Abstract
The zero-energy astrophysical S-factor for the fusion process \(p+p\rightarrow d+e^{+}+\nu _{e}\) is calculated using pionless effective field theory (\(~/\!\!\pi \)EFT). In the present study, the order-by-order results are reproduced according to a new suggested power counting. The short range interactions in the S-wave pp scattering at leading order and the corrections in the next-to-leading order including the Coulomb interaction are introduced. In addition the Coulomb interaction between the incoming protons has been considered. The new nuclear \(~/\!\!\pi \)EFT amplitude is compatible with renormalization-group invariance.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: In the tables data is referenced.]
References
X. Kong, F. Ravndal, Nucl. Phys. A 656, 421 (1999)
X. Kong, F. Ravndal, Phys. Lett. B 470, 1 (1999)
X. Kong, F. Ravndal, Phys. Rev. C 64, 044002 (2001)
C.-J. Yang, C. Elster, D.R. Phillips, Phys. Rev. C 80, 044002 (2009)
B. Long, C.-J. Yang, Phys. Rev. C 86, 024001 (2012)
D.B. Kaplan, Nucl. Phys. B 494, 471 (1997)
T.D. Cohen, J.M. Hansen, Phys. Lett. B 440, 233 (1998)
J.V. Steele, R.J. Furnstahl, Nucl. Phys. A 645, 439 (1999)
T. Mehen, I.W. Stewart, Phys. Rev. C 59, 2365 (1999)
T. Frederico, V.S. Timóteo, L. Tomio, Nucl. Phys. A 653, 209 (1999)
J. Gegelia, Phys. Lett. B 463, 133 (1999)
D.B. Kaplan, J.V. Steele, Phys. Rev. C 60, 064002 (1999)
C.H. Hyun, D.-P. Min, T.-S. Park, Phys. Lett. B 473, 6 (2000)
M. Lutz, Nucl. Phys. A 677, 241 (2000)
S.R. Beane, P.F. Bedaque, M.J. Savage, U. van Kolck, Nucl. Phys. A 700, 377 (2002)
J.M. Nieves, Phys. Lett. B 568, 109 (2003)
J.A. Oller, Nucl. Phys. A 725, 85 (2003)
M. Pavón Valderrama, and E. Ruiz Arriola, Phys. Lett. B 580 (2004) 149
M. Pavón Valderrama, and E. Ruiz Arriola, Phys. Rev. C 70 (2004) 044006
V.S. Timóteo, T. Frederico, A. Delno, L. Tomio, Phys. Lett. B 621, 109 (2005)
M. Pavón Valderrama, and E. Ruiz Arriola, Phys. Rev. C 74 (2006) 054001
J.-F. Yang, J.-H. Huang, Commun. Theor. Phys. 47, 699 (2007)
D.R. Entem, E. Ruiz Arriola, M. Pavón Valderrama, and R. Machleidt, Phys. Rev. C 77 (2008) 044006
J. Soto, J. Tarrús, Phys. Rev. C 78, 024003 (2008)
D. Shukla, D.R. Phillips, E. Mortenson, J. Phys. G 35, 115009 (2008)
C.-J. Yang, Ch. Elster, D.R. Phillips, Phys. Rev. C 77, 014002 (2008)
M.C. Birse, Eur. Phys. J. A 46, 231 (2010)
K. Harada, H. Kubo, Y. Yamamoto, Phys. Rev. C 83, 034002 (2011)
S.-I. Ando, C.H. Hyun, Phys. Rev. C 86, 024002 (2012)
S. Szpigel, V.S. Timóteo, J. Phys. G 39, 105102 (2012)
B. Long, Phys. Rev. C 88, 014002 (2013)
K. Harada, H. Kubo, T. Sakaeda, and Y. Yamamoto, arXiv:1311.3063 [nucl-th]
E. Epelbaum, A.M. Gasparyan, J. Gegelia, H. Krebs, Eur. Phys. J. A 51, 71 (2015)
X.-L. Ren, K.-W. Li, L.-S. Geng, B.-W. Long, P. Ring, J. Meng, Chin. Phys. C 42, 014103 (2018)
U. van Kolck, Lect. Notes Phys. 513, 62 (1998)
U. van Kolck, Nucl. Phys. A 645, 273 (1999)
M. Butler, J.-W. Chen, Phys. Lett. B 520, 87 (2001)
J.W. Chen, C.-P. Liu, S.H. Yu, Phys. Lett. B 720, 385 (2013)
B. Acharya, L. Platter, G. Rupak, Phys. Rev. C 100, 021001 (2019)
S. Weinberg, Phys. Lett. B 251, 288 (1990)
S. Weinberg, Nucl. Phys. B 363, 3 (1991)
M. Rho, Phys. Rev. Lett. 66, 1275 (1991)
A. Manohar, H. Georgi, Nucl. Phys. B 234, 189 (1984)
H. Georgi, Phys. Lett. B 298, 187 (1993)
D.B. Kaplan, M.J. Savag, M.B. Wise, Nucl. Phys. B 478, 629 (1996)
L.E. Marcucci, R. Schiavilla, M. Viviani, Phys. Rev. Lett. 110, 192503 (2013)
C.-J. Yang, Eur. Phys. J. A 56, 96 (2020)
A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C 72, 054006 (2005)
M.P. Valderrama, Phys. Rev. C 83, 024003 (2011)
Bingwei Long and C. J. Yang, Phys. Rev. C 84 (2011) 057001
M.C. Birse, Phys. Rev. C 74, 014003 (2006)
J.R. Bergervoet, P.C. van Campen, R.A.M. Klomp, J.L. de Kok, T.A. Rijken, V.G.J. Stoks, J.J. de Swart, Phys. Rev. C 41, 1435 (1990)
V.G.J. Stoks, R.A.M. Klomp, M.C.M. Rentmeester, J.J. de Swart, Phys. Rev. C 48, 792 (1993)
NN-OnLine, http://www.nn-online.org/
M. Sánchez Sánchez, C.-J. Yang, B. Long and U. van Kolck, Phys. Rev. C 97 (2018) 024001
E.G. Adelberger, A.B. Balantekin, D. Bemmerer, C.A. Bertulani, J.-W. Chen, H. Costantini, M. Couder, R. Cyburt et al., Rev. Mod. Phys. 83, 195 (2011)
S.R. Beane, M.J. Savage, Nucl. Phys. A 694, 511 (2001)
S. i. Ando, and C. H. Hyun, Phys. Rev. C 72 (2005) 014008
S. Ando, J.W. Shin, C.H. Hyun, S.W. Hong, K. Kubodera, Phys. Lett. B 668, 187 (2008)
V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen, J.J. de Swart, Phys. Rev. C 49, 2950 (1994)
H.A. Bethe, Phys. Rev. 76, 38 (1949)
I. Stetcu, B.R. Barrett, U. van Kolck, Phys. Lett. B 653, 358 (2007)
J.R. Bergervoet, P.C. van Campen, W.A. van der Sanden, J.J. de Swart, Phys. Rev. C 38, 15 (1988)
E. Lomon, R. Wilson, Phys. Rev. C 9, 1329 (1974)
V.A. Babenko, N.M. Petrov, Phys. At. Nucl. 73, 1499 (2010)
S. König, H.W. Grießhammer, H.W. Hammer, U. van Kolck, J. Phys. G 43, 055106 (2016)
S. Ando, M.C. Birse, J. Phys. G: Nucl. Part. Phys. 37, 105108 (2010)
B. Long, C.-J. Yang, Phys. Rev. C 85, 034002 (2012)
J.N. Bahcall, R.M. May, Astrophys. J. 155, 501 (1969)
M. Fukugita, T. Kubota, Phys. Rev. D 72, 071301 (2005)
R.B. Wiringa, V.G.J. Stokes, R. Schiavilla, Phys. Rev. C 51, 38 (1995)
T.-S. Park et al., Phys. Rev. C 67, 055206 (2003)
R. Schiavilla et al., Phys. Rev. C 58, 1263 (1998)
L. Koester, W. Nistler, Z. Physik 272, 189 (1975)
S. König, H.W. Grießhammer, H.W. Hammer, U. van Kolck, Phys. Rev. Lett. 118, 202501 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vittorio Somà
Appendix A: The calculation of the LO amplitude
Appendix A: The calculation of the LO amplitude
We present here a brief LO calculation of \(^{1}S_{0}\) state of np scattering amplitude. The LO inverse amplitude by taking the two-dibaryon fields derives as below,
If here k is the smallest scale than the other scales, at large cutoff, Eq. (A.1) is equal to ERE
such that, for np scattering, The scattering length is \(a \simeq \)\(-23.7\) fm \(\simeq \) − (8 MeV)\(^{-1}\) [74]. and the effective range is \(r_{0}\simeq \) 2.7 fm \(\simeq \)(73 MeV)\(^{-1}\) [64], Also, The shape parameter is \(P_{0}\simeq \)2.0 \(\hbox {fm}^{3}\)\(\simeq \) (158 MeV)\(^{-3}\) [65] and \(k_{0} \simeq \)340 MeV [54] is the scattering momentum. Because of the anomalously large value of \(\mid a \mid \) that results a virtual bound state too close to threshold, its binding momentum corresponds to small scale \(\aleph \sim \) 10 MeV. The suggestion of an enlarged range of validity of \(~/\!\!\pi \)EFT is introduced, with the low momentum scale \(M_{lo}\sim m_{\pi }\sim Q\) and the EFT breakdown scale, \(M_{hi} \le M_{QCD}\sim \)1 GeV , so that \(\aleph \) appears at NLO [66, 75]. Therefore phenomenological parameters of the theory rescaled as
the amplitude can be taken in the following form:
so that the high indexes (inside the bracket) show the order of the observables and,
Since \(1{/}\mid a \mid \) is very small, it imposes an expansion of the NN \(^{1}S_{0}\) amplitude around the unitary limit [66, 75] and it takes as below:
According to Eq. (A.1), for reproducing the zero amplitude at LO requires a minimum of three non-vanishing bare parameters and scales the following form
for relating three non-vanishing LO bare parameters of the two dibaryon fields, \({\varDelta }_{1}^{[0]}(\varLambda )\), \({\varDelta }_{2}^{[0]}(\varLambda ),\) and \({c}_{2}^{[0]}(\varLambda )\), to found observables, we introduce
and impose three renormalization conditions on it the following form:
so that, we derive three parameters of two-dibaryon fields as below
where two renormalized parameters are
Also, the prediction for shape parameter at this order is given as below [55]
where \(P_{0}^{[0]}k_{0}^{2}/(2r_{0})=1.03\pm 0.3\) that it has about 30\(\%\) error from a careful analysis in Ref. [65].
Rights and permissions
About this article
Cite this article
Behzadmoghaddam, B., Bayegan, S. & Arani, M.M. Proton–proton fusion in new pionless EFT power counting. Eur. Phys. J. A 56, 158 (2020). https://doi.org/10.1140/epja/s10050-020-00166-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epja/s10050-020-00166-0