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Title: Determination of $${{\rm{\Lambda }}}_{\overline{{\rm{MS}}}}$$ at five loops from holographic QCD

Abstract

Here, the recent determination of the $$\beta$$--function of the QCD running coupling $$\alpha_{\overline{MS}}(Q^2)$$ to five-loops, provides a verification of the convergence of a novel method for determining the fundamental QCD parameter $$\Lambda_s$$ based on the Light-Front Holographic approach to nonperturbative QCD. The new 5-loop analysis, together with improvements in determining the holographic QCD nonperturbative scale parameter $$\kappa$$ from hadronic spectroscopy, leads to an improved precision of the value of $$\Lambda_s$$ in the $${\overline{MS}}$$ scheme close to a factor of two; we find $$\Lambda^{(3)}_{\overline{MS}}=0.339\pm0.019$$ GeV for $$n_{f}=3$$, in excellent agreement with the world average, $$\Lambda_{\overline{MS}}^{(3)}=0.332\pm0.017$$ GeV. Lastly, we also discuss the constraints imposed on the scale dependence of the strong coupling in the nonperturbative domain by superconformal quantum mechanics and its holographic embedding in anti-de Sitter space.

Authors:
 [1];  [2];  [3]
  1. Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)
  2. SLAC National Accelerator Lab., Menlo Park, CA (United States); Stanford Univ., CA (United States)
  3. Universidad de Costa Rica, San Jose (Costa Rica)
Publication Date:
Research Org.:
Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States); SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)
OSTI Identifier:
1379486
Alternate Identifier(s):
OSTI ID: 1418322
Report Number(s):
JLAB-PHY-16-2312; DOE/OR/23177-3913; arXiv:1608.04933
Journal ID: ISSN 0954-3899
Grant/Contract Number:  
AC05-06OR23177; AC02-76SF00515
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Physics. G, Nuclear and Particle Physics
Additional Journal Information:
Journal Volume: 44; Journal Issue: 10; Journal ID: ISSN 0954-3899
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; strong interaction; QCD; nonperturbative; strong coupling

Citation Formats

Deur, Alexandre, Brodsky, Stanley J., and de Téramond, Guy F. Determination of ${{\rm{\Lambda }}}_{\overline{{\rm{MS}}}}$ at five loops from holographic QCD. United States: N. p., 2017. Web. doi:10.1088/1361-6471/aa888a.
Deur, Alexandre, Brodsky, Stanley J., & de Téramond, Guy F. Determination of ${{\rm{\Lambda }}}_{\overline{{\rm{MS}}}}$ at five loops from holographic QCD. United States. doi:10.1088/1361-6471/aa888a.
Deur, Alexandre, Brodsky, Stanley J., and de Téramond, Guy F. Fri . "Determination of ${{\rm{\Lambda }}}_{\overline{{\rm{MS}}}}$ at five loops from holographic QCD". United States. doi:10.1088/1361-6471/aa888a. https://www.osti.gov/servlets/purl/1379486.
@article{osti_1379486,
title = {Determination of ${{\rm{\Lambda }}}_{\overline{{\rm{MS}}}}$ at five loops from holographic QCD},
author = {Deur, Alexandre and Brodsky, Stanley J. and de Téramond, Guy F.},
abstractNote = {Here, the recent determination of the $\beta$--function of the QCD running coupling $\alpha_{\overline{MS}}(Q^2)$ to five-loops, provides a verification of the convergence of a novel method for determining the fundamental QCD parameter $\Lambda_s$ based on the Light-Front Holographic approach to nonperturbative QCD. The new 5-loop analysis, together with improvements in determining the holographic QCD nonperturbative scale parameter $\kappa$ from hadronic spectroscopy, leads to an improved precision of the value of $\Lambda_s$ in the ${\overline{MS}}$ scheme close to a factor of two; we find $\Lambda^{(3)}_{\overline{MS}}=0.339\pm0.019$ GeV for $n_{f}=3$, in excellent agreement with the world average, $\Lambda_{\overline{MS}}^{(3)}=0.332\pm0.017$ GeV. Lastly, we also discuss the constraints imposed on the scale dependence of the strong coupling in the nonperturbative domain by superconformal quantum mechanics and its holographic embedding in anti-de Sitter space.},
doi = {10.1088/1361-6471/aa888a},
journal = {Journal of Physics. G, Nuclear and Particle Physics},
number = 10,
volume = 44,
place = {United States},
year = {2017},
month = {8}
}

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<span class="author">/SLAC /Southern Denmark U., CP3-Origins</span> ; <span class="author">de Teramond, Guy F</span> ; <span class="author">...</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">The relation between the hadronic short-distance constituent quark and gluon particle limit and the long-range confining domain is yet one of the most challenging aspects of particle physics due to the strong coupling nature of Quantum Chromodynamics, the fundamental theory of the strong interactions. The central question is how one can compute hadronic properties from first principles; i.e., directly from the QCD Lagrangian. The most successful theoretical approach thus far has been to quantize QCD on discrete lattices in Euclidean space-time. Lattice numerical results follow from computation of frame-dependent moments of distributions in Euclidean space and dynamical observables in Minkowski<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> spacetime, such as the time-like hadronic form factors, are not amenable to Euclidean lattice computations. The Dyson-Schwinger methods have led to many important insights, such as the infrared fixed point behavior of the strong coupling constant, but in practice, the analyses are limited to ladder approximation in Landau gauge. Baryon spectroscopy and the excitation dynamics of nucleon resonances encoded in the nucleon transition form factors can provide fundamental insight into the strong-coupling dynamics of QCD. New theoretical tools are thus of primary interest for the interpretation of the results expected at the new mass scale and kinematic regions accessible to the JLab 12 GeV Upgrade Project. The AdS/CFT correspondence between gravity or string theory on a higher-dimensional anti-de Sitter (AdS) space and conformal field theories in physical space-time has led to a semiclassical approximation for strongly-coupled QCD, which provides physical insights into its nonperturbative dynamics. The correspondence is holographic in the sense that it determines a duality between theories in different number of space-time dimensions. This geometric approach leads in fact to a simple analytical and phenomenologically compelling nonperturbative approximation to the full light-front QCD Hamiltonian 'Light-Front Holography'. Light-Front Holography is in fact one of the most remarkable features of the AdS/CFT correspondence. The Hamiltonian equation of motion in the light-front (LF) is frame independent and has a structure similar to eigenmode equations in AdS space. This makes a direct connection of QCD with AdS/CFT methods possible. Remarkably, the AdS equations correspond to the kinetic energy terms of the partons inside a hadron, whereas the interaction terms build confinement and correspond to the truncation of AdS space in an effective dual gravity approximation. One can also study the gauge/gravity duality starting from the bound-state structure of hadrons in QCD quantized in the light-front. The LF Lorentz-invariant Hamiltonian equation for the relativistic bound-state system is P{sub {mu}}P{sup {mu}}|{psi}(P)> = (P{sup +}P{sup -} - P{sub {perpendicular}}{sup 2})|{psi}(P)> = M{sup 2}|{psi}(P)>, P{sup {+-}} = P{sup 0} {+-} P{sup 3}, where the LF time evolution operator P{sup -} is determined canonically from the QCD Lagrangian. To a first semiclassical approximation, where quantum loops and quark masses are not included, this leads to a LF Hamiltonian equation which describes the bound-state dynamics of light hadrons in terms of an invariant impact variable {zeta} which measures the separation of the partons within the hadron at equal light-front time {tau} = x{sup 0} + x{sup 3}. This allows us to identify the holographic variable z in AdS space with an impact variable {zeta}. The resulting Lorentz-invariant Schroedinger equation for general spin incorporates color confinement and is systematically improvable. Light-front holographic methods were originally introduced by matching the electromagnetic current matrix elements in AdS space with the corresponding expression using LF theory in physical space time. It was also shown that one obtains identical holographic mapping using the matrix elements of the energy-momentum tensor by perturbing the AdS metric around its static solution. A gravity dual to QCD is not known, but the mechanisms of confinement can be incorporated in the gauge/gravity correspondence by modifying the AdS geometry in the large infrared (IR) domain z {approx} 1 = {Lambda}{sub QCD}, which also sets the scale of the strong interactions. In this simplified approach we consider the propagation of hadronic modes in a fixed effective gravitational background asymptotic to AdS space, which encodes salient properties of the QCD dual theory, such as the ultraviolet (UV) conformal limit at the AdS boundary, as well as modifications of the background geometry in the large z IR region to describe confinement. The modified theory generates the point-like hard behavior expected from QCD, instead of the soft behavior characteristic of extended objects.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/servlets/purl/1035097" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1035097" data-product-type="Conference" data-product-subtype="" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1314949-qcd-running-coupling" itemprop="url">The QCD running coupling</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Deur, Alexandre</span> ; <span class="author">Brodsky, Stanley J.</span> ; <span class="author">de Téramond, Guy F.</span> <span class="text-muted pubdata"> - Progress in Particle and Nuclear Physics</span> </span> </div> <div class="abstract">Here, we review present knowledge on<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> $$\alpha_{s}$$, the Quantum Chromodynamics (QCD) running coupling. The dependence of $$\alpha_s(Q^2)$$ on momentum transfer $Q$ encodes the underlying dynamics of hadron physics --from color confinement in the infrared domain to asymptotic freedom at short distances. We will survey our present theoretical and empirical knowledge of $$\alpha_s(Q^2)$$, including constraints at high $Q^2$ predicted by perturbative QCD, and constraints at small $Q^2$ based on models of nonperturbative dynamics. In the first, introductory, part of this review, we explain the phenomenological meaning of the coupling, the reason for its running, and the challenges facing a complete understanding of its analytic behavior in the infrared domain. In the second, more technical, part of the review, we discuss $$\alpha_s(Q^2)$$ in the high momentum transfer domain of QCD. We review how $$\alpha_s$$ is defined, including its renormalization scheme dependence, the definition of its renormalization scale, the utility of effective charges, as well as `` Commensurate Scale Relations" which connect the various definitions of the QCD coupling without renormalization scale ambiguity. We also report recent important experimental measurements and advanced theoretical analyses which have led to precise QCD predictions at high energy. As an example of an important optimization procedure, we discuss the ``Principle of Maximum Conformality" which enhances QCD's predictive power by removing the dependence of the predictions for physical observables on the choice of the gauge and renormalization scheme. In last part of the review, we discuss $$\alpha_s(Q^2)$$ in the low momentum transfer domain, where there has been no consensus on how to define $$\alpha_s(Q^2)$$ or its analytic behavior. We will discuss the various approaches used for low energy calculations. Among them, we will discuss the light-front holographic approach to QCD in the strongly coupled regime and its prediction for the analytic form of $$\alpha_s(Q^2)$$. The AdS/QCD light-front holographic analysis predicts the color confinement potential underlying hadron spectroscopy and dynamics, and it gives a remarkable connection between the perturbative QCD scale $$\Lambda$$ and hadron masses. One can also identify a specific scale $$Q_0$$ which demarcates the division between perturbative and nonperturbative QCD. We also review other important methods for computing the QCD coupling, including Lattice QCD, Schwinger-Dyson equations and the Gribov-Zwanziger analysis. After describing these approaches and enumerating conflicting results, we provide a partial discussion on the origin of these discrepancies and how to remedy them. Our aim is not only to review the advances on this difficult subject, but also to suggest what could be the best definition of $$\alpha_s(Q^2)$$ in order to bring better unity to the subject.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 28<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1016/j.ppnp.2016.04.003" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1314949" data-product-type="Journal Article" data-product-subtype="AM" >10.1016/j.ppnp.2016.04.003</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1314949" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1314949" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1443973-hadron-spectroscopy-dynamics-from-light-front-holography-superconformal-algebra" itemprop="url">Hadron Spectroscopy and Dynamics from Light-Front Holography and Superconformal Algebra</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Brodsky, Stanley J.</span> <span class="text-muted pubdata"> - Few-Body Systems</span> </span> </div> <div class="abstract">QCD is not supersymmetrical in the traditional sense – the QCD Lagrangian is based on quark and gluonic fields, not squarks nor gluinos. However, its hadronic eigensolutions conform to a representation of superconformal algebra, reflecting the underlying conformal symmetry of chiral QCD and its Pauli matrix representation. The eigensolutions of superconformal algebra provide a unified Regge spectroscopy of meson, baryon, and tetraquarks of the same parity and twist as equal-mass members of the same 4-plet representation with a universal Regge slope. The pion<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> $$q\bar{q}$$ eigenstate has zero mass for m <sub>q</sub> = 0. The superconformal relations also can be extended to heavy-light quark mesons and baryons. The combined approach of light-front holography and superconformal algebra also provides insight into the origin of the QCD mass scale and color confinement. A key observation is the remarkable dAFF principle which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale κ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling, and the mass parameter underlying the Gaussian functional form of the nonperturbative QCD running coupling: α <sub>s</sub>(Q <sup>2</sup>) ∝ exp-Q <sup>2</sup>/4κ <sup>2</sup>, in agreement with the effective charge determined from measurements of the Bjorken sum rule. The mass scale κ underlying hadron masses can be connected to the parameter Λ <sub>$$\overline{MS}$$</sub> in the QCD running coupling by matching its predicted nonperturbative form to the perturbative QCD regime. The result is an effective coupling α <sub>s</sub>(Q <sup>2</sup>) defined at all momenta. In conclusion, one also obtains empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1007/s00601-018-1409-4" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1443973" data-product-type="Journal Article" data-product-subtype="AM" >10.1007/s00601-018-1409-4</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1443973" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1443973" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1490464-supersymmetric-conformal-features-hadron-physics" itemprop="url">Supersymmetric and Conformal Features of Hadron Physics</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Brodsky, Stanley</span> <span class="text-muted pubdata"> - Universe</span> </span> </div> <div class="abstract">The QCD Lagrangian is based on quark and gluonic fields—not squarks nor gluinos. However, one can show that its hadronic eigensolutions conform to a representation of superconformal algebra, reflecting the underlying conformal symmetry of chiral QCD. The eigensolutions of superconformal algebra provide a unified Regge spectroscopy of meson, baryon, and tetraquarks of the same parity and twist as equal-mass members of the same 4-plet representation with a universal Regge slope. The predictions from light-front holography and superconformal algebra can also be extended to mesons, baryons, and tetraquarks with strange, charm and bottom quarks. The pion<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> $$q\overline{q}$$ eigenstate has zero mass for m <sub>q</sub> = 0 . A key tool is the remarkable observation of de Alfaro, Fubini, and Furlan (dAFF) which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale κ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling. One also predicts the form of the nonperturbative QCD running coupling: α <sub>s</sub> (Q <sup>2</sup>) ∝ e <sup>-Q<sup>2</sup>/4κ<sup>2</sup></sup>, in agreement with the effective charge determined from measurements of the Bjorken sum rule. One also obtains viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. The combination of conformal symmetry, light-front dynamics, its holographic mapping to AdS <sub>5</sub> space, and the dAFF procedure thus provide new insights, not only into the physics underlying color confinement, but also the nonperturbative QCD coupling and the QCD mass scale.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.3390/universe4110120" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1490464" data-product-type="Journal Article" data-product-subtype="AM" >10.3390/universe4110120</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1490464" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1490464" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="6" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1223370-light-front-schrodinger-equation-determination-perturbative-qcd-scale-from-color-confinement" itemprop="url">The Light-Front Schrödinger Equation and Determination of the Perturbative QCD Scale from Color Confinement</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Conference</small><span class="authors"> <span class="author">Brodsky, Stanley J.</span> ; <span class="author">de Teramond, Guy F.</span> ; <span class="author">Deur, Alexandre P.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Few Body Syst.</span> </span> </div> <div class="abstract">The valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a relativistic equation of motion with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. If one requires that the effective action which underlies the QCD Lagrangian remains conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to light front Hamiltonian theory, the potential U has a unique form of a harmonic oscillator potential, and a mass gap arises. The result is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates color confinement and other essential spectroscopic and<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the same slope in the radial quantum number n and orbital angular momentum L. Only one mass parameter κ appears. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. We also show how the mass scale κ underlying confinement and hadron masses determines the scale Λ <sub>{ovr MS}</sub> controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the light-front and its embedding in AdS space, to the perturbative QCD regime computed to four-loop order. The result is an effective coupling defined at all momenta. The predicted value Λ <sub>{ovr MS</sub>}=0.328±0.034 GeV is in agreement with the world average 0.339±0.010 GeV. The analysis applies to any renormalization scheme.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1007/s00601-015-0964-1" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1223370" data-product-type="Conference" data-product-subtype="" >10.1007/s00601-015-0964-1</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/servlets/purl/1223370" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1223370" data-product-type="Conference" data-product-subtype="" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9; /* padding-top: 0.5rem; */"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator" /> <div class="text-center" style="margin-top:1.25rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="/"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="text-center small" style="margin-top:0.5em;margin-bottom:2.0rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="/disclaim" class="pure-menu-link"><span class="fa fa-institution"></span> Website Policies <span class="hidden-xs">/ Important Links</span></a></li> <li class="pure-menu-item"><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span> Contact Us</a></li> <li class="d-block d-md-none"></li> <li class="pure-menu-item"><a href="https://www.facebook.com/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-facebook" style=""></span></a></li> <li class="pure-menu-item"><a href="https://twitter.com/OSTIgov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-twitter" style=""></span></a></li> <li class="pure-menu-item"><a href="https://www.youtube.com/user/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-youtube-play" style=""></span></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.200423.1532.css" rel="stylesheet"> <script src="/pages/js/pages.200423.1532.js"></script><noscript></noscript> <script src='https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_CHTML' async> MathJax.Hub.Config({ tex2jax: {displayMath: [], inlineMath: [['$$','$$'], ['\\(','\\)']]}, asciimath2jax: {delimiters: []}, skipStartupTypeset: true, webFont: null, CommonHTML: { scale: 84, // mathjx wants to scale everything to 119% by default? mtextFontInherit: true } }); MathJax.Hub.Queue(["Typeset",MathJax.Hub,"item-list"]); MathJax.Hub.Queue(["Typeset",MathJax.Hub,"citation-pagetitle"]); MathJax.Hub.Queue(["Typeset",MathJax.Hub,"citation-abstract"]); </script><noscript></noscript><script defer src="/pages/js/pages.biblio.200423.1532.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript><script async type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript></body> <!-- DOE PAGES v.200423.1532 --> </html>