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DOE OSTI2019

Nucleon-to-Roper electromagnetic transition form factors at large Q 2

High-precision nucleon-resonance electroproduction data on a large kinematic domain of energy and momentum transfer have proven crucial in revealing novel features of strong interactions within the Standard Model and unfolding structural details of baryon excited states. Thus, in anticipation of new data reaching to unprecedented photon virtuality, we employ a quark-diquark approximation to the three-valence-quark bound-state problem to compute γ*p → R + and γ*n → R 0 transition form factors on Q 2 /m 2 N ϵ[0,12], where m N is the nucleon mass. Having simultaneously analyzed both charged and neutral channels, we also provide a quark-flavor separation of the transition form factors. Furthermore, the results should be useful in planning new-generation experiments.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Observation of D s + → η ′ μ + ν μ , Precision Test of Lepton Flavor Universality with D s + → η ( ′ ) l + ν l , and First Measurements of D s + → η ( ′ ) μ + ν μ Decay Dynamics

By analyzing 7.33 fb − 1 of e + e − annihilation data collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector, we report the observation of the semileptonic decay D s + → η ′ μ + ν μ , with a statistical significance larger than 10 σ , and the measurements of the D s + → η μ + ν μ and D s + → η ′ μ + ν μ decay dynamics for the first time. The branching fractions of D s + → η μ + ν μ and D s + → η ′ μ + ν μ are determined to be ( 2.235 ± 0.05 1 stat ± 0.05 2 syst ) % and ( 0.801 ± 0.05 5 stat ± 0.02 8 syst ) % , respectively, with precision improved by factors of 6.0 and 6.6 compared to the previous best measurements. Combined with the results for the decays D s + → η e + ν e and D s + → η ′ e + ν e , the ratios of the decay widths are examined both inclusively and in several ℓ + ν ℓ four-momentum transfer ranges. No evidence for lepton flavor universality violation is found within the current statistics. The products of the hadronic form factors f + , 0 η ( ′ ) ( 0 ) and the c → s Cabibbo-Kobayashi-Maskawa matrix element | V c s | are determined. The results based on the two-parameter series expansion are f + , 0 η ( 0 ) | V c s | = 0.452 ± 0.01 0 stat ± 0.00 7 syst and f + , 0 η ′ ( 0 ) | V c s | = 0.504 ± 0.03 7 stat ± 0.01 2 syst , which help to constrain present models on f + , 0 η ( ′ ) ( 0 ) . The forward-backward asymmetries are determined to be ⟨ A FB η ⟩ = − 0.059 ± 0.03 1 stat ± 0.00 5 syst and ⟨ A FB η ′ ⟩ = − 0.064 ± 0.07 9 stat ± 0.00 6 syst for the first time, which are consistent with the theoretical calculation. Published by the American Physical Society 2024

Physics↗

DOE OSTI2023

Measurement of electrons from open heavy-flavor hadron decays in Au+Au collisions at $ \sqrt{s_{\textrm{NN}}} $ = 200 GeV with the STAR detector

We report a new measurement of the production of electrons from open heavy-flavor hadron decays (HFEs) at mid-rapidity (|y| < 0.7) in Au+Au collisions at $ \sqrt{s_{\textrm{NN}}} $ = 200 GeV. Invariant yields of HFEs are measured for the transverse momentum range of 3.5 < p T < 9 GeV/c in various configurations of the collision geometry. The HFE yields in head-on Au+Au collisions are suppressed by approximately a factor of 2 compared to that in p + p collisions scaled by the average number of binary collisions, indicating strong interactions between heavy quarks and the hot and dense medium created in heavy-ion collisions. Comparison of these results with models provides additional tests of theoretical calculations of heavy quark energy loss in the quark-gluon plasma.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2018

Coping with silent and fail-stop errors at scale by combining replication and checkpointing

This paper provides a model and an analytical study of replication as a technique to cope with silent errors, as well as a mixture of both silent and fail-stop errors on large-scale platforms. Compared with fail-stop errors that are immediately detected when they occur, silent errors require a detection mechanism. To detect silent errors, many application-specific techniques are available, either based on algorithms (e.g., ABFT), invariant preservation or data analytics, but replication remains the most transparent and least intrusive technique. We explore the right level (duplication, triplication or more) of replication for two frameworks: (i) when the platform is subject to only silent errors, and (ii) when the platform is subject to both silent and fail-stop errors. A higher level of replication is more expensive in terms of resource usage but enables to tolerate more errors and to even correct some errors, hence there is a trade-off to be found. Replication is combined with checkpointing and comes with two flavors: process replication and group replication. Process replication applies to message-passing applications with communicating processes. Each process is replicated, and the platform is composed of process pairs, or triplets. Group replication applies to black-box applications, whose parallel execution is replicated several times. The platform is partitioned into two halves (or three thirds). In both scenarios, results are compared before each checkpoint, which is taken only when both results (duplication) or two out of three results (triplication) coincide. Otherwise, one or more silent errors have been detected, and the application rolls back to the last checkpoint, as well as when fail-stop errors have struck. We provide a detailed analytical study for all of these scenarios, with formulas to decide, for each scenario, the optimal parameters as a function of the error rate, checkpoint cost, and platform size. We also report a set of extensive simulation results that nicely corroborates the analytical model.

Checkpointing↗

DOE OSTI2023

Search for supersymmetry in final states with a single electron or muon using angular correlations and heavy-object identification in proton-proton collisions at $ \sqrt{s} $ = 13 TeV

A search for supersymmetry is presented in events with a single charged lepton, electron or muon, and multiple hadronic jets. The data correspond to an integrated luminosity of 138 fb –1 of proton-proton collisions at a center-of-mass energy of 13 TeV, recorded by the CMS experiment at the CERN LHC. The search targets gluino pair production, where the gluinos decay into final states with the lightest supersymmetric particle (LSP) and either a top quark-antiquark ($\text{t}\bar{\text{t}}$ pair, or a light-flavor quark-antiquark ($\text{q}\bar{\text{q}}$ pair and a virtual or on-shell W boson. The main backgrounds, $\text{t}\bar{\text{t}}$ pair and W+jets production, are suppressed by requirements on the azimuthal angle between the momenta of the lepton and of its reconstructed parent W boson candidate, and by top quark and W boson identification based on a machine-learning technique. The number of observed events is consistent with the expectations from standard model processes. Limits are evaluated on supersymmetric particle masses in the context of two simplified models of gluino pair production. Exclusions for gluino masses reach up to 2120 (2050) GeV at 95% confidence level for a model with gluino decay to a $\text{t}\bar{\text{t}}$ pair (a $\text{q}\bar{\text{q}}$ pair and a W boson) and the LSP. For the same models, limits on the mass of the LSP reach up to 1250 (1070) GeV.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Chiral-odd generalized parton distributions in the large-𝑁 𝑐 limit of QCD: Spin-flavor structure, polynomiality, and sum rules

We study the nonperturbative properties of the nucleon’s chiral-odd generalized parton distributions (transversity GPDs) in the large-𝑁 𝑐 limit of QCD. This includes the parametric ordering of the spin-flavor components, the polynomiality property of the moments, and the sum rules connecting the GPDs with the tensor form factors. A multipole expansion in the transverse momentum transfer is used to enumerate and interpret the structures in the nucleon matrix element of the chiral-odd partonic operator, including monopole, dipole and quadrupole terms. The 1/𝑁 𝑐 expansion of the GPDs is performed using the abstract mean-field picture of baryons in the large-𝑁 𝑐 limit and its symmetries. We derive a large-𝑁 𝑐 relation between the flavor-nonsinglet GPDs 𝐸$^{𝑢−𝑑}_𝑇$ and $\tilde{𝐻}^{𝑢−𝑑}_𝑇$ and test it with recent lattice QCD results. We show that the polynomiality property and sum rules of the GPDs are fulfilled with the restricted realization of translational and rotational invariance in the mean-field picture. The results provide a basis for the phenomenological analysis of chiral-odd GPDs and hard exclusive processes in the large-𝑁 𝑐 limit, and for calculations in specific dynamical models.

generalized parton distributions↗

DOE OSTI2017

Measurement of the b -Quark Production Cross Section in 7 and 13 TeV p p Collisions

Measurements of the cross section for producing b quarks in the reaction pp→bbX are reported in 7 and 13 TeV collisions at the LHC as a function of the pseudorapidity η in the range 2 < η < 5 covered by the acceptance of the LHCb experiment. The measurements are done using semileptonic decays of b-flavored hadrons decaying into a ground-state charmed hadron in association with a muon. The cross sections in the covered η range are 72.0±0.3±6.8 and 154.3±1.5±14.3 μb for 7 and 13 TeV. The ratio is 2.14±0.02±0.13, where the quoted uncertainties are statistical and systematic, respectively. The agreement with theoretical expectation is good at 7 TeV, but differs somewhat at 13 TeV. The measured ratio of cross sections is larger at lower η than the model prediction.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2011

Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality

A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale {mu} of the running coupling {alpha}{sub s}({mu}{sup 2}): The purpose of the running coupling in any gauge theory is to sum all terms involving the {beta} function; in fact, when the renormalization scale is set properly, all non-conformal {beta} {ne} 0 terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with {beta} = 0. The resulting scale-fixed predictions using the 'principle of maximum conformality' (PMC) are independent of the choice of renormalization scheme - a key requirement of renormalization group invariance. The results avoid renormalon resummation and agree with QED scale-setting in the Abelian limit. The PMC is also the theoretical principle underlying the BLM procedure, commensurate scale relations between observables, and the scale-setting method used in lattice gauge theory. The number of active flavors nf in the QCD {beta} function is also correctly determined. We discuss several methods for determining the PMC/BLM scale for QCD processes. We show that a single global PMC scale, valid at leading order, can be derived from basic properties of the perturbative QCD cross section. The elimination of the renormalization scheme ambiguity using the PMC will not only increase the precision of QCD tests, but it will also increase the sensitivity of collider experiments to new physics beyond the Standard Model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2015

Spectroscopy of doubly charmed baryons from lattice QCD

This study presents the ground and excited state spectra of doubly charmed baryons from lattice QCD with dynamical quark fields. Calculations are performed on anisotropic lattices of size 16³ × 128, with inverse spacing in temporal direction a t ⁻¹=5.67(4) GeV and with a pion mass of about 390 MeV. A large set of baryonic operators that respect the symmetries of the lattice yet which retain a memory of their continuum analogues are used. These operators transform as irreducible representations of SU(3) F symmetry for flavor, SU(4) symmetry for Dirac spins of quarks and O(3) for spatial symmetry. The distillation method is utilized to generate baryon correlation functions which are analyzed using the variational fitting method to extract excited states. The lattice spectra obtained have baryonic states with well-defined total spins up to 7/2 and the pattern of low-lying states does not support the diquark picture for doubly charmed baryons. On the contrary the calculated spectra are remarkably similar to the expectations from models with an SU(6)×O(3) symmetry. Various spin-dependent energy splittings between the extracted states are also evaluated.

Padmanath, M.↗

NASA NTRS2020

Integrated Modeling and Development of Component-Based Embedded Software in Scala

Programming of embedded systems is challenging due to the low-level design patterns normally applied in the implementation of such systems. Furthermore, programming languages normally considered suitable for this level of programming, such as C and C++, are themselves low-level compared to more modern programming languages. We report on an effort exploring modeling and programming of embedded systems in modern high-level programming languages combining object-oriented and functional programming. We present an integration of four separate internal DSLs (libraries), considered useful for embedded program- ming, in the Scala programming language, for programming and testing component-based systems. These include a DSL for defining components and their connections, and a DSL for programming the individual components as hierarchical state machines. Two additional DSLs support testing, and include a DSL for writing temporal logic flavored test oracles for monitoring program executions, and a DSL for rule-based test input generation. The paper discusses the gap between Scala as used here and the needs for embedded systems programming.

Bocchino, Robert↗

DOE OSTI2018

Neutrino Oscillations in Supernovae

A (core-collapse) supernova is the final death call of a massive star during which a gigantic amount of a special kind of elementary particles known as neutrinos are released for a brief period of time. An earlier work by the PI and his collaborators has shown that the dense neutrino gas deep inside the supernova can experience "flavor transformation" or "oscillations" in a collective fashion, a macroscopic quantum phenomenon spanning over distances of 100-1000 kilometers. During this process neutrinos of different energies and propagation directions change their identities simultaneously. This phenomenon may be a key to understanding the mechanism of supernova explosion and the origin of many of the chemical elements found on the earth which were produced in the supernovae exploded in the past. The previous understanding of neutrino oscillations in supernovae was based on a stationary 3-dimensional model with 1 spatial dimension and 2 momentum dimensions. In this "neutrino bulb model" neutrinos are emitted from a spherically symmetric surface and they are decoupled from the motion of matter and the production of chemical elements. In order to accurately model the neutrino oscillations in supernovae and determine its physical consequences, sophisticated 7-dimensional (1 temporal dimension + 3 spatial dimensions + 3 momentum dimensions) modelling of neutrino transport must be performed self-consistently together with the hydrodynamic simulation of the flow of energy and matter inside the supernova and with the calculation of the production and destruction of chemical elements through a large nucleosynthesis network. Although it will be long journey to achieve the lofty goal, our project has made important progress in three different fronts towards this goal. In the first front we explored new computing hardware on which the next generation simulation codes may be run. We ported the neutrino bulb code to the platform of Graphics Processing Unit (GPU) and then later developed a new bulb code that can be run on supercomputers equipped with the Intel Xeon CPUs and/or the Xeon Phi coprocessors. In the second front we developed new algorithms that can be used towards computing neutrino oscillations in multi-dimensional models. We have developed a moment expansion method which is approximately 10 times more efficient than the old algorithm. We have also developed numerical codes to solve neutrino oscillations in toy models with two spatial dimensions. In the last front we used analytic methods to study neutrino oscillations in more sophisticated models, e.g. of multiple spatial dimensions and/or with time dependence. By analyzing the neutrino oscillations in toy models with two spacial dimensions we found that the approximate spacial translation symmetry, which is similar to the spherical symmetry in the neutrino bulb model, can be broken spontaneously as neutrino oscillations begin. By analyzing a time-dependent neutrino bulb model we found that certain modes of neutrino oscillations that are suppressed in the stationary model can become active in the time-dependent model. We have also developed theories which give insights to how ordinary matter may affect neutrino oscillations in supernovae and other astrophysical environments.

Duan, Huaiyu↗

DOE OSTI2019

Peccei-Quinn symmetry and nucleon decay in renormalizable SUSY SO(10)

We suggest simple ways of implementing Peccei-Quinn (PQ) symmetry to solve the strong CP problem in renormalizable SUSY SO(10) models with a minimal Yukawa sector. Realistic fermion mass generation requires that a second pair of Higgs doublets survive down to the PQ scale. We show how unification of gauge couplings can be achieved in this context. Higgsino mediated proton decay rate is strongly suppressed by a factor of (M PQ /M GUT ) 2 , which enables all SUSY particles to have masses of order TeV. With TeV scale SUSY spectrum, $p → \bar{ν}K^+$ decay rate is expected to be in the observable range. Lepton flavor violating processes µ → eγ decay and µ - e conversion in nuclei, induced by the Dirac neutrino Yukawa couplings, are found to be within reach of forthcoming experiments.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

The Future Circular Collider: a Summary for the US 2021 Snowmass Process

In this white paper for the 2021 Snowmass process, we give a description of the proposed Future Circular Collider (FCC) project and its physics program. The paper summarizes and updates the discussion submitted to the European Strategy on Particle Physics. After construction of an approximately 90 km tunnel, an electron-positron collider based on established technologies allows world-record instantaneous luminosities at center-of-mass energies from the Z resonance up to tt thresholds, enabling a rich set of fundamental measurements including Higgs couplings determinations at the sub percent level, precision tests of the weak and strong forces, and searches for new particles, including dark matter, both directly and via virtual corrections or mixing. Among other possibilities, the FCC-ee will be able to (i) indirectly discover new particles coupling to the Higgs and/or electroweak bosons up to scales around 7 and 50 TeV, respectively; (ii) perform competitive SUSY tests at the loop level in regions not accessible at the LHC; (iii) study heavy-flavor and tau physics in ultra-rare decays beyond the LHC reach, and (iv) achieve the best potential in direct collider searches for dark matter, sterile neutrinos, and axion-like particles with masses up to around 90 GeV. The tunnel can then be reused for a proton-proton collider, establishing record center-of-mass collision energy, allowing unprecedented reach for direct searches for new particles up to the around 50 TeV scale, and a diverse program of measurements of the Standard Model and Higgs boson, including a precision measurement of the Higgs self-coupling, and conclusively testing weakly-interacting massive particle scenarios of thermal relic dark matter.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2018

A light Higgs at the LHC and the B-anomalies

After the Higgs discovery, the LHC has been looking for new resonances, decaying into pairs of Standard Model (SM) particles. Recently, the CMS experimen observed an excess in the di-photon channel, with a di-photon invariant mass of about 96 GeV. This mass range is similar to the one of an excess observed in the search for the associated production of Higgs bosons with the Z neutral gauge boson at LEP, with the Higgs bosons decaying to bottom quark pairs. On the other hand, the LHCb experiment observed a discrepancy with respect to the SM expectations of the ratio of the decay of B -mesons to K -mesons and a pair of leptons, $ {R}_{K^{\left(\ast \right)}}=BR\left(B\to {K}^{\left(\ast \right)}{\mu}^{+}{\mu}^{-}\right)/BR\left(B\to {K}^{\left(\ast \right)}{e}^{+}{e}^{-}\right) $ . This observation provides a hint of the violation of lepton-flavor universality in the charged lepton sector and may be explained by the existence of a vector boson originating form a $ U{(1)_{L_{\mu}}}_{-{L}_{{}_{\tau }}} $ symmetry and heavy quarks that mix with the left-handed down quarks. Since the coupling to heavy quarks could lead to sizable Higgs di-photon rates in the gluon fusion channel, in this article we propose a common origin of these anomalies identifying a Higgs associated with the breakdown of the $ U{(1)_{L_{\mu}}}_{-{L}_{{}_{\tau }}} $ symmetry and at the same time responsible to the quark mixing, with the one observed at the LHC. We also discuss the constraints on the identification of the same Higgs with the one associated with the bottom quark pair excess observed at LEP.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

The Palomar Transient Factory Core-collapse Supernova Host-galaxy Sample. I. Host-galaxy Distribution Functions and Environment Dependence of Core-collapse Supernovae

Several thousand core-collapse supernovae (CCSNe) of different flavors have been discovered so far. However, identifying their progenitors has remained an outstanding open question in astrophysics. Studies of SN host galaxies have proven to be powerful in providing constraints on the progenitor populations. In this paper, we present all CCSNe detected between 2009 and 2017 by the Palomar Transient Factory. This sample includes 888 SNe of 12 distinct classes out to redshift z ≈ 1. We present the photometric properties of their host galaxies from the far-ultraviolet to the mid-infrared and model the host-galaxy spectral energy distributions to derive physical properties. The galaxy mass function of Type Ic, Ib, IIb, II, and IIn SNe ranges from 105 to 1011.5 M o˙, probing the entire mass range of star-forming galaxies down to the least-massive star-forming galaxies known. Moreover, the galaxy mass distributions are consistent with models of star-formation-weighted mass functions. Regular CCSNe are hence direct tracers of star formation. Small but notable differences exist between some of the SN classes. Type Ib/c SNe prefer galaxies with slightly higher masses (i.e., higher metallicities) and star formation rates than Type IIb and II SNe. These differences are less pronounced than previously thought. H-poor superluminous supernovae (SLSNe) and SNe Ic-BL are scarce in galaxies above 1010 M o˙. Their progenitors require environments with metallicities of < 0.4 and < 1 solar, respectively. In addition, the hosts of H-poor SLSNe are dominated by a younger stellar population than all other classes of CCSNe. Our findings corroborate the notion that low metallicity and young age play an important role in the formation of SLSN progenitors.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2006

Effects of gamma-Radiation on Select Lipids and Antioxidants

Radiation encountered on an extended duration space mission (estimates of 3 Sieverts for a mission to Mars) poses a threat not only to human health, but also to the quality, nutritional value, and palatability of the food system. Free radicals generated by radiation interaction with foods may initiate many unwanted reactions including: 1) autoxidation in lipids that alters flavor, odor, and concentrations of essential fatty acids, and 2) depletion of antioxidants food products and dietary supplements. Studies have shown that antioxidants may provide long term health protection from oxidative stress caused by radiation exposure; therefore, consumption of antioxidants will be important. Stability of essential fatty acids is also important for astronauts long-term health status. The objectives of this study were to characterize the effects of low dose gamma-radiation on lipids and antioxidants by monitoring oxidation and reducing power, respectively, in model systems. Select oils and antioxidants were exposed to levels of gamma-radiation ranging from 0 to 1000 Gy (1 Gy = 1 Sv) using a Gammacell 220 and stored at ambient or elevated temperatures (65 C) for up to 3 months prior to analysis. A Fricke dosimeter was used to verify differences between the radiation doses administered. Primary and secondary products of lipid oxidation in soybean and peanut oils were monitored using conjugated diene and 2-thiobarbituric acid (TBARs) assays. Changes in fatty acid composition and formation and vitamin E levels were also measured. The reducing power of antioxidant compounds, including vitamins C and E and beta-carotene, was determined using the ferric reducing antioxidant power (FRAP) assay. Significant differences (alpha =0.05) were present between all radiation doses tested using the Fricke dosimeter. Increasing radiation doses above 3 Sv resulted in significantly (alpha =0.05) elevated levels of oxidation and free fatty acids in soybean and peanut oils. Decreases in concentrations of essential fatty acids upon increasing radiation doses were also observed. Increasing radiation doses caused significant (alpha =0.05) decreases in reducing power and hence the effectiveness of vitamins C and E as well as beta-carotene. This work establishes a need for quantifying the effects of space relevant radiation doses in the development of a food system for an extended duration mission and for identifying threshold radiation levels that will impact the useful shelf-life of the variety of foods that will be sent. Eventual rancidity of lipids and the loss of antioxidant bioprotective effects are major concerns for the acceptability and nutritional profile of a food system.

Gandolph, Jacob↗

DOE OSTI2023

Different flavors of exact-factorization-based mixed quantum-classical methods for multistate dynamics

Here, the exact factorization approach has led to the development of new mixed quantum-classical methods for simulating coupled electron–ion dynamics. We compare their performance for dynamics when more than two electronic states are occupied at a given time, and analyze: (1) the use of coupled versus auxiliary trajectories in evaluating the electron–nuclear correlation terms, (2) the approximation of using these terms within surface-hopping and Ehrenfest frameworks, and (3) the relevance of the exact conditions of zero population transfer away from nonadiabatic coupling regions and total energy conservation. Dynamics through the three-state conical intersection in the uracil radical cation as well as polaritonic models in one dimension are studied.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

MUonE sensitivity to new physics explanations of the muon anomalous magnetic moment

The MUonE experiment aims at a precision measurement of the hadronic vacuum polarization contribution to the muon g – 2, via elastic muon-electron scattering. Since the current muon g – 2 anomaly hints at the potential existence of new physics (NP) related to the muon, the question then arises as to whether the measurement of hadronic vacuum polarization in MUonE could be affected by the same NP as well. In this work, we address this question by investigating a variety of NP explanations of the muon g – 2 anomaly via either vector or scalar mediators with either flavor-universal, non-universal or even flavor-violating couplings to electrons and muons. We derive the corresponding MUonE sensitivity in each case and find that the measurement of hadronic vacuum polarization at the MUonE is not vulnerable to any of these NP scenarios.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗