We present a coupled system of integral equations for the pi pi --> Nbar N
and Kbar K --> Nbar N S-waves derived from Roy-Steiner equations for
pion-nucleon scattering. We discuss the solution of the corresponding
two-channel Muskhelishvili-Omnes problem and apply the results to a dispersive
analysis of the scalar form factor of the nucleon fully including Kbar K
intermediate states. In particular, we determine the corrections Delta_sigma
and Delta_D, which are needed for the extraction of the pion-nucleon sigma term
from pi N scattering, and show that the difference Delta_D - Delta_sigma=(-1.8
+/- 0.2) MeV is insensitive to the input pi N parameters.Comment: 6 pages, 3 figures, Proceedings for the 7th International Workshop on
  Chiral Dynamics, Newport News, Virginia, USA, August 6-10, 201

Ditsche, Christoph

Hoferichter, Martin

Kubis, Bastian

Meißner, Ulf-G.

English

arXiv

We present a coupled system of integral equations for the pp → ¯NN and ¯K K → ¯N N S-waves derived from Roy–Steiner equations for pion–nucleon scattering. We discuss the solution of the corresponding two-channel Muskhelishvili–Omnès problem and apply the results to a dispersive analysis of the scalar form factor of the nucleon fully including ¯KK intermediate states. In particular, we determine the corrections Ds and DD, which are needed for the extraction of the pion–

nucleon s term from pN scattering, and show that the difference DD −Ds = (−1.8±0.2)MeV is insensitive to the input pN parameters

Meissner, Ulf-G.

Bern Open Repository and Information System (BORIS)

PoS(CD12)069Improved dispersive analysis of the scalar formfactor of the nucleonMartin Hoferichter∗, ab Christoph Ditsche, a Bastian Kubis, a and Ulf-G. Meißneraca Helmholtz-Institut für Strahlen- und Kernphysik (Theorie), Bethe Center for TheoreticalPhysics, Universität Bonn, D-53115 Bonn, Germanyb Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, UniversitätBern, CH-3012 Bern, Switzerlandc Forschungszentrum Jülich, Institut für Kernphysik, Jülich Center for Hadron Physics andInstitute for Advanced Simulation, D-52425 Jülich, GermanyE-mail: hoferichter@itp.unibe.chWe present a coupled system of integral equations for the pipi → ¯NN and ¯KK → ¯NN S-wavesderived from Roy–Steiner equations for pion–nucleon scattering. We discuss the solution of thecorresponding two-channel Muskhelishvili–Omnès problem and apply the results to a dispersiveanalysis of the scalar form factor of the nucleon fully including ¯KK intermediate states. In par-ticular, we determine the corrections ∆σ and ∆D, which are needed for the extraction of the pion–nucleon σ term from piN scattering, and show that the difference ∆D −∆σ = (−1.8± 0.2)MeVis insensitive to the input piN parameters.The 7th International Workshop on Chiral Dynamics,August 6 -10, 2012Jefferson Lab, Newport News, Virginia, USA∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/source: https://doi.org/10.7892/boris.42582 | downloaded: 13.3.2017PoS(CD12)069Improved dispersive analysis of the scalar form factor of the nucleon Martin Hoferichter1. IntroductionThe pion–nucleon σ term σpiN measures the contribution of the light quarks to the nucleonmass m, and is directly related to the form factor of the scalar currentσ(t) =12m〈N(p′)|mˆ(u¯u+ ¯dd)|N(p)〉, t = (p′− p)2, mˆ = mu +md2, (1.1)at vanishing momentum transfer σ(0) = σpiN . The standard procedure for its extraction from pion–nucleon (piN) scattering relies on the low-energy theorem [1, 2]F2pi ¯D+(s = m2, t = 2M2pi)= σ(2M2pi)+∆R, (1.2)which relates the Born-term-subtracted isoscalar piN scattering amplitude at the Cheng–Dashenpoint ¯D+(s = m2, t = 2M2pi) to the scalar form factor evaluated at 2M2pi . The remainder ∆R is freeof chiral logarithms at full one-loop order in chiral perturbation theory (ChPT) [3, 4], and has beenestimated as [3]|∆R|. 2MeV. (1.3)Rewriting (1.2) in terms of∆D = F2pi{¯D+(s = m2, t = 2M2pi)−d+00−2M2pid+01}, ∆σ = σ(2M2pi)−σpiN , (1.4)the extraction of the σ term reduces to the determination of the subthreshold parameters d+00 andd+01 as well as the combination ∆D−∆σ −∆R. The first two corrections can be calculated using adispersive approach [5]∆D−∆σ = (−3.3±0.2)MeV, (1.5)where the error only covers the uncertainties in the pipi phase shifts available at that time. Here, weupdate the determination of ∆D and ∆σ using modern pipi phases, fully including ¯KK intermediatestates, and carefully studying the dependence of the results on piN subthreshold parameters as wellas the piN coupling constant.2. Scalar pion and kaon form factorsWe first consider the case of the scalar pion and kaon form factors FSpi (t) and FSK (t), whichserve both to illustrate the method and as input for the scalar form factor of the nucleon. Unitarityin the pipi/ ¯KK system intertwines both form factors according to [6]ImFS(t) =(T (t))∗Σ(t)FS(t), FS(t) =(FSpi (t)2√3FSK(t)), (2.1)with the phase-space factorΣ(t) = diag(σ pit θ(t− tpi),σ Kt θ(t− tK)), σ it =√1− tit, ti = 4M2i i ∈ {pi,K}, (2.2)2PoS(CD12)069Improved dispersive analysis of the scalar form factor of the nucleon Martin HoferichterFigure 1: Modulus and phase of the scalar pion form factor. The solid, dashed, and dot-dashed lines refer toFSK (0) = M2pi/2, 0.4M2pi , and 0.6M2pi . The phase of FSpi (t) is compared to δ 00 , as indicated by the dotted line.and the T -matrixT (t) =η00 (t)e2iδ 00 (t)−12iσpit|g(t)|eiψ00 (t)|g(t)|eiψ00 (t) η00 (t)e2i(ψ00 (t)−δ 00 (t))−12iσKt , (2.3)expressed in terms of the pipi and pipi → ¯KK phase shifts δ 00 and ψ00 as well as the inelasticity pa-rameter η00 =√1−4σ pit σ Kt |g(t)|2θ(t− tpi). The two-channel Muskhelishvili–Omnès (MO) prob-lem [7, 8] defined by the unitarity relation (2.1) permits two linearly independent solutions Ω1,Ω2 [7], which may be combined in the Omnès matrix Ω(t). In general, there is no analyticalsolution for Ω(t), we follow here the discretization method of [9] for its numerical calculation.Since the form factors are devoid of a left-hand cut, they are related directly to the solutions ofthe MO problem with coefficients determined by FSpi (0) and FSK(0) [6]. Using ChPT at O(p4) andthe low-energy constants from [10] we findFSpi (0) = (0.984±0.006)M2pi , FSK(0) = (0.4 . . .0.6)M2pi , (2.4)which, together with δ 00 and η00 from an extended Roy-equation analysis of pipi scattering [11], ψ00from partial-wave analyses [12], and |g(t)| from a Roy–Steiner (RS) analysis of piK scattering [13],yield the results for FSpi (t) depicted in Fig. 1. The strong dependence of FSpi (t) near tK on FSK (0) at-tests to the inherent two-channel nature of the problem and implies that an effective single-channeldescription in terms of the phase of FSpi (t) only works for sufficiently large FSK(0).3. From Roy–Steiner equations to the scalar form factorUnitarity couples the pipi → ¯NN and ¯KK → ¯NN S-waves f 0+(t) and h0+(t) analogously to (2.1)Im f (t) = (T (t))∗Σ(t) f (t), f (t) =(f 0+(t)2√3h0+(t)), (3.1)but due to the presence of the left-hand cut the solution of the corresponding MO problem involvesinhomogeneous contributions, which may be derived from RS equations, cf. [13 – 15]. Generically,3PoS(CD12)069Improved dispersive analysis of the scalar form factor of the nucleon Martin HoferichterFigure 2: Results for the real and imaginary part of f 0+(t). The solid, dashed, and dot-dashed lines refer tothe input RS1, RS2, and RS3 as described in the main text. The black crosses indicate the results of [17].the integral equation takes the formf (t) = ∆(t)+ (a+bt)(t−4m2)+ t2(t−4m2)pi∞∫tpidt ′ Im f (t′)t ′2(t ′−4m2)(t ′− t) , (3.2)where ∆(t) includes Born terms, s-channel integrals, and higher t-channel partial waves, while aand b subsume subthreshold parameters that emerge as subtraction constants. The main difficultyin the evaluation of the formal solutionf (t) = ∆(t)+ (t−4m2)Ω(t)(1− t ˙Ω(0))a+ t(t−4m2)Ω(t)b (3.3)− t2(t−4m2)piΩ(t)tm∫tpidt ′ ImΩ−1(t ′)∆(t ′)t ′2(t ′−4m2)(t ′− t) +t2(t−4m2)piΩ(t)∞∫tmdt ′ Ω−1(t ′)Im f (t ′)t ′2(t ′−4m2)(t ′− t) ,concerns the construction of the Omnès matrix for a finite matching point tm [14].In the numerical analysis we put Im f (t) = 0 above tm, which we choose as tm = 4m2 (thusexploiting a kinematical zero of f (t)), take the piN and KN s-channel partial waves from [16],and use the KH80 piN coupling constant and subthreshold parameters as reference point [17].In order to assess the uncertainties for higher energies we consider the following variants of theinput: first, we keep the phase shifts δ 00 and ψ00 constant above√t0 = 1.3GeV (“RS1”), where 4piintermediate states become important and thus the two-channel approximation will break down,and second, guide both phase shifts smoothly to their asymptotic value of 2pi as for the mesonform factors (“RS2”). Finally, we amend RS1 in such a way that ∆2(t), the KN component ofthe inhomogeneity, is put to zero in order to assess the uncertainty in the KN input (“RS3”). Thecorresponding results for f 0+(t) depicted in Fig. 2 show that the largest uncertainty is induced bythe high-energy phase shifts.4. ResultsThe scalar form factor of the nucleon fulfills the unitarity relationImσ(t) =24m2− t{34σ pit(FSpi (t))∗ f 0+(t)θ(t− tpi)+σ Kt (FSK(t))∗h0+(t)θ(t− tK)}, (4.1)4PoS(CD12)069Improved dispersive analysis of the scalar form factor of the nucleon Martin HoferichterFigure 3: σpiN and ∆σ as a function of the integral cutoff Λ.so that, based on the results of the previous sections, the un- and once-subtracted dispersion rela-tionsσ(t) =1pi∞∫tpidt ′ Imσ(t′)t ′− t = σpiN +tpi∞∫tpidt ′ Imσ(t′)t ′(t ′− t) (4.2)evaluated at t = 0 and t = 2M2pi in principle determine σpiN and ∆σ provided the two-channel ap-proximation for the spectral function is sufficiently accurate in the energy range dominating thedispersive integral. Fig. 3 shows that, while the dispersion relation converges too slowly for theσ term itself, the result for ∆σ becomes stable for Λ & 1GeV. Adding the uncertainties from thespectral function and the variation of the integral cutoff between Λ = 1.3GeV and Λ = 2m, we find∆σ = (13.9±0.3)MeV+Z1(g24pi−14.28)+Z2(d+00 Mpi +1.46)+Z3(d+01 M3pi −1.14)+Z4(b+00 M3pi +3.54),Z1 = 0.36MeV, Z2 = 0.57MeV, Z3 = 12.0MeV, Z4 =−0.81MeV, (4.3)where we have made the dependence on the piN parameters explicit (note that more modern deter-minations point to lower values of the piN coupling constant around g2/4pi ∼ 13.7 [18 – 20]).Similarly, the correction ∆D follows from the t-channel expansion¯D+(s = m2, t) = d+00 +d+01t−16t2∞∫tpidt ′Im f 0+(t ′)t ′2(t ′−4m2)(t ′− t) +{J ≥ 2}+{s-channel integrals}(4.4)evaluated at t = 2M2pi , which gives∆D = (12.1±0.3)MeV+ ˜Z1(g24pi−14.28)+ ˜Z2(d+00 Mpi +1.46)+ ˜Z3(d+01 M3pi −1.14)+ ˜Z4(b+00 M3pi +3.54),˜Z1 = 0.42MeV, ˜Z2 = 0.67MeV, ˜Z3 = 12.0MeV, ˜Z4 =−0.77MeV. (4.5)5PoS(CD12)069Improved dispersive analysis of the scalar form factor of the nucleon Martin HoferichterComparison with (4.3) shows that the dependence on the piN parameters cancels nearly completelyin the difference∆D−∆σ = (−1.8±0.2)MeV. (4.6)This cancellation can be explained by the observation that the spectral function in both dispersionrelations involves f 0+ in a very similar manner, so that both integrals are equally affected by thedependence on the piN parameters inherited from f 0+. In the same way, part of the uncertaintiesdiscussed in Sect. 3 drop out, so that the final error estimate for ∆D−∆σ even decreases comparedto the uncertainty in both corrections individually.References[1] T. P. Cheng and R. F. Dashen, Phys. Rev. Lett. 26 (1971) 594.[2] L. S. Brown, W. J. Pardee and R. D. Peccei, Phys. Rev. D 4 (1971) 2801.[3] V. Bernard, N. Kaiser and U.-G. Meißner, Phys. Lett. B 389 (1996) 144 [hep-ph/9607245].[4] T. Becher and H. Leutwyler, JHEP 0106 (2001) 017 [hep-ph/0103263].[5] J. Gasser, H. Leutwyler and M. E. Sainio, Phys. Lett. B 253 (1991) 260.[6] J. F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B 343 (1990) 341.[7] N. I. Muskhelishvili, Singular Integral Equations, Wolters-Noordhoff Publishing, Groningen, 1953.[8] R. Omnès, Nuovo Cim. 8 (1958) 316.[9] B. Moussallam, Eur. Phys. J. C 14 (2000) 111 [hep-ph/9909292].[10] G. Colangelo et al., Eur. Phys. J. C 71 (2011) 1695 [arXiv:1011.4408 [hep-lat]].[11] I. Caprini, G. Colangelo and H. Leutwyler, Eur. Phys. J. C 72 (2012) 1860 [arXiv:1111.7160[hep-ph]]; in preparation.[12] D. H. Cohen, D. S. Ayres, R. Diebold, S. L. Kramer, A. J. Pawlicki and A. B. Wicklund, Phys. Rev. D22 (1980) 2595; A. Etkin et al., Phys. Rev. D 25 (1982) 1786.[13] P. Büttiker, S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C 33 (2004) 409 [hep-ph/0310283].[14] C. Ditsche, M. Hoferichter, B. Kubis and U.-G. Meißner, JHEP 1206 (2012) 043 [arXiv:1203.4758[hep-ph]]; JHEP 1206 (2012) 063 [arXiv:1204.6251 [hep-ph]].[15] M. Hoferichter, D. R. Phillips and C. Schat, Eur. Phys. J. C 71 (2011) 1743 [arXiv:1106.4147[hep-ph]].[16] R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, and R. L. Workman, Eur. Phys. J. A 35 (2008) 311;J. S. Hyslop, R. A. Arndt, L. D. Roper, and R. L. Workman, Phys. Rev. D 46 (1992) 961.[17] G. Höhler, Pion–Nukleon-Streuung: Methoden und Ergebnisse, in Landolt-Börnstein, 9b2, ed.H. Schopper, Springer Verlag, Berlin, 1983.[18] J. J. de Swart, M. C. M. Rentmeester and R. G. E. Timmermans, PiN Newslett. 13 (1997) 96[nucl-th/9802084].[19] R. A. Arndt, W. J. Briscoe, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 74 (2006) 045205[nucl-th/0605082].[20] V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga and D. R. Phillips, Phys. Lett. B 694 (2011)473 [arXiv:1003.4444 [nucl-th]]; Nucl. Phys. A 872 (2011) 69 [arXiv:1107.5509 [nucl-th]].6

Improved dispersive analysis of the scalar form factor of the nucleon

Abstract

Similar works

Full text

Available Versions

Bern Open Repository and Information System (BORIS)

Juelich Shared Electronic Resources