Skip to main content
SpringerLink
Log in

Symmetries of the Nucleon–Nucleon S-Matrix and Effective Field Theory Expansions

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

The s-wave nucleon–nucleon scattering matrix (S-matrix) exhibits UV/IR symmetries which are hidden in the effective field theory (EFT) action and scattering amplitudes, and which explain some generic features of the phase shifts. These symmetries offer clarifying interpretations of existing pionless EFT expansions, and suggest starting points for novel expansions. The leading-order (LO) S-matrix obtained in the pionless EFT with scattering lengths treated exactly is shown to have a UV/IR symmetry which leaves the sum of s-wave phase shifts invariant. A new scheme, which treats effective range corrections exactly, and which possesses a distinct UV/IR symmetry at LO, is developed up to NLO (next-to-LO) and compared with data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The literature is too vast to cite in its entirety. For helpful discussions, see Refs. [11,12,13].

  2. For an alternate point of view that consistent renormalization has been achieved see Refs. [13, 16,17,18,19,20,21].

  3. Halo nuclei have also been described successfully by an extension of the pionless EFT technology [27, 28].

  4. Note however that scattering with s-wave resonances can be described with zero-range forces [35].

  5. This fixed point is not to be confused with the fixed points of the RG at which the EFT exhibits Schrödinger symmetry and there is no scattering.

  6. The general case is considered in Ref. [47].

  7. Note that the potential considered in the previous section is phase equivalent only in the limiting sense of \(\varLambda \rightarrow \infty \) and \(r < 0\).

  8. The linearly divergent integral \({{\mathbb {J}}}_2\) in the \({\overline{MS}}\) scheme can be obtained by setting \(\omega = 0\). Similarly, cutoff regularization, as in Eq. (31), is obtained by replacing \(\omega \) with \(\varLambda \) (for \(\varLambda \) large).

  9. Note that one can also decompose \(a=a_{\scriptscriptstyle {LO}}+a_{\scriptscriptstyle {NLO}}\) in which case this condition follows from \(a_{\scriptscriptstyle {NLO}}=0\).

References

  1. S. Weinberg, Physica A 96(1–2), 327 (1979). https://doi.org/10.1016/0378-4371(79)90223-1

    Article  ADS  Google Scholar 

  2. J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984). https://doi.org/10.1016/0003-4916(84)90242-2

    Article  ADS  Google Scholar 

  3. S. Weinberg, Phys. Lett. B 251, 288 (1990). https://doi.org/10.1016/0370-2693(90)90938-3

    Article  ADS  Google Scholar 

  4. S. Weinberg, Nucl. Phys. B 363, 3 (1991). https://doi.org/10.1016/0550-3213(91)90231-L

    Article  ADS  Google Scholar 

  5. S. Weinberg, Phys. Lett. B 295, 114 (1992). https://doi.org/10.1016/0370-2693(92)90099-P

    Article  ADS  Google Scholar 

  6. C. Ordonez, U. van Kolck, Phys. Lett. B 291, 459 (1992). https://doi.org/10.1016/0370-2693(92)91404-W

    Article  ADS  Google Scholar 

  7. C. Ordoñez, L. Ray, U. van Kolck, Phys. Rev. Lett. 72, 1982 (1994). https://doi.org/10.1103/PhysRevLett.72.1982

    Article  ADS  Google Scholar 

  8. C. Ordoñez, L. Ray, U. van Kolck, Phys. Rev. C 53, 2086 (1996). https://doi.org/10.1103/PhysRevC.53.2086

    Article  ADS  Google Scholar 

  9. D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 478, 629 (1996). https://doi.org/10.1016/0550-3213(96)00357-4

    Article  ADS  Google Scholar 

  10. A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C 72, 054006 (2005). https://doi.org/10.1103/PhysRevC.72.054006

    Article  ADS  Google Scholar 

  11. S. Beane, P.F. Bedaque, M. Savage, U. van Kolck, Nucl. Phys. A 700, 377 (2002). https://doi.org/10.1016/S0375-9474(01)01324-0

    Article  ADS  Google Scholar 

  12. M. Pavón Valderrama, M. Sánchez Sánchez, C.J. Yang, B. Long, J. Carbonell, U. van Kolck, Phys. Rev. C 95(5), 054001 (2017). https://doi.org/10.1103/PhysRevC.95.054001

    Article  ADS  Google Scholar 

  13. U. van Kolck, Front. Phys. 8, 79 (2020). https://doi.org/10.3389/fphy.2020.00079

    Article  Google Scholar 

  14. E. Epelbaum, H.W. Hammer, U.G. Meißner, Rev. Mod. Phys. 81, 1773 (2009). https://doi.org/10.1103/RevModPhys.81.1773

    Article  ADS  Google Scholar 

  15. R. Machleidt, D.R. Entem, Phys. Rep. 503, 1 (2011). https://doi.org/10.1016/j.physrep.2011.02.001

    Article  ADS  Google Scholar 

  16. E. Epelbaum, J. Gegelia, U.G. Meißner, Nucl. Phys. B 925, 161 (2017). https://doi.org/10.1016/j.nuclphysb.2017.10.008

    Article  ADS  Google Scholar 

  17. E. Epelbaum, J. Gegelia, U.G. Meißner, Commun. Theor. Phys. 69(3), 303 (2018). https://doi.org/10.1088/0253-6102/69/3/303

    Article  ADS  MathSciNet  Google Scholar 

  18. E. Epelbaum, A.M. Gasparyan, J. Gegelia, U.G. Meißner, Eur. Phys. J. A 54(11), 186 (2018). https://doi.org/10.1140/epja/i2018-12632-1

    Article  ADS  Google Scholar 

  19. E. Epelbaum, A.M. Gasparyan, J. Gegelia, U.G. Meißner, Eur. Phys. J. A 55, 56 (2019). https://doi.org/10.1140/epja/i2019-12751-1

    Article  ADS  Google Scholar 

  20. E. Epelbaum, A.M. Gasparyan, J. Gegelia, U.G. Meißner, X.L. Ren, Eur. Phys. J. A 56(5), 152 (2020). https://doi.org/10.1140/epja/s10050-020-00162-4

    Article  ADS  Google Scholar 

  21. M. Pavon Valderrama, Eur. Phys. J. A 55(4), 55 (2019). https://doi.org/10.1140/epja/i2019-12703-9

    Article  ADS  Google Scholar 

  22. D.B. Kaplan, M.J. Savage, M.B. Wise, Phys. Lett. B 424, 390 (1998). https://doi.org/10.1016/S0370-2693(98)00210-X

    Article  ADS  Google Scholar 

  23. D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 534, 329 (1998). https://doi.org/10.1016/S0550-3213(98)00440-4

    Article  ADS  Google Scholar 

  24. S. Fleming, T. Mehen, I.W. Stewart, Nucl. Phys. A 677, 313 (2000). https://doi.org/10.1016/S0375-9474(00)00221-9

    Article  ADS  Google Scholar 

  25. U. van Kolck, Nucl. Phys. A 645, 273 (1999). https://doi.org/10.1016/S0375-9474(98)00612-5

    Article  ADS  Google Scholar 

  26. M.C. Birse, J.A. McGovern, K.G. Richardson, Phys. Lett. B 464, 169 (1999). https://doi.org/10.1016/S0370-2693(99)00991-0

    Article  ADS  MathSciNet  Google Scholar 

  27. C.A. Bertulani, H.W. Hammer, U. Van Kolck, Nucl. Phys. A 712, 37 (2002). https://doi.org/10.1016/S0375-9474(02)01270-8

    Article  ADS  Google Scholar 

  28. H.W. Hammer, C. Ji, D.R. Phillips, J. Phys. G 44(10), 103002 (2017). https://doi.org/10.1088/1361-6471/aa83db

    Article  ADS  Google Scholar 

  29. S.R. Beane, T.D. Cohen, D.R. Phillips, Nucl. Phys. A 632, 445 (1998). https://doi.org/10.1016/S0375-9474(98)00007-4

    Article  ADS  Google Scholar 

  30. D.R. Phillips, T.D. Cohen, Phys. Lett. B 390, 7 (1997). https://doi.org/10.1016/S0370-2693(96)01411-6

    Article  ADS  Google Scholar 

  31. K.A. Scaldeferri, D.R. Phillips, C.W. Kao, T.D. Cohen, Phys. Rev. C 56, 679 (1997). https://doi.org/10.1103/PhysRevC.56.679

    Article  ADS  Google Scholar 

  32. D.R. Phillips, S.R. Beane, T.D. Cohen, Annals Phys. 263, 255 (1998). https://doi.org/10.1006/aphy.1997.5771

    Article  ADS  Google Scholar 

  33. H.W. Hammer, D. Lee, Ann. Phys. 325, 2212 (2010). https://doi.org/10.1016/j.aop.2010.06.006

    Article  ADS  Google Scholar 

  34. E.P. Wigner, Phys. Rev. 98, 145 (1955). https://doi.org/10.1103/PhysRev.98.145

    Article  ADS  MathSciNet  Google Scholar 

  35. J.B. Habashi, S. Sen, S. Fleming, U. van Kolck, Ann. Phys. 422, 168283 (2020). https://doi.org/10.1016/j.aop.2020.168283

    Article  Google Scholar 

  36. J.W. Chen, G. Rupak, M.J. Savage, Nucl. Phys. A 653, 386 (1999). https://doi.org/10.1016/S0375-9474(99)00298-5

    Article  ADS  Google Scholar 

  37. H.W. Hammer, S. König, U. van Kolck, Rev. Mod. Phys. 92(2), 025004 (2020). https://doi.org/10.1103/RevModPhys.92.025004

    Article  ADS  Google Scholar 

  38. J. Gegelia, Phys. Lett. B 429, 227 (1998). https://doi.org/10.1016/S0370-2693(98)00460-2

    Article  ADS  Google Scholar 

  39. D.R. Phillips, G. Rupak, M.J. Savage, Phys. Lett. B 473, 209 (2000). https://doi.org/10.1016/S0370-2693(99)01496-3

    Article  ADS  Google Scholar 

  40. J. Gegelia, G. Japaridze, Phys. Lett. B 517, 476 (2001). https://doi.org/10.1016/S0370-2693(01)00981-9

    Article  ADS  Google Scholar 

  41. M. Sánchez Sánchez, C.J. Yang, B. Long, U. van Kolck, Phys. Rev. C 97(2), 024001 (2018). https://doi.org/10.1103/PhysRevC.97.024001

    Article  ADS  Google Scholar 

  42. D.B. Kaplan, J.V. Steele, Phys. Rev. C 60, 064002 (1999). https://doi.org/10.1103/PhysRevC.60.064002

    Article  ADS  Google Scholar 

  43. D.B. Kaplan, Nucl. Phys. B 494, 471 (1997). https://doi.org/10.1016/S0550-3213(97)00178-8

    Article  ADS  Google Scholar 

  44. S.R. Beane, M.J. Savage, Nucl. Phys. A 694, 511 (2001). https://doi.org/10.1016/S0375-9474(01)01088-0

    Article  ADS  Google Scholar 

  45. T. Mehen, I.W. Stewart, M.B. Wise, Phys. Lett. B 474, 145 (2000). https://doi.org/10.1016/S0370-2693(00)00006-X

    Article  ADS  MathSciNet  Google Scholar 

  46. S.R. Beane, R.C. Farrell, Ann. Phys. 433, 168581 (2021). https://doi.org/10.1016/j.aop.2021.168581

    Article  Google Scholar 

  47. S.R. Beane, R.C. Farrell, Causality and dimensionality in geometric scattering (2021)

  48. S.R. Beane, R.C. Farrell. UV/IR symmetries of the \(S\)-matrix and RG flow (2021)

  49. S.R. Beane, D.B. Kaplan, N. Klco, M.J. Savage, Phys. Rev. Lett. 122(10), 102001 (2019). https://doi.org/10.1103/PhysRevLett.122.102001

    Article  ADS  Google Scholar 

  50. J.J. de Swart, C.P.F. Terheggen, V.G.J. Stoks, in 3rd International Symposium on Dubna Deuteron 95 (1995)

  51. Y. Yamaguchi, Phys. Rev. 95, 1628 (1954). https://doi.org/10.1103/PhysRev.95.1628

    Article  ADS  Google Scholar 

  52. D.R. Phillips, I.R. Afnan, A.G. Henry-Edwards, Phys. Rev. C 61, 044002 (2000). https://doi.org/10.1103/PhysRevC.61.044002

    Article  ADS  Google Scholar 

  53. R.U. Nijmegen. NN-online (2005), http://nn-online.org/. Accessed 2018-12-01

Download references

Acknowledgements

We would like to thank Daniel R. Phillips for a careful reading of the manuscript and many useful comments and suggestions. In addition we are grateful to Bira van Kolck and Ulf G. Meißner for interesting comments and for pointing out important missing references. This work was supported by the U. S. Department of Energy grants DE-FG02-97ER-41014 (UW Nuclear Theory, NT@UW-21-16) and DE-SC0020970 (InQubator for Quantum Simulation, IQuS@UW-21-017)

Author information

Authors and Affiliations

  1. Department of Physics, University of Washington, Seattle, WA, 98195, USA

    Silas R. Beane & Roland C. Farrell

Authors
  1. Silas R. Beane
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roland C. Farrell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Silas R. Beane.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beane, S.R., Farrell, R.C. Symmetries of the Nucleon–Nucleon S-Matrix and Effective Field Theory Expansions. Few-Body Syst 63, 45 (2022). https://doi.org/10.1007/s00601-022-01748-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00601-022-01748-y

Search

Navigation