Pion–nucleon scattering: from chiral perturbation theory to Roy–Steiner equations Bastian Kubis Helmholtz-Institut f¨ ur Strahlen- und Kernphysik (Theorie) Bethe Center for Theoretical Physics Universit¨ at Bonn, Germany B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 1
Outline Why is pion–nucleon scattering important? Chiral perturbation theory • phase shift analyses with chiral amplitudes A new dispersive analysis: Roy–Steiner equations • phase shifts, σ -term, and low-energy constants in collaboration with M. Hoferichter, J. Ruiz de Elvira, and U.-G. Meißner PRL 115 (2015) 092301, PRL 115 (2015) 192301, Phys. Rept. 625 (2016) 1, arXiv:1602.07688 B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 2
Chiral pion–nucleon interaction • simplest process for chiral pion interaction with nucleons π π π π N N N N g πN Weinberg– Tomozawa • leading-order O ( p ) = O ( M π ) predictions for πN : scattering lengths: M π m N a − = � � a + = O � � M 3 M 2 + O π π 8 π ( m N + M π ) F 2 π Weinberg 1966 g πN = g A m N Goldberger–Treiman relation: F π B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 3
Chiral pion–nucleon interaction • simplest process for chiral pion interaction with nucleons π π π π ∼ ∆ N N N N c 1 – 4 • next-to-leading order O ( p 2 ) : low-energy constants (LECs) c 1 – 4 effectively incorporate effects of the ∆(1232) resonance: low mass m ∆ − m N ≈ 2 M π and strong couplings • determination of c i very important for nuclear physics: πN important for NN / determines longest-range 3 N forces B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 3
The pion–nucleon σ -term • scalar form factor of the nucleon: uu + ¯ t = ( p − p ′ ) 2 � N ( p ′ ) | ˆ u ( p ′ ) u ( p ) m (¯ dd ) | N ( p ) � = σ ( t )¯ m ˆ m = m u + m d uu + ¯ σ πN ≡ σ (0) = � N | ¯ dd | N � ˆ 2 m N 2 • σ πN determines light quark contribution to nucleon mass: Feynman–Hellmann theorem m∂m N m = − 4 c 1 M 2 π + O ( M 3 σ πN = ˆ π ) ∂ ˆ − → at leading order, related to the chiral coupling c 1 • σ πN determines scalar couplings wanted for direct-detection dark matter searches e.g. Ellis et al. 2008 B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 4
Extracting LECs from pion–nucleon scattering Mojžiš 1997, Fettes et al. 1998–2000, Ellis et al. 1998–2003 Alarcón et al. 2011–2013, Chen et al. 2013, Krebs et al. 2012, Gasparyan, Lutz 2010 Strategy: • fit results of phase shift analyses: ⊲ Karlsruhe–Helsinki (KH) − → dispersion theory based Koch, Pietarinen 1980, Höhler 1983 ⊲ GWU/SAID (GW) − → modern data input Workman et al. 2012 ⊲ Matsinos et al. (EM) Matsinos et al. 2006 B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 5
Extracting LECs from pion–nucleon scattering Mojžiš 1997, Fettes et al. 1998–2000, Ellis et al. 1998–2003 Alarcón et al. 2011–2013, Chen et al. 2013, Krebs et al. 2012, Gasparyan, Lutz 2010 Strategy: • fit results of phase shift analyses: ⊲ Karlsruhe–Helsinki (KH) − → dispersion theory based Koch, Pietarinen 1980, Höhler 1983 ⊲ GWU/SAID (GW) − → modern data input Workman et al. 2012 ⊲ Matsinos et al. (EM) Matsinos et al. 2006 • # of parameters: O ( p 2 ) : 4 c i O ( p 3 ) : 4 d i (+ d 18 from GT discrepancy) O ( p 4 ) : 5 e i • use chiral low-energy theorems to extract σ πN B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 5
Extracting LECs from pion–nucleon scattering Mojžiš 1997, Fettes et al. 1998–2000, Ellis et al. 1998–2003 Alarcón et al. 2011–2013, Chen et al. 2013, Krebs et al. 2012, Gasparyan, Lutz 2010 Strategy: • fit results of phase shift analyses: ⊲ Karlsruhe–Helsinki (KH) − → dispersion theory based Koch, Pietarinen 1980, Höhler 1983 ⊲ GWU/SAID (GW) − → modern data input Workman et al. 2012 ⊲ Matsinos et al. (EM) Matsinos et al. 2006 • # of parameters: O ( p 2 ) : 4 c i O ( p 3 ) : 4 d i (+ d 18 from GT discrepancy) O ( p 4 ) : 5 e i • use chiral low-energy theorems to extract σ πN • ChPT obeys unitarity only perturbatively de-facto unitarisation to calculate phase shifts from real parts: � | p | � | p | 8 π √ s Re T 8 π √ s Re T δ = arctan ≈ B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 5
Convergence of the chiral expansion 0 2 S 11 P 11 δ [ degree] 10 -5 0 S 31 5 -2 -10 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 30 0 δ [ degree] 0 P 31 P 13 P 33 -1 15 -1 -2 -2 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 O ( p 2 ) O ( p 3 ) O ( p 4 ) vs. vs. Krebs, Gasparyan, Epelbaum 2012 = √ s ≈ 1 . 13 GeV, • fitted up to p Lab = 150 MeV ˆ = √ s ≈ 1 . 17 GeV maximum energy shown p Lab = 200 MeV ˆ • convergence assessed using LECs from highest-order fit • D-waves also fitted B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 6
ChPT with and without explicit ∆(1232) O ( p 3 ) O ( δ 3 ) / Alarcón, Martin Camalich, Oller 2013 fit range: √ s max = 1 . 13 GeV / √ s max = 1 . 20 GeV B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 7
On the chiral extractions of σ πN The Cheng–Dashen theorem • isoscalar amplitude at CD point related to scalar form factor π ¯ F 2 D + ( s = u, t = 2 M 2 π ) � �� � 2 s = ( m + M ) π ( d + π d + � = 0 F 2 00 +2 M 2 01 )+∆ D 2 u = ( m + M ) CD−point = σ (2 M 2 2 π ) +∆ R t = 4 M � �� � * σ πN +∆ σ u -c hannel s -c hannel | ∆ R | � 2 MeV Bernard et al. 1996 threshold B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 8
On the chiral extractions of σ πN The Cheng–Dashen theorem • isoscalar amplitude at CD point related to scalar form factor π ¯ F 2 D + ( s = u, t = 2 M 2 π ) � �� � 2 s = ( m + M ) π ( d + π d + � = 0 F 2 00 +2 M 2 01 )+∆ D 2 u = ( m + M ) CD−point = σ (2 M 2 2 π ) +∆ R t = 4 M � �� � * σ πN +∆ σ u -c hannel s -c hannel | ∆ R | � 2 MeV Bernard et al. 1996 threshold • ChPT fulfils all these relations perturbatively only is known to fail at one loop for ∆ D , ∆ σ : Gasser, Leutwyler, Sainio 1991 curvature d + 02 not reproduced at one loop Alarcón et al. 2013 • we’re lucky: ∆ D − ∆ σ = ( − 1 . 8 ± 0 . 2) MeV cancels to large extent − → one-loop ChPT does not describe pion–nucleon scattering accurately in the whole low-energy region B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 8
On the chiral extractions of σ πN The Cheng–Dashen theorem • isoscalar amplitude at CD point related to scalar form factor π ¯ F 2 D + ( s = u, t = 2 M 2 π ) � �� � 2 s = ( m + M ) π ( d + π d + � = 0 F 2 00 +2 M 2 01 )+∆ D 2 u = ( m + M ) CD−point = σ (2 M 2 2 π ) +∆ R t = 4 M � �� � * σ πN +∆ σ u -c hannel s -c hannel | ∆ R | � 2 MeV Bernard et al. 1996 threshold • ChPT fulfils all these relations perturbatively only is known to fail at one loop for ∆ D , ∆ σ : Gasser, Leutwyler, Sainio 1991 curvature d + 02 not reproduced at one loop Alarcón et al. 2013 • we’re lucky: ∆ D − ∆ σ = ( − 1 . 8 ± 0 . 2) MeV cancels to large extent − → update dispersive analysis, Roy–Steiner equations Hoferichter, Ruiz de Elvira, BK, Meißner B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 8
The well-known paradigm: ππ Roy equations Roy equations = coupled system of partial-wave dispersion relations + crossing symmetry + unitarity • twice-subtracted fixed- t dispersion relation: � ∞ � � s 2 u 2 T ( s, t ) = c ( t ) + 1 ds ′ Im T ( s ′ , t ) + s ′ 2 ( s ′ − s ) s ′ 2 ( s ′ − u ) π 4 M 2 π � �� � � �� � s -channel cut u -channel cut • subtraction function c ( t ) determined from crossing symmetry B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 9
The well-known paradigm: ππ Roy equations Roy equations = coupled system of partial-wave dispersion relations + crossing symmetry + unitarity • twice-subtracted fixed- t dispersion relation: � ∞ � � s 2 u 2 T ( s, t ) = c ( t ) + 1 ds ′ Im T ( s ′ , t ) + s ′ 2 ( s ′ − s ) s ′ 2 ( s ′ − u ) π 4 M 2 π � �� � � �� � s -channel cut u -channel cut • subtraction function c ( t ) determined from crossing symmetry • project onto partial waves t I J ( s ) (angular momentum J , isospin I ) expand Im T ( s ′ , t ) in partial waves 2 � ∞ ∞ � � ds ′ K II ′ JJ ′ ( s, s ′ )Im t I ′ t I J ( s ) = polynomial ( a 0 0 , a 2 J ′ ( s ′ ) 0 ) + 4 M 2 I ′ =0 J ′ =0 π kernel functions K II ′ JJ ′ ( s, s ′ ) known analytically Roy 1971 B. Kubis, Roy–Steiner analysis of pion–nucleon scattering – p. 9
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