Colloquium: Gravitational Form Factors of the Proton

The physics of the gravitational form factors of the proton, and their understanding within quantum chromodynamics, has advanced significantly in the past two decades through both theory and experiment. This Colloquium provides an overview of this progress, highlights the physical insights unveiled by studies of gravitational form factors, and reviews their interpretation in terms of the mechanical properties of the proton.

[1]  William Focillon,et al.  Energy-momentum tensor in the scalar diquark model , 2023, 2304.10339.

[2]  Y. Hatta,et al.  Gravitational form factors of nuclei in the Skyrme model , 2023, Physical Review D.

[3]  Bo Liu,et al.  Measurement of the J/$\psi $ photoproduction cross section over the full near-threshold kinematic region , 2023, 2304.03845.

[4]  L. Pentchev,et al.  Determining the gluonic gravitational form factors of the proton , 2023, Nature.

[5]  A. Mukherjee,et al.  Gluon contribution to the mechanical properties of a dressed quark in light-front Hamiltonian QCD , 2023, Physical Review D.

[6]  Yi Chen,et al.  Nucleon relativistic polarization and magnetization distributions , 2023, Physical Review D.

[7]  A. Freese,et al.  The gravitational form factors of the electron in quantum electrodynamics , 2022, Physics Letters B.

[8]  Kazuhiro Tanaka.  Twist-four gravitational form factor at NNLO QCD from trace anomaly constraints , 2022, Journal of High Energy Physics.

[9]  John R. Terry,et al.  Precision studies of QCD in the low energy domain of the EIC , 2022, Progress in Particle and Nuclear Physics.

[10]  A. Alaoui,et al.  First CLAS12 measurement of DVCS beam-spin asymmetries in the extended valence region , 2022, 2211.11274.

[11]  U. Yakhshiev,et al.  Medium modification of singly heavy baryons in a pion-mean-field approach , 2022, Physical Review D.

[12]  J. Kim,et al.  Gravitational form factors of the baryon octet with flavor SU(3) symmetry breaking , 2022, Physical Review D.

[13]  Yi Chen,et al.  Pion and nucleon relativistic electromagnetic four-current distributions , 2022, Physical Review D.

[14]  Qin-Tao Song,et al.  Kinematical higher-twist corrections in $\gamma^* \gamma \to M \bar M $ , 2022, 2209.11140.

[15]  E. Epelbaum,et al.  Gravitational form factors of the delta resonance in chiral EFT , 2022, The European Physical Journal C.

[16]  M. Czakon,et al.  A note on quark and gluon energy-momentum tensors , 2022, Journal of High Energy Physics.

[17]  Dongyan Fu,et al.  Mechanical structure of a spin-1 particle , 2022, Physical Review D.

[18]  W. Cosyn,et al.  Spatial densities of momentum and forces in spin-one hadrons , 2022, Physical Review D.

[19]  W. Cosyn,et al.  Spatial densities of the photon on the light front , 2022, Physical Review D.

[20]  A. Mukherjee,et al.  Gravitational form factors and mechanical properties of the proton: Connections between distributions in 2D and 3D , 2022, Physical Review D.

[21]  J. Qiu,et al.  Exclusive production of a pair of high transverse momentum photons in pion-nucleon collisions for extracting generalized parton distributions , 2022, Journal of High Energy Physics.

[22]  J. Kim,et al.  Medium modification of the nucleon mechanical properties: Abel tomography case , 2022, The European Physical Journal C.

[23]  I. Zahed,et al.  $J/\psi$ near threshold in holographic QCD: A and D gravitational form factors , 2022, 2204.08857.

[24]  O. Grocholski,et al.  Phenomenology of diphoton photoproduction at next-to-leading order , 2022, Physical Review D.

[25]  F. Yuan,et al.  Perturbative calculations of gravitational form factors at large momentum transfer , 2022, Journal of High Energy Physics.

[26]  P. Schweitzer,et al.  2D energy-momentum tensor distributions of nucleon in a large- Nc quark model from ultrarelativistic to nonrelativistic limit , 2022, Physical Review D.

[27]  Dongyan Fu,et al.  Electromagnetic and gravitational form factors of Δ resonance in a covariant quark-diquark approach , 2022, Physical Review D.

[28]  A. Krutov,et al.  Relativistic composite-particle theory of the gravitational form factors of the pion: Quantitative results , 2022, Physical Review D.

[29]  A. Mukherjee,et al.  Gravitational form factors and mechanical properties of a quark at one loop in light-front Hamiltonian QCD , 2021, Physical Review D.

[30]  J. Napolitano,et al.  Physics with CEBAF at 12 GeV and future opportunities , 2021, Progress in Particle and Nuclear Physics.

[31]  Jian Liang,et al.  Proton momentum and angular momentum decompositions with overlap fermions , 2021, Physical Review D.

[32]  P. Sun,et al.  Near threshold heavy quarkonium photoproduction at large momentum transfer , 2021, Physical Review D.

[33]  X. Ji,et al.  Momentum-Current Gravitational Multipoles of Hadrons , 2021, Physical Review D.

[34]  A. Metz,et al.  Energy-momentum tensor in QCD: nucleon mass decomposition and mechanical equilibrium , 2021, Journal of High Energy Physics.

[35]  J. Rodríguez-Quintero,et al.  Revealing pion and kaon structure via generalised parton distributions , 2021, Chinese Physics C.

[36]  E. Epelbaum,et al.  Chiral theory of ρ -meson gravitational form factors , 2021, Physical Review D.

[37]  L. Jenkovszky,et al.  On Matter and Pressure Distribution in Nucleons , 2021, Physics of Particles and Nuclei Letters.

[38]  A. Alaoui,et al.  First Measurement of Timelike Compton Scattering. , 2021, Physical review letters.

[39]  K. Jansen,et al.  Transversity GPDs of the proton from lattice QCD , 2021, Physical Review D.

[40]  A. Freese,et al.  Unified formalism for electromagnetic and gravitational probes: Densities , 2021, Physical Review D.

[41]  P. Shanahan,et al.  Gluon gravitational structure of hadrons of different spin , 2021, Physical Review D.

[42]  A. Thomas,et al.  Effect of the pion field on the distributions of pressure and shear in the proton , 2021, Physics Letters B.

[43]  J. Kim,et al.  Energy-momentum tensor of the nucleon on the light front: Abel tomography case , 2021, Physical Review D.

[44]  J. Gegelia,et al.  A bound on the nucleon Druck-term from chiral EFT in curved space-time and mechanical stability conditions , 2021, 2104.13954.

[45]  Xurong Chen,et al.  Determination of the Gluonic D-term and Mechanical Radii of Proton from Experimental data , 2021, 2104.12962.

[46]  A. Metz,et al.  The gravitational form factor D(t) of the electron , 2021, 2104.04207.

[47]  V. Bertone,et al.  Deconvolution problem of deeply virtual Compton scattering , 2021, 2104.03836.

[48]  V. Burkert,et al.  Determination of shear forces inside the proton , 2021, 2104.02031.

[49]  Keh-Fei Liu.  Proton mass decomposition and hadron cosmological constant , 2021, Physical Review D.

[50]  P. Cole,et al.  Beam charge asymmetries for deeply virtual Compton scattering off the proton , 2021, The European Physical Journal A.

[51]  P. Sun,et al.  Perturbative QCD analysis of near threshold heavy quarkonium photoproduction at large momentum transfer , 2021, Physics Letters B.

[52]  X. Ji,et al.  QCD analysis of near-threshold photon-proton production of heavy quarkonium , 2021, Physical Review D.

[53]  C. Lorc'e.  Relativistic spin sum rules and the role of the pivot , 2021, The European Physical Journal C.

[54]  R. Perry,et al.  Parton physics from a heavy-quark operator product expansion: Formalism and Wilson coefficients , 2021, Physical Review D.

[55]  A. S. Nunes,et al.  Science Requirements and Detector Concepts for the Electron-Ion Collider , 2021, Nuclear Physics A.

[56]  I. Zahed,et al.  Nucleon mass radii and distribution: Holographic QCD, lattice QCD, and GlueX data , 2021, 2103.03186.

[57]  M. Strikman,et al.  ϕ-meson lepto-production near threshold and the strangeness D-term , 2021, 2102.12631.

[58]  M. Polyakov,et al.  Forces inside the nucleon on the light front from 3D Breit frame force distributions: Abel tomography case , 2021, Physical Review D.

[59]  X. Ji.  Proton mass decomposition: Naturalness and interpretations , 2021, Frontiers of Physics.

[60]  A. Freese,et al.  Forces within hadrons on the light front , 2021, 2102.01683.

[61]  D. Kharzeev.  Mass radius of the proton , 2021, Physical Review D.

[62]  J. Kim,et al.  Strong force fields and stabilities of the nucleon and singly heavy baryon Σc , 2021 .

[63]  H. Dutrieux,et al.  Phenomenological assessment of proton mechanical properties from deeply virtual Compton scattering , 2021, The European Physical Journal C.

[64]  F. Yuan,et al.  Gluon gravitational form factors at large momentum transfer , 2021, Physics Letters B.

[65]  J. Kim,et al.  Gravitational form factors of a baryon with spin-3/2 , 2021, The European Physical Journal C.

[66]  K. Azizi,et al.  Gravitational form factors of $N(1535)$ in QCD , 2020, 2012.06895.

[67]  Jake Grigsby,et al.  Deep learning analysis of deeply virtual exclusive photoproduction , 2020, Physical Review D.

[68]  X. Ji,et al.  What we know and what we don’t know about the proton spin after 30 years , 2020, Nature Reviews Physics.

[69]  J. Kim,et al.  Gravitational form factors of a baryon with spin-3/2 , 2020, 2011.00292.

[70]  A. Krutov,et al.  Pion gravitational form factors in a relativistic theory of composite particles , 2020, 2010.11640.

[71]  C. Mondal,et al.  Gravitational form factors and mechanical properties of proton in a light-front quark-diquark model , 2020, 2010.04215.

[72]  K. Orginos,et al.  Lattice QCD Constraints on the Parton Distribution Functions of ^{3}He. , 2020, Physical review letters.

[73]  Huey-Wen Lin.  Nucleon Tomography and Generalized Parton Distribution at Physical Pion Mass from Lattice QCD. , 2020, Physical review letters.

[74]  K. Jansen,et al.  Unpolarized and Helicity Generalized Parton Distributions of the Proton within Lattice QCD. , 2020, Physical review letters.

[75]  T. Barakat,et al.  Gravitational formfactors of the ρ , π , and K mesons in the light-cone QCD sum rules , 2020, Physical Review D.

[76]  S. Meinel,et al.  From Ji to Jaffe-Manohar orbital angular momentum in lattice QCD using a direct derivative method , 2020, 2008.03660.

[77]  C. Lorc'e.  Charge Distributions of Moving Nucleons. , 2020, Physical review letters.

[78]  A. Metz,et al.  Revisiting the proton mass decomposition , 2020, Physical Review D.

[79]  T. Giani,et al.  Parton distributions and lattice-QCD calculations: Toward 3D structure , 2020, Progress in Particle and Nuclear Physics.

[80]  P. Schweitzer,et al.  Effects of long-range forces on the D -term and the energy-momentum structure , 2020, Physical Review D.

[81]  J. Gegelia,et al.  Chiral theory of nucleons and pions in the presence of an external gravitational field , 2020, Physical Review D.

[82]  A. Metz,et al.  Mass sum rules of the electron in quantum electrodynamics , 2020, Journal of High Energy Physics.

[83]  M. Polyakov,et al.  Quadrupole pressure and shear forces inside baryons in the large N limit , 2020, 2004.02912.

[84]  W. Brooks,et al.  The CLAS12 Spectrometer at Jefferson Laboratory , 2020, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment.

[85]  K. Azizi,et al.  Gravitational form factors of hyperons in light-cone QCD , 2020, 2003.12588.

[86]  K. Jansen,et al.  Complete flavor decomposition of the spin and momentum fraction of the proton using lattice QCD simulations at physical pion mass , 2020, Physical Review D.

[87]  L. Szymanowski,et al.  Electroproduction of a large invariant mass photon pair , 2020, 2003.03263.

[88]  Yubing Dong,et al.  Gravitational form factors of ρ meson with a light-cone constituent quark model , 2020, 2002.02648.

[89]  P. Lowdon,et al.  Covariant multipole expansion of local currents for massive states of any spin , 2019, Physical Review D.

[90]  M. Neubelt,et al.  Energy momentum tensor and the D term in the bag model , 2019, Physical Review D.

[91]  I. Zahed,et al.  Diffractive photoproduction of J/ψ and ϒ using holographic QCD: Gravitational form factors and GPD of gluons in the proton , 2019, Physical Review D.

[92]  K. Jansen,et al.  Nucleon axial, tensor, and scalar charges and σ -terms in lattice QCD , 2019, Physical Review D.

[93]  W. Wang,et al.  Electron-ion collider in China , 2019, Frontiers of Physics.

[94]  K. Azizi,et al.  Nucleon’s energy–momentum tensor form factors in light-cone QCD , 2019, The European Physical Journal C.

[95]  P. Lowdon,et al.  Universality of the Poincaré gravitational form factor constraints , 2019, The European Physical Journal C.

[96]  Lucy Rosenbloom.  arXiv , 2019, The Charleston Advisor.

[97]  K. Kumerički.  Measurability of pressure inside the proton , 2019, Nature.

[98]  A. S. Nunes,et al.  Transverse extension of partons in the proton probed in the sea-quark range by measuring the DVCS cross section , 2019, Physics Letters B.

[99]  P. Lowdon,et al.  Poincaré constraints on the gravitational form factors for massive states with arbitrary spin , 2019, Physical Review D.

[100]  G. Zhao,et al.  First Measurement of Near-Threshold J/ψ Exclusive Photoproduction off the Proton. , 2019, Physical review letters.

[101]  I. Cloet,et al.  Gravitational form factors of light mesons , 2019, Physical Review C.

[102]  M. Polyakov,et al.  Gravitational form factors of a spin one particle , 2019, Physical Review D.

[103]  S. Cotogno,et al.  The energy-momentum tensor of spin-1 hadrons: formalism , 2019, The European Physical Journal C.

[104]  I. Anikin.  Gravitational form factors within light-cone sum rules at leading order , 2019, Physical Review D.

[105]  Kazuhiro Tanaka.  Three-loop formula for quark and gluon contributions to the QCD trace anomaly , 2018, Journal of High Energy Physics.

[106]  H. Moutarde,et al.  Revisiting the mechanical properties of the nucleon , 2018, The European Physical Journal C.

[107]  W. Detmold,et al.  Pressure Distribution and Shear Forces inside the Proton. , 2018, Physical review letters.

[108]  Kazuhiro Tanaka,et al.  Quark and gluon contributions to the QCD trace anomaly , 2018, Journal of High Energy Physics.

[109]  W. Detmold,et al.  Gluon gravitational form factors of the nucleon and the pion from lattice QCD , 2018, Physical Review D.

[110]  S. Wallon,et al.  Probing axial quark generalized parton distributions through exclusive photoproduction of a γπ± pair with a large invariant mass , 2018, Journal of High Energy Physics.

[111]  C. Lorcé.  The relativistic center of mass in field theory with spin , 2018, The European Physical Journal C.

[112]  Jian Liang,et al.  Proton Mass Decomposition from the QCD Energy Momentum Tensor. , 2018, Physical review letters.

[113]  Y. Hatta,et al.  Holographic J/ψ production near threshold and the proton mass problem , 2018, Physical Review D.

[114]  M. Polyakov,et al.  Nucleon gravitational form factors from instantons: forces between quark and gluon subsystems , 2018, Journal of High Energy Physics.

[115]  K. Jansen,et al.  Nucleon spin structure from lattice QCD , 2018, Proceedings of XXVI International Workshop on Deep-Inelastic Scattering and Related Subjects — PoS(DIS2018).

[116]  Kazuhiro Tanaka.  Operator relations for gravitational form factors of a spin-0 hadron , 2018, Physical Review D.

[117]  P. Schweitzer,et al.  Forces inside hadrons: Pressure, surface tension, mechanical radius, and all that , 2018, International Journal of Modern Physics A.

[118]  V. Burkert,et al.  The pressure distribution inside the proton , 2018, Nature.

[119]  C. Lorcé.  The relativistic center of mass in field theory with spin , 2018, The European Physical Journal C.

[120]  P. Shanahan,et al.  Nonperturbatively renormalized glue momentum fraction at the physical pion mass from lattice QCD , 2018, Physical Review D.

[121]  P. Schweitzer,et al.  D-term, strong forces in the nucleon, and their applications , 2018, 1801.05858.

[122]  P. Schweitzer,et al.  D term and the structure of pointlike and composed spin-0 particles , 2017, 1712.05316.

[123]  C. Mondal,et al.  Gravitational form factors and angular momentum densities in light-front quark-diquark model , 2017, 1712.02110.

[124]  J. Qiu,et al.  Exploring Partonic Structure of Hadrons Using ab initio Lattice QCD Calculations. , 2017, Physical review letters.

[125]  S. Brodsky,et al.  Rigorous constraints on the matrix elements of the energy-momentum tensor , 2017, 1707.06313.

[126]  C. Lorcé.  On the hadron mass decomposition , 2017, 1706.05853.

[127]  A. Chambers,et al.  Nucleon Structure Functions from Operator Product Expansion on the Lattice. , 2017, Physical review letters.

[128]  K. Jansen,et al.  Nucleon Spin and Momentum Decomposition Using Lattice QCD Simulations. , 2017, Physical review letters.

[129]  A. Radyushkin.  Quasi-parton distribution functions, momentum distributions, and pseudo-parton distribution functions , 2017 .

[130]  C. Lorc'e,et al.  Spatial distribution of angular momentum inside the nucleon , 2017, 1704.08557.

[131]  W. Detmold,et al.  Off-forward gluonic structure of vector mesons , 2017, 1703.08220.

[132]  A. Alexandru,et al.  Glue Spin and Helicity in the Proton from Lattice QCD. , 2016, Physical review letters.

[133]  S. Wallon,et al.  Exclusive photoproduction of a γ ρ pair with a large invariant mass , 2016, 1609.03830.

[134]  C. Mondal,et al.  Charge and longitudinal momentum distributions in transverse coordinate space , 2016, 1608.01095.

[135]  P. Schweitzer,et al.  LHCb pentaquarks as a baryon- ψ ( 2 S ) bound state: Prediction of isospin- 3 2 pentaquarks with hidden charm , 2016, 1607.07008.

[136]  S. Niccolai,et al.  Experimental overview of Deeply Virtual Compton Scattering , 2016 .

[137]  O. Dadoun,et al.  Production of Highly Polarized Positrons Using Polarized Electrons at MeV Energies. , 2016, Physical review letters.

[138]  P. Freire,et al.  Masses, Radii, and the Equation of State of Neutron Stars , 2016, 1603.02698.

[139]  T. Horn,et al.  The pion: an enigma within the Standard Model , 2016, 1602.04016.

[140]  G. Bali,et al.  Nucleon generalized form factors from lattice QCD with nearly physical quark masses , 2016, 1601.04818.

[141]  R. Roedl.  Nucleon generalized form factors from lattice QCD with nearly physical quark masses , 2016 .

[142]  M. Guidal,et al.  PARTONS: PARtonic Tomography Of Nucleon Software. A computing framework for the phenomenology of Generalized Parton Distributions , 2015 .

[143]  C. Mondal.  Longitudinal momentum densities in transverse plane for nucleons , 2015, 1511.01736.

[144]  M. Mai,et al.  The energy-momentum tensor and D-term of Q-clouds , 2015, 1510.08015.

[145]  M. Hoferichter,et al.  Roy-Steiner-equation analysis of pion-nucleon scattering , 2015, 1510.06039.

[146]  Jae Hyun Kim,et al.  Study of $\pi^0$ pair production in single-tag two-photon collisions , 2015, 1508.06757.

[147]  C. Mondal,et al.  Gravitational form factors and transverse spin sum rule in a light front quark-diquark model in AdS/QCD , 2015, 1505.02013.

[148]  A. Alaoui,et al.  Cross Sections for the Exclusive Photon Electroproduction on the Proton and Generalized Parton Distributions. , 2015, Physical review letters.

[149]  S. Pisano,et al.  Single and double spin asymmetries for deeply virtual Compton scattering measured with CLAS and a longitudinally polarized proton target , 2015, 1501.07052.

[150]  W. Brooks,et al.  Longitudinal target-spin asymmetries for deeply virtual compton scattering. , 2014, Physical review letters.

[151]  Hyun-Chul Kim,et al.  Stability of the pion and the pattern of chiral symmetry breaking , 2014, 1410.1420.

[152]  T. K. Hemmick,et al.  A White Paper on SoLID (Solenoidal Large Intensity Device) , 2014, 1409.7741.

[153]  B. Pasquini,et al.  Dispersive evaluation of the D-term form factor in deeply virtual Compton scattering , 2014, 1407.5960.

[154]  F. Englert.  Nobel lecture: The BEH mechanism and its scalar boson , 2014 .

[155]  P. W. Higgs.  Nobel Lecture: Evading the Goldstone theorem* , 2014 .

[156]  M. Wakamatsu.  Is gauge-invariant complete decomposition of the nucleon spin possible? , 2014, 1402.4193.

[157]  U. Yakhshiev,et al.  Energy–momentum tensor form factors of the nucleon within a π–ρ–ω soliton model , 2013, 1310.8064.

[158]  T. Lautenschlager,et al.  Towards a fitting procedure to deeply virtual meson production - the next-to-leading order case , 2013, 1310.5394.

[159]  E. Leader,et al.  The angular momentum controversy: What’s it all about and does it matter? , 2013, Physics of Particles and Nuclei.

[160]  X. Ji.  Parton physics on a Euclidean lattice. , 2013, Physical review letters.

[161]  Hiroshi Suzuki,et al.  Energy-momentum tensor from the Yang-Mills gradient flow , 2013, 1304.0533.

[162]  C. Lorcé.  Wilson lines and orbital angular momentum , 2012, 1210.2581.

[163]  M. Mai,et al.  Radial excitations of Q -balls, and their D -term , 2012, 1206.2930.

[164]  M. Mai,et al.  Energy momentum tensor, stability, and the D-term of Q-balls , 2012, 1206.2632.

[165]  C. Lorcé.  Geometrical approach to the proton spin decomposition , 2012, 1205.6483.

[166]  U. Yakhshiev,et al.  Energy–momentum tensor form factors of the nucleon in nuclear matter , 2012, 1205.5228.

[167]  O. Teryaev,et al.  Analyticity and sum rules for photon GPDs , 2012, 1204.6471.

[168]  Y. Hatta.  Notes on the orbital angular momentum of quarks in the nucleon , 2011, 1111.3547.

[169]  D. Ryckbosch,et al.  Measurement of double-spin asymmetries associated with deeply virtual Compton scattering on a transversely polarized hydrogen target , 2011, 1106.2990.

[170]  K. Jansen,et al.  Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass , 2011, Physical Review D.

[171]  L. Szymanowski,et al.  NLO corrections to timelike, spacelike and double deeply virtual Compton scattering , 2011, 1101.0555.

[172]  B. Kopeliovich,et al.  DDVCS on nucleons and nuclei , 2010, 1005.4621.

[173]  C. Carlson,et al.  Nucleon electromagnetic and gravitational form factors from holography , 2009, 0903.4818.

[174]  T. Goldman,et al.  Spin and orbital angular momentum in gauge theories: nucleon spin structure and multipole radiation revisited. , 2008, Physical review letters.

[175]  S. Brodsky,et al.  Light-Front Dynamics and AdS/QCD Correspondence: Gravitational Form Factors of Composite Hadrons , 2008, 0804.0452.

[176]  C. Carlson,et al.  Gravitational Form Factors in the Axial Sector from an AdS/QCD Model , 2008, 0804.0214.

[177]  W. Brooks,et al.  Measurement of deeply virtual compton scattering beam-spin asymmetries. , 2007, Physical review letters.

[178]  O. Teryaev,et al.  Dispersion relations and QCD factorization in hard reactions , 2007, 0710.4211.

[179]  D. Hwang,et al.  Implication of the overlap representation for modelling generalized parton distributions , 2007, 0710.1567.

[180]  M. Diehl,et al.  Dispersion representations for hard exclusive processes: beyond the Born approximation , 2007, 0707.0351.

[181]  K. Orginos,et al.  Nucleon Generalized Parton Distributions from Full Lattice QCD , 2007, 0705.4295.

[182]  S. Boffi,et al.  Nucleon spin densities in a light-front constituent quark model , 2007, 0705.4345.

[183]  K. Goeke,et al.  Nucleon form factors of the energy-momentum tensor in the chiral quark-soliton model , 2007 .

[184]  K. Goeke,et al.  The nucleon form-factors of the energy–momentum tensor in the Skyrme model , 2007, hep-ph/0703025.

[185]  K. Goeke,et al.  Pion mass dependence of the nucleon form factors of the energy-momentum tensor in the chiral quark-soliton model , 2007, hep-ph/0702031.

[186]  M. Wakamatsu.  On the D-term of the nucleon generalized parton distributions , 2007, hep-ph/0701057.

[187]  L. Szymanowski,et al.  Deeply virtual compton scattering on a photon and generalized parton distributions in the photon , 2006, hep-ph/0611176.

[188]  A. Schäfer,et al.  Chiral perturbation theory for nucleon generalized parton distributions , 2006, hep-ph/0608113.

[189]  X. Jiang,et al.  Scaling tests of the cross section for deeply virtual Compton scattering. , 2006, Physical review letters.

[190]  J. Napolitano,et al.  Measurement of deeply virtual compton scattering with a polarized-proton target. , 2006, Physical review letters.

[191]  D. Pleiter,et al.  Structure of the Pion from Full Lattice QCD , 2005, hep-lat/0509133.

[192]  V. Guzey,et al.  On the A-dependence of nuclear generalized parton distributions , 2005, hep-ph/0509158.

[193]  W. Detmold,et al.  Deep-inelastic scattering and the operator product expansion in lattice QCD , 2005, hep-lat/0507007.

[194]  S. Liuti,et al.  Nuclear medium modifications of hadrons from generalized parton distributions , 2005, hep-ph/0504027.

[195]  Emmy Noether,et al.  Invariant Variation Problems , 2005, physics/0503066.

[196]  X. Ji,et al.  GENERALIZED PARTON DISTRIBUTIONS , 2004 .

[197]  P.Schweitzer,et al.  The generalized parton distribution function (E^u+E^d)(x,xi,t) of the nucleon in the chiral quark soliton model , 2004, hep-ph/0411172.

[198]  B. Bakker,et al.  Critique of the angular momentum sum rules and a new angular momentum sum rule , 2004, hep-ph/0406139.

[199]  A. Schaefer,et al.  Exclusive photoproduction of a heavy vector meson in QCD , 2004, hep-ph/0401131.

[200]  V. Braun,et al.  The Uses of conformal symmetry in QCD , 2003, hep-ph/0306057.

[201]  The H1 Collaboration,et al.  Measurement of deeply virtual Compton scattering at HERA , 2005, hep-ex/0505061.

[202]  R. C. Cuevas,et al.  The CEBAF large acceptance spectrometer (CLAS) , 2003 .

[203]  D. Müller,et al.  Exclusive electroproduction of lepton pairs as a probe of nucleon structure. , 2002, Physical review letters.

[204]  M. Polyakov.  Generalized parton distributions and strong forces inside nucleons and nuclei , 2002, hep-ph/0210165.

[205]  O. Teryaev,et al.  Probing chiral-odd GPDs in diffractive electroproduction of two vector mesons , 2002, hep-ph/0209300.

[206]  M. Guidal,et al.  Double deeply virtual Compton scattering off the nucleon. , 2002, Physical review letters.

[207]  S. Boffi,et al.  Polynomiality of unpolarized off forward distribution functions and the D term in the chiral quark soliton model , 2002, hep-ph/0207230.

[208]  A. Belitsky,et al.  Chiral structure of nucleon gravitational form factors , 2002, hep-ph/0203276.

[209]  Y. Obukhov.  On Gravitational Interaction of Fermions , 2001, gr-qc/0112080.

[210]  B. Garbrecht,et al.  Quantum corrections to the Reissner-Nordstrom and Kerr-Newman metrics (vol 529, pg 132, 2002) , 2001, hep-th/0112237.

[211]  A. Belitsky,et al.  Theory of deeply virtual Compton scattering on the nucleon , 2001, hep-ph/0112108.

[212]  M. Diehl,et al.  Timelike Compton scattering: exclusive photoproduction of lepton pairs , 2001, hep-ph/0110062.

[213]  E. al.,et al.  Observation of exclusive deeply virtual Compton scattering in polarized electron beam asymmetry measurements , 2001, hep-ex/0107043.

[214]  E. al.,et al.  Measurement of the Beam-Spin Azimuthal Asymmetry Associated with Deeply-Virtual Compton Scattering , 2001, hep-ex/0106068.

[215]  K. Goeke,et al.  Hard exclusive reactions and the structure of hadrons , 2001, hep-ph/0106012.

[216]  O. Teryaev,et al.  Crossing and Radon tomography for generalized parton distributions , 2001, hep-ph/0102303.

[217]  J. Lattimer,et al.  Evolution of a Neutron Star from Its Birth to Old Age , 2000, astro-ph/0012136.

[218]  S. Brodsky,et al.  Light-cone representation of the spin and orbital angular momentum of relativistic composite systems , 2000, hep-th/0003082.

[219]  U. Meißner,et al.  Virtual photons in the pion form-factors and the energy momentum tensor , 1999, hep-ph/9908261.

[220]  M. Guidal,et al.  Deeply virtual electroproduction of photons and mesons on the nucleon , 1999, hep-ph/9905372.

[221]  O. Teryaev.  Spin Structure of Nucleon and Equivalence Principle , 1999, hep-ph/9904376.

[222]  C. Weiss,et al.  Skewed and double distributions in pion and nucleon , 1999, hep-ph/9902451.

[223]  F. Dittes,et al.  Wave functions, evolution equations and evolution kernels from light ray operators of QCD , 1998, hep-ph/9812448.

[224]  A. Freund,et al.  Proof of factorization for deeply virtual Compton scattering in QCD , 1998, hep-ph/9801262.

[225]  X. Ji.  Lorentz symmetry and the internal structure of the nucleon , 1997, hep-ph/9710290.

[226]  C. Weiss,et al.  Off - forward quark distributions of the nucleon in the large N(c) limit , 1997, hep-ph/9710270.

[227]  X. Ji,et al.  Study of off-forward parton distributions , 1997, hep-ph/9702379.

[228]  J. Collins,et al.  Factorization for hard exclusive electroproduction of mesons in QCD , 1996, hep-ph/9611433.

[229]  X. Ji.  Deeply Virtual Compton Scattering , 1996, hep-ph/9609381.

[230]  Alan D. Martin,et al.  Review of Particle Physics , 2000, Physical Review D.

[231]  A. Radyushkin.  Scaling limit of deeply virtual compton scattering , 1996, hep-ph/9604317.

[232]  X. Ji.  Gauge-Invariant Decomposition of Nucleon Spin , 1996, hep-ph/9603249.

[233]  D. Kharzeev.  Quarkonium Interactions in QCD , 1996, nucl-th/9601029.

[234]  Ji,et al.  Breakup of hadron masses and the energy-momentum tensor of QCD. , 1995, Physical review. D, Particles and fields.

[235]  X. Ji,et al.  QCD analysis of the mass structure of the nucleon. , 1994, Physical review letters.

[236]  Jenkins,et al.  1/Nc expansion for baryons. , 1993, Physical review. D, Particles and fields.

[237]  C. Aidala,et al.  The Spin Structure of the Nucleon , 2012, 1209.2803.

[238]  R. Dashen,et al.  The 1/N$_{c}$ expansion for baryons , 1993 .

[239]  I. Bialynicki-Birula.  Simple relativistic model of a finite-size particle , 1993, nucl-th/9306006.

[240]  H. Leutwyler,et al.  Energy and momentum in chiral theories , 1991 .

[241]  H. Leutwyler,et al.  The decay of a light Higgs boson , 1990 .

[242]  H. Leutwyler,et al.  Goldstone bosons generate peculiar conformal anomalies , 1989 .

[243]  V. Novikov,et al.  Comment on the ψ′→J/ψππ decay , 1981 .

[244]  F. J. Kirschfink,et al.  Evidence for a spin-1 gluon in three-jet events , 1980 .

[245]  E. Witten.  Baryons in the 1/n Expansion , 1979 .

[246]  A. Vainshtein,et al.  Remarks on Higgs-Boson Interactions with Nucleons , 1978 .

[247]  A. Duncan,et al.  Trace and Dilatation Anomalies in Gauge Theories , 1977 .

[248]  N. Nielsen.  The energy-momentum tensor in a non-Abelian quark gluon theory , 1977 .

[249]  V. Weisskopf,et al.  A New Extended Model of Hadrons , 1974 .

[250]  M. Gell-Mann,et al.  Advantages of the color octet gluon picture , 1973 .

[251]  G. Hooft,et al.  Regularization and Renormalization of Gauge Fields , 1972 .

[252]  Richard Phillips Feynman,et al.  Very high-energy collisions of hadrons , 1969 .

[253]  Richard E. Taylor,et al.  High-Energy Inelastic e p Scattering at 6-Degrees and 10-Degrees , 1969 .

[254]  J. Bjorken.  Asymptotic Sum Rules at Infinite Momentum , 1969 .

[255]  H. Pagels.  ENERGY-MOMENTUM STRUCTURE FORM FACTORS OF PARTICLES , 1966 .

[256]  M. Gell-Mann.  A Schematic Model of Baryons and Mesons , 1964, Resonance.

[257]  G. Zweig.  An SU(3) model for strong interaction symmetry and its breaking. Version 1 , 1964 .

[258]  F. Belinfante.  Consequences of the Postulate of a Complete Commuting Set of Observables in Quantum Electrodynamics , 1962 .

[259]  R. Sachs.  High-Energy Behavior of Nucleon Electromagnetic Form Factors , 1962 .

[260]  G. Jona-Lasinio,et al.  Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II , 1961 .

[261]  R. Hofstadter,et al.  Elastic Scattering of 188-Mev Electrons from the Proton and the Alpha Particle , 1956 .

[262]  M. Shapiro.  Mesons and Hyperons , 1956 .

[263]  F. Bloch,et al.  A Quantitative Determination of the Neutron Moment in Absolute Nuclear Magnetons , 1940 .

[264]  C. D. Ellis,et al.  Structure of Atomic Nuclei , 1932, Nature.

[265]  J. CHADWICK,et al.  Possible Existence of a Neutron , 1932, Nature.

[266]  E. Rutherford,et al.  Collision of α particles with light atoms. IV. An anomalous effect in nitrogen , 1919 .

[267]  Fancong Zeng,et al.  2022 Gluon gravitational form factors and mechanical properties of proton , 2022 .

[268]  O. Teryaev,et al.  Hadron tomography by generalized distribution amplitudes in pion-pair production process γ ∗ γ → π 0 π 0 and gravitational form factors for pion , 2021 .

[269]  A. Biselli,et al.  The CLAS12 Spectrometer at Jefferson Laboratory , 2021 .

[270]  K. McGee.  Anomaly , 2020, Encyclopedia of Wireless Networks.

[271]  H. Suzuki.  Energy–momentum tensor from the Yang–Mills gradient flow , 2013 .

[272]  P. Cai,et al.  Reliable Perturbative Results for Strong Interactions ? , 2011 .

[273]  A. Fridman.  Asymptotic Sum Rules at Infinite Momentum , 2011 .

[274]  D. Brömmel.  Pion Structure from the Lattice , 2007 .

[275]  M. Voloshin,et al.  Gravitational interaction of Goldstone bosons , 1982 .

[276]  J. Fregeau.  Nucleon Structure. , 1963, Science.

[277]  O. Stern,et al.  Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I , 1933 .

[278]  M. Laue.  Zur Dynamik der Relativittstheorie , 1911 .

[279]  In-medium modified energy-momentum tensor form factors of the nucleon within the framework of a π - ρ - ω soliton model , 2022 .