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A New Non-Inheriting Homogeneous Solution Of The Einstein-Maxwell Equations With Cosmological Term, Charles G. Torre 2022 Department of Physics, Utah State University

A New Non-Inheriting Homogeneous Solution Of The Einstein-Maxwell Equations With Cosmological Term, Charles G. Torre

Research Vignettes

No abstract provided.


Study Of The Decay B^± → K^0_S Π^± Π^0 At The Belle Experiment, Suravinda Janaka Kumara Kospalage 2022 Wayne State University

Study Of The Decay B^± → K^0_S Π^± Π^0 At The Belle Experiment, Suravinda Janaka Kumara Kospalage

Wayne State University Dissertations

Belle is a particle physics experiment based at the KEK laboratory in Tsukuba Japanwhich ran from 1999 to 2010 and collected 1ab−1 of data. The Belle experiment is focused on studying the properties of particles called B mesons which are produced by accelerating and colliding electron and positron beams. These B mesons show the biggest differences between the properties of matter and anti-matter of any known particles. One of the main goals of the Belle experiments is to understand the differences between matter and anti-matter, specifically violations of charge-parity symmetry (CP violation) and how anti-matter vanished and we come to …


Jet Substructure Measurements In Proton-Proton And Proton-Gold Collisions, Isaac Arthur Mooney 2022 Wayne State University

Jet Substructure Measurements In Proton-Proton And Proton-Gold Collisions, Isaac Arthur Mooney

Wayne State University Dissertations

Observing a modification of high-energy probes in the hot nuclear environment created by ultrarelativistic heavy-ion collisions is critical for understanding the microscopic structure of the QCD medium. However, before a modification can be observed, the probes must be calibrated. For this reason, measurements of jet observables in vacuum (pp collisions) and in the presence of cold nuclear matter (pAu collisions) are presented, which will allow for future comparison of jet substructure measurements in AuAu to these baseline measurements. \parData from the STAR detector on the Relativistic Heavy Ion Collider at Brookhaven National Laboratory are used for both analyses. The pp …


Toward The Determination Of The Gluon Helicity Distribution In The Nucleon From Lattice Quantum Chromodynamics, Colin Egerer, Bálint Joó, Joseph Karpie, Nikhil Karthik, Tanjib Khan, Christopher J. Monahan, Wayne Morris, Kostas Orginos, Anatoly Radyushkin, David G. Richards, Eloy Romero, Raza Sabbir Sufian, Savvas Zafeiropoulos 2022 Old Dominion University

Toward The Determination Of The Gluon Helicity Distribution In The Nucleon From Lattice Quantum Chromodynamics, Colin Egerer, Bálint Joó, Joseph Karpie, Nikhil Karthik, Tanjib Khan, Christopher J. Monahan, Wayne Morris, Kostas Orginos, Anatoly Radyushkin, David G. Richards, Eloy Romero, Raza Sabbir Sufian, Savvas Zafeiropoulos

Physics Faculty Publications

We present the first exploratory lattice quantum chromodynamics (QCD) calculation of the polarized gluon Ioffe-time pseudodistribution in the nucleon. The Ioffe-time pseudodistribution provides a frame-independent and gauge-invariant framework to determine the gluon helicity in the nucleon from first principles. We employ a high-statistics computation using a 323 × 64 lattice ensemble characterized by a 358 MeV pion mass and a 0.094 fm lattice spacing. We establish the pseudodistribution approach as a feasible method to address the proton spin puzzle with successive improvements in statistical and systematic uncertainties anticipated in the future. Within the statistical precision of our data, we …


Non-Perturbative Processes In Standard Model \& New Physics: From Standard Calculations To Machine Learning, Cody Michael Grant 2022 Wayne State University

Non-Perturbative Processes In Standard Model \& New Physics: From Standard Calculations To Machine Learning, Cody Michael Grant

Wayne State University Dissertations

I calculated non-perturbative processes to study the Standard Model and probe areas of new physics. I found that invisible decays of heavy mesons are not dominated by the leading order contribution in the Standard Model. I argue that this affects neutrino mass constraints and avenues for discovering new physics, such as dark photons. In addition to this, I used a novel approach to approximate a semileptonic form factor with an artificial neural network (ANN). Comparing the results from the ANN to common phenomenological parameterizations, and using it to set a lower bound constraint on the CKM matrix element, $V_{cd}$. Finally, …


Cooling Performance In A Dual Energy Storage Ring Cooler, B. Dhital, Y. S. Derbenev, D. Douglas, G. A. Krafft, H. Zhang, F. Lin, V. S. Morozov, Y. Zhang 2022 Old Dominion University

Cooling Performance In A Dual Energy Storage Ring Cooler, B. Dhital, Y. S. Derbenev, D. Douglas, G. A. Krafft, H. Zhang, F. Lin, V. S. Morozov, Y. Zhang

Physics Faculty Publications

The longitudinal and transverse emittance growth in hadron beams due to intra-beam scattering (IBS) and other heating sources deteriorate the luminosity in a collider. Hence, a strong hadron beam cooling is required to reduce and preserve the emittance. The cooling of high energy hadron beam is challenging. We propose a dual energy storage ring-based electron cooler that uses an electron beam to extract heat away from hadron beam in the cooler ring while the electron beam is cooled by synchrotron radiation damping in the high energy damping ring. In this paper, we present a design of a dual energy storage …


Defining Entanglement Without Tensor Factoring: A Euclidean Hourglass Prescription, Takanori Anegawa, Norihiro Iizuka, Daniel Kabat 2022 Osaka University

Defining Entanglement Without Tensor Factoring: A Euclidean Hourglass Prescription, Takanori Anegawa, Norihiro Iizuka, Daniel Kabat

Publications and Research

We consider entanglement across a planar boundary in flat space. Entanglement entropy is usually thought of as the von Neumann entropy of a reduced density matrix, but it can also be thought of as half the von Neumann entropy of a product of reduced density matrices on the left and right. The latter form allows a natural regulator in which two cones are smoothed into a Euclidean hourglass geometry. Since there is no need to tensor factor the Hilbert space, the regulated entropy is manifestly gauge invariant and has a manifest state-counting interpretation. We explore this prescription for scalar fields, …


Spin(7) Orientifolds And 2d N = (0, 1) Triality, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu 2022 CUNY City College

Spin(7) Orientifolds And 2d N = (0, 1) Triality, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu

Publications and Research

We present a new, geometric perspective on the recently proposed triality of 2d N = (0, 1) gauge theories, based on its engineering in terms of D1-branes probing Spin(7) orientifolds. In this context, triality translates into the fact that multiple gauge theories correspond to the same underlying orientifold. We show how Spin(7) orientifolds based on a particular involution, which we call the universal involution, give rise to precisely the original version of N = (0, 1) triality. Interestingly, our work also shows that the space of possibilities is significantly richer. Indeed, general Spin(7) orientifolds extend triality to theories that can …


2d N = (0, 1) Gauge Theories And Spin(7) Orientifolds, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu 2022 CUNY Graduate Center

2d N = (0, 1) Gauge Theories And Spin(7) Orientifolds, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu

Publications and Research

We initiate the geometric engineering of 2d N = (0, 1) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds. Spin(7) orientifolds explicitly realize the perspective on 2d N = (0, 1) theories as real slices of N = (0, 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by antiholomorphic …


Superluminal Propagation On A Moving Braneworld, Brian Greene, Daniel Kabat, Janna Levin, Arjun S. Menon 2022 Columbia University

Superluminal Propagation On A Moving Braneworld, Brian Greene, Daniel Kabat, Janna Levin, Arjun S. Menon

Publications and Research

We consider a braneworld scenario in the simplest setting, M4 × S1, with a four-dimensional (4D) Minkowski metric induced on the brane, and establish the possibility of superluminal propagation. If the brane is at rest, the 4D Lorentz symmetry of the brane is exact, but if the brane is in motion, it is broken globally by the compactification. By measuring bulk fields, an observer on the brane sees a slice through a higher-dimensional field profile, which carries an imprint of the extra dimensions even when the brane is at rest. If the brane is in motion, we …


Two-Loop Four-Fermion Scattering Amplitude In Qed, R. Bonciani, A. Broggio, S. Di Vita, A. Ferroglia, M. K. Mandal, P. Mastrolia, L. Mattiazzi, A. Primo, J. Ronca, U. Schubert, W. J. Torres Bobadilla, F. Tramontano 2022 Sapienza University of Rome

Two-Loop Four-Fermion Scattering Amplitude In Qed, R. Bonciani, A. Broggio, S. Di Vita, A. Ferroglia, M. K. Mandal, P. Mastrolia, L. Mattiazzi, A. Primo, J. Ronca, U. Schubert, W. J. Torres Bobadilla, F. Tramontano

Publications and Research

We present the first fully analytic evaluation of the transition amplitude for the scattering of a massless into a massive pair of fermions at the two-loop level in quantum electrodynamics. Our result is an essential ingredient for the determination of the electromagnetic coupling within scattering reactions, beyond the currently known accuracy, which has a crucial impact on the evaluation of the anomalous magnetic moment of the muon. It will allow, in particular, for a precise determination of the leading hadronic contribution to the (g − 2)μ in the MUonE experiment at CERN, and therefore can be used to …


A Template Method To Measure The $T \Bar T$ Polarisation, J. A. Aguilar-Saavedra, M. C. N. Fiolhais, P. Martín-Ramiro, J. M. Moreno, António Onofre 2022 Instituto de Física Teórica, IFT-UAM/CSIC

A Template Method To Measure The $T \Bar T$ Polarisation, J. A. Aguilar-Saavedra, M. C. N. Fiolhais, P. Martín-Ramiro, J. M. Moreno, António Onofre

Publications and Research

We develop a template method for the measurement of the polarisation of $t \bar t$ pairs produced in hadron collisions. The method would allow to extract the individual fractions of $t_L \bar t_L$, $t_R \bar t_R$, $t_L \bar t_R$ and $t_R \bar t_L$ pairs with a fit to data, where L, R refer to the polarisation along any axis. These polarisation fractions have not been independently measured at present. Secondarily, the method also provides the net polarisation of t and $\bar t$, as well as their spin correlation for arbitrary axes.


Short-Distance Structure Of Unpolarized Gluon Pseudodistributions, Ian Balitsky, Wayne Morris, Anatoly Radyushkin 2022 Old Dominion University

Short-Distance Structure Of Unpolarized Gluon Pseudodistributions, Ian Balitsky, Wayne Morris, Anatoly Radyushkin

Physics Faculty Publications

We present the results that form the basis for calculations of the unpolarized gluon parton distributions (PDFs) using the pseudo-PDF approach. We give the results for the most complicated box diagram both for gluon bilocal operators with arbitrary indices and for combinations of indices corresponding to three matrix elements that are most convenient to extract the twist-2 invariant amplitude. We also present detailed results for the gluon-quark transition diagram. The additional results for the box diagram and the gluon-quark contribution may be used for extractions of the gluon PDF from different matrix elements, with a possible cross-check of the results …


Polarized Gluon Pseudodistributions At Short Distances, Ian Balitsky, Wayne Morris, Anatoly Radyushkin 2022 Old Dominion University

Polarized Gluon Pseudodistributions At Short Distances, Ian Balitsky, Wayne Morris, Anatoly Radyushkin

Physics Faculty Publications

We formulate the basic points of the pseudo-PDF approach to the lattice calculation of polarized gluon PDFs. We present the results of our calculations of the one-loop corrections for the bilocal Gμα(z)G̃λβ(0) correlator of gluonic fields. Expressions are given for a general situation when all four indices are arbitrary, and also for specific combinations of indices corresponding to three matrix elements that contain the twist-2 invariant amplitude related to the polarized PDF. We study the evolution properties of these matrix elements, and derive matching relations between Euclidean and light-cone Ioffe-time distributions. These relations are necessary for …


Search For Photoproduction Of Axionlike Particles At Gluex, Shankar Adhikari, C.S. Akondi, M. Albrecht, Moskov Amaryan, Tyler P. Viducic, Z. Zhang, Z. Zhao, J. Zhou, The GLUEX Collaboration, et al. 2022 Old Dominion University

Search For Photoproduction Of Axionlike Particles At Gluex, Shankar Adhikari, C.S. Akondi, M. Albrecht, Moskov Amaryan, Tyler P. Viducic, Z. Zhang, Z. Zhao, J. Zhou, The Gluex Collaboration, Et Al.

Physics Faculty Publications

We present a search for axionlike particles, a, produced in photon-proton collisions at a center-of-mass energy of approximately 4 GeV, focusing on the scenario where the a-gluon coupling is dominant. The search uses a → γγ and a → π+π−π0 decays, and a data sample corresponding to an integrated luminosity of 168  pb−1 collected with the GlueX detector. The search for a → γγ decays is performed in the mass range of 180 < ma <480  MeV, while the search for a → π+π−π0 decays explores the 600 < ma …


Cebaf Injector Model For KL Beam Conditions, Sunil Pokharel, Geoffrey A. Krafft, A. S. Hofler, R. Kazimi, M. Bruker, J. Grames, S. Zhang 2022 Old Dominion University

Cebaf Injector Model For KL Beam Conditions, Sunil Pokharel, Geoffrey A. Krafft, A. S. Hofler, R. Kazimi, M. Bruker, J. Grames, S. Zhang

Physics Faculty Publications

The Jefferson Lab KL experiment will run at the Continuous Electron Beam Accelerator Facility with a much lower bunch repetition rate (7.80 or 15.59 MHz) than nominally used (249.5 or 499 MHz). While the proposed average current of 2.5 - 5.0 µA is relatively low compared to the maximum CEBAF current of approximately 180 µA, the corresponding bunch charge is atypically high for CEBAF injector operation. In this work, we investigated the evolution and transmission of low-rep-rate, high-bunch-charge (0.32 to 0.64 pC) beams through the CEBAF injector. Using the commercial software General Particle Tracer, we have simulated and analyzed the …


Study Of The Λ/Σ⁰ Electroproduction In The Low-Q² Region At Jlab, K. Okuyama, K. Itabashi, S. Nagao, S. N. Nakamura, K. N. Suzuki, T. Gogami, B. Pandey, L. Tang, D. Abrams, T. Akiyama, D. Androic, K. Aniol, C. Ayerbe Gayoso, J. Bane, S. Barcus, J. Barrow, V. Bellini, H. Bhatt, D. Bhetuwal, D. Biswas, A. Camsonne, J. Castellanos, J.- P. Chen, J. Chen, S. Covrig, D. Chrisman, R. Cruz-Torres, R. Das, E. Fuchey, K. Gnanvo, F. Garibaldi, T. Gautam, J. Gomez, P. Gueye, T. J. Hague, O. Hansen, W. Henry, Florian Hauenstein, D. W. Higinbotham, Charles E. Hyde, M. Kaneta, C. Keppel, T. Kutz, N. Lashley-Colthirst, S. Li, H. Liu, J. Mammei, P. Markowitz, R. E. McClellan, F. Meddi, D. Meekins, R. Michaels, M. Mihovilovic, A. Moyer, D. Nguyen, M. Nycz, V. Owen, C. Palatchi, S. Park, T. Petkovic, S. Premathilake, P. E. Reimer, J. Reinhold, S. Riordan, V. Rodriguez, C. Samanta, S. N. Santiesteban, B. Sawatzky, S. Širca, K. Slifer, T. Su, Y. Tian, Y. Toyama, K. Uehara, G. M. Urciuoli, D. Votaw, J. Williamson, B. Wojtsekhowski, S. A. Wood, B. Yale, Z. Ye, J. Zhang, X. Zheng 2022 Old Dominion University

Study Of The Λ/Σ⁰ Electroproduction In The Low-Q² Region At Jlab, K. Okuyama, K. Itabashi, S. Nagao, S. N. Nakamura, K. N. Suzuki, T. Gogami, B. Pandey, L. Tang, D. Abrams, T. Akiyama, D. Androic, K. Aniol, C. Ayerbe Gayoso, J. Bane, S. Barcus, J. Barrow, V. Bellini, H. Bhatt, D. Bhetuwal, D. Biswas, A. Camsonne, J. Castellanos, J.- P. Chen, J. Chen, S. Covrig, D. Chrisman, R. Cruz-Torres, R. Das, E. Fuchey, K. Gnanvo, F. Garibaldi, T. Gautam, J. Gomez, P. Gueye, T. J. Hague, O. Hansen, W. Henry, Florian Hauenstein, D. W. Higinbotham, Charles E. Hyde, M. Kaneta, C. Keppel, T. Kutz, N. Lashley-Colthirst, S. Li, H. Liu, J. Mammei, P. Markowitz, R. E. Mcclellan, F. Meddi, D. Meekins, R. Michaels, M. Mihovilovic, A. Moyer, D. Nguyen, M. Nycz, V. Owen, C. Palatchi, S. Park, T. Petkovic, S. Premathilake, P. E. Reimer, J. Reinhold, S. Riordan, V. Rodriguez, C. Samanta, S. N. Santiesteban, B. Sawatzky, S. Širca, K. Slifer, T. Su, Y. Tian, Y. Toyama, K. Uehara, G. M. Urciuoli, D. Votaw, J. Williamson, B. Wojtsekhowski, S. A. Wood, B. Yale, Z. Ye, J. Zhang, X. Zheng

Physics Faculty Publications

We performed an experiment using tritium and hydrogen cryogenic gas targets at Thomas Jefferson National Accelerator Facility (JLab) in 2018 (E12-17-003)[1, 2]. In this article, we discuss the Λ/Σ0 hyperon electroproduction from hydrogen target. Elementary Λ/Σ0 hyperon production processes are important not only for an absolute mass scale calibration in our experiment, but also for the study of the electroproduction mechanisms themselves. In this article, we reported the results of the differential cross section for the p(e, e’K+)Λ/Σ0 reaction at Q2 ∼ 0.5 (GeV/c)2.


Study Of ⋀N Fsi With ⋀ Quasi-Free Productions On The ³H (E, E'K⁺) X Reaction At Jlab, K. Itabashi, K. N. Suzuki, B. Pandey, K. Okuyama, T. Gogami, S. Nagao, S. N. Nakamura, L. Tang, D. Abrams, T. Akiyama, D. Androic, K. Aniol, C. Ayerbe Gayoso, J. Bane, S. Barcus, J. Barrow, V. Bellini, H. Bhatt, D. Bhetuwal, D. Biswas, A. Camsonne, J. Castellanos, J.-P. Chen, J. Chen, S. Covrig, D. Chrisman, R. Cruz-Torres, R. Das, E. Fuchey, K. Gnanvo, F. Garibaldi, T. Gautam, J. Gomez, P. Gueye, T. J. Hague, O. Hansen, W. Henry, F. Hauenstein, D. W. Higinbotham, C. E. Hyde, M. Kaneta, C. Keppel, T. Kutz, N. Lashley-Colthirst, S. Li, H. Liu, J. Mammei, P. Markowitz, R. E. McClellan, F. Meddi, D. Meekins, R. Michaels, M. Milhovilovic, A. Moyer, D. Nguyen, M. Nycz, V. Owen, C. Palatchi, S. Park, T. Petkovic, S. Premathilake, P. E. Reimer, J. Reinhold, S. Riordan, V. Rodriguez, C. Samanta, S. N. Santiesteban, B. Sawatzky, S. Širca, K. Slifer, T. Su, Y. Tian, Y. Toyama, K. Uehara, G. M. Urciuoli, D. Votaw, J. Williamson, B. Wojtsekhowski, S. A. Wood, B. Yale, Z. Ye, J. Zhang, X. Zheng 2022 Old Dominion University

Study Of ⋀N Fsi With ⋀ Quasi-Free Productions On The ³H (E, E'K⁺) X Reaction At Jlab, K. Itabashi, K. N. Suzuki, B. Pandey, K. Okuyama, T. Gogami, S. Nagao, S. N. Nakamura, L. Tang, D. Abrams, T. Akiyama, D. Androic, K. Aniol, C. Ayerbe Gayoso, J. Bane, S. Barcus, J. Barrow, V. Bellini, H. Bhatt, D. Bhetuwal, D. Biswas, A. Camsonne, J. Castellanos, J.-P. Chen, J. Chen, S. Covrig, D. Chrisman, R. Cruz-Torres, R. Das, E. Fuchey, K. Gnanvo, F. Garibaldi, T. Gautam, J. Gomez, P. Gueye, T. J. Hague, O. Hansen, W. Henry, F. Hauenstein, D. W. Higinbotham, C. E. Hyde, M. Kaneta, C. Keppel, T. Kutz, N. Lashley-Colthirst, S. Li, H. Liu, J. Mammei, P. Markowitz, R. E. Mcclellan, F. Meddi, D. Meekins, R. Michaels, M. Milhovilovic, A. Moyer, D. Nguyen, M. Nycz, V. Owen, C. Palatchi, S. Park, T. Petkovic, S. Premathilake, P. E. Reimer, J. Reinhold, S. Riordan, V. Rodriguez, C. Samanta, S. N. Santiesteban, B. Sawatzky, S. Širca, K. Slifer, T. Su, Y. Tian, Y. Toyama, K. Uehara, G. M. Urciuoli, D. Votaw, J. Williamson, B. Wojtsekhowski, S. A. Wood, B. Yale, Z. Ye, J. Zhang, X. Zheng

Physics Faculty Publications

An nnΛ is a neutral baryon system with no charge. The study of the pure Λ-neutron system such as nnΛ gives us information on the Λn interaction. The nnΛ search experiment (E12-17-003) was performed at JLab Hall A in 2018. In this article, the Λn FSI was investigated by a shape analysis of the 3H(e, e′K+)X missing mass spectrum, and a preliminary result for the Λn FSI study is given.


X17 Boson And The ³H(P, E⁺E⁻) ⁴He And ³He(N,E⁺E⁻) ⁴He Processes: A Theoretical Analysis, M. Viviani, E. Filandri, C. Gustavino, A. Kievsky, L. E. Marcucci, Rocco Schiavilla 2022 Old Dominion University

X17 Boson And The ³H(P, E⁺E⁻) ⁴He And ³He(N,E⁺E⁻) ⁴He Processes: A Theoretical Analysis, M. Viviani, E. Filandri, C. Gustavino, A. Kievsky, L. E. Marcucci, Rocco Schiavilla

Physics Faculty Publications

The present work deals with e+−e− pair production in the four-nucleon system. We first analyze the process as a purely electromagnetic one in the context of a state-of-the-art approach to nuclear strong-interaction dynamics and nuclear electromagnetic currents, derived from chiral effective field theory (χEFT). Next, we examine how the exchange of a hypothetical low-mass boson would impact the cross section for such a process. We consider several possibilities, that this boson is either a scalar, pseudoscalar, vector, or axial particle. The ab initio calculations use exact hyperspherical-harmonics methods to describe the bound state and low-energy spectrum of …


Transversity Parton Distribution Function Of The Nucleon Using The Pseudodistribution Approach, Colin Egerer, Christos Kallidonis, Joseph Karpie, Nikhil Karthik, Christopher J. Monahan, Wayne Morris, Kostas Orginos, Anatoly Radyushkin, Eloy Romero, Raza Sabbir Sufian, Savvas Zafeiropoulos, On behalf of the HadStruc Collaboration 2022 Old Dominion University

Transversity Parton Distribution Function Of The Nucleon Using The Pseudodistribution Approach, Colin Egerer, Christos Kallidonis, Joseph Karpie, Nikhil Karthik, Christopher J. Monahan, Wayne Morris, Kostas Orginos, Anatoly Radyushkin, Eloy Romero, Raza Sabbir Sufian, Savvas Zafeiropoulos, On Behalf Of The Hadstruc Collaboration

Physics Faculty Publications

We present a determination of the nonsinglet transversity parton distribution function (PDF) of the nucleon, normalized with respect to the tensor charge at μ2 ¼ 2 GeV2 from lattice quantum chromodynamics. We apply the pseudodistribution approach, using a gauge ensemble with a lattice spacing of 0.094 fm and the light quark mass tuned to a pion mass of 358 MeV. We extract the transversity PDF from the analysis of the short-distance behavior of the Ioffe-time pseudodistribution using the leading-twist nextto-leading order (NLO) matching coefficients calculated for transversity. We reconstruct the x-dependence of the transversity PDF through an expansion in a …


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