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Tensor form factors of the Δ+ baryon induced by isovector and isoscalar currents in QCD

Z. Asmaee1, N. Hajirasouliha1, and K. Azizi1,2,*

  • *Contact author: kazem.azizi@ut.ac.ir
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Phys. Rev. D 113, 054002 – Published 2 March, 2026

DOI: https://doi.org/10.1103/mlt1-gvnw

Abstract

The tensor form factors of the Δ+ baryon are defined through the matrix element of the tensor current and describe its internal structure and spin distribution. We present the full Lorentz decomposition for the Δ+→Δ+ tensor current matrix element, including all independent structures consistent with Lorentz covariance, the Rarita–Schwinger constraints, and the discrete symmetries of Hermiticity, time-reversal, and parity invariance. By investigating the tensor form factors corresponding to both the isovector and isoscalar tensor currents, we observe differences that reflect the distinct contributions of up and down quark components in the Δ+ baryon.

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References (79)

  1. T. M. Aliev, K. Azizi, A. Ozpineci, and M. Savci, Nucleon electromagnetic form factors in QCD, Phys. Rev. D 77, 114014 (2008).
  2. N. Er and K. Azizi, Spectroscopic parameters and electromagnetic form factor of kaon in vacuum and a dense medium, Eur. Phys. J. C 82, 397 (2022).
  3. H. Sundu, B. Barsbay, S. S. Agaev, and K. Azizi, Probing an axial-vector tetraquark Zs via its semileptonic decay Zs→ X(4274) l¯ νl, Eur. Phys. J. A 54, 124 (2018).
  4. B. D. Sun and Y. B. Dong, Gravitational form factors of ρ meson with a light-cone constituent quark model, Phys. Rev. D 101, 096008 (2020).
  5. D. Fu, B. D. Sun, and Y. Dong, Generalized parton distributions of Δ resonance in a diquark spectator approach, Phys. Rev. D 107, 116021 (2023).
  6. S. Kumano, Q. T. Song, and O. V. Teryaev, Hadron tomography by generalized distribution amplitudes in pion-pair production process γ*γ→π0π0 and gravitational form factors for pion, Phys. Rev. D 97, 014020 (2018).
  7. A. M. Bincer, Electromagnetic structure of the nucleon, Phys. Rev. 118, 855 (1960).
  8. V. Keiner, A Covariant diquark—quark model of the nucleon in the Salpeter approach, Phys. Rev. C 54, 3232 (1996).
  9. H. C. Kim, P. Schweitzer, and U. Yakhshiev, Energy-momentum tensor form factors of the nucleon in nuclear matter, Phys. Lett. B 718, 625 (2012).
  10. W. Cosyn, A. Freese, and B. Pire, Polynomiality sum rules for generalized parton distributions of spin-1 targets, Phys. Rev. D 99, 094035 (2019).
  11. B. D. Sun and Y. B. Dong, ρ meson unpolarized generalized parton distributions with a light-front constituent quark model, Phys. Rev. D 96, 036019 (2017).
  12. Y. Dong and C. Liang, Generalized parton distribution functions of a deuteron in a phenomenological Lagrangian approach, J. Phys. G 40, 025001 (2013).
  13. M. V. Polyakov and B. D. Sun, Gravitational form factors of a spin one particle, Phys. Rev. D 100, 036003 (2019).
  14. C. Alexandrou, T. Korzec, G. Koutsou, T. Leontiou, C. Lorce, J. W. Negele, V. Pascalutsa, A. Tsapalis, and M. Vanderhaeghen, Delta-baryon electromagnetic form factors in lattice QCD, Phys. Rev. D 79, 014507 (2009).
  15. C. Alexandrou, T. Korzec, G. Koutsou, J. W. Negele, and Y. Proestos, The electromagnetic form factors of the Ω− in lattice QCD, Phys. Rev. D 82, 034504 (2010).
  16. S. Boinepalli, D. B. Leinweber, P. J. Moran, A. G. Williams, J. M. Zanotti, and J. B. Zhang, Precision electromagnetic structure of decuplet baryons in the chiral regime, Phys. Rev. D 80, 054505 (2009).
  17. C. Aubin, K. Orginos, V. Pascalutsa, and M. Vanderhaeghen, Lattice calculation of the magnetic moments of and baryons with dynamical clover fermions, Phys. Rev. D 79, 051502 (2009).
  18. D. B. Leinweber, T. Draper, and R. M. Woloshyn, Decuplet baryon structure from lattice QCD, Phys. Rev. D 46, 3067 (1992).
  19. F. X. Lee, R. Kelly, L. Zhou, and W. Wilcox, Baryon magnetic moments in the background field method, Phys. Lett. B 627, 71 (2005).
  20. Y. Oh, Electric quadrupole moments of the decuplet baryons in the Skyrme model, Mod. Phys. Lett. A 10, 1027 (1995).
  21. L. S. Geng, J. Martin Camalich, and M. J. Vicente Vacas, Electromagnetic structure of the lowest-lying decuplet resonances in covariant chiral perturbation theory, Phys. Rev. D 80, 034027 (2009).
  22. H. S. Li, Z. W. Liu, X. L. Chen, W. Z. Deng, and S. L. Zhu, Magnetic moments and electromagnetic form factors of the decuplet baryons in chiral perturbation theory, Phys. Rev. D 95, 076001 (2017).
  23. M. I. Krivoruchenko and M. M. Giannini, Quadrupole moments of the decuplet baryons, Phys. Rev. D 43, 3763 (1991).
  24. K. Berger, R. F. Wagenbrunn, and W. Plessas, Covariant baryon charge radii and magnetic moments in a chiral constituent quark model, Phys. Rev. D 70, 094027 (2004).
  25. F. Schlumpf, Magnetic moments of the baryon decuplet in a relativistic quark model, Phys. Rev. D 48, 4478 (1993).
  26. T. M. Aliev, K. Azizi, and A. Ozpineci, Radiative decays of the heavy flavored baryons in light cone QCD sum rules, Phys. Rev. D 79, 056005 (2009).
  27. T. M. Aliev, K. Azizi, and M. Savci, Analysis of the Λb→Λℓ+ℓ− decay in QCD, Phys. Rev. D 81, 056006 (2010).
  28. T. M. Aliev, K. Azizi, and M. Savci, Electric quadrupole and magnetic octupole moments of the light decuplet baryons within light cone QCD sum rules, Phys. Lett. B 681, 240 (2009).
  29. F. X. Lee, Determination of decuplet baryon magnetic moments from QCD sum rules, Phys. Rev. D 57, 1801 (1998).
  30. K. Azizi, Magnetic dipole, electric quadrupole and magnetic octupole moments of the delta baryons in light cone QCD sum rules, Eur. Phys. J. C 61, 311 (2009).
  31. G. Wagner, A. J. Buchmann, and A. Faessler, Electromagnetic properties of decuplet hyperons in a chiral quark model with exchange currents, J. Phys. G 26, 267 (2000).
  32. J. Y. Kim and H. C. Kim, Electromagnetic form factors of the baryon decuplet with flavor SU(3) symmetry breaking, Eur. Phys. J. C 79, 570 (2019).
  33. J. Y. Kim and B. D. Sun, Gravitational form factors of a baryon with spin-3/2, Eur. Phys. J. C 81, 85 (2021).
  34. J. Wang, D. Fu, and Y. Dong, Form factors of decuplet baryons in a covariant quark–diquark approach, Eur. Phys. J. C 84, 79 (2024).
  35. Z. Dehghan, K. Azizi, and U. Özdem, Gravitational form factors of the Δ baryon via QCD sum rules, Phys. Rev. D 108, 094037 (2023).
  36. Z. Dehghan, F. Almaksusi, and K. Azizi, Mechanical properties of proton using flavor-decomposed gravitational form factors, J. High Energy Phys. 06 (2025) 025.
  37. Z. Dehghan and K. Azizi, Mechanical properties of the Ω- baryon from gravitational form factors, Phys. Rev. D 112, 054014 (2025).
  38. A. Kucukarslan, U. Ozdem, and A. Ozpineci, Tensor form factors of the octet hyperons in QCD, Phys. Rev. D 94, 094010 (2016).
  39. T. M. Aliev, K. Azizi, and M. Savci, Nucleon tensor form factors induced by isovector and isoscalar currents in QCD, Phys. Rev. D 84, 076005 (2011).
  40. T. Gutsche, M. A. Ivanov, J. G. Korner, S. Kovalenko, and V. E. Lyubovitskij, Nucleon tensor form factors in a relativistic confined quark model, Phys. Rev. D 94, 114030 (2016).
  41. K. Azizi and U. Özdem, Nucleon’s energy–momentum tensor form factors in light-cone QCD, Eur. Phys. J. C 80, 104 (2020).
  42. U. Özdem, Isovector and isoscalar tensor form factors of N(1535)→N transition in light-cone QCD, Phys. Rev. D 102, 014001 (2020).
  43. U. Özdem, Tensor form factors of N(1535) state via light-cone QCD, Chin. J. Phys. 72, 93 (2021).
  44. J. P. Ralston and D. E. Soper, Production of dimuons from high-energy polarized proton proton collisions, Nucl. Phys. B152, 109 (1979).
  45. R. L. Jaffe and X. D. Ji, Chiral odd parton distributions and polarized Drell-Yan, Phys. Rev. Lett. 67, 552 (1991).
  46. R. L. Jaffe and X. D. Ji, Chiral odd parton distributions and Drell-Yan processes, Nucl. Phys. B375, 527 (1992).
  47. V. Barone, A. Drago, and P. G. Ratcliffe, Transverse polarisation of quarks in hadrons, Phys. Rep. 359, 1 (2002).
  48. M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, and C. Turk, Transversity and Collins functions from SIDIS and e+e− data, Phys. Rev. D 75, 054032 (2007).
  49. M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, and S. Melis, Update on transversity and Collins functions from SIDIS and e+e− data, Nucl. Phys. B, Proc. Suppl. 191, 98 (2009).
  50. M. Anselmino, M. Boglione, U. D’Alesio, S. Melis, F. Murgia, and A. Prokudin, Simultaneous extraction of transversity and Collins functions from new SIDIS and e+e− data, Phys. Rev. D 87, 094019 (2013).
  51. H. He and X. D. Ji, The Nucleon’s tensor charge, Phys. Rev. D 52, 2960 (1995).
  52. B. Pasquini, M. Pincetti, and S. Boffi, Chiral-odd generalized parton distributions in constituent quark models, Phys. Rev. D 72, 094029 (2005).
  53. L. P. Gamberg and G. R. Goldstein, Flavor spin symmetry estimate of the nucleon tensor charge, Phys. Rev. Lett. 87, 242001 (2001).
  54. H. x. He and X. D. Ji, QCD sum rule calculation for the tensor charge of the nucleon, Phys. Rev. D 54, 6897 (1996).
  55. D. Fu, Y. Dong, and S. Kumano, Transversity generalized parton distributions in spin-3/2 particles, Phys. Rev. D 109, 096006 (2024).
  56. M. Diehl and P. Hagler, Spin densities in the transverse plane and generalized transversity distributions, Eur. Phys. J. C 44, 87 (2005).
  57. M. Burkardt, Impact parameter dependent parton distributions and off forward parton distributions for ζ—>0, Phys. Rev. D 62, 071503 (2000).
  58. M. Burkardt, Impact parameter space interpretation for generalized parton distributions, Int. J. Mod. Phys. A 18, 173 (2003).
  59. K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Hard exclusive reactions and the structure of hadrons, Prog. Part. Nucl. Phys. 47, 401 (2001).
  60. M. Diehl, Generalized parton distributions with helicity flip, Eur. Phys. J. C 19, 485 (2001).
  61. M. Diehl, Generalized parton distributions, Phys. Rep. 388, 41 (2003).
  62. A. V. Belitsky and A. V. Radyushkin, Unraveling hadron structure with generalized parton distributions, Phys. Rep. 418, 1 (2005).
  63. R. L. Workman et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  64. B. L. Ioffe, Calculation of baryon masses in quantum chromodynamics, Nucl. Phys. B188, 317 (1981).
  65. T. M. Aliev, A. Ozpineci, and M. Savci, Octet baryon magnetic moments in light cone QCD sum rules, Phys. Rev. D 66, 016002 (2002).
  66. K. Azizi and N. Er, Properties of nucleon in nuclear matter: Once more, Eur. Phys. J. C 74, 2904 (2014).
  67. U. Ozdem and K. Azizi, Magnetic and quadrupole moments of the Zc(3900), Phys. Rev. D 96, 074030 (2017).
  68. K. Azizi, A. R. Olamaei, and S. Rostami, Beautiful mathematics for beauty-full and other multi-heavy hadronic systems, Eur. Phys. J. A 54, 162 (2018).
  69. K. Azizi and N. Er, X (3872): Propagating in a dense medium, Nucl. Phys. B936, 151 (2018).
  70. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  71. V. M. Belyaev and B. L. Ioffe, Determination of baryon and baryonic resonance masses from QCD sum rules. 1. Nonstrange baryons, Sov. Phys. JETP 56, 493 (1982).
  72. V. M. Belyaev and B. L. Ioffe, Determination of the baryon mass and baryon resonances from the quantum-chromodynamics sum rule. Strange baryons, Sov. Phys. JETP 57, 716 (1983).
  73. T. M. Aliev, K. Azizi, and A. Ozpineci, Light cone QCD sum rules analysis of the axial N→Δ transition form-factors, Nucl. Phys. A799, 105 (2008).
  74. P. Abreu et al. (DELPHI Collaboration), Determination of alpha-s from the scaling violation in the fragmentation functions in e+e− annihilation, Phys. Lett. B 311, 408 (1993).
  75. M. Pospelov and A. Ritz, Electric dipole moments as probes of new physics, Ann. Phys. (Amsterdam) 318, 119 (2005).
  76. Q. W. Wang, S. X. Qin, C. D. Roberts, and S. M. Schmidt, Proton tensor charges from a Poincaré-covariant Faddeev equation, Phys. Rev. D 98, 054019 (2018).
  77. S. Cotogno, C. Lorcé, P. Lowdon, and M. Morales, Covariant multipole expansion of local currents for massive states of any spin, Phys. Rev. D 101, 056016 (2020).
  78. A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Korner, and V. E. Lyubovitskij, Semileptonic decays of double heavy baryons in a relativistic constituent three-quark model, Phys. Rev. D 80, 034025 (2009).
  79. S. Agaev, K. Azizi, and H. Sundu, Four-quark exotic mesons, Turk. J. Phys. 44, 95 (2020).

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Phys. Rev. D 113, 054002– Published 2 March, 2026
  • Received 17 November 2025
  • Accepted 11 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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