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Title: Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra

Abstract

The QCD light-front Hamiltonian equation H L F Ψ = M 2 Ψ derived from quantization at fixed LF time τ = t + z / c provides a causal, frame-independent method for computing hadron spectroscopy as well as dynamical observables such as structure functions, transverse momentum distributions, and distribution amplitudes. The QCD Lagrangian with zero quark mass has no explicit mass scale. de Alfaro, Fubini, and Furlan (dAFF) have made an important observation that a mass scale can appear in the equations of motion without affecting the conformal invariance of the action if one adds a term to the Hamiltonian proportional to the dilatation operator or the special conformal operator. If one applies the dAFF procedure to the QCD light-front Hamiltonian, it leads to a color-confining potential κ 4 ζ 2 for mesons, where ζ 2 is the LF radial variable conjugate to the q q ¯ invariant mass squared. The same result, including spin terms, is obtained using light-front holography, the duality between light-front dynamics and A d S 5 , if one modifies the A d S 5 action by the dilaton e κ 2 z 2 in the fifth dimension z . When one generalizes this procedure using superconformal algebra, the resulting light-front eigensolutions provide a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons and a universal Regge slope. The pion q q ¯ eigenstate has zero mass at m q = 0 . The superconformal relations also can be extended to heavy-light quark mesons and baryons. This approach also leads to insights into the physics underlying hadronization at the amplitude level. I will also discuss the remarkable features of the Poincaré invariant, causal vacuum defined by light-front quantization and its impact on the interpretation of the cosmological constant. AdS/QCD also predicts the analytic form of the nonperturbative running coupling α s ( Q 2 ) e - Q 2 / 4 κ 2 . The mass scale κ underlying hadron masses can be connected to the parameter Λ M S ¯ in the QCD running coupling by matching the nonperturbative dynamics to the perturbative QCD regime. The result is an effective coupling α s ( Q 2 ) defined at all momenta. One obtains empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. Finally, I address the interesting question of whether the momentum sum rule is valid for nuclear structure functions.

Authors:
ORCiD logo [1]
  1. SLAC National Accelerator Laboratory, Stanford University, Stanford, CA, USA
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1433298
Alternate Identifier(s):
OSTI ID: 1443762
Report Number(s):
SLAC-PUB-17012
Journal ID: ISSN 1687-7357; PII: 7236382; 7236382
Grant/Contract Number:  
AC02-76SF00515
Resource Type:
Published Article
Journal Name:
Advances in High Energy Physics
Additional Journal Information:
Journal Name: Advances in High Energy Physics Journal Volume: 2018; Journal ID: ISSN 1687-7357
Publisher:
Hindawi
Country of Publication:
Egypt
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 97 MATHEMATICS AND COMPUTING

Citation Formats

Brodsky, Stanley J. Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra. Egypt: N. p., 2018. Web. doi:10.1155/2018/7236382.
Brodsky, Stanley J. Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra. Egypt. doi:10.1155/2018/7236382.
Brodsky, Stanley J. Mon . "Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra". Egypt. doi:10.1155/2018/7236382.
@article{osti_1433298,
title = {Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra},
author = {Brodsky, Stanley J.},
abstractNote = {The QCD light-front Hamiltonian equation H L F Ψ = M 2 Ψ derived from quantization at fixed LF time τ = t     +     z / c provides a causal, frame-independent method for computing hadron spectroscopy as well as dynamical observables such as structure functions, transverse momentum distributions, and distribution amplitudes. The QCD Lagrangian with zero quark mass has no explicit mass scale. de Alfaro, Fubini, and Furlan (dAFF) have made an important observation that a mass scale can appear in the equations of motion without affecting the conformal invariance of the action if one adds a term to the Hamiltonian proportional to the dilatation operator or the special conformal operator. If one applies the dAFF procedure to the QCD light-front Hamiltonian, it leads to a color-confining potential κ 4 ζ 2 for mesons, where ζ 2 is the LF radial variable conjugate to the q q ¯ invariant mass squared. The same result, including spin terms, is obtained using light-front holography, the duality between light-front dynamics and A d S 5 , if one modifies the A d S 5 action by the dilaton e κ 2 z 2 in the fifth dimension z . When one generalizes this procedure using superconformal algebra, the resulting light-front eigensolutions provide a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons and a universal Regge slope. The pion q q ¯ eigenstate has zero mass at m q = 0 . The superconformal relations also can be extended to heavy-light quark mesons and baryons. This approach also leads to insights into the physics underlying hadronization at the amplitude level. I will also discuss the remarkable features of the Poincaré invariant, causal vacuum defined by light-front quantization and its impact on the interpretation of the cosmological constant. AdS/QCD also predicts the analytic form of the nonperturbative running coupling α s ( Q 2 ) ∝ e - Q 2 / 4 κ 2 . The mass scale κ underlying hadron masses can be connected to the parameter Λ M S ¯ in the QCD running coupling by matching the nonperturbative dynamics to the perturbative QCD regime. The result is an effective coupling α s ( Q 2 ) defined at all momenta. One obtains empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. Finally, I address the interesting question of whether the momentum sum rule is valid for nuclear structure functions.},
doi = {10.1155/2018/7236382},
journal = {Advances in High Energy Physics},
number = ,
volume = 2018,
place = {Egypt},
year = {2018},
month = {1}
}

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DOI: 10.1155/2018/7236382

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Works referenced in this record:

Threefold complementary approach to holographic QCD
journal, February 2014


Confinement contains condensates
journal, June 2012


Commensurate scale relations in quantum chromodynamics
journal, April 1995


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font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-02-01">February 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Reinhardt, H.</span> </li> <li> Physical Review D, Vol. 95, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.95.045015" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.95.045015<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.95.014011" target="_blank" rel="noopener noreferrer" class="name">Analysis of nucleon electromagnetic form factors from light-front holographic QCD: The spacelike region<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-01-01">January 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Sufian, Raza Sabbir; de Téramond, Guy F.; Brodsky, Stanley J.</span> </li> <li> Physical Review D, Vol. 95, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.95.014011" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.95.014011<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.ppnp.2015.07.002" target="_blank" rel="noopener noreferrer" class="name">Scattering amplitudes in the light-front formalism<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-11-01">November 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Cruz-Santiago, C.; Kotko, P.; Staśto, A. 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E.</span> ; <span class="author">Adachi, I.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review. D, Particles, Fields, Gravitation and Cosmology</span> </span> </div> <div class="abstract">Here, we repor<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> t the results of a study of the exclusive semileptonic decays <math> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <msup> <mi> π </mi> <mn> 0 </mn> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> </math>, <math> <msup> <mover> <mi> B </mi> <mo> ¯ </mo> </mover> <mn> 0 </mn> </msup> <mo> → </mo> <msup> <mi> π </mi> <mo> + </mo> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> </math>, <math> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <msup> <mi> ρ </mi> <mn> 0 </mn> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> </math>, <math> <msup> <mover> <mi> B </mi> <mo> ¯ </mo> </mover> <mn> 0 </mn> </msup> <mo> → </mo> <msup> <mi> ρ </mi> <mo> + </mo> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> </math> and <math> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <mi> ω </mi> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> </math>, where <math> <mi> ℓ </mi> </math> represents an electron or a muon. The events are tagged by fully reconstructing a second <math> <mi> B </mi> </math> meson in the event in a hadronic decay mode. The measured branching fractions are <math> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <msup> <mi> π </mi> <mn> 0 </mn> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> <mo> ) </mo> <mo> = </mo> <mo> ( </mo> <mn> 0.80 </mn> <mo> ± </mo> <mn> 0.08 </mn> <mo> ± </mo> <mn> 0.04 </mn> <mo> ) </mo> <mo> × </mo> <msup> <mn> 10 </mn> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> </math>, <math> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msup> <mover> <mi> B </mi> <mo> ¯ </mo> </mover> <mn> 0 </mn> </msup> <mo> → </mo> <msup> <mi> π </mi> <mo> + </mo> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> <mo> ) </mo> <mo> = </mo> <mo> ( </mo> <mn> 1.49 </mn> <mo> ± </mo> <mn> 0.09 </mn> <mo> ± </mo> <mn> 0.07 </mn> <mo> ) </mo> <mo> × </mo> <msup> <mn> 10 </mn> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> </math>, <math> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <msup> <mi> ρ </mi> <mn> 0 </mn> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> <mo> ) </mo> <mo> = </mo> <mo> ( </mo> <mn> 1.83 </mn> <mo> ± </mo> <mn> 0.10 </mn> <mo> ± </mo> <mn> 0.10 </mn> <mo> ) </mo> <mo> × </mo> <msup> <mn> 10 </mn> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> </math>, <math> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msup> <mover> <mi> B </mi> <mo> ¯ </mo> </mover> <mn> 0 </mn> </msup> <mo> → </mo> <msup> <mi> ρ </mi> <mo> + </mo> </msup> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> <mo> ) </mo> <mo> = </mo> <mo> ( </mo> <mn> 3.22 </mn> <mo> ± </mo> <mn> 0.27 </mn> <mo> ± </mo> <mn> 0.24 </mn> <mo> ) </mo> <mo> × </mo> <msup> <mn> 10 </mn> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> </math>, and <math> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <mi> ω </mi> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> <mo> ) </mo> <mo> = </mo> <mo> ( </mo> <mn> 1.07 </mn> <mo> ± </mo> <mn> 0.16 </mn> <mo> ± </mo> <mn> 0.07 </mn> <mo> ) </mo> <mo> × </mo> <msup> <mn> 10 </mn> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> </math>, where the first error is statistical and the second one is systematic. The obtained branching fractions are inclusive of soft photon emission. Furthermore, we determine the branching fractions as a function of the 4-momentum transfer squared to the leptonic system <math> <msup> <mi> q </mi> <mn> 2 </mn> </msup> <mo> = </mo> <mo> ( </mo> <msub> <mi> p </mi> <mi> ℓ </mi> </msub> <mo> + </mo> <msub> <mi> p </mi> <mi> ν </mi> </msub> <msup> <mo> ) </mo> <mn> 2 </mn> </msup> </math>, where <math> <msub> <mi> p </mi> <mi> ℓ </mi> </msub> </math> and <math> <msub> <mi> p </mi> <mi> ν </mi> </msub> </math> are the lepton and neutrino 4-momenta, respectively. Using the pion modes, a recent light cone sum rule calculation, lattice QCD results and a model-independent description of the hadronic form factor, a value of the Cabibbo-Kobayashi-Maskawa matrix element <math> <mo> | </mo> <msub> <mi> V </mi> <mrow> <mi> u </mi> <mi> b </mi> </mrow> </msub> <mo> | </mo> <mo> = </mo> <mo> ( </mo> <mn> 3.52 </mn> <mo> ± </mo> <mn> 0.29 </mn> <mo> ) </mo> <mo> × </mo> <msup> <mn> 10 </mn> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> </msup> </math> is extracted. A structure in the two-pion invariant mass distribution near <math> <mn> 1.3 </mn> <mtext></mtext> <mtext></mtext> <mi> GeV </mi> <mo> / </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </math>, which might be dominated by the decay <math> <msup> <mi> B </mi> <mo> - </mo> </msup> <mo> → </mo> <msub> <mi> f </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mn> 1270 </mn> <mo> ) </mo> <msup> <mi> ℓ </mi> <mo> - </mo> </msup> <msub> <mover> <mi> ν </mi> <mo> ¯ </mo> </mover> <mi> ℓ </mi> </msub> </math>, <math> <msub> <mi> f </mi> <mn> 2 </mn> </msub> <mo> → </mo> <msup> <mi> π </mi> <mo> + </mo> </msup> <msup> <mi> π </mi> <mo> - </mo> </msup> </math>, is seen. These results are obtained from a <math> <mn> 711 </mn> <mtext></mtext> <mtext></mtext> <msup> <mi> fb </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </math> data sample that contains <math> <mn> 772 </mn> <mo> × </mo> <msup> <mn> 10 </mn> <mn> 6 </mn> </msup> </math> <math> <mi> B </mi> <mover> <mi> B </mi> <mo> ¯ </mo> </mover> </math> pairs, collected near the <math> <mi> Υ </mi> <mo> ( </mo> <mn> 4 </mn> <mi> S </mi> <mo> ) </mo> </math> resonance with the Belle detector at the KEKB asymmetric-energy <math> <msup> <mi> e </mi> <mo> + </mo> </msup> <msup> <mi> e </mi> <mo> - </mo> </msup> </math> collider.</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 55<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/10.1103/PhysRevD.88.032005" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1594532" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevD.88.032005</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1594532" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1594532" 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="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1481037-direct-coupling-light-quarks-heavy-di-quarks" itemprop="url">The direct coupling of light quarks to heavy di-quarks</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">An, Haipeng</span> ; <span class="author">Wise, Mark B.</span> <span class="text-muted pubdata"> - Physics Letters B</span> </span> </div> <div class="abstract">In<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> the limit <math> <msub> <mrow> <mi> m </mi> </mrow> <mrow> <mi> Q </mi> </mrow> </msub> <mo> > </mo> <msub> <mrow> <mi> m </mi> </mrow> <mrow> <mi> Q </mi> </mrow> </msub> <msub> <mrow> <mi> v </mi> </mrow> <mrow> <mi mathvariant="normal"> rel </mi> </mrow> </msub> <mo> > </mo> <msub> <mrow> <mi> m </mi> </mrow> <mrow> <mi> Q </mi> </mrow> </msub> <msup> <mrow> <msub> <mrow> <mi> v </mi> </mrow> <mrow> <mi mathvariant="normal"> rel </mi> </mrow> </msub> </mrow> <mrow> <mn> 2 </mn> </mrow> </msup> <mo> >> </mo> <msub> <mrow> <mi mathvariant="normal"> Λ </mi> </mrow> <mrow> <mi> Q </mi> <mi> C </mi> <mi> D </mi> </mrow> </msub> </math> hadronic states with two heavy quarks <em>Q</em> should be describable by a version of HQET where the heavy quark is replaced by a di-quark degree of freedom. this limit the di-quark is a small (compared with <math> <mn> 1 </mn> <mo> / </mo> <msub> <mrow> <mi mathvariant="normal"> Λ </mi> </mrow> <mrow> <mi> Q </mi> <mi> C </mi> <mi> D </mi> </mrow> </msub> </math>) color anti-triplet, bound primarily by a color Coulomb potential. The excited Coulombic states and color six states are much heavier than the color anti-triplet ground state. The low lying spectrum of hadrons containing two heavy quarks is then determined by the coupling of the light quarks and gluons with momentum of order <math> <msub> <mrow> <mi mathvariant="normal"> Λ </mi> </mrow> <mrow> <mi> Q </mi> <mi> C </mi> <mi> D </mi> </mrow> </msub> </math> to this ground state di-quark. this short paper we calculate the coefficient of leading local operator <math> <mrow> <mo> ( </mo> <msubsup> <mrow> <mi> S </mi> </mrow> <mrow> <mi> v </mi> </mrow> <mrow> <mi mathvariant="normal"> † </mi> </mrow> </msubsup> <msub> <mrow> <mi> S </mi> </mrow> <mrow> <mi> v </mi> </mrow> </msub> <mo> ) </mo> </mrow> <mrow> <mo> ( </mo> <mover> <mrow> <mi> q </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mi> μ </mi> </mrow> </msup> <msub> <mrow> <mi> v </mi> </mrow> <mrow> <mi> μ </mi> </mrow> </msub> <mi> q </mi> <mo> ) </mo> </mrow> </math> that couples this color anti-triplet di-quark field <math> <msub> <mrow> <mi> S </mi> </mrow> <mrow> <mi> v </mi> </mrow> </msub> </math> (with four-velocity <em>v</em>) directly to the light quarks <em>q</em> in the low energy effective theory. It is <math> <mi mathvariant="script"> O </mi> <mo> ( </mo> <mn> 1 </mn> <mo> / </mo> <mo> ( </mo> <msub> <mrow> <mi> α </mi> </mrow> <mrow> <mi> s </mi> </mrow> </msub> <mo> ( </mo> <msub> <mrow> <mi> m </mi> </mrow> <mrow> <mi> Q </mi> </mrow> </msub> <msub> <mrow> <mi> v </mi> </mrow> <mrow> <mi mathvariant="normal"> rel </mi> </mrow> </msub> <mo> ) </mo> <msubsup> <mrow> <mi> m </mi> </mrow> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> <mo> ) </mo> </math>. While our work is mostly of pedagogical value we make an estimate of the contribution of this operator to the masses of <math> <msub> <mrow> <mi mathvariant="normal"> Ξ </mi> </mrow> <mrow> <mi> b </mi> <mi> b </mi> <mi> q </mi> </mrow> </msub> </math> baryon and <math> <msub> <mrow> <mi> T </mi> </mrow> <mrow> <mi> Q </mi> <mi> Q </mi> <mover> <mrow> <mi> q </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <mover> <mrow> <mi> q </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> </mrow> </msub> </math> tetraquark using the non-relativistic constituent quark model.</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 2<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.physletb.2018.11.004" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1481037" data-product-type="Journal Article" data-product-subtype="PA" >10.1016/j.physletb.2018.11.004</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/1594386-first-observation-ppk+k-search-exotic-baryons-pk-systems" itemprop="url">First observation of <math> <mi> γ </mi> <mi> γ </mi> <mo> → </mo> <mi> p </mi> <mover> <mi> p </mi> <mo> ¯ </mo> </mover> <msup> <mi> K </mi> <mo> + </mo> </msup> <msup> <mi> K </mi> <mo> - </mo> </msup> </math> and search for exotic baryons in <math> <mi> p </mi> <mi> K </mi> </math> systems</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">Shen, C. P.</span> ; <span class="author">Yuan, C. Z.</span> ; <span class="author">Adachi, I.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">T<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> he process <math> <mrow> <mi> γ </mi> <mi> γ </mi> <mo> → </mo> <mi> p </mi> <mover> <mrow> <mi> p </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> </mrow> </math> and its intermediate processes are measured for the first time using a <math> <mrow> <mn> 980 </mn> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi> fb </mi> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </math> data sample collected with the Belle detector at the KEKB asymmetric-energy <math> <msup> <mi> e </mi> <mo> + </mo> </msup> <msup> <mi> e </mi> <mo> - </mo> </msup> </math> collider. he production of <math> <mi> p </mi> <mover> <mi> p </mi> <mo> ¯ </mo> </mover> <msup> <mi> K </mi> <mo> + </mo> </msup> <msup> <mi> K </mi> <mo> - </mo> </msup> </math> and a <math> <mrow> <mi mathvariant="normal"> Λ </mi> <mo> ( </mo> <mn> 1520 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mn> 0 </mn> </mrow> </msup> </mrow> </math> <math> <mrow> <mo> ( </mo> <mover> <mrow> <mi mathvariant="normal"> Λ </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <mo> ( </mo> <mn> 1520 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mn> 0 </mn> </mrow> </msup> <mo> ) </mo> </mrow> </math> signal in the <math> <mi> p </mi> <msup> <mi> K </mi> <mo> - </mo> </msup> </math> ( <math> <mover> <mi> p </mi> <mo> ¯ </mo> </mover> <msup> <mi> K </mi> <mo> + </mo> </msup> </math>) invariant mass spectrum are clearly observed. However, no evidence for an exotic baryon near <math> <mrow> <mn> 1540 </mn> <mtext> </mtext> <mtext> </mtext> <mi> MeV </mi> <mo> / </mo> <msup> <mrow> <mi> c </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> </msup> </mrow> </math>, denoted as <math> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mo> ) </mo> <mn> 0 </mn> </msup> </math> ( <math> <mover> <mi mathvariant="normal"> Θ </mi> <mo> ¯ </mo> </mover> <mo> ( </mo> <mn> 1540 </mn> <msup> <mo> ) </mo> <mn> 0 </mn> </msup> </math>) or <math> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mo> ) </mo> <mrow> <mo> + </mo> <mo> + </mo> </mrow> </msup> </math> ( <math> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mo> ) </mo> <mrow> <mo> - </mo> <mo> -- </mo> </mrow> </msup> </math>), is seen in the <math> <mi> p </mi> <msup> <mi> K </mi> <mo> - </mo> </msup> </math> ( <math> <mover> <mi> p </mi> <mo> ¯ </mo> </mover> <msup> <mi> K </mi> <mo> + </mo> </msup> </math>) or <math> <mi> p </mi> <msup> <mi> K </mi> <mo> + </mo> </msup> </math> ( <math> <mover> <mi> p </mi> <mo> ¯ </mo> </mover> <msup> <mi> K </mi> <mo> - </mo> </msup> </math>) invariant mass spectra. Cross sections for <math> <mi> γ </mi> <mi> γ </mi> <mo> → </mo> <mi> p </mi> <mover> <mi> p </mi> <mo> ¯ </mo> </mover> <msup> <mi> K </mi> <mo> + </mo> </msup> <msup> <mi> K </mi> <mo> - </mo> </msup> </math>, <math> <mrow> <mi mathvariant="normal"> Λ </mi> <mo> ( </mo> <mn> 1520 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mn> 0 </mn> </mrow> </msup> <mover> <mrow> <mi> p </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mo> + </mo> <mi mathvariant="normal"> c </mi> <mo> . </mo> <mi mathvariant="normal"> c </mi> <mo> . </mo> </mrow> </math> and the products <math> <mrow> <mi> σ </mi> <mo> ( </mo> <mi> γ </mi> <mi> γ </mi> <mo> → </mo> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mn> 0 </mn> </mrow> </msup> <mover> <mrow> <mi> p </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mo> + </mo> <mi mathvariant="normal"> c </mi> <mo> . </mo> <mi mathvariant="normal"> c </mi> <mo> . </mo> <mo> ) </mo> <mi mathvariant="script"> B </mi> <mo> ( </mo> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mn> 0 </mn> </mrow> </msup> <mo> → </mo> <mi> p </mi> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> ) </mo> </mrow> </math> and <math> <mrow> <mi> σ </mi> <mo> ( </mo> <mi> γ </mi> <mi> γ </mi> <mo> → </mo> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mo> + </mo> <mo> + </mo> </mrow> </msup> <mover> <mrow> <mi> p </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> + </mo> <mi mathvariant="normal"> c </mi> <mo> . </mo> <mi mathvariant="normal"> c </mi> <mo> . </mo> <mo> ) </mo> <mi mathvariant="script"> B </mi> <mo> ( </mo> <mi mathvariant="normal"> Θ </mi> <mo> ( </mo> <mn> 1540 </mn> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mo> + </mo> <mo> + </mo> </mrow> </msup> <mo> → </mo> <mi> p </mi> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mo> ) </mo> </mrow> </math> are measured. We also determine upper limits on the products of the <math> <mrow> <msub> <mrow> <mi> χ </mi> </mrow> <mrow> <mi> c </mi> <mn> 0 </mn> </mrow> </msub> </mrow> </math> and <math> <mrow> <msub> <mrow> <mi> χ </mi> </mrow> <mrow> <mi> c </mi> <mn> 2 </mn> </mrow> </msub> </mrow> </math> two-photon decay widths and their branching fractions to <math> <mrow> <mi> p </mi> <mover> <mrow> <mi> p </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> K </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> </mrow> </math> at the 90% credibility level.</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 1<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/10.1103/PhysRevD.93.112017" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1594386" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevD.93.112017</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1594386" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1594386" 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="5" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1362011-event-patterns-extracted-from-transverse-momentum-rapidity-spectra-bosons-quarkonium-states-produced-pp-pb-pb-collisions-lhc" itemprop="url">Event Patterns Extracted from Transverse Momentum and Rapidity Spectra of <math> <mrow> <mi> Z </mi> </mrow> </math> Bosons and Quarkonium States Produced in pp and Pb-Pb Collisions at LHC</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">Chen, Ya-Hui</span> ; <span class="author">Liu, Fu-Hu</span> ; <span class="author">Lacey, Roy A.</span> <span class="text-muted pubdata"> - Advances in High Energy Physics</span> </span> </div> <div class="abstract">Tr<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> ansverse momentum ( <math> <mrow> <msub> <mrow> <mi> p </mi> </mrow> <mrow> <mi> T </mi> </mrow> </msub> </mrow> </math>) and rapidity ( <math> <mrow> <mi> y </mi> </mrow> </math>) spectra of <math> <mrow> <mi> Z </mi> </mrow> </math> bosons and quarkonium states (some charmonium <math> <mi> c </mi> <mover> <mrow> <mi> c </mi> </mrow> <mo> ¯ </mo> </mover> </math> mesons such as <math> <mi> J </mi> <mo> / </mo> <mi> ψ </mi> </math> and <math> <mi> ψ </mi> <mo> ( </mo> <mn> 2 </mn> <mi> S </mi> <mo> ) </mo> </math> and some bottomonium <math> <mi> b </mi> <mover> <mrow> <mi> b </mi> </mrow> <mo> ¯ </mo> </mover> </math> mesons such as <math> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 1 </mn> <mi> S </mi> <mo> ) </mo> </math>, <math> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 2 </mn> <mi> S </mi> <mo> ) </mo> </math>, and <math> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 3 </mn> <mi> S </mi> <mo> ) </mo> </math>) produced in proton-proton (pp) and lead-lead (Pb-Pb) collisions at the large hadron collider (LHC) are uniformly described by a hybrid model of two-component Erlang distribution for <math> <mrow> <msub> <mrow> <mi> p </mi> </mrow> <mrow> <mi> T </mi> </mrow> </msub> </mrow> </math> spectrum and two-component Gaussian distribution for <math> <mrow> <mi> y </mi> </mrow> </math> spectrum. The former distribution results from a multisource thermal model, and the latter one results from the revised Landau hydrodynamic model. The modelling results are in agreement with the experimental data measured in pp collisions at center-of-mass energies <math> <msqrt> <mi> s </mi> </msqrt> <mo> = </mo> <mn> 2.76 </mn> </math> and 7 TeV and in Pb-Pb collisions at center-of-mass energy per nucleon pair <math> <msqrt> <msub> <mrow> <mi> s </mi> </mrow> <mrow> <mi mathvariant="normal"> N </mi> <mi mathvariant="normal"> N </mi> </mrow> </msub> </msqrt> <mo> = </mo> <mn> 2.76 </mn> </math> TeV. Based on the parameter values extracted from <math> <mrow> <msub> <mrow> <mi> p </mi> </mrow> <mrow> <mi> T </mi> </mrow> </msub> </mrow> </math> and <math> <mrow> <mi> y </mi> </mrow> </math> spectra, the event patterns (particle scatter plots) in two-dimensional <math> <mrow> <msub> <mrow> <mi> p </mi> </mrow> <mrow> <mi> T </mi> </mrow> </msub> </mrow> </math>- <math> <mrow> <mi> y </mi> </mrow> </math> space and in three-dimensional velocity space are obtained.</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 2<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/10.1155/2016/9876253" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1362011" data-product-type="Journal Article" data-product-subtype="AM" >10.1155/2016/9876253</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1362011" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1362011" 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="/pages/biblio/1507811-doubly-virtual-transition-form-factors-light-front-quark-model" itemprop="url">Doubly virtual <math> <mo> ( </mo> <msup> <mi> π </mi> <mn> 0 </mn> </msup> <mo> , </mo> <mi> η </mi> <mo> , </mo> <msup> <mi> η </mi> <mo> ' </mo> </msup> <mo> ) </mo> <mo> → </mo> <msup> <mi> γ </mi> <mo> * </mo> </msup> <msup> <mi> γ </mi> <mo> * </mo> </msup> </math> transition form factors in the light-front quark model</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">Choi, Ho-Meoyng</span> ; <span class="author">Ryu, Hui-Young</span> ; <span class="author">Ji, Chueng-Ryong</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">We repor<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> t our investigation on the doubly virtual transition form factors (TFFs) <math> <mrow> <msub> <mrow> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> P </mi> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mo> * </mo> </mrow> </msup> </mrow> </msub> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> </mrow> </math> for the <math> <mrow> <mi mathvariant="normal"> P </mi> <mo> → </mo> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mo> * </mo> </mrow> </msup> <mo> ( </mo> <msub> <mrow> <mi> q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mo> * </mo> </mrow> </msup> <mo> ( </mo> <msub> <mrow> <mi> q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> </msub> <mo> ) </mo> <mo> ( </mo> <mi mathvariant="normal"> P </mi> <mo> = </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mn> 0 </mn> </mrow> </msup> <mo> , </mo> <mi> η </mi> <mo> , </mo> <msup> <mrow> <mi> η </mi> </mrow> <mrow> <mo> ' </mo> </mrow> </msup> <mo> ) </mo> </mrow> </math> transitions using the light-front quark model (LFQM). Performing a LF calculation in the exactly solvable manifestly covariant Bethe-Salpeter (BS) model as the first illustration, we use the <math> <msubsup> <mi> q </mi> <mn> 1 </mn> <mo> + </mo> </msubsup> <mo> = </mo> <mn> 0 </mn> </math> frame and find that both LF and manifestly covariant calculations produce exactly the same results for <math> <mrow> <msub> <mrow> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> P </mi> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mo> * </mo> </mrow> </msup> </mrow> </msub> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> </mrow> </math>. This confirms the absence of the LF zero mode in the doubly virtual TFFs. We then map this covariant BS model to the standard LFQM using the more phenomenologically accessible Gaussian wave function provided by the LFQM analysis of meson mass spectra. For the numerical analyses of <math> <msub> <mi> F </mi> <mrow> <mi mathvariant="normal"> P </mi> <msup> <mi> γ </mi> <mo> * </mo> </msup> </mrow> </msub> <mo> ( </mo> <msubsup> <mi> Q </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> , </mo> <msubsup> <mi> Q </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> ) </mo> </math>, we compare our LFQM results with the available experimental data and the perturbative QCD (pQCD) and vector meson dominance (VMD) model predictions. As <math> <mo> ( </mo> <msubsup> <mi> Q </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> , </mo> <msubsup> <mi> Q </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> ) </mo> <mo> → </mo> <mi> ∞ </mi> </math>, our LFQM result for doubly virtual TFF is consistent with the pQCD prediction, i.e., <math> <mrow> <msub> <mrow> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> P </mi> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mo> * </mo> </mrow> </msup> </mrow> </msub> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> <mo> ~ </mo> <mn> 1 </mn> <mo> / </mo> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> </mrow> </math>, while it differs greatly from the result of the VMD model, which behaves as <math> <mrow> <msubsup> <mrow> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> P </mi> <msup> <mrow> <mi> γ </mi> </mrow> <mrow> <mo> * </mo> </mrow> </msup> </mrow> <mrow> <mi> VMD </mi> </mrow> </msubsup> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> <mo> ~ </mo> <mn> 1 </mn> <mo> / </mo> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> </mrow> </math>. Our LFQM prediction for <math> <msub> <mi> F </mi> <mrow> <msup> <mi> η </mi> <mo> ' </mo> </msup> <msup> <mi> γ </mi> <mo> * </mo> </msup> </mrow> </msub> <mo> ( </mo> <msubsup> <mi> Q </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> , </mo> <msubsup> <mi> Q </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> ) </mo> </math> shows an agreement with the very recent experimental data obtained from the <i>BABAR</i> Collaboration for the ranges of <math> <mrow> <mn> 2 </mn> <mo> < </mo> <mo> ( </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow> <mi> Q </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> </mrow> </msubsup> <mo> ) </mo> <mo> < </mo> <mn> 60 </mn> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi> GeV </mi> </mrow> <mrow> <mn> 2 </mn> </mrow> </msup> </mrow> </math>.</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/10.1103/PhysRevD.99.076012" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1507811" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.99.076012</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? 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