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Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Unpolarized quarks at nonzero skewness

Min-Huan Chu1,*, Manuel Colaço1, Shohini Bhattacharya2, Krzysztof Cichy1, Martha Constantinou3, Andreas Metz3, and Fernanda Steffens4

  • *Contact author: minhuan.chu@amu.edu.pl
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Phys. Rev. D 112, 094510 – Published 20 November, 2025

DOI: https://doi.org/10.1103/ts5s-hvb1

Abstract

We extend the formalism of asymmetric frames of reference for generalized parton distributions (GPDs) to the case of nonzero skewness, i.e., including longitudinal momentum transfer. The framework, based on Lorentz-invariant amplitudes and previously developed and numerically implemented for unpolarized, helicity and transversity GPDs at zero skewness, gives efficient access to a broad range of kinematics, making full mapping of GPDs from the lattice realistic. The general-skewness formalism is tested using lattice data with both transverse and longitudinal or only longitudinal momentum transfer, the latter being a special case with a reduced number of independent amplitudes. We extract the amplitudes in coordinate space and express the GPDs H and E in terms of these amplitudes. This is followed by reconstruction of quasidistributions and their matching to the light cone. We further identify and discuss the principal challenges for nonzero skewness GPDs.

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References (202)

  1. D. Müller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hořejši, Fortschr. Phys. 42, 101 (1994).
  2. X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997).
  3. A. V. Radyushkin, Phys. Lett. B 380, 417 (1996).
  4. M. Burkardt, Phys. Rev. D 62, 071503 (2000); 66, 119903(E) (2002).
  5. J. P. Ralston and B. Pire, Phys. Rev. D 66, 111501 (2002).
  6. M. Diehl, Eur. Phys. J. C 25, 223 (2002); 31, 277(E) (2003).
  7. M. Burkardt, Int. J. Mod. Phys. A 18, 173 (2003).
  8. M. V. Polyakov and A. G. Shuvaev, arXiv:hep-ph/0207153.
  9. M. V. Polyakov, Phys. Lett. B 555, 57 (2003).
  10. M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33, 1830025 (2018).
  11. K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47, 401 (2001).
  12. M. Diehl, Phys. Rep. 388, 41 (2003).
  13. X. Ji, Annu. Rev. Nucl. Part. Sci. 54, 413 (2004).
  14. A. V. Belitsky and A. V. Radyushkin, Phys. Rep. 418, 1 (2005).
  15. S. Boffi and B. Pasquini, Riv. Nuovo Cimento 30, 387 (2007).
  16. M. Guidal, H. Moutarde, and M. Vanderhaeghen, Rep. Prog. Phys. 76, 066202 (2013).
  17. D. Mueller, Few Body Syst. 55, 317 (2014).
  18. K. Kumericki, S. Liuti, and H. Moutarde, Eur. Phys. J. A 52, 157 (2016).
  19. C. Lorcé, A. Metz, B. Pasquini, and P. Schweitzer, arXiv:2507.12664.
  20. Y. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977).
  21. V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972).
  22. G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
  23. A. V. Efremov and A. V. Radyushkin, Phys. Lett. B 94B, 245 (1980).
  24. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
  25. X.-D. Ji, Phys. Rev. D 55, 7114 (1997).
  26. J. C. Collins and A. Freund, Phys. Rev. D 59, 074009 (1999).
  27. A. V. Radyushkin, Phys. Lett. B 385, 333 (1996).
  28. J. C. Collins, L. Frankfurt, and M. Strikman, Phys. Rev. D 56, 2982 (1997).
  29. L. Mankiewicz, G. Piller, and T. Weigl, Eur. Phys. J. C 5, 119 (1998).
  30. V. Bertone, H. Dutrieux, C. Mezrag, H. Moutarde, and P. Sznajder, Phys. Rev. D 103, 114019 (2021).
  31. E. Moffat, A. Freese, I. Cloët, T. Donohoe, L. Gamberg, W. Melnitchouk, A. Metz, A. Prokudin, and N. Sato, Phys. Rev. D 108, 036027 (2023).
  32. A. V. Radyushkin, Phys. Rev. D 56, 5524 (1997).
  33. A. V. Radyushkin, Phys. Rev. D 59, 014030 (1999).
  34. M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Eur. Phys. J. C 39, 1 (2005).
  35. P. Schweitzer, S. Boffi, and M. Radici, Nucl. Phys. A711, 207 (2002).
  36. K. A. Mamo and I. Zahed, Phys. Rev. Lett. 133, 241901 (2024).
  37. K. A. Mamo and I. Zahed, Phys. Rev. D 110, 114016 (2024).
  38. F. Hechenberger, K. A. Mamo, and I. Zahed, arXiv:2507.18615.
  39. F. Hechenberger, K. A. Mamo, and I. Zahed, Phys. Rev. D 112, 074018 (2025).
  40. P. Hagler, J. W. Negele, D. B. Renner, W. Schroers, T. Lippert, and K. Schilling (LHPC and SESAM Collaborations), Phys. Rev. D 68, 034505 (2003).
  41. D. Brommel et al. (QCDSF-UKQCD Collaboration), Proc. Sci., LATTICE2007 (2007) 158 [arXiv:0710.1534].
  42. C. Alexandrou, J. Carbonell, M. Constantinou, P. A. Harraud, P. Guichon, K. Jansen, C. Kallidonis, T. Korzec, and M. Papinutto, Phys. Rev. D 83, 114513 (2011).
  43. C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, and G. Koutsou, Phys. Rev. D 88, 014509 (2013).
  44. M. Constantinou, Proc. Sci., LATTICE2014 (2015) 001 [arXiv:1411.0078].
  45. J. R. Green, J. W. Negele, A. V. Pochinsky, S. N. Syritsyn, M. Engelhardt, and S. Krieg, Phys. Rev. D 90, 074507 (2014).
  46. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Phys. Rev. D 96, 034503 (2017).
  47. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Phys. Rev. D 96, 054507 (2017).
  48. N. Hasan, J. Green, S. Meinel, M. Engelhardt, S. Krieg, J. Negele, A. Pochinsky, and S. Syritsyn, Phys. Rev. D 97, 034504 (2018).
  49. R. Gupta, Y.-C. Jang, H.-W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D 96, 114503 (2017).
  50. S. Capitani, M. Della Morte, D. Djukanovic, G. M. von Hippel, J. Hua, B. Jäger, P. M. Junnarkar, H. B. Meyer, T. D. Rae, and H. Wittig, Int. J. Mod. Phys. A 34, 1950009 (2019).
  51. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, Phys. Rev. D 100, 014509 (2019).
  52. E. Shintani, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, and T. Yamazaki, Phys. Rev. D 99, 014510 (2019); 102, 019902(E) (2020).
  53. Y.-C. Jang, T. Bhattacharya, R. Gupta, H.-W. Lin, and B. Yoon (PNDME Collaboration), Proc. Sci., LATTICE2018 (2018) 123 [arXiv:1901.00060].
  54. G. S. Bali, S. Collins, M. Gruber, A. Schäfer, P. Wein, and T. Wurm, Phys. Lett. B 789, 666 (2019).
  55. G. S. Bali, S. Collins, M. Göckeler, R. Rödl, A. Schäfer, and A. Sternbeck, Phys. Rev. D 100, 014507 (2019).
  56. C. Alexandrou et al., Phys. Rev. D 101, 034519 (2020).
  57. Y.-C. Jang, R. Gupta, H.-W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D 100, 014507 (2020).
  58. M. Constantinou et al., Prog. Part. Nucl. Phys. 121, 103908 (2021).
  59. C. Alexandrou et al., Phys. Rev. D 107, 054504 (2023).
  60. R. Tsuji, Y. Aoki, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, K. Sato, E. Shintani, H. Watanabe, and T. Yamazaki (PACS Collaboration), Phys. Rev. D 109, 094505 (2024).
  61. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Phys. Rev. D 109, 094510 (2024).
  62. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Phys. Rev. Lett. 132, 211901 (2024).
  63. D. C. Hackett, D. A. Pefkou, and P. E. Shanahan, Phys. Rev. Lett. 132, 251904 (2024).
  64. C. Alexandrou, S. Bacchio, G. Koutsou, B. Prasad, and G. Spanoudes, arXiv:2507.20910.
  65. K.-F. Liu and S.-J. Dong, Phys. Rev. Lett. 72, 1790 (1994).
  66. V. Braun, P. Gornicki, and L. Mankiewicz, Phys. Rev. D 51, 6036 (1995).
  67. U. Aglietti, M. Ciuchini, G. Corbo, E. Franco, G. Martinelli, and L. Silvestrini, Phys. Lett. B 441, 371 (1998).
  68. W. Detmold and C. J. D. Lin, Phys. Rev. D 73, 014501 (2006).
  69. V. Braun and D. Müller, Eur. Phys. J. C 55, 349 (2008).
  70. X. Ji, Phys. Rev. Lett. 110, 262002 (2013).
  71. X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014).
  72. X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, Rev. Mod. Phys. 93, 035005 (2021).
  73. X. Xiong, X. Ji, J.-H. Zhang, and Y. Zhao, Phys. Rev. D 90, 014051 (2014).
  74. Y.-Q. Ma and J.-W. Qiu, Phys. Rev. D 98, 074021 (2018).
  75. T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao, Phys. Rev. D 98, 056004 (2018).
  76. A. J. Chambers, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, K. Somfleth, R. D. Young, and J. M. Zanotti, Phys. Rev. Lett. 118, 242001 (2017).
  77. A. V. Radyushkin, Phys. Rev. D 96, 034025 (2017).
  78. A. V. Radyushkin, Int. J. Mod. Phys. A 35, 2030002 (2020).
  79. Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett. 120, 022003 (2018).
  80. W. Detmold, A. V. Grebe, I. Kanamori, C. J. D. Lin, R. J. Perry, and Y. Zhao (HOPE Collaboration), Phys. Rev. D 104, 074511 (2021).
  81. H.-W. Lin, J.-W. Chen, S. D. Cohen, and X. Ji, Phys. Rev. D 91, 054510 (2015).
  82. C. Alexandrou, K. Cichy, V. Drach, E. Garcia-Ramos, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Phys. Rev. D 92, 014502 (2015).
  83. J.-W. Chen, S. D. Cohen, X. Ji, H.-W. Lin, and J.-H. Zhang, Nucl. Phys. B911, 246 (2016).
  84. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Phys. Rev. D 96, 014513 (2017).
  85. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, H. Panagopoulos, and F. Steffens, Nucl. Phys. B923, 394 (2017).
  86. J.-W. Chen, T. Ishikawa, L. Jin, H.-W. Lin, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Phys. Rev. D 97, 014505 (2018).
  87. K. Orginos, A. Radyushkin, J. Karpie, and S. Zafeiropoulos, Phys. Rev. D 96, 094503 (2017).
  88. H.-W. Lin, J.-W. Chen, T. Ishikawa, and J.-H. Zhang (LP3 Collaboration), Phys. Rev. D 98, 054504 (2018).
  89. C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. Lett. 121, 112001 (2018).
  90. H.-W. Lin, J.-W. Chen, X. Ji, L. Jin, R. Li, Y.-S. Liu, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Phys. Rev. Lett. 121, 242003 (2018).
  91. C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D 98, 091503 (2018).
  92. Y.-S. Liu et al. (Lattice Parton Collaboration), Phys. Rev. D 101, 034020 (2020).
  93. J.-H. Zhang, J.-W. Chen, L. Jin, H.-W. Lin, A. Schäfer, and Y. Zhao, Phys. Rev. D 100, 034505 (2019).
  94. R. S. Sufian, J. Karpie, C. Egerer, K. Orginos, J.-W. Qiu, and D. G. Richards, Phys. Rev. D 99, 074507 (2019).
  95. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D 99, 114504 (2019).
  96. T. Izubuchi, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, and S. Syritsyn, Phys. Rev. D 100, 034516 (2019).
  97. B. Joó, J. Karpie, K. Orginos, A. Radyushkin, D. Richards, and S. Zafeiropoulos, J. High Energy Phys. 12 (2019) 081.
  98. B. Joó, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, R. S. Sufian, and S. Zafeiropoulos, Phys. Rev. D 100, 114512 (2019).
  99. K. Cichy, L. Del Debbio, and T. Giani, J. High Energy Phys. 10 (2019) 137.
  100. J. Bringewatt, N. Sato, W. Melnitchouk, J.-W. Qiu, F. Steffens, and M. Constantinou, Phys. Rev. D 103, 016003 (2021).
  101. L. Del Debbio, T. Giani, J. Karpie, K. Orginos, A. Radyushkin, and S. Zafeiropoulos, J. High Energy Phys. 02 (2021) 138.
  102. Y. Chai et al., Phys. Rev. D 102, 014508 (2020).
  103. B. Joó, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, and S. Zafeiropoulos, Phys. Rev. Lett. 125, 232003 (2020).
  104. M. Bhat, K. Cichy, M. Constantinou, and A. Scapellato, Phys. Rev. D 103, 034510 (2021).
  105. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, and F. Manigrasso, Phys. Rev. Lett. 126, 102003 (2021).
  106. C. Alexandrou, K. Cichy, M. Constantinou, J. R. Green, K. Hadjiyiannakou, K. Jansen, F. Manigrasso, A. Scapellato, and F. Steffens, Phys. Rev. D 103, 094512 (2021).
  107. H.-W. Lin, J.-W. Chen, Z. Fan, J.-H. Zhang, and R. Zhang, Phys. Rev. D 103, 014516 (2021).
  108. Z. Fan, X. Gao, R. Li, H.-W. Lin, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, Y.-B. Yang, and R. Zhang, Phys. Rev. D 102, 074504 (2020).
  109. X. Gao, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, S. Syritsyn, and Y. Zhao, Phys. Rev. D 102, 094513 (2020).
  110. H.-W. Lin, J.-W. Chen, and R. Zhang, arXiv:2011.14971.
  111. J. Karpie, K. Orginos, A. Radyushkin, and S. Zafeiropoulos (HadStruc Collaboration), J. High Energy Phys. 11 (2021) 024.
  112. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, and F. Manigrasso, Phys. Rev. D 104, 054503 (2021).
  113. C. Egerer, R. G. Edwards, C. Kallidonis, K. Orginos, A. V. Radyushkin, D. G. Richards, E. Romero, and S. Zafeiropoulos (HadStruc Collaboration), J. High Energy Phys. 11 (2021) 148.
  114. C. Egerer et al. (HadStruc Collaboration), Phys. Rev. D 105, 034507 (2022).
  115. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Phys. Rev. Lett. 128, 142003 (2022).
  116. X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Shi, S. Syritsyn, Y. Zhao, and K. Zhou, Phys. Rev. D 106, 114510 (2022).
  117. F. Yao et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 131, 261901 (2023).
  118. C. Egerer et al. (HadStruc Collaboration), Phys. Rev. D 106, 094511 (2022).
  119. R. G. Edwards et al. (HadStruc Collaboration), J. High Energy Phys. 03 (2023) 086.
  120. X. Gao, A. D. Hanlon, J. Holligan, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, and Y. Zhao, Phys. Rev. D 107, 074509 (2023).
  121. J. Delmar, C. Alexandrou, K. Cichy, M. Constantinou, and K. Hadjiyiannakou, Phys. Rev. D 108, 094515 (2023).
  122. J. Holligan and H.-W. Lin, J. Phys. G 51, 065101 (2024).
  123. J. Holligan and H.-W. Lin, Phys. Lett. B 854, 138731 (2024).
  124. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 102, 111501 (2020).
  125. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 102, 034005 (2020); 108, 039901(E) (2023).
  126. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 102, 114025 (2020).
  127. S. Bhattacharya and A. Metz, Phys. Rev. D 105, 054027 (2022).
  128. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 104, 114510 (2021).
  129. J.-H. Zhang, L. Jin, H.-W. Lin, A. Schäfer, P. Sun, Y.-B. Yang, R. Zhang, Y. Zhao, and J.-W. Chen (LP3 Collaboration), Nucl. Phys. B939, 429 (2019).
  130. J.-H. Zhang, J.-W. Chen, X. Ji, L. Jin, and H.-W. Lin, Phys. Rev. D 95, 094514 (2017).
  131. G. S. Bali, V. M. Braun, B. Gläßle, M. Göckeler, M. Gruber, F. Hutzler, P. Korcyl, A. Schäfer, P. Wein, and J.-H. Zhang, Phys. Rev. D 98, 094507 (2018).
  132. J. Xu, Q.-A. Zhang, and S. Zhao, Phys. Rev. D 97, 114026 (2018).
  133. R. Zhang, C. Honkala, H.-W. Lin, and J.-W. Chen, Phys. Rev. D 102, 094519 (2020).
  134. J. Hua, M.-H. Chu, P. Sun, W. Wang, J. Xu, Y.-B. Yang, J.-H. Zhang, and Q.-A. Zhang (Lattice Parton Collaboration), Phys. Rev. Lett. 127, 062002 (2021).
  135. W. Detmold, A. V. Grebe, I. Kanamori, C. J. D. Lin, S. Mondal, R. J. Perry, and Y. Zhao (HOPE Collaboration), Phys. Rev. D 105, 034506 (2022).
  136. J. Hua et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 129, 132001 (2022).
  137. X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Phys. Rev. D 106, 074505 (2022).
  138. J. Holligan, X. Ji, H.-W. Lin, Y. Su, and R. Zhang, Nucl. Phys. B993, 116282 (2023).
  139. M.-H. Chu et al. (Lattice Parton Collaboration), Phys. Rev. D 111, 034510 (2025).
  140. X.-Y. Han et al. (Lattice Parton Collaboration), Phys. Rev. D 111, 034503 (2025).
  141. Q.-A. Zhang et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 125, 192001 (2020).
  142. P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 102, 014511 (2020).
  143. Q.-A. Zhang et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 125, 192001 (2020).
  144. Y. Li et al., Phys. Rev. Lett. 128, 062002 (2022).
  145. P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 104, 114502 (2021).
  146. M. Schlemmer, A. Vladimirov, C. Zimmermann, M. Engelhardt, and A. Schäfer, J. High Energy Phys. 08 (2021) 004.
  147. M.-H. Chu et al. (Lattice Parton (LPC) Collaboration), Phys. Rev. D 106, 034509 (2022).
  148. J.-C. He, M.-H. Chu, J. Hua, X. Ji, A. Schäfer, Y. Su, W. Wang, Y.-B. Yang, J.-H. Zhang, and Q.-A. Zhang (Lattice Parton Collaboration (LPC)), Phys. Rev. D 109, 114513 (2024).
  149. K. Zhang, X. Ji, Y.-B. Yang, F. Yao, and J.-H. Zhang (Lattice Parton (LPC) Collaboration), Phys. Rev. Lett. 129, 082002 (2022).
  150. C. Alexandrou et al., Phys. Rev. D 108, 114503 (2023).
  151. M.-H. Chu et al. (Lattice Parton Collaboration), Phys. Rev. D 109, L091503 (2024).
  152. M.-H. Chu et al. (Lattice Parton (LPC) Collaboration), J. High Energy Phys. 08 (2023) 172.
  153. L. Walter et al. (Lattice Parton, LPC Collaboration), Phys. Rev. D 111, 094507 (2025).
  154. A. Shindler, Phys. Rev. D 110, L051503 (2024).
  155. X. Gao, W.-Y. Liu, and Y. Zhao, Phys. Rev. D 109, 094506 (2024).
  156. K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019).
  157. M. Constantinou, Eur. Phys. J. A 57, 77 (2021).
  158. K. Cichy, Proc. Sci., LATTICE2021 (2022) 017 [arXiv:2110.07440].
  159. K. Cichy, EPJ Web Conf. 258, 01005 (2022).
  160. X. Ji, A. Schäfer, X. Xiong, and J.-H. Zhang, Phys. Rev. D 92, 014039 (2015).
  161. X. Xiong and J.-H. Zhang, Phys. Rev. D 92, 054037 (2015).
  162. Y.-S. Liu, W. Wang, J. Xu, Q.-A. Zhang, J.-H. Zhang, S. Zhao, and Y. Zhao, Phys. Rev. D 100, 034006 (2019).
  163. A. V. Radyushkin, Phys. Rev. D 100, 116011 (2019).
  164. J. P. Ma, Z. Y. Pang, and G. P. Zhang, J. High Energy Phys. 08 (2022) 130.
  165. F. Yao, Y. Ji, and J.-H. Zhang, J. High Energy Phys. 11 (2023) 021.
  166. S. Bhattacharya, C. Cocuzza, and A. Metz, Phys. Lett. B 788, 453 (2019).
  167. S. Bhattacharya, C. Cocuzza, and A. Metz, Phys. Rev. D 102, 054021 (2020).
  168. Z.-L. Ma, J.-Q. Zhu, and Z. Lu, Phys. Rev. D 101, 114005 (2020).
  169. X. Luo and H. Sun, Eur. Phys. J. C 80, 828 (2020).
  170. V. Shastry, W. Broniowski, and E. Ruiz Arriola, Phys. Rev. D 106, 114035 (2022).
  171. J.-W. Chen, H.-W. Lin, and J.-H. Zhang, Nucl. Phys. B952, 114940 (2020).
  172. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. Lett. 125, 262001 (2020).
  173. H.-W. Lin, Phys. Rev. Lett. 127, 182001 (2021).
  174. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D 105, 034501 (2022).
  175. A. Hannaford-Gunn, K. U. Can, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, H. Stüben, G. Schierholz, R. D. Young, and J. M. Zanotti (CSSM/QCDSF/UKQCD Collaboration), Phys. Rev. D 105, 014502 (2022).
  176. H.-W. Lin, Phys. Lett. B 824, 136821 (2022).
  177. S. Bhattacharya, K. Cichy, M. Constantinou, J. Dodson, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 108, 054501 (2023).
  178. S. Bhattacharya, K. Cichy, M. Constantinou, J. Dodson, X. Gao, A. Metz, S. Mukherjee, A. Scapellato, F. Steffens, and Y. Zhao, Phys. Rev. D 106, 114512 (2022).
  179. V. M. Braun, J. High Energy Phys. 10 (2023) 134.
  180. K. Cichy et al., Acta Phys. Pol. B Proc. Suppl. 16, 7 (2023).
  181. S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, Phys. Rev. D 108, 014507 (2023).
  182. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, N. Nurminen, and F. Steffens, Phys. Rev. D 110, 054502 (2024).
  183. H.-T. Ding, X. Gao, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, and Y. Zhao, J. High Energy Phys. 02 (2025) 056.
  184. K. Cichy, M. Constantinou, P. Sznajder, and J. Wagner, Phys. Rev. D 110, 114025 (2024).
  185. S. Bhattacharya et al., Phys. Rev. D 109, 034508 (2024).
  186. S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, J. High Energy Phys. 01 (2025) 146.
  187. S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, J. Miller, P. Petreczky, and F. Steffens, arXiv:2505.11288.
  188. J. Holligan and H.-W. Lin, Phys. Rev. D 110, 034503 (2024).
  189. X. Ji, Y. Liu, A. Schäfer, W. Wang, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Nucl. Phys. B964, 115311 (2021).
  190. H. Dutrieux, R. G. Edwards, C. Egerer, J. Karpie, C. Monahan, K. Orginos, A. Radyushkin, D. Richards, E. Romero, and S. Zafeiropoulos (HadStruc Collaboration), J. High Energy Phys. 08 (2024) 162.
  191. A. V. Radyushkin, Phys. Rev. D 109, 014514 (2024).
  192. C. Alexandrou et al., Phys. Rev. D 98, 054518 (2018).
  193. G. S. Bali, B. Lang, B. U. Musch, and A. Schäfer, Phys. Rev. D 93, 094515 (2016).
  194. G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, Nucl. Phys. B445, 81 (1995).
  195. J. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropoulos, J. High Energy Phys. 04 (2019) 057.
  196. G. Backus and F. Gilbert, Geophys. J. Int. 16, 169 (1968).
  197. M.-H. Chu, K. Cichy, M. Constantinou, P. Sznajder, and J. Wagner, arXiv:2509.15931.
  198. J. Holligan, H.-W. Lin, R. Zhang, and Y. Zhao, J. High Energy Phys. 07 (2025) 241.
  199. V. Y. Petrov, P. V. Pobylitsa, M. V. Polyakov, I. Bornig, K. Goeke, and C. Weiss, Phys. Rev. D 57, 4325 (1998).
  200. X. Ji, Research 8, 0695 (2025).
  201. A. Frommer, K. Kahl, S. Krieg, B. Leder, and M. Rottmann, SIAM J. Sci. Comput. 36, A1581 (2014).
  202. C. Alexandrou, S. Bacchio, J. Finkenrath, A. Frommer, K. Kahl, and M. Rottmann, Phys. Rev. D 94, 114509 (2016).

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Phys. Rev. D 112, 094510– Published 20 November, 2025
  • Received 30 August 2025
  • Accepted 28 October 2025
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