{"title":"关于两个具有微扰介子的接近统一的核子","authors":"Yu-Ping Teng, Harald W. Grießhammer","doi":"10.1140/epja/s10050-025-01675-6","DOIUrl":null,"url":null,"abstract":"<div><p>We explore the impact of perturbative pions on the Unitarity Expansion in the two-nucleon S-waves of Chiral Effective Field Theory at next-to-next-to leading order (<span>\\(\\textrm{N}^{2}\\)</span>LO). Pion exchange explicitly breaks the nontrivial fixed point’s universality, i.e. invariance of S waves under both conformal and Wigner’s combined <span>\\(\\textrm{SU}(4)\\)</span> spin–isospin transformations. On the other hand, Unitarity explicitly breaks chiral symmetry. The two seem incompatible in their respective exact-symmetry limits. <span>\\({\\upchi }\\textrm{EFT}\\)</span> with Perturbative Pions in the Unitarity Expansion resolves the apparent conflict in the Unitarity Window (phase shifts <span>\\(45^\\circ \\lesssim \\delta (k)\\lesssim 135^\\circ \\)</span>), i.e. around momenta <span>\\(k\\approx m_\\pi \\)</span> most relevant for low-energy nuclear systems. Its only LO scale is the scattering momentum; NLO adds only scattering length, effective range and non-iterated one-pion exchange (OPE); and <span>\\(\\textrm{N}^{2}\\)</span>LO only once-iterated OPE. Agreement in the <span>\\({\\phantom {0}}^{1}\\textrm{S}_{0}\\)</span> channel is very good. Apparently large discrepancies in the <span>\\({\\phantom {0}}^{3}\\textrm{S}_{1}\\)</span> channel even at <span>\\(k\\approx 100\\;\\textrm{MeV}\\)</span> are remedied by taking at <span>\\(\\textrm{N}^{2}\\)</span>LO only the central part of OPE. In contradistinction to the tensor part, it is identical in the <span>\\({\\phantom {0}}^{1}\\textrm{S}_{0}\\)</span> and <span>\\({\\phantom {0}}^{3}\\textrm{S}_{1}\\)</span> channels. Both channels then match empirical phase shifts and pole parameters well within mutually consistent quantitative theory uncertainty estimates. Pionic effects are small, even for <span>\\(k\\gtrsim m_\\pi \\)</span>. Empirical breakdown scales are consistent with <span>\\(\\overline{\\Lambda }_{\\textrm{NN}}=\\frac{16\\pi f_\\pi ^2}{g_A^2M}\\approx 300\\;\\textrm{MeV}\\)</span>, where iterated OPE is not suppressed. We therefore conjecture: Both conformal and Wigner symmetry in the Unitarity Expansion show <i>persistence</i>, i.e. the footprint of both combined dominates even for <span>\\(k\\gtrsim m_\\pi \\)</span> and is more relevant than chiral symmetry, so that the tensor/Wigner-<span>\\(\\textrm{SU}(4)\\)</span> symmetry-breaking part of OPE does not enter before <span>\\(\\textrm{N}^{3}\\)</span>LO. We also discuss the potential relevance of entanglement and possible resolution of a conflict with the strength of the tensor interaction in the large-<span>\\(N_C\\)</span> expansion.</p></div>","PeriodicalId":786,"journal":{"name":"The European Physical Journal A","volume":"61 9","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On two nucleons near unitarity with perturbative pions\",\"authors\":\"Yu-Ping Teng, Harald W. Grießhammer\",\"doi\":\"10.1140/epja/s10050-025-01675-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We explore the impact of perturbative pions on the Unitarity Expansion in the two-nucleon S-waves of Chiral Effective Field Theory at next-to-next-to leading order (<span>\\\\(\\\\textrm{N}^{2}\\\\)</span>LO). Pion exchange explicitly breaks the nontrivial fixed point’s universality, i.e. invariance of S waves under both conformal and Wigner’s combined <span>\\\\(\\\\textrm{SU}(4)\\\\)</span> spin–isospin transformations. On the other hand, Unitarity explicitly breaks chiral symmetry. The two seem incompatible in their respective exact-symmetry limits. <span>\\\\({\\\\upchi }\\\\textrm{EFT}\\\\)</span> with Perturbative Pions in the Unitarity Expansion resolves the apparent conflict in the Unitarity Window (phase shifts <span>\\\\(45^\\\\circ \\\\lesssim \\\\delta (k)\\\\lesssim 135^\\\\circ \\\\)</span>), i.e. around momenta <span>\\\\(k\\\\approx m_\\\\pi \\\\)</span> most relevant for low-energy nuclear systems. Its only LO scale is the scattering momentum; NLO adds only scattering length, effective range and non-iterated one-pion exchange (OPE); and <span>\\\\(\\\\textrm{N}^{2}\\\\)</span>LO only once-iterated OPE. Agreement in the <span>\\\\({\\\\phantom {0}}^{1}\\\\textrm{S}_{0}\\\\)</span> channel is very good. Apparently large discrepancies in the <span>\\\\({\\\\phantom {0}}^{3}\\\\textrm{S}_{1}\\\\)</span> channel even at <span>\\\\(k\\\\approx 100\\\\;\\\\textrm{MeV}\\\\)</span> are remedied by taking at <span>\\\\(\\\\textrm{N}^{2}\\\\)</span>LO only the central part of OPE. In contradistinction to the tensor part, it is identical in the <span>\\\\({\\\\phantom {0}}^{1}\\\\textrm{S}_{0}\\\\)</span> and <span>\\\\({\\\\phantom {0}}^{3}\\\\textrm{S}_{1}\\\\)</span> channels. Both channels then match empirical phase shifts and pole parameters well within mutually consistent quantitative theory uncertainty estimates. Pionic effects are small, even for <span>\\\\(k\\\\gtrsim m_\\\\pi \\\\)</span>. Empirical breakdown scales are consistent with <span>\\\\(\\\\overline{\\\\Lambda }_{\\\\textrm{NN}}=\\\\frac{16\\\\pi f_\\\\pi ^2}{g_A^2M}\\\\approx 300\\\\;\\\\textrm{MeV}\\\\)</span>, where iterated OPE is not suppressed. We therefore conjecture: Both conformal and Wigner symmetry in the Unitarity Expansion show <i>persistence</i>, i.e. the footprint of both combined dominates even for <span>\\\\(k\\\\gtrsim m_\\\\pi \\\\)</span> and is more relevant than chiral symmetry, so that the tensor/Wigner-<span>\\\\(\\\\textrm{SU}(4)\\\\)</span> symmetry-breaking part of OPE does not enter before <span>\\\\(\\\\textrm{N}^{3}\\\\)</span>LO. We also discuss the potential relevance of entanglement and possible resolution of a conflict with the strength of the tensor interaction in the large-<span>\\\\(N_C\\\\)</span> expansion.</p></div>\",\"PeriodicalId\":786,\"journal\":{\"name\":\"The European Physical Journal A\",\"volume\":\"61 9\",\"pages\":\"\"},\"PeriodicalIF\":2.8000,\"publicationDate\":\"2025-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The European Physical Journal A\",\"FirstCategoryId\":\"4\",\"ListUrlMain\":\"https://link.springer.com/article/10.1140/epja/s10050-025-01675-6\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, NUCLEAR\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The European Physical Journal A","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1140/epja/s10050-025-01675-6","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, NUCLEAR","Score":null,"Total":0}
On two nucleons near unitarity with perturbative pions
We explore the impact of perturbative pions on the Unitarity Expansion in the two-nucleon S-waves of Chiral Effective Field Theory at next-to-next-to leading order (\(\textrm{N}^{2}\)LO). Pion exchange explicitly breaks the nontrivial fixed point’s universality, i.e. invariance of S waves under both conformal and Wigner’s combined \(\textrm{SU}(4)\) spin–isospin transformations. On the other hand, Unitarity explicitly breaks chiral symmetry. The two seem incompatible in their respective exact-symmetry limits. \({\upchi }\textrm{EFT}\) with Perturbative Pions in the Unitarity Expansion resolves the apparent conflict in the Unitarity Window (phase shifts \(45^\circ \lesssim \delta (k)\lesssim 135^\circ \)), i.e. around momenta \(k\approx m_\pi \) most relevant for low-energy nuclear systems. Its only LO scale is the scattering momentum; NLO adds only scattering length, effective range and non-iterated one-pion exchange (OPE); and \(\textrm{N}^{2}\)LO only once-iterated OPE. Agreement in the \({\phantom {0}}^{1}\textrm{S}_{0}\) channel is very good. Apparently large discrepancies in the \({\phantom {0}}^{3}\textrm{S}_{1}\) channel even at \(k\approx 100\;\textrm{MeV}\) are remedied by taking at \(\textrm{N}^{2}\)LO only the central part of OPE. In contradistinction to the tensor part, it is identical in the \({\phantom {0}}^{1}\textrm{S}_{0}\) and \({\phantom {0}}^{3}\textrm{S}_{1}\) channels. Both channels then match empirical phase shifts and pole parameters well within mutually consistent quantitative theory uncertainty estimates. Pionic effects are small, even for \(k\gtrsim m_\pi \). Empirical breakdown scales are consistent with \(\overline{\Lambda }_{\textrm{NN}}=\frac{16\pi f_\pi ^2}{g_A^2M}\approx 300\;\textrm{MeV}\), where iterated OPE is not suppressed. We therefore conjecture: Both conformal and Wigner symmetry in the Unitarity Expansion show persistence, i.e. the footprint of both combined dominates even for \(k\gtrsim m_\pi \) and is more relevant than chiral symmetry, so that the tensor/Wigner-\(\textrm{SU}(4)\) symmetry-breaking part of OPE does not enter before \(\textrm{N}^{3}\)LO. We also discuss the potential relevance of entanglement and possible resolution of a conflict with the strength of the tensor interaction in the large-\(N_C\) expansion.
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