SIMP using chiral perturbation theory. We find that in the WIMP scenario the gravitational wave background spectra can be observed by future space gravitational wave antennas. On the other hand, when the dark pion is the SIMP dark matter with the constraints for the chiral perturbative limit and pion-pion scattering cross section, the chiral phase transition becomes crossover and then the gravitational waves are not produced.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Off-shell pion properties: Electromagnetic form factors and light-front wave functions

The off-shell pion electromagnetic form factors are explored with corresponding off-shell light-front wave functions modeled by constituent quark and antiquark. We apply the Mandelstam approach for the microscopic computation of the form factors relating the model parameters with the pion decay constant and charge radius. Analyzing the existing data on the cross sections for the Sullivan process, 1 H(e,e',π + )⁢n, Charged pion form factor between Q 2 = 0.60, and 2.45 GeV 2 . I. Measurements of the cross section for the 1 H⁡(e,e'⁢π + )⁢n reaction, we extract the off-shell pion form factor using the relation derived from the generalized Ward-Takahashi identity for the pion electromagnetic current. They are compared with our previous results from exactly solvable manifestly covariant model of a (3+1)-dimensional fermion field theory. We find that the adopted constituent quark model reproduces the extracted off-shell form factor F 1 ⁡(Q 2 ,t) from the experimental data within a few percent difference and matches well with our previous theoretical simulation which exhibits a variation of about 10% for the extracted off-shell pion form factor g(Q 2 ,t). We also identify the pion valence parton distribution function (PDF) and transverse momentum distribution (TMD) in terms of the light-front wave function and discuss their off-shell properties.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Bulk viscosity of nuclear matter with pions in the neutrino-trapped regime

Recent work has shown that a population of thermal pions could modify the equation of state and transport properties of hot and dense neutron-rich matter and introduce new reaction pathways to change the proton fraction. In this article we study their impact on the bulk viscosity of dense matter, focusing on the neutrino-trapped regime that would be realized in neutron star mergers and supernovae. We find that the presence of a thermal population of pions alters the bulk viscosity by modifying the EoS (via the susceptibilities) and by providing new reaction pathways to achieve beta equilibrium. In neutron star merger conditions, the bulk viscosity in neutrino-trapped npeμ matter (without pions) has its peak at temperatures of at most a couple MeV and is quite small at temperatures of tens of MeV. We find that thermal pions enhance the low-temperature peak of the bulk viscosity by a factor of a few and shift it to slightly lower temperatures. At higher temperatures, where the pion abundance is large but the bulk viscosity is traditionally small, pions can increase the bulk viscosity by an order of magnitude or more, although it is still orders of magnitude smaller than its peak value.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2020

Pion and kaon parton distribution functions from basis light front quantization and QCD evolution

We investigate the parton distribution functions (PDFs) of the pion and the kaon by combining quantum chromodynamics (QCD) evolution with the basis light front quantization. The initial PDFs result from the light front wave functions obtained by diagonalizing the effective Hamiltonian consisting of the holographic QCD confinement potential, a complementary longitudinal confinement potential, and the color-singlet Nambu–Jona-Lasinio interactions. The valence-quark PDF of the pion, after QCD evolution, is consistent with the result from the E-0615 experiment at Fermilab. Meanwhile, the pion structure function calculated from the PDFs agrees with the ZEUS and the H1 experiments at DESY-HERA for large x. Additionally, the ratio of the up quark PDF of the kaon to that of the pion is in agreement with the NA-003 experiment at CERN. Finally, we also present the cross section for the pion-nucleus induced Drell-Yan process with the obtained pion PDFs supplemented by the PDFs of the target nuclei.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Analysis of parton distributions in a pion with Bézier parametrizations

We explore the role of parametrizations for nonperturbative QCD functions in global analyses, with a specific application to extending a phenomenological analysis of the parton distribution functions (PDFs) in the charged pion realized in the xFitter fitting framework. The parametrization dependence of PDFs in our pion fits substantially enlarges the uncertainties from the experimental sources estimated in the previous analyses. We systematically explore the parametrization dependence by employing a novel technique to automate generation of polynomial parametrizations for PDFs that makes use of Bézier curves. This technique is implemented in a ++ module that is included in the xFitter program. Our analysis reveals that the sea and gluon distributions in the pion are not well disentangled, even when considering measurements in leading-neutron deep inelastic scattering. For example, the pion PDF solutions with a vanishing gluon and large quark sea are still experimentally allowed, which elevates the importance of ongoing lattice and nonperturbative QCD calculations, together with the planned pion scattering experiments, for conclusive studies of the pion structure. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Gluon parton distribution of the pion from lattice QCD

We present the first determination of the x-dependent pion gluon distribution from lattice QCD using the pseudo-PDF approach. We use lattice ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, at two lattice spacings a ≈ 0.12 and 0.15 fm and three pion masses M π ≈ 220, 310 and 690 MeV. We use clover fermions for the valence action and momentum smearing to achieve pion boost momentum up to 2.29 GeV. We find that the dependence of the pion gluon parton distribution on lattice spacing and pion mass is mild. We compare our results from the lightest pion mass ensemble with the determination by JAM and xFitter global fits.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

First simultaneous global QCD analysis of kaon and pion parton distributions with lattice QCD constraints

We perform the first simultaneous global QCD analysis of pion and kaon parton distribution functions (PDFs) constrained by pion- and kaon-induced Drell-Yan (DY) and leading neutron electroproduction data together with lattice QCD data on pion and kaon PDF moments. The analysis indicates a softer valence u ¯ distribution in the K − than in the π − and a significantly more peaked valence s -quark density in K − compared with the u ¯ . The effective exponent governing the high- x behavior of the PDF is found to be larger for u ¯ in the kaon, β u ¯ K − = 1.6 ( 2 ) , than in the pion, β u ¯ π − = 1.16 ( 4 ) , in the range 0.7 ≤ x ≤ 0.95 . From the gluon momentum fractions, we find the pion’s gluon content accounts for ≈ 1 / 3 of the mass budget of the pion at μ = 2 GeV , but only ≈ 1 / 4 for the kaon.

Barry, P. C. [Argonne National Laboratory] (ORCID:↗

DOE OSTI2019

Electroweak pion production on nuclei within the extended factorization scheme

We have applied the extended factorization scheme to investigate the electroweak pion production on nuclei. The ANL-Osaka model, which was obtained by analyzing the data of πN, γN, N(e,e'π), and N(ν,μπ) reactions up to invariant mass W=2GeV, is used to generate the matrix elements of current operators relevant to pion production off the nucleon. Medium effects on the Δ (1232) component of the meson-exchange current are included by using a Δ-nucleus potential determined from the previous Δ-hole model studies of pion-nucleus reactions. Nuclear correlations in the initial target state and in the spectator system(s) are modeled using realistic hole spectral functions. As a first step, we show that the data of 12 C(e,e') up to the Δ (1232) region can be described reasonably well. The interplay between the pion production and two-body meson-exchange mechanisms is shown to be essential in improving the agreement with the data in the “dip” region, between the quasielastic and the Δ (1232) peaks. Predictions for 12 C(ν,μπ) have also been made. They can be used to estimate pion-emission rates in neutrino-nucleus cross section, which constitutes an important systematic uncertainty to the reconstructed neutrino energy. With further improvements of the Metropolis Monte Carlo techniques to account for final states comprised of more than two particles, our approach can be employed up to W=2 GeV, where two-pion production and higher mass nucleon resonances must be included for analyzing the data from accelerator-based neutrino-oscillation experiments.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Pions in hot dense matter and their astrophysical implications

Here we study the role of pions in hot dense matter encountered in astrophysics. We find that strong interactions enhance the number density of negatively charged pions and that this enhancement can be calculated reliably for a relevant range of density and temperature by using the virial expansion. We assess the influence of pions and muons on the equation of state (EOS) and weak-interaction rates in hot dense matter. We find that thermal pions increase the proton fraction and soften the EOS. We also find that charged current weak reactions involving pions and muons $\bar{ν}_ μ + μ^– → π$ and $ν_μ + π^– → μ^–$ make an important contribution to the opacity of muon neutrinos. This could influence the dynamics of core-collapse supernovae and neutron-star mergers. Finally, we note that pion-nucleon reactions can alter the evolution of the proton fraction when weak interactions are not in equilibrium.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Photoproduction of pion pairs at high energy and small angles

Here, we explore the photoproduction mechanisms for charged and neutral pion pairs off a heavy nucleus at threshold. We calculate the production of charged pairs in the Coulomb field of the nucleus in the Born approximation using explicit expressions for the differential cross sections and their connection with the total cross section σ(γγ→π + π - ). The production of σ mesons at threshold and their subsequent decay into pions is an important mechanism and is investigated quantitatively in a framework that can be used for both charged and neutral pions. σ mesons can be produced via electromagnetic (photon exchange) or strong (ω exchange) interactions. For neutral pions, another production mechanism consists of the two-step process where charged pions are produced first and then charge exchange into two neutral pions. Differential and total cross sections are computed for each production mechanism.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2020

Spatial and momentum imaging of the pion and kaon

We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. Using these LFWFs we study the multidimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: ( b T 2 ) u π = 0.11 fm 2 , ( b T 2 ) s K = 0.08 fm 2 , and ( b T 2 ) u K = 0.13 fm 2 , and therefore the s quark in the kaon is much closer to the center of transverse momentum than the u quark. From the electromagnetic and gravitational form factors we find that the light-cone energy radii are about 60% smaller than the light-cone charge radii for each quark sector in the pion and kaon. A quantitative measure of the importance of the leading Fock state is obtained via comparison with a full Dyson-Schwinger equations calculation (containing an infinite tower of Fock states) for the pion form factor.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Lattice QCD calculation of 𝑥-dependent meson distribution amplitudes at physical pion mass with threshold logarithm resummation

We present a lattice quantum chromodynamics (QCD) calculation of the 𝑥-dependent pion and kaon distribution amplitudes (DA) in the framework of large momentum effective theory. This calculation is performed on a fine lattice of 𝑎 = 0.076 fm at physical pion mass, with the pion boosted to 1.8 GeV and kaon boosted to 2.3 GeV. We renormalize the matrix elements in the hybrid scheme and match to Math output error with a subtraction of the leading renormalon in the Wilson-line mass. The perturbative matching is improved by resumming the large logarithms related to the small quark and gluon momenta in the soft-gluon limit. After resummation, we demonstrate that we are able to calculate a range of 𝑥 ∈[𝑥 0 ,1 − 𝑥 0 ] with 𝑥 0 = 0.25 for pion and 𝑥 0 = 0.2 for kaon with theoretical systematic errors under control. The kaon DA is shown to be slighted skewed, and narrower than pion DA. Although the 𝑥-dependence cannot be direct calculated beyond these ranges, we estimate higher moments of the pion and kaon DAs by complementing our calculation with short-distance factorization.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗