Jefferson Lab E97-110 Collaboration

# Measurement of the generalized spin polarizabilities of the neutron in the
low $Q^{2}$ region

Vincent Sulkosky  William & Mary, Williamsburg, Virginia 23187-8795, USA
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606,
USA  University of Virginia, Charlottesville, Virginia 22904, USA  Chao Peng
Duke University, Durham, North Carolina 27708, USA  Argonne National
Laboratory, Lemont, Illinois 60439, USA  Jian-ping Chen  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  Alexandre
Deur111Corresponding author; E-mail<EMAIL_ADDRESS>University of Virginia,
Charlottesville, Virginia 22904, USA  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606, USA  Sergey Abrahamyan  Yerevan
Physics Institute, Yerevan 375036, Armenia  Konrad A. Aniol  California State
University, Los Angeles, Los Angeles, California 90032, USA  David S.
Armstrong  William & Mary, Williamsburg, Virginia 23187-8795, USA  Todd
Averett  William & Mary, Williamsburg, Virginia 23187-8795, USA  Stephanie L.
Bailey  William & Mary, Williamsburg, Virginia 23187-8795, USA  Arie Beck
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Pierre Bertin  LPC Clermont-Ferrand, Université Blaise Pascal, CNRS/IN2P3,
F-63177 Aubière, France  Florentin Butaru  Temple University, Philadelphia,
Pennsylvania 19122, USA  Werner Boeglin  Florida International University,
Miami, Florida 33199, USA  Alexandre Camsonne  LPC Clermont-Ferrand,
Université Blaise Pascal, CNRS/IN2P3, F-63177 Aubière, France  Gordon D. Cates
University of Virginia, Charlottesville, Virginia 22904, USA  Chia-Cheh Chang
University of Maryland, College Park, Maryland 20742, USA  Seonho Choi  Temple
University, Philadelphia, Pennsylvania 19122, USA  Eugene Chudakov  Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
Luminita Coman  Florida International University, Miami, Florida 33199, USA
Juan C. Cornejo  California State University, Los Angeles, Los Angeles,
California 90032, USA  Brandon Craver  University of Virginia,
Charlottesville, Virginia 22904, USA  Francesco Cusanno  Istituto Nazionale di
Fisica Nucleare, Sezione di Roma, I-00185 Rome, Italy  Raffaele De Leo
Istituto Nazionale di Fisica Nucleare, Sezione di Bari and University of Bari,
I-70126 Bari, Italy  Cornelis W. de Jager222Deceased. Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  Joseph D.
Denton  Longwood University, Farmville, VA 23909, USA  Seema Dhamija
University of Kentucky, Lexington, Kentucky 40506, USA  Robert Feuerbach
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606,
USA  John M. Finn† William & Mary, Williamsburg, Virginia 23187-8795, USA
Salvatore Frullani† Istituto Nazionale di Fisica Nucleare, Sezione di Roma,
I-00185 Rome, Italy  Istituto Superiore di Sanità, I-00161 Rome, Italy
Kirsten Fuoti  William & Mary, Williamsburg, Virginia 23187-8795, USA  Haiyan
Gao  Duke University and Triangle Universities Nuclear Laboratory, Durham, NC,
USA  Franco Garibaldi  Istituto Nazionale di Fisica Nucleare, Sezione di Roma,
I-00185 Rome, Italy  Istituto Superiore di Sanità, I-00161 Rome, Italy
Olivier Gayou  Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Ronald Gilman  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606, USA  Rutgers, The State University of New
Jersey, Piscataway, New Jersey 08855, USA  Alexander Glamazdin  Kharkov
Institute of Physics and Technology, Kharkov 310108, Ukraine  Charles
Glashausser  Rutgers, The State University of New Jersey, Piscataway, New
Jersey 08855, USA  Javier Gomez  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606, USA  Jens-Ole Hansen  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  David Hayes
Old Dominion University, Norfolk, Virginia 23529, USA  F. William Hersman
University of New Hampshire, Durham, New Hamphsire 03824, USA  Douglas W.
Higinbotham  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  Timothy Holmstrom  William & Mary, Williamsburg, Virginia
23187-8795, USA  Longwood University, Farmville, VA 23909, USA  Thomas B.
Humensky  University of Virginia, Charlottesville, Virginia 22904, USA
Charles E. Hyde  Old Dominion University, Norfolk, Virginia 23529, USA  Hassan
Ibrahim  Old Dominion University, Norfolk, Virginia 23529, USA  Cairo
University, Cairo, Giza 12613, Egypt  Mauro Iodice  Istituto Nazionale di
Fisica Nucleare, Sezione di Roma, I-00185 Rome, Italy  Xiandong Jiang
Rutgers, The State University of New Jersey, Piscataway, New Jersey 08855, USA
Lisa J. Kaufman  University of Massachusetts-Amherst, Amherst, Massachusetts
01003, USA  Aidan Kelleher  William & Mary, Williamsburg, Virginia 23187-8795,
USA  Kathryn E. Keister  William & Mary, Williamsburg, Virginia 23187-8795,
USA  Wooyoung Kim  Kyungpook National University, Taegu City, South Korea
Ameya Kolarkar  University of Kentucky, Lexington, Kentucky 40506, USA  Norm
Kolb  University of Saskatchewan, Saskatoon, SK S7N 5E2, Canada  Wolfgang
Korsch  University of Kentucky, Lexington, Kentucky 40506, USA  Kevin Kramer
William & Mary, Williamsburg, Virginia 23187-8795, USA  Duke University,
Durham, North Carolina 27708, USA  Gerfried Kumbartzki  Rutgers, The State
University of New Jersey, Piscataway, New Jersey 08855, USA  Luigi Lagamba
Istituto Nazionale di Fisica Nucleare, Sezione di Bari and University of Bari,
I-70126 Bari, Italy  Vivien Lainé  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606, USA  LPC Clermont-Ferrand, Université
Blaise Pascal, CNRS/IN2P3, F-63177 Aubière, France  Geraud Laveissiere  LPC
Clermont-Ferrand, Université Blaise Pascal, CNRS/IN2P3, F-63177 Aubière,
France  John J. Lerose  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606, USA  David Lhuillier  DAPNIA/SPhN, CEA Saclay,
F-91191 Gif-sur-Yvette, France  Richard Lindgren  University of Virginia,
Charlottesville, Virginia 22904, USA  Nilanga Liyanage  University of
Virginia, Charlottesville, Virginia 22904, USA  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  Hai-Jiang Lu
Department of Modern Physics, University of Science and Technology of China,
Hefei 230026, China  Bin Ma  Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Demetrius J. Margaziotis  California State
University, Los Angeles, Los Angeles, California 90032, USA  Peter Markowitz
Florida International University, Miami, Florida 33199, USA  Kathleen R.
McCormick  Rutgers, The State University of New Jersey, Piscataway, New Jersey
08855, USA  Mehdi Meziane  Duke University, Durham, North Carolina 27708, USA
Zein-Eddine Meziani  Temple University, Philadelphia, Pennsylvania 19122, USA
Robert Michaels  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  Bryan Moffit  William & Mary, Williamsburg, Virginia
23187-8795, USA  Peter Monaghan  Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Sirish Nanda  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606, USA  Jennifer Niedziela
University of Massachusetts-Amherst, Amherst, Massachusetts 01003, USA
Mikhail Niskin  Florida International University, Miami, Florida 33199, USA
Ronald Pandolfi  Randolph-Macon College, Ashland, Virginia 23005, USA  Kent D.
Paschke  University of Massachusetts-Amherst, Amherst, Massachusetts 01003,
USA  Milan Potokar† Institut Jozef Stefan, University of Ljubljana, Ljubljana,
Slovenia  Andrew Puckett  University of Virginia, Charlottesville, Virginia
22904, USA  Vina A. Punjabi  Norfolk State University, Norfolk, Virginia
23504, USA  Yi Qiang  Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Ronald D. Ransome  Rutgers, The State University of
New Jersey, Piscataway, New Jersey 08855, USA  Bodo Reitz  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606, USA  Rikki Roché
Florida State University, Tallahassee, Florida 32306, USA  Arun Saha† Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
Alexander Shabetai  Rutgers, The State University of New Jersey, Piscataway,
New Jersey 08855, USA  Simon Širca  Faculty of Mathematics and Physics,
University of Ljubljana, Slovenia  Jaideep T. Singh  University of Virginia,
Charlottesville, Virginia 22904, USA  Karl Slifer  Temple University,
Philadelphia, Pennsylvania 19122, USA  Ryan Snyder  University of Virginia,
Charlottesville, Virginia 22904, USA  Patricia Solvignon† Temple University,
Philadelphia, Pennsylvania 19122, USA  Ronald Stringer  Duke University,
Durham, North Carolina 27708, USA  Ramesh Subedi† Kent State University, Kent,
Ohio 44242, USA  William A. Tobias  University of Virginia, Charlottesville,
Virginia 22904, USA  Ngyen Ton  University of Virginia, Charlottesville,
Virginia 22904, USA  Paul E. Ulmer  Old Dominion University, Norfolk, Virginia
23529, USA  Guido Maria Urciuoli  Istituto Nazionale di Fisica Nucleare,
Sezione di Roma, I-00185 Rome, Italy  Antonin Vacheret  DAPNIA/SPhN, CEA
Saclay, F-91191 Gif-sur-Yvette, France  Eric Voutier  LPSC, Université Joseph
Fourier, CNRS/IN2P3, INPG, F-38026 Grenoble, France  Kebin Wang  University of
Virginia, Charlottesville, Virginia 22904, USA  Lu Wan  Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Bogdan
Wojtsekhowski  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  Seungtae Woo  Kyungpook National University, Taegu City,
South Korea  Huan Yao  Temple University, Philadelphia, Pennsylvania 19122,
USA  Jing Yuan  Rutgers, The State University of New Jersey, Piscataway, New
Jersey 08855, USA  Xiaohui Zhan  Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Xiaochao Zheng  Argonne National
Laboratory, Lemont, Illinois 60439, USA  University of Virginia,
Charlottesville, Virginia 22904, USA  Lingyan Zhu  Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

Understanding the nucleon spin structure in the regime where the strong
interaction becomes truly strong poses a challenge to both experiment and
theory. At energy scales below the nucleon mass of about 1 GeV, the intense
interaction among the quarks and gluons inside the nucleon makes them highly
correlated. Their coherent behaviour causes the emergence of effective degrees
of freedom, requiring the application of non-perturbative techniques, such as
chiral effective field theory [1]. Here, we present measurements of the
neutron’s generalized spin-polarizabilities that quantify the neutron’s spin
precession under electromagnetic fields at very low energy-momentum transfer
squared down to 0.035 GeV2. In this regime, chiral effective field theory
calculations [2, 3, 4] are expected to be applicable. Our data, however, show
a strong discrepancy with these predictions, presenting a challenge to the
current description of the neutron’s spin properties.

The nucleon is the basic building block of nature, accounting for about 99% of
the universe’s visible mass. Understanding its properties, e.g., mass and
spin, is thus crucial. Those are mainly determined by the Strong Interaction,
which is described by Quantum Chromodynamics (QCD) with quarks and gluons as
the fundamental degrees of freedom. The nucleon structure is satisfactorily
understood at high $Q^{2}$ (short space-time scales, see Fig. 1 for the
definition of kinematic variables), since there QCD is calculable using
perturbation methods (perturbative QCD) and tested by numerous experimental
measurements. At lower $Q^{2}$, the strong coupling $\alpha_{s}$ becomes too
large for perturbative QCD to be applicable [5]. Yet, calculations are
critically needed since the Strong Interaction’s chiral symmetry breaks in
this region. Chiral symmetry and its breaking is one of the most important
properties of the Strong Interaction and is believed to lead to the emergence
of the nucleon’s global properties. To understand how the underlying structure
leads to the emergence of these global properties, non-perturbative methods
must be used. A method using the fundamental quark and gluon degrees of
freedom is lattice QCD. However, calculations from this method are often
intractable for spin observables at low $Q^{2}$ [6]. Another solution is to
employ effective theories. Chiral effective field theory ($\chi$EFT)
capitalizes on QCD’s approximate chiral symmetry and uses the emergent
hadronic degrees of freedom. Therein lies $\chi$EFT’s strengths and
challenges: while the nucleon and the pion are used for first-order
calculations, this is often insufficient to describe the data, and heavier
hadrons, such as the nucleon’s first excited state $\Delta(1232)$, become
needed. This complicates $\chi$EFT calculations, and theorists are still
seeking the best way to include the $\Delta(1232)$ in their calculations. It
is therefore crucial to perform precision measurements at low enough $Q^{2}$
to test $\chi$EFT calculations. Spin observables, among them the generalized
spin-polarizabilities that are reported here, provide an extensive set of
tests to benchmark $\chi$EFT calculations [6].

Polarizabilities describe how the components of an object collectively react
to external electromagnetic fields. In particular, spin-polarizabilities
quantify the object’s spin precession under an electromagnetic field. The
spin-polarizabilities, initially defined with real photons, can be generalized
to virtual photons such as those used to probe the neutron in our experiment.
Accordingly, generalized spin-polarizabilities are extracted by scattering
polarized electrons off polarized nucleons and measuring how the cross-section
changes when the relative orientation between the electron and nucleon spins
is varied (see Fig. 1).

Figure 1:  Electron scattering off a neutron by the one-photon exchange
process. The 4-momenta of the incident and the scattered electrons are
$k^{\mu}=(E,\mathbf{k})$ and $k^{\prime\mu}=(E^{\prime},\mathbf{k^{\prime}})$,
respectively, and that of the photon is $q^{\mu}=(\nu,\mathbf{q})$. The
neutron, at rest in the laboratory frame, has a 4-momentum
$P^{\mu}=(M,\mathbf{0})$ The arrows $\uparrow\downarrow$ represent the spin
direction of the incident electron and $\Uparrow$ that of the neutron. The
generalized spin-polarizabilities of the neutron can be measured when both the
incident electron and the neutron are polarized.

The energy-momentum transferred between the electron and neutron is
$(\nu,\bm{q})$, with $Q^{2}=\bm{q}^{2}-\nu^{2}$ characterizing the space-time
scale at which we probe the neutron. While real photons ($Q^{2}=0$) only have
transverse polarizations, mediating virtual photons ($Q^{2}\neq 0$) are
transversely ($\mathrm{T}$) or longitudinally ($\mathrm{L}$) polarized. Thus,
two contributions to the spin-polarizability arise: one from the transverse-
transverse ($\mathrm{TT}$) interference called the forward spin-polarizability
$\gamma_{0}(Q^{2})$, and the other from the longitudinal-transverse
($\mathrm{LT}$) interference, called the Longitudinal-Transverse interference
polarizability $\delta_{\mathrm{LT}}(Q^{2})$, which is available only with
virtual photons. The additional longitudinal polarization direction and the
ensuing interference term offer extra latitude to test theories describing the
Strong Interaction.

The theoretical basis to measure $\delta_{\mathrm{LT}}(Q^{2})$ originates from
a work of Gell-Mann, Goldberger and Thirring [7, 8]. This work led to
relations between the cross-sections measured in polarized electron-nucleon
scattering (Fig. 1) and the spin-polarizabilities:

$\displaystyle\gamma_{0}(Q^{2})=\frac{1}{2\pi^{2}}\int_{\nu_{0}}^{\infty}\frac{\kappa_{\gamma}}{\nu^{2}}\frac{\sigma_{\mathrm{TT}}(\nu,Q^{2})}{\nu^{2}}d\nu,$
(1)
$\displaystyle\delta_{\mathrm{LT}}(Q^{2})=\left(\frac{1}{2\pi^{2}}\right)\int_{\nu_{0}}^{\infty}\frac{\kappa_{\gamma}}{\nu
Q}\frac{\sigma_{\mathrm{LT}}(\nu,Q^{2})}{\nu^{2}}d\nu,$ (2)

where $\kappa_{\gamma}=\nu-\nicefrac{{Q^{2}}}{{2M}}$ [9] is the photon flux
factor, $\nu_{0}$ the photoproduction threshold, and $\sigma_{\mathrm{TT}}$
and $\sigma_{\mathrm{LT}}$ are respectively the $\mathrm{TT}$ and
$\mathrm{LT}$ interference cross-sections. They are obtained from [6, 10]:

$\displaystyle\sigma_{TT}(\nu,Q^{2})=\frac{\pi^{2}EQ^{2}(1-\epsilon)}{\alpha\kappa_{\gamma}E^{\prime}(1-\epsilon
E^{\prime}/E)(1+\eta\zeta)}\left(\sqrt{\frac{2\epsilon}{1+\epsilon}}\Delta\sigma_{\parallel}(\nu,Q^{2})-\eta\Delta\sigma_{\perp}(\nu,Q^{2})\right),$
(3)
$\displaystyle\sigma_{LT}(\nu,Q^{2})=\frac{\pi^{2}EQ^{2}(1-\epsilon)}{\alpha\kappa_{\gamma}E^{\prime}(1-\epsilon
E^{\prime}/E)(1+\eta\zeta)}\left(\sqrt{\frac{2\epsilon}{1+\epsilon}}\zeta\Delta\sigma_{\parallel}(\nu,Q^{2})+\Delta\sigma_{\perp}(\nu,Q^{2})\right),$
(4)

where $\Delta\sigma_{||}$ ($\Delta\sigma_{\bot}$) is the difference between
the cross sections when the beam and target spin directions are parallel and
antiparallel (perpendicular), $\alpha$ is the electromagnetic coupling
constant,
$\epsilon=1/[1+2(1+\nicefrac{{Q^{2}}}{{4M^{2}x^{2}}})\tan^{2}(\nicefrac{{\theta}}{{2}})]$
with $x=\nicefrac{{Q^{2}}}{{2m\nu}}$ the Bjorken scaling variable and $\theta$
the electron scattering angle in the laboratory frame,
$\eta=\nicefrac{{\epsilon Q}}{{(E-E^{\prime}\epsilon)}}$ and
$\zeta=\nicefrac{{\eta(1+\epsilon)}}{{2\epsilon}}$. The $\sigma_{\mathrm{TT}}$
and $\sigma_{\mathrm{LT}}$, shown in Figs. 2 and 3,

Figure 2:  The transverse-transverse cross-section
$\sigma_{\mathrm{TT}}(\nu,Q^{2})$ for 3He. The data are displayed at the
$Q^{2}$ values at which they are integrated to form $\gamma_{0}$ (Eq. 1). The
error bars, sometimes too small to be visible, represent the statistical
uncertainties. The systematic uncertainty is indicated by the band at the
bottom of each panel. The nuclear corrections providing the neutron
information from the 3He data are applied after the integration. The prominent
negative peak at small-$\nu$ is the $\Delta(1232)$ contribution.  Figure 3:
The longitudinal-transverse interference cross-section
$\sigma_{\mathrm{LT}}(\nu,Q^{2})$ for 3He. The data are displayed at the
$Q^{2}$ values at which they are integrated into $\delta_{\mathrm{LT}}(Q^{2})$
(Eq. 2) or $I_{\mathrm{LT}}(Q^{2})$ (Eq. 5). The error bars represent the
statistical uncertainties. The systematic uncertainty is indicated by the band
at the bottom of each panel. The nuclear corrections [18] necessary to obtain
the neutron information from the 3He data are applied after the integration.
The prominent $\Delta(1232)$ contribution seen for
$\sigma_{\mathrm{TT}}(\nu,Q^{2})$ in Fig. 2 is not present here, in agreement
with the expectation that the role of $\Delta(1232)$ is suppressed in
$\mathrm{LT}$-interference quantities.

were integrated according to Eqs. (1) and (2) to obtain $\gamma_{0}(Q^{2})$
and $\delta_{\mathrm{LT}}(Q^{2})$. The unmeasured part of the integrals at
large $\nu$ is often negligible due to the $\nu$-weighting.

An outstanding feature of $\delta_{\mathrm{LT}}(Q^{2})$ at low $Q^{2}$ is that
the $\Delta(1232)$ is not expected to appreciably contribute to the LT-
interference cross section, since exciting the $\Delta(1232)$ overwhelmingly
involves transverse photons. This should alleviate the difficulty of including
the $\Delta(1232)$ in $\chi$EFT calculations, making them more robust.
However, the first measurement of $\delta_{\mathrm{LT}}(Q^{2})$ from JLab
experiment E94-010 [11] done at $Q^{2}\geq 0.1$ GeV2 strongly disagreed with
$\chi$EFT calculations [12, 13]. This surprising result, known as the
“$\delta_{\mathrm{LT}}$ puzzle” [10], triggered improved $\chi$EFT
calculations [14] which now explicitly include the $\Delta(1232)$ [2, 3, 4],
and measurements of $\delta_{\mathrm{LT}}$ at lower $Q^{2}$ where $\chi$EFT
can be best tested. New data of $\delta_{\mathrm{LT}}$ on the neutron at very
low $Q^{2}$ are presented next, which were taken during experiment JLab
E97-110.

Eq. (2) allows measuring $\delta_{\mathrm{LT}}^{n}(Q^{2})$ (the superscript
$n$ indicates neutron quantities) by scattering polarized electrons off
polarized neutrons in 3He nuclei. The data were acquired in Hall A [15] of
Jefferson Lab (JLab) during experiment E97-110 [16]. The probing virtual
photons were produced by a longitudinally polarized electron beam during its
scattering off a polarized 3He target [15]. The beam polarization, flipped
pseudo-randomly at 30 Hz and monitored by Møller and Compton polarimeters, was
(75.0 $\pm$ 2.3)%. The beam energies ranged from 1.1 to 4.4 GeV, and the beam
current was typically a few $\mu$A. Since free neutrons are unstable, we used
3He nuclei as an effective polarized neutron target. To first-order, polarized
3He nuclei can be treated as effective polarized neutrons together with
unpolarized protons because the 3He’s nucleons (two protons and one neutron)
are mostly in an $S$-state, and so the Pauli exclusion principle dictates that
in the S-state the proton spins point oppositely, yielding no net contribution
to the 3He spin. The gaseous ($\approx 12$ atm) 3He was contained in a 40 cm-
long glass cylinder and polarized by spin-exchange optical pumping of Rubidium
atoms. Helmholtz coils provided a longitudinal or transverse 2.5 mT field used
to maintain the polarization, to orient it longitudinally or transversely (in-
plane) to the beam direction, and to aid in performing polarimetry. The
average target polarization in-beam was (39.0 $\pm$ 1.6)%. The scattered
electrons from the reaction
$\vec{\rm{{}^{3}He}}$($\vec{\rm{e}},\rm{e^{\prime}}$) were detected by a High
Resolution Spectrometer (HRS) [15] supplemented by a dipole magnet [17]
allowing us to detect electrons scattered at angles down to 6∘. Behind the
HRS, drift chambers provided particle tracking, scintillator planes enabled
the data acquisition trigger, and a gas Cherenkov counter and electromagnetic
calorimeters ensured the identification of the particle type.

The measured $\sigma_{\mathrm{TT}}$ ($\sigma_{\mathrm{LT}}$) on 3He is shown
in Fig. 2 (Fig. 3). Its values with their uncertainties are available in the
Supplementary Data Files. While polarized 3He nuclei are effectively polarized
neutrons to good approximation, nuclear corrections are needed to obtain
genuine neutron information. The prescription of Ref. [18] was used for the
correction. The effect of the nuclear correction, which can be obtained from
Tables I-III in the Supplementary Data Files, is relatively small. In
particular it does not appreciably affect the $Q^{2}$ trend seen for the
uncorrected 3He integrals. The relative uncertainty on this correction is
estimated to be 6 to 14% relative to the correction, the higher uncertainties
corresponding to our lowest $Q^{2}$ values. The quasi-elastic contamination
was corrected following the procedure described in [16]. The correction is
small for $\delta_{\mathrm{LT}}^{n}$, but important for $\gamma_{0}^{n}$ and
was estimated using [19]. No calculation uncertainty is provided in [19] and
using another quasi-elastic calculation [20] may shift the lowest-$Q^{2}$
$\gamma_{0}^{n}$ data points by as much as our total systematic uncertainty.
The other main systematic uncertainties come from the absolute cross-sections
(3.5 to 4.5%), target and beam polarizations (3 to 5% and 3.5%, respectively),
and radiative corrections (3 to 7%).

Our $\delta_{\mathrm{LT}}^{n}(Q^{2})$ data are shown in the left panel of Fig.
4. They agree with earlier data from E94-010 at larger $Q^{2}$ [11] while
reaching much lower $Q^{2}$ where the $\chi$EFT is expected to work well. The
measurement can be compared to $\chi$EFT calculations [12, 13, 2, 4] and a
model parameterization of the world photo- and electro-production data called
MAID [21].

Figure 4:  The generalized spin polarizabilities
$\delta_{\mathrm{LT}}^{n}(Q^{2})$ and $\gamma_{0}^{n}(Q^{2})$. Left: The
generalized spin-polarizability $\delta_{\mathrm{LT}}^{n}(Q^{2})$. The circles
represent the results from experiment E97-110. They can be compared to earlier
E94-010 data [11] (triangles) and theoretical calculations: the older
$\chi$EFT calculations [12] (dot-dashed line) and [13] (dashed line) in which
the $\Delta$ resonance contribution is not included or included
phenomenologically, the state-of-the-art calculations [2] (cyan band) and [4]
(magenta band) that include the $\Delta$, as well as the MAID model [21]
(black curve) which is a fit to world resonance data. For the E97-110 data,
the inner error bars, sometimes too small to be visible, represent the
statistical uncertainties. The outer error bars show the statistical and
uncorrelated systematic uncertainties. The correlated systematic uncertainty
is indicated by the band at the bottom. For the other experimental data, the
error bars show the statistical and systematic uncertainties added in
quadrature. Right: The generalized forward spin-polarizability
$\gamma_{0}^{n}(Q^{2})$, using the same symbols as in the left panel. The
asterisks represent the CLAS data [22].

Earlier $\chi$EFT calculations [12, 13] used different approaches (Heavy
Baryon and Relativistic Baryon chiral perturbation theory: HB$\chi$PT and
RB$\chi$PT, respectively), and furthermore either neglected the $\Delta(1232)$
degrees of freedom, or included it approximately. Newer calculations [2, 3,
4], which are all fully relativistic, account for the $\Delta(1232)$
explicitly by using a perturbative expansion, but they differ in their choice
of expansion parameter. Despite this theoretical improvement and the
small-$Q^{2}$ reach that places our data well in the validity domain of
$\chi$EFT, our $\delta_{\mathrm{LT}}^{n}(Q^{2})$ starkly disagrees with the
predictions. This is even more surprising because the latest $\chi$EFT
calculations of $\delta_{\mathrm{LT}}^{n}$ agree with each other, suggesting
that calculations for this particular observable should be under control.
However, our data reveal an opposite trend with $Q^{2}$ to that of all the
$\chi$EFT calculations.

This startling discrepancy demanded further scrutinization of our data. They
are compatible with the E94-010 data where they overlap. This is also true for
$\gamma_{0}^{n}(Q^{2})$, which we measured concurrently and show in the right
panel of Fig. 4. The measured $\gamma_{0}^{n}(Q^{2})$ also agrees with data
from CLAS experiment EG1 [22], which used a target and detectors that are very
different from E97-110 and E94-010. Our $\gamma_{0}^{n}(Q^{2})$ data generally
disagree with $\chi$EFT calculations. Since $\gamma_{0}(Q^{2})$ does not
benefit from the suppression of the $\Delta(1232)$ contribution, and since
$\gamma^{n}_{0}(Q^{2})$ predictions do not reach a consensus, this
disagreement is not entirely surprising, in contrast to the unexpected
$\delta_{\mathrm{LT}}^{n}(Q^{2})$ disagreement. Interestingly, we can also
study with our data the Schwinger relation [23], which has a similar
definition but without $\nu^{-2}$ weighting in its integrand:

$\displaystyle I_{\mathrm{LT}}(Q^{2})$ $\displaystyle\equiv$
$\displaystyle\left(\frac{M^{2}}{\alpha\pi^{2}}\right)\int_{\nu_{0}}^{\infty}\Bigl{[}\kappa_{\gamma}\frac{\sigma_{\mathrm{LT}}(\nu,Q^{2})}{Q\nu}\Bigr{]}_{Q=0}d\nu.$
(5)

Schwinger predicted that $I_{\mathrm{LT}}(Q^{2})\xrightarrow[Q^{2}\to
0]{~{}}\kappa e_{t}$, with $\kappa$ the anomalous magnetic moment of the
target particle and $e_{t}$ its electric charge. This prediction is general,
e.g. it does not use $\chi$EFT. $I_{\mathrm{LT}}(Q^{2})$ having no
$\nicefrac{{1}}{{\nu^{2}}}$-weighting, the large $\nu$ contribution to the
integral is not negligible. Since this contribution to the integral cannot be
measured, a parameterization based on the model described in [24] completed by
a Regge-based parameterization [25] for the largest $\nu$ part was used to
extrapolate it. Our measurement of $I^{n}_{\mathrm{LT}}(Q^{2})$ is shown in
Fig. 5. Our measurement of $I^{n}_{\mathrm{LT}}(Q^{2})$ without the Regge-
based parameterization [25] for the large-$\nu$ part (open symbols), which is
suppressed in $\delta_{\mathrm{LT}}(Q^{2})$, displays a similar pattern as
$\delta_{\mathrm{LT}}^{n}(Q^{2})$. The Gerasimov-Drell-Hearn (GDH) relation
[28, 29] can be used to extrapolate our $I^{n}_{\mathrm{LT}}(Q^{2})$ to
$Q^{2}=0$; and provided that the GDH relation is valid, which is widely
expected and supported by dedicated experimental studies [30], our data
satisfy Schwinger’s prediction that $I^{n}_{\mathrm{LT}}(0)=0$ [23]. Our trend
contrasts with the MAID model and presumably the $\chi$EFT calculations, since
MAID tracks those (see Fig. 4). This suggests that the problem lies in the
theoretical description of the neutron structure.

Figure 5:  The Schwinger integral $I_{\mathrm{LT}}^{n}(Q^{2})$. The open
symbols are our results without the large $\nu$ part of $I_{\mathrm{LT}}$. The
filled blue circles are our results for the full $I_{\mathrm{LT}}$, using an
estimate for the large $\nu$ contribution. The inner error bars represent the
statistical uncertainties. The outer error bars show the combined statistical
and uncorrelated systematic uncertainties. The correlated systematic
uncertainty is indicated by the band. The Schwinger relation [23] for the
neutron predicts $I_{\mathrm{LT}}^{n}(0)=0$ at $Q^{2}=0$. The plain line shows
the MAID model [21], which is a fit to world resonance data (to be compared to
the open symbols). The dashed line uses the GDH ($\Gamma_{1}$) [28, 29] and
Burkhardt-Cottingham [26] relations, together with an elastic form factor (FF)
parameterization [27], to obtain $I_{\mathrm{LT}}^{n}(Q^{2})$ for $Q^{2}\to
0$.

The measured $I_{\mathrm{LT}}(Q^{2})$ displays a similar $Q^{2}$-behavior as
$\delta_{\mathrm{LT}}$, irrespective of the different $\nu$-weighting. Other
integrals without $\nu^{-2}$ weighting formed using our data and reported in
[16] did not display the surprisingly strong disagreement with the predictions
seen here. The values of $\gamma_{0}^{n}$, $\delta_{\mathrm{LT}}^{n}$ and
$I_{\mathrm{LT}}^{n}$ with their uncertainties are available in the
Supplementary Data Files.

Our data indicate that both the TT and LT interferences of the electromagnetic
field’s components induce a clear spin precession of the neutron. While it was
predicted by all calculations and models that the LT term influence should
intensify at small $Q^{2}$, our data reveal the opposite trend. This notable
disagreement is perplexing since our measurements were done well into the
domain where $\chi$EFT is expected to describe reliably the nucleon
properties, especially the “gold-plated” $\delta_{\mathrm{LT}}$. Lattice QCD
calculations of $\delta_{\mathrm{LT}}(Q^{2})$ are possible [31], but not yet
available. Our data motivate such calculations since the measured generalized
spin-polarizabilities underline a current lack of reliable quantitative
descriptions of the Strong Interaction at the nucleon-size scale.

Data availability All experimental data that support the findings of this
study are provided in the Supplementary Data Files or are available from J.P.
Chen (jpchen@jlab.org), A. Deur (deurpam@jlab.org), C. Peng<EMAIL_ADDRESS>or V. Sulkosky<EMAIL_ADDRESS>upon request.

Code availability The computer codes that support the plots within this paper
and the findings of this study are available from J.P. Chen (jpchen@jlab.org),
A. Deur (deurpam@jlab.org), C. Peng<EMAIL_ADDRESS>or V. Sulkosky
<EMAIL_ADDRESS>upon request.

Author contributions The members of the Jefferson Lab E97-110 Collaboration
constructed and operated the experimental equipment used in this experiment.
All authors contributed to the data collection, experiment design and
commissioning, data processing, data analysis or Monte Carlo simulations. The
following authors especially contributed to the main data analysis: J.P Chen,
A. Deur, C. Peng and V. Sulkosky.

Competing interests The authors declare no competing interests.

###### Acknowledgements.

We acknowledge the outstanding support of the Jefferson Lab Hall A technical
staff and the Physics and Accelerator Divisions that made this work possible.
We thank A. Deltuva, J. Golak, F. Hagelstein, H. Krebs, V. Lensky, U.-G.
Meißner, V. Pascalutsa, G. Salmè, S. Scopetta and M. Vanderhaeghen for useful
discussions and for sharing their calculations. We are grateful to V.
Pascalutsa and M. Vanderhaeghen for suggesting to compare the data to the
Schwinger relation. This material is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics under
contract DE-AC05-06OR23177, and by the NSF under grant PHY-0099557.

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Data tables

$Q^{2}$ |  $\nu_{max}$ ($W_{max}$) |  $\delta_{\mathrm{LT}}^{{}^{3}He,~{}res}(Q^{2})\pm$(stat)$\pm$(syst) |  $\delta_{\mathrm{LT}}^{n,~{}res}(Q^{2})\pm$(stat)$\pm$(syst) |  $\delta_{\mathrm{LT}}^{n}(Q^{2})\pm$(stat)$\pm$(syst)
---|---|---|---|---
$[$GeV${}^{2}]$ | [GeV] |  [$10^{-4}$ fm4] |  [$10^{-4}$ fm4] |  [$10^{-4}$ fm4]
0.035 | 1.690 (2.00) | $-0.356\pm 0.280\pm 0.583$ | $-0.379\pm 0.326\pm 0.677$ | $-0.383\pm 0.326\pm 0.677$
0.057 | 1.700 (2.00) | $0.174\pm 0.061\pm 0.169$ | $0.229\pm 0.071\pm 0.197$ | $0.225\pm 0.071\pm 0.197$
0.079 | 1.710 (2.00) | $0.360\pm 0.084\pm 0.168$ | $0.439\pm 0.098\pm 0.195$ | $0.435\pm 0.098\pm 0.195$
0.100 | 2.885 (2.49) | $0.410\pm 0.072\pm 0.180$ | $0.493\pm 0.083\pm 0.209$ | $0.491\pm 0.083\pm 0.209$
0.150 | 2.910 (2.48) | $0.178\pm 0.045\pm 0.149$ | $0.216\pm 0.053\pm 0.173$ | $0.215\pm 0.053\pm 0.173$
0.200 | 2.655 (2.38) | $0.091\pm 0.024\pm 0.078$ | $0.112\pm 0.028\pm 0.091$ | $0.111\pm 0.028\pm 0.091$
0.240 | 2.320 (2.23) | $0.090\pm 0.017\pm 0.041$ | $0.110\pm 0.020\pm 0.043$ | $0.108\pm 0.020\pm 0.043$
Table 1:  Data and kinematics for $\delta_{\mathrm{LT}}(Q^{2})$. From left to right: Four-momentum transfer squared; Maximum $\nu$ value experimentally covered (equivalent maximum invariant $W$ ($W=(M^{2}+2M\nu-Q^{2})^{\nicefrac{{1}}{{2}}}$; $\delta_{\mathrm{LT}}^{{}^{3}He,~{}res}(Q^{2})$ measured over the $\nu$ ($W$) range from pion threshold up to maximum $\nu$ ($W$) covered experimentally (mostly the nucleon resonance region) and before applying the nuclear corrections to extract the neutron information. (stat) denotes the statistical uncertainty and (syst) the systematic uncertainty; Extracted neutron $\delta_{\mathrm{LT}}^{n,~{}res}$ (resonance) after applying nuclear corrections to the previous column; Total $\delta_{\mathrm{LT}}^{n}(Q^{2})$. Comparing the two last columns shows that the unmeasured parts of $\delta_{\mathrm{LT}}^{n}(Q^{2})$, i.e, the contributions for $\nu>\nu_{max}$, are negligible. $Q^{2}$ |  $\nu_{max}$ ($W_{max}$) |  $\gamma_{0}^{{}^{3}He,~{}res}(Q^{2})\pm$(stat)$\pm$(syst) |  $\gamma_{0}^{n,~{}res}(Q^{2})\pm$(stat)$\pm$(syst) |  $\gamma_{0}^{n}(Q^{2})\pm$(stat)$\pm$(syst)
---|---|---|---|---
$[$GeV${}^{2}]$ | [GeV] |  [$10^{-4}$ fm4] |  [$10^{-4}$ fm4] |  [$10^{-4}$ fm4]
0.035 | 1.690 (2.00) | $-2.590\pm 0.111\pm 0.225$ | $-3.092\pm 0.129\pm 0.270$ | $-3.094\pm 0.129\pm 0.270$
0.057 | 1.700 (2.00) | $-2.613\pm 0.121\pm 0.215$ | $-3.115\pm 0.141\pm 0.259$ | $-3.117\pm 0.141\pm 0.259$
0.079 | 1.710 (2.00) | $-2.274\pm 0.121\pm 0.226$ | $-2.715\pm 0.140\pm 0.270$ | $-2.717\pm 0.140\pm 0.270$
0.100 | 2.885 (2.49) | $-1.725\pm 0.063\pm 0.143$ | $-2.070\pm 0.074\pm 0.170$ | $-2.070\pm 0.074\pm 0.170$
0.150 | 2.910 (2.48) | $-1.135\pm 0.044\pm 0.105$ | $-1.370\pm 0.051\pm 0.125$ | $-1.370\pm 0.051\pm 0.125$
0.200 | 2.655 (2.38) | $-0.798\pm 0.027\pm 0.056$ | $-0.964\pm 0.032\pm 0.065$ | $-0.965\pm 0.032\pm 0.065$
0.240 | 2.320 (2.23) | $-0.612\pm 0.022\pm 0.043$ | $-0.740\pm 0.026\pm 0.050$ | $-0.742\pm 0.026\pm 0.050$
Table 2:  Data and kinematics for $\gamma_{0}(Q^{2})$. From left to right: Four-momentum transfer squared; Maximum $\nu$ value experimentally covered (equivalent maximum invariant $W$ ($W=(M^{2}+2M\nu-Q^{2})^{\nicefrac{{1}}{{2}}}$; $\gamma_{0}^{{}^{3}He,~{}res}(Q^{2})$ measured over the $\nu$ ($W$) range from pion threshold up to maximum $\nu$ ($W$) covered experimentally (mostly the nucleon resonance region) and before applying the nuclear corrections to extract the neutron information. (stat) denotes the statistical uncertainty and (syst) the systematic uncertainty; Extracted neutron $\gamma_{0}^{n,~{}res}$ (resonance) after applying nuclear corrections to the previous column; Total $\gamma_{0}^{n}(Q^{2})$; Comparing the two last columns shows that the unmeasured parts of $\gamma_{0}^{n}(Q^{2})$, i.e, the contributions for $\nu>\nu_{max}$, are negligible.  $Q^{2}$ |  $\nu_{max}$ ($W_{max}$) |  $I_{\mathrm{LT}}^{{}^{3}He,~{}res}(Q^{2})\pm$(stat)$\pm$(syst) |  $I_{\mathrm{LT}}^{n,~{}res}(Q^{2})\pm$(stat)$\pm$(syst) |  $I_{\mathrm{LT}}(Q^{2})^{n}\pm$(stat)$\pm$(syst)
---|---|---|---|---
$[$GeV${}^{2}]$ | [GeV] |  |  |
0.035 | 1.690 (2.00) | $-0.326\pm 0.272\pm 0.520$ | $-0.294\pm 0.316\pm 0.604$ | $-1.112\pm 0.316\pm 0.606$
0.057 | 1.700 (2.00) | $0.285\pm 0.117\pm 0.333$ | $0.413\pm 0.136\pm 0.387$ | $-0.862\pm 0.136\pm 0.389$
0.079 | 1.710 (2.00) | $0.610\pm 0.128\pm 0.268$ | $0.786\pm 0.149\pm 0.312$ | $-0.721\pm 0.149\pm 0.314$
0.100 | 2.885 (2.49) | $0.879\pm 0.098\pm 0.280$ | $1.092\pm 0.114\pm 0.327$ | $-0.126\pm 0.114\pm 0.329$
0.150 | 2.910 (2.48) | $0.339\pm 0.049\pm 0.198$ | $0.458\pm 0.057\pm 0.231$ | $-0.266\pm 0.057\pm 0.233$
0.200 | 2.655 (2.38) | $0.289\pm 0.047\pm 0.228$ | $0.389\pm 0.055\pm 0.265$ | $-0.345\pm 0.055\pm 0.267$
0.240 | 2.320 (2.23) | $0.391\pm 0.057\pm 0.163$ | $0.511\pm 0.067\pm 0.190$ | $-0.267\pm 0.067\pm 0.192$
Table 3:  Data and kinematics for $I_{\mathrm{LT}}(Q^{2})$. From left to right: Four-momentum transfer squared; Maximum $\nu$ value experimentally covered (equivalent maximum invariant $W$ ($W=(M^{2}+2M\nu-Q^{2})^{\nicefrac{{1}}{{2}}}$; $I_{\mathrm{LT}}^{{}^{3}He,~{}res}(Q^{2})$ measured over the $\nu$ ($W$) range from pion threshold up to maximum $\nu$ ($W$) covered experimentally (mostly the nucleon resonance region) and before applying the nuclear corrections to extract the neutron information. (stat) denotes the statistical uncertainty and (syst) the systematic uncertainty; Extracted neutron $I_{\mathrm{LT}}^{n,~{}res}$ (resonance) after applying nuclear corrections to the previous column; Total $I_{\mathrm{LT}}^{n}(Q^{2})$, including an estimate for the unmeasured contribution above $\nu_{max}$.  $Q^{2}$ | $\nu$ | $W$ | $x$ | $\sigma_{\mathrm{LT}}$ | Stat. | Uncor. syst. | Cor. syst. | $\sigma_{\mathrm{TT}}$ | Stat. | Uncor. syst. | Cor. syst.
---|---|---|---|---|---|---|---|---|---|---|---
$[$GeV${}^{2}]$ | [MeV] | [MeV] |  |  [$\mu$b] |  [$\mu$b] |  [$\mu$b] |  [$\mu$b] |  [$\mu$b] |  [$\mu$b] |  [$\mu$b] |  [$\mu$b]
0.035 | 167.5 | 1076.9 | 0.1114 | -29.593 | 7.821 | 4.430 | 6.567 | -50.779 | 8.849 | 4.803 | 7.200
0.035 | 210.0 | 1113.3 | 0.0888 | -9.283 | 8.641 | 4.463 | 2.718 | -60.029 | 9.257 | 4.828 | 3.067
0.035 | 250.0 | 1146.5 | 0.0746 | 5.012 | 7.507 | 7.804 | 4.201 | -70.097 | 7.608 | 10.592 | 5.505
0.035 | 290.0 | 1178.8 | 0.0643 | -4.189 | 8.133 | 4.768 | 4.598 | -144.587 | 7.398 | 6.008 | 5.937
0.035 | 330.0 | 1210.2 | 0.0565 | -11.970 | 7.757 | 5.264 | 4.145 | -180.197 | 6.794 | 8.238 | 5.752
0.035 | 370.0 | 1240.8 | 0.0504 | 3.328 | 7.025 | 5.624 | 3.348 | -159.284 | 6.028 | 7.449 | 5.188
0.035 | 410.0 | 1270.7 | 0.0455 | 7.802 | 6.144 | 4.136 | 2.339 | -102.038 | 5.192 | 5.252 | 3.146
0.035 | 450.0 | 1299.9 | 0.0414 | 2.569 | 6.034 | 1.754 | 1.333 | -56.856 | 4.689 | 3.063 | 1.734
0.035 | 490.0 | 1328.5 | 0.0381 | -2.776 | 6.006 | 1.077 | 1.003 | -36.599 | 4.210 | 2.488 | 1.797
0.035 | 530.0 | 1356.4 | 0.0352 | -0.849 | 5.748 | 0.699 | 2.120 | -22.279 | 3.827 | 1.881 | 2.340
0.035 | 570.0 | 1383.8 | 0.0327 | 4.796 | 6.483 | 0.646 | 0.845 | -17.020 | 4.317 | 1.184 | 2.026
0.035 | 610.0 | 1410.7 | 0.0306 | 7.627 | 9.229 | 0.833 | 0.899 | -19.965 | 7.244 | 1.079 | 2.128
0.035 | 650.0 | 1437.0 | 0.0287 | 11.711 | 10.295 | 0.878 | 0.757 | -13.732 | 8.808 | 0.690 | 1.336
0.035 | 690.0 | 1462.9 | 0.0270 | 22.036 | 10.449 | 1.301 | 1.731 | -18.433 | 8.192 | 0.832 | 1.271
0.035 | 730.0 | 1488.4 | 0.0256 | 10.433 | 10.676 | 1.255 | 3.585 | -17.674 | 6.646 | 0.939 | 1.543
0.035 | 770.0 | 1513.4 | 0.0242 | -16.164 | 9.879 | 0.690 | 2.002 | -9.317 | 5.897 | 0.830 | 1.388
0.035 | 810.0 | 1538.0 | 0.0230 | -3.384 | 10.144 | 0.905 | 2.704 | -13.691 | 5.873 | 1.137 | 1.231
0.035 | 850.0 | 1562.2 | 0.0219 | 6.799 | 11.135 | 0.939 | 3.434 | -16.569 | 6.153 | 1.448 | 2.080
0.035 | 890.0 | 1586.0 | 0.0210 | -5.700 | 10.532 | 0.600 | 3.009 | -9.035 | 5.791 | 0.515 | 2.530
0.035 | 930.0 | 1609.5 | 0.0201 | 17.363 | 10.209 | 0.852 | 1.794 | -18.818 | 5.628 | 0.690 | 1.982
0.035 | 970.0 | 1632.7 | 0.0192 | 8.494 | 9.638 | 0.846 | 1.477 | -21.969 | 4.806 | 1.291 | 0.844
0.035 | 1010.0 | 1655.5 | 0.0185 | -8.975 | 8.213 | 0.660 | 1.582 | -25.754 | 4.428 | 1.639 | 1.197
0.035 | 1050.0 | 1678.0 | 0.0178 | -14.309 | 7.307 | 0.924 | 1.935 | -15.582 | 4.476 | 1.424 | 1.432
0.035 | 1090.0 | 1700.2 | 0.0171 | 3.170 | 8.109 | 0.537 | 2.119 | -13.869 | 4.743 | 0.862 | 0.833
0.035 | 1130.0 | 1722.2 | 0.0165 | -8.033 | 9.470 | 0.818 | 2.159 | -8.614 | 5.365 | 0.660 | 0.765
0.035 | 1170.0 | 1743.8 | 0.0159 | -15.669 | 9.088 | 0.909 | 2.006 | -14.402 | 5.160 | 2.370 | 1.013
0.035 | 1210.0 | 1765.2 | 0.0154 | -3.340 | 9.120 |  0.257 |  0.488 | -20.220 | 5.268 |  1.405 |  2.551
---|---|---|---|---|---|---|---|---|---|---|---
0.035 | 1250.0 | 1786.3 | 0.0149 | 8.626 | 9.670 | 0.657 | 1.584 | -17.874 | 5.217 | 0.934 | 1.967
0.035 | 1290.0 | 1807.2 | 0.0145 | 6.658 | 9.509 | 0.504 | 0.796 | -22.003 | 5.059 | 1.209 | 1.012
0.035 | 1330.0 | 1827.9 | 0.0140 | 10.562 | 9.728 | 0.755 | 1.773 | -17.263 | 5.284 | 0.894 | 0.558
0.035 | 1370.0 | 1848.3 | 0.0136 | -6.227 | 10.036 | 0.190 | 1.003 | -15.983 | 5.941 | 0.877 | 0.438
0.035 | 1410.0 | 1868.5 | 0.0132 | -0.496 | 10.253 | 0.396 | 1.494 | -17.223 | 6.110 | 0.925 | 0.429
0.035 | 1450.0 | 1888.5 | 0.0129 | 9.730 | 10.555 | 0.963 | 2.563 | -17.723 | 6.335 | 0.921 | 0.561
0.035 | 1490.0 | 1908.2 | 0.0125 | -16.688 | 10.365 | 0.758 | 1.789 | -15.680 | 6.315 | 0.938 | 0.887
0.035 | 1530.0 | 1927.8 | 0.0122 | -6.735 | 10.420 | 0.298 | 0.434 | -26.546 | 6.354 | 1.474 | 1.010
0.035 | 1570.0 | 1947.2 | 0.0119 | -3.131 | 11.567 | 0.146 | 0.590 | -16.390 | 6.927 | 0.895 | 0.547
0.035 | 1610.0 | 1966.4 | 0.0116 | -12.388 | 12.041 | 0.576 | 1.025 | -15.621 | 7.166 | 0.876 | 0.443
0.035 | 1650.0 | 1985.4 | 0.0113 | -25.057 | 12.833 | 1.492 | 2.250 | -11.943 | 7.797 | 0.939 | 0.488
0.035 | 1690.0 | 2004.2 | 0.0110 | 12.734 | 13.111 | 0.708 | 1.519 | -16.559 | 8.754 | 2.147 | 0.546
0.035 | 1730.0 | 2022.8 | 0.0108 | -3.404 | 12.912 | 0.180 | 0.339 | -23.472 | 8.469 | 1.405 | 0.496
0.035 | 1770.0 | 2041.3 | 0.0105 | -18.275 | 10.605 | 0.945 | 0.752 | -10.813 | 8.329 | 0.977 | 0.719
0.035 | 1810.0 | 2059.6 | 0.0103 | -6.756 | 11.453 | 0.312 | 1.048 | -23.516 | 7.914 | 1.303 | 1.315
0.035 | 1850.0 | 2077.7 | 0.0101 | -18.357 | 14.142 | 1.001 | 0.658 | -1.761 | 10.219 | 0.197 | 0.559
0.057 | 177.5 | 1075.4 | 0.1712 | 7.220 | 15.736 | 3.537 | 8.714 | -92.373 | 19.571 | 4.724 | 11.368
0.057 | 220.0 | 1111.8 | 0.1381 | 8.333 | 7.607 | 3.298 | 3.970 | -85.011 | 9.007 | 3.795 | 4.741
0.057 | 260.0 | 1145.1 | 0.1168 | 4.145 | 6.307 | 3.452 | 5.860 | -85.636 | 6.975 | 3.884 | 6.973
0.057 | 300.0 | 1177.4 | 0.1013 | -11.320 | 6.279 | 4.864 | 6.505 | -136.987 | 6.600 | 5.833 | 7.964
0.057 | 340.0 | 1208.9 | 0.0893 | 3.996 | 6.344 | 5.770 | 5.386 | -179.319 | 6.447 | 7.269 | 6.118
0.057 | 380.0 | 1239.5 | 0.0799 | 16.750 | 5.973 | 6.692 | 6.489 | -159.668 | 5.916 | 7.136 | 6.346
0.057 | 420.0 | 1269.5 | 0.0723 | 14.794 | 5.623 | 4.770 | 4.507 | -104.990 | 5.277 | 4.722 | 3.989
0.057 | 460.0 | 1298.7 | 0.0660 | 16.454 | 5.864 | 2.330 | 2.421 | -64.416 | 5.024 | 3.446 | 3.120
0.057 | 500.0 | 1327.3 | 0.0608 | 13.597 | 5.883 | 1.659 | 4.063 | -36.795 | 4.674 | 2.493 | 2.566
0.057 | 540.0 | 1355.2 | 0.0562 | 4.128 | 5.647 |  0.697 |  2.973 | -15.701 | 4.370 |  0.971 |  2.064
---|---|---|---|---|---|---|---|---|---|---|---
0.057 | 580.0 | 1382.7 | 0.0524 | -3.473 | 5.804 | 0.581 | 4.200 | -11.937 | 4.380 | 0.789 | 2.328
0.057 | 620.0 | 1409.5 | 0.0490 | -3.249 | 6.656 | 0.632 | 1.316 | -20.045 | 4.979 | 1.075 | 1.765
0.057 | 660.0 | 1435.9 | 0.0460 | 2.172 | 7.360 | 0.474 | 2.020 | -12.806 | 5.690 | 0.642 | 2.699
0.057 | 700.0 | 1461.8 | 0.0434 | 13.961 | 7.318 | 0.966 | 1.970 | -11.075 | 5.491 | 0.483 | 2.133
0.057 | 740.0 | 1487.3 | 0.0410 | 14.570 | 7.427 | 1.118 | 2.889 | -20.931 | 5.619 | 0.900 | 1.405
0.057 | 780.0 | 1512.3 | 0.0389 | 3.221 | 7.617 | 0.675 | 2.554 | -16.318 | 5.599 | 0.791 | 1.357
0.057 | 820.0 | 1536.9 | 0.0370 | 5.563 | 7.585 | 0.637 | 1.174 | -14.062 | 5.286 | 0.659 | 0.750
0.057 | 860.0 | 1561.2 | 0.0353 | 0.817 | 7.583 | 0.307 | 0.947 | -12.846 | 5.180 | 0.696 | 0.753
0.057 | 900.0 | 1585.0 | 0.0338 | 4.921 | 7.582 | 0.588 | 1.167 | -17.092 | 5.121 | 0.806 | 1.163
0.057 | 940.0 | 1608.5 | 0.0323 | 16.944 | 8.508 | 0.796 | 1.911 | -22.979 | 5.428 | 1.003 | 1.584
0.057 | 980.0 | 1631.7 | 0.0310 | 11.996 | 9.503 | 1.179 | 2.745 | -15.319 | 5.418 | 1.004 | 1.096
0.057 | 1020.0 | 1654.5 | 0.0298 | -23.536 | 9.559 | 1.079 | 1.610 | -9.656 | 5.365 | 1.281 | 0.666
0.057 | 1060.0 | 1677.0 | 0.0287 | -26.072 | 9.871 | 1.414 | 3.111 | -5.177 | 5.518 | 0.649 | 0.900
0.057 | 1100.0 | 1699.3 | 0.0276 | -0.363 | 10.045 | 0.119 | 1.423 | -8.255 | 5.946 | 0.430 | 0.809
0.057 | 1140.0 | 1721.2 | 0.0266 | -1.114 | 10.216 | 0.145 | 0.423 | -13.182 | 6.421 | 0.699 | 0.462
0.057 | 1180.0 | 1742.9 | 0.0257 | 0.422 | 10.433 | 0.162 | 0.878 | -13.824 | 6.438 | 0.720 | 0.435
0.057 | 1220.0 | 1764.3 | 0.0249 | 3.185 | 10.857 | 0.291 | 0.661 | -13.149 | 6.460 | 0.667 | 0.883
0.057 | 1260.0 | 1785.4 | 0.0241 | -0.032 | 11.082 | 0.142 | 0.533 | -9.820 | 6.050 | 0.562 | 0.857
0.057 | 1300.0 | 1806.3 | 0.0234 | 3.530 | 10.636 | 0.393 | 0.546 | -19.872 | 5.708 | 1.107 | 1.144
0.057 | 1340.0 | 1827.0 | 0.0227 | 7.619 | 10.962 | 0.596 | 1.217 | -16.451 | 5.826 | 0.862 | 1.034
0.057 | 1380.0 | 1847.4 | 0.0220 | -3.622 | 11.395 | 0.246 | 0.806 | -15.794 | 6.415 | 0.868 | 1.215
0.057 | 1420.0 | 1867.6 | 0.0214 | -0.549 | 11.656 | 0.357 | 1.057 | -16.219 | 6.442 | 0.876 | 0.915
0.057 | 1460.0 | 1887.6 | 0.0208 | 7.762 | 11.952 | 0.710 | 1.369 | -16.017 | 6.482 | 0.825 | 0.515
0.057 | 1500.0 | 1907.4 | 0.0203 | -10.832 | 11.640 | 0.617 | 0.986 | -13.847 | 6.253 | 0.836 | 0.717
0.057 | 1540.0 | 1927.0 | 0.0197 | -5.680 | 11.515 | 0.254 | 0.371 | -21.179 | 6.064 | 1.181 | 0.650
0.057 | 1580.0 | 1946.3 | 0.0192 | -1.858 | 12.504 |  0.162 |  0.468 | -14.862 | 6.311 |  0.812 |  0.678
---|---|---|---|---|---|---|---|---|---|---|---
0.057 | 1620.0 | 1965.5 | 0.0188 | -5.926 | 12.726 | 0.461 | 0.789 | -14.826 | 6.284 | 0.829 | 0.619
0.057 | 1660.0 | 1984.5 | 0.0183 | -15.284 | 13.152 | 1.151 | 2.330 | -11.164 | 6.518 | 0.793 | 0.594
0.057 | 1700.0 | 2003.4 | 0.0179 | 5.253 | 13.024 | 0.574 | 1.241 | -16.632 | 6.962 | 1.697 | 1.296
0.057 | 1740.0 | 2022.0 | 0.0175 | -2.795 | 12.474 | 0.177 | 0.293 | -21.069 | 6.512 | 1.227 | 0.648
0.057 | 1780.0 | 2040.5 | 0.0171 | -11.946 | 9.985 | 0.612 | 1.899 | -11.633 | 6.119 | 0.893 | 0.878
0.057 | 1820.0 | 2058.8 | 0.0167 | -4.724 | 10.712 | 0.237 | 0.985 | -18.421 | 5.839 | 1.025 | 1.079
0.057 | 1860.0 | 2076.9 | 0.0163 | -11.979 | 13.047 | 0.712 | 2.817 | -4.903 | 7.290 | 0.348 | 0.752
0.079 | 187.5 | 1073.9 | 0.2246 | 34.143 | 18.241 | 3.088 | 7.411 | -64.904 | 24.672 | 5.429 | 10.679
0.079 | 230.0 | 1110.4 | 0.1830 | 17.772 | 12.184 | 3.035 | 3.783 | -92.997 | 15.691 | 4.322 | 5.277
0.079 | 270.0 | 1143.7 | 0.1559 | -2.744 | 8.866 | 3.869 | 4.971 | -111.705 | 10.617 | 4.587 | 5.878
0.079 | 310.0 | 1176.0 | 0.1358 | -12.400 | 7.942 | 5.274 | 5.393 | -154.179 | 9.071 | 6.282 | 6.244
0.079 | 350.0 | 1207.5 | 0.1203 | 11.408 | 8.144 | 6.164 | 5.841 | -177.788 | 9.089 | 7.241 | 6.483
0.079 | 390.0 | 1238.2 | 0.1079 | 34.764 | 7.864 | 6.817 | 4.885 | -157.234 | 8.585 | 7.005 | 4.887
0.079 | 430.0 | 1268.2 | 0.0979 | 33.358 | 7.078 | 4.791 | 2.725 | -105.957 | 7.208 | 4.616 | 2.363
0.079 | 470.0 | 1297.4 | 0.0896 | 22.314 | 6.960 | 2.869 | 1.047 | -66.394 | 6.410 | 3.677 | 1.380
0.079 | 510.0 | 1326.0 | 0.0825 | 18.464 | 6.282 | 1.899 | 1.063 | -36.957 | 5.451 | 2.386 | 1.736
0.079 | 550.0 | 1354.0 | 0.0765 | 13.599 | 5.490 | 1.138 | 0.957 | -17.115 | 4.625 | 0.778 | 1.702
0.079 | 590.0 | 1381.5 | 0.0714 | -5.347 | 6.480 | 0.826 | 1.173 | -11.003 | 5.262 | 0.910 | 1.139
0.079 | 630.0 | 1408.4 | 0.0668 | -9.027 | 6.974 | 0.677 | 2.375 | -17.658 | 5.457 | 1.287 | 1.825
0.079 | 670.0 | 1434.8 | 0.0628 | -3.163 | 7.671 | 0.457 | 2.089 | -15.417 | 5.870 | 0.967 | 2.565
0.079 | 710.0 | 1460.7 | 0.0593 | 6.803 | 7.476 | 0.640 | 2.526 | -7.954 | 5.633 | 0.483 | 2.015
0.079 | 750.0 | 1486.2 | 0.0561 | 14.630 | 7.207 | 1.060 | 2.058 | -18.739 | 5.307 | 1.055 | 1.122
0.079 | 790.0 | 1511.2 | 0.0533 | 9.856 | 7.011 | 0.892 | 1.908 | -16.028 | 4.978 | 1.029 | 1.266
0.079 | 830.0 | 1535.9 | 0.0507 | 9.177 | 6.570 | 0.857 | 1.488 | -13.424 | 4.470 | 1.107 | 0.843
0.079 | 870.0 | 1560.1 | 0.0484 | 0.695 | 6.005 | 0.333 | 1.137 | -9.367 | 4.057 | 0.889 | 0.881
0.079 | 910.0 | 1584.0 | 0.0463 | 5.882 | 5.417 |  0.564 |  0.961 | -12.198 | 3.555 |  1.055 |  0.818
---|---|---|---|---|---|---|---|---|---|---|---
0.079 | 950.0 | 1607.5 | 0.0443 | 11.564 | 6.316 | 4.244 | 0.788 | -13.436 | 4.176 | 1.958 | 1.268
0.079 | 990.0 | 1630.7 | 0.0425 | 12.303 | 10.534 | 1.106 | 2.812 | -12.655 | 5.667 | 0.773 | 1.305
0.079 | 1030.0 | 1653.5 | 0.0409 | -21.372 | 10.231 | 0.955 | 2.364 | -6.450 | 5.427 | 1.054 | 1.287
0.079 | 1070.0 | 1676.1 | 0.0393 | -26.240 | 10.201 | 1.367 | 3.819 | -3.708 | 5.380 | 0.610 | 1.147
0.079 | 1110.0 | 1698.3 | 0.0379 | -2.104 | 10.257 | 0.121 | 1.186 | -7.792 | 5.619 | 0.419 | 1.100
0.079 | 1150.0 | 1720.3 | 0.0366 | 0.965 | 10.455 | 0.200 | 2.524 | -14.070 | 5.978 | 0.712 | 1.339
0.079 | 1190.0 | 1742.0 | 0.0354 | 2.804 | 10.560 | 0.224 | 3.987 | -14.248 | 5.945 | 0.702 | 1.186
0.079 | 1230.0 | 1763.4 | 0.0342 | 5.423 | 10.901 | 0.392 | 1.310 | -13.352 | 5.970 | 0.643 | 2.508
0.079 | 1270.0 | 1784.5 | 0.0331 | 2.133 | 11.226 | 0.318 | 1.516 | -10.841 | 5.678 | 0.639 | 0.856
0.079 | 1310.0 | 1805.4 | 0.0321 | 1.789 | 9.723 | 0.412 | 0.904 | -23.168 | 4.838 | 1.335 | 2.538
0.079 | 1350.0 | 1826.1 | 0.0312 | 5.894 | 9.835 | 0.557 | 1.305 | -19.972 | 4.818 | 1.087 | 2.263
0.079 | 1390.0 | 1846.5 | 0.0303 | 0.030 | 10.081 | 0.415 | 1.938 | -19.555 | 5.131 | 1.049 | 2.583
0.079 | 1430.0 | 1866.8 | 0.0294 | 1.654 | 10.405 | 0.479 | 1.437 | -19.315 | 5.292 | 1.035 | 1.709
0.079 | 1470.0 | 1886.8 | 0.0286 | 6.673 | 10.523 | 0.579 | 1.348 | -16.243 | 5.254 | 0.834 | 0.761
0.079 | 1510.0 | 1906.5 | 0.0279 | -4.686 | 10.171 | 0.596 | 1.309 | -13.888 | 5.054 | 0.809 | 0.828
0.079 | 1550.0 | 1926.1 | 0.0272 | -4.896 | 9.978 | 0.225 | 0.786 | -17.706 | 4.908 | 0.988 | 0.511
0.079 | 1590.0 | 1945.5 | 0.0265 | -0.354 | 10.333 | 0.202 | 0.455 | -14.538 | 4.777 | 0.786 | 0.801
0.079 | 1630.0 | 1964.7 | 0.0258 | -0.897 | 10.428 | 0.442 | 1.642 | -15.400 | 4.767 | 0.838 | 0.931
0.079 | 1670.0 | 1983.7 | 0.0252 | -6.199 | 10.704 | 0.993 | 2.901 | -11.479 | 4.933 | 0.721 | 0.804
0.079 | 1710.0 | 2002.6 | 0.0246 | -1.247 | 10.500 | 0.496 | 1.681 | -16.864 | 5.130 | 1.390 | 1.502
0.079 | 1750.0 | 2021.2 | 0.0241 | -2.692 | 9.746 | 0.187 | 0.367 | -19.326 | 4.745 | 1.113 | 0.728
0.079 | 1790.0 | 2039.7 | 0.0235 | -6.890 | 8.195 | 0.376 | 1.963 | -13.116 | 4.471 | 0.863 | 0.913
0.079 | 1830.0 | 2058.0 | 0.0230 | -3.057 | 9.057 | 0.200 | 0.908 | -14.979 | 4.587 | 0.833 | 0.936
0.079 | 1870.0 | 2076.2 | 0.0225 | -7.990 | 10.082 | 0.521 | 2.570 | -7.945 | 5.094 | 0.497 | 0.747
0.100 | 202.5 | 1077.2 | 0.2632 | 15.456 | 8.022 | 3.954 | 3.871 | -76.998 | 11.594 | 6.669 | 7.076
0.100 | 245.0 | 1113.6 | 0.2175 | 19.384 | 7.195 |  4.030 |  3.331 | -65.123 | 9.686 |  5.755 |  5.130
---|---|---|---|---|---|---|---|---|---|---|---
0.100 | 285.0 | 1146.8 | 0.1870 | 26.126 | 5.169 | 3.594 | 3.967 | -101.549 | 6.431 | 4.671 | 5.009
0.100 | 325.0 | 1179.1 | 0.1640 | 9.181 | 5.267 | 5.300 | 4.472 | -145.831 | 6.295 | 6.330 | 5.300
0.100 | 365.0 | 1210.5 | 0.1460 | 15.938 | 5.842 | 8.753 | 4.948 | -185.663 | 6.703 | 9.835 | 5.651
0.100 | 405.0 | 1241.1 | 0.1316 | 9.510 | 5.673 | 5.847 | 4.029 | -138.361 | 6.112 | 6.834 | 4.482
0.100 | 445.0 | 1271.0 | 0.1198 | 21.309 | 5.085 | 3.680 | 1.449 | -84.572 | 5.254 | 4.195 | 1.821
0.100 | 485.0 | 1300.2 | 0.1099 | 21.822 | 4.835 | 2.374 | 1.142 | -45.411 | 4.819 | 2.345 | 1.578
0.100 | 525.0 | 1328.7 | 0.1015 | 5.873 | 5.348 | 0.828 | 3.050 | -17.056 | 5.058 | 0.906 | 2.513
0.100 | 565.0 | 1356.7 | 0.0943 | 11.819 | 4.088 | 1.243 | 2.111 | -17.313 | 3.736 | 1.101 | 1.720
0.100 | 605.0 | 1384.1 | 0.0881 | 10.628 | 3.602 | 1.331 | 2.131 | -20.914 | 3.192 | 1.446 | 1.485
0.100 | 645.0 | 1410.9 | 0.0826 | 6.652 | 4.049 | 1.041 | 1.367 | -22.418 | 3.365 | 1.561 | 1.095
0.100 | 685.0 | 1437.3 | 0.0778 | 10.810 | 4.650 | 1.263 | 0.900 | -21.760 | 3.758 | 1.737 | 1.250
0.100 | 725.0 | 1463.2 | 0.0735 | 7.994 | 4.797 | 0.884 | 0.550 | -14.369 | 3.857 | 1.624 | 0.831
0.100 | 765.0 | 1488.6 | 0.0697 | 4.631 | 4.649 | 0.766 | 0.824 | -13.315 | 3.650 | 1.297 | 0.902
0.100 | 805.0 | 1513.6 | 0.0662 | 8.032 | 4.672 | 1.190 | 0.631 | -12.766 | 3.494 | 1.563 | 0.877
0.100 | 845.0 | 1538.2 | 0.0631 | 6.553 | 4.473 | 1.124 | 0.330 | -11.851 | 3.606 | 1.709 | 0.640
0.100 | 885.0 | 1562.4 | 0.0602 | 4.063 | 4.603 | 0.570 | 0.354 | -7.306 | 3.889 | 1.272 | 0.615
0.100 | 925.0 | 1586.2 | 0.0576 | 3.901 | 4.887 | 0.682 | 0.339 | -7.039 | 3.701 | 1.351 | 0.521
0.100 | 965.0 | 1609.7 | 0.0552 | 7.498 | 10.407 | 9.152 | 0.687 | -3.231 | 7.243 | 3.668 | 0.847
0.100 | 1005.0 | 1632.9 | 0.0530 | 8.199 | 6.938 | 0.583 | 0.580 | -8.522 | 4.879 | 0.443 | 0.958
0.100 | 1045.0 | 1655.7 | 0.0510 | 3.330 | 6.326 | 1.012 | 2.855 | -19.256 | 4.404 | 2.984 | 1.133
0.100 | 1085.0 | 1678.2 | 0.0491 | -5.996 | 6.643 | 1.041 | 3.084 | -13.311 | 4.487 | 1.578 | 1.626
0.100 | 1125.0 | 1700.4 | 0.0474 | 5.958 | 5.880 | 0.558 | 0.716 | -14.852 | 3.817 | 0.716 | 0.856
0.100 | 1165.0 | 1722.4 | 0.0457 | 5.983 | 6.420 | 0.522 | 3.161 | -21.348 | 3.982 | 1.217 | 1.772
0.100 | 1205.0 | 1744.0 | 0.0442 | -1.774 | 6.282 | 0.583 | 2.308 | -19.196 | 3.834 | 2.291 | 1.536
0.100 | 1245.0 | 1765.4 | 0.0428 | 0.367 | 6.886 | 0.344 | 0.767 | -16.672 | 4.099 | 1.209 | 2.685
0.100 | 1285.0 | 1786.5 | 0.0415 | 8.289 |  6.606 |  0.901 |  2.520 |  -28.115 | 3.851 |  1.665 |  1.862
---|---|---|---|---|---|---|---|---|---|---|---
0.100 | 1325.0 | 1807.4 | 0.0402 | -2.222 | 9.184 | 0.268 | 3.484 | -10.811 | 4.707 | 0.642 | 1.507
0.100 | 1365.0 | 1828.1 | 0.0390 | 1.185 | 9.278 | 0.396 | 1.450 | -9.927 | 4.728 | 0.561 | 0.632
0.100 | 1405.0 | 1848.5 | 0.0379 | -3.194 | 9.330 | 0.228 | 0.502 | -12.630 | 4.945 | 0.778 | 0.957
0.100 | 1445.0 | 1868.7 | 0.0369 | -3.062 | 9.625 | 0.281 | 1.343 | -4.770 | 4.941 | 0.309 | 3.315
0.100 | 1485.0 | 1888.7 | 0.0359 | 1.817 | 9.708 | 0.369 | 1.398 | -10.809 | 4.950 | 0.625 | 0.566
0.100 | 1525.0 | 1908.4 | 0.0349 | -14.830 | 9.162 | 0.684 | 2.861 | -19.837 | 4.816 | 1.353 | 3.762
0.100 | 1565.0 | 1928.0 | 0.0341 | -2.055 | 9.173 | 0.206 | 0.802 | -12.951 | 4.850 | 0.779 | 0.856
0.100 | 1605.0 | 1947.4 | 0.0332 | -4.243 | 9.530 | 0.193 | 1.135 | -9.340 | 4.905 | 0.600 | 0.611
0.100 | 1645.0 | 1966.5 | 0.0324 | 2.988 | 9.658 | 0.687 | 1.475 | -17.478 | 5.030 | 1.073 | 1.227
0.100 | 1685.0 | 1985.5 | 0.0316 | 0.209 | 7.303 | 0.857 | 1.807 | -12.793 | 3.697 | 0.748 | 0.948
0.100 | 1725.0 | 2004.3 | 0.0309 | -6.303 | 7.199 | 0.399 | 1.489 | -17.072 | 3.758 | 1.158 | 0.984
0.100 | 1765.0 | 2023.0 | 0.0302 | -2.988 | 7.012 | 0.190 | 0.278 | -17.894 | 3.704 | 1.000 | 0.489
0.100 | 1805.0 | 2041.5 | 0.0295 | -2.567 | 6.152 | 0.222 | 0.989 | -14.275 | 3.386 | 0.827 | 0.351
0.100 | 1845.0 | 2059.8 | 0.0289 | -2.509 | 7.388 | 0.206 | 0.900 | -12.301 | 3.916 | 0.688 | 0.376
0.100 | 1885.0 | 2077.9 | 0.0283 | -4.084 | 7.543 | 0.436 | 1.750 | -9.504 | 3.958 | 0.560 | 0.562
0.100 | 2005.0 | 2131.4 | 0.0266 | 3.842 | 4.597 | 0.274 | 0.536 | -12.950 | 2.610 | 0.675 | 0.322
0.100 | 2045.0 | 2148.9 | 0.0261 | 3.536 | 4.544 | 0.289 | 0.070 | -12.664 | 2.563 | 0.671 | 0.119
0.100 | 2085.0 | 2166.3 | 0.0256 | 3.312 | 4.672 | 0.275 | 0.081 | -12.602 | 2.601 | 0.669 | 0.123
0.100 | 2125.0 | 2183.6 | 0.0251 | 3.729 | 4.758 | 0.291 | 0.203 | -12.564 | 2.613 | 0.666 | 0.132
0.100 | 2165.0 | 2200.7 | 0.0246 | 3.467 | 6.179 | 0.425 | 1.015 | -12.002 | 3.365 | 0.643 | 0.216
0.100 | 2205.0 | 2217.7 | 0.0242 | -7.082 | 9.019 | 0.380 | 0.589 | -11.031 | 4.931 | 0.664 | 0.175
0.100 | 2245.0 | 2234.6 | 0.0237 | 3.104 | 7.672 | 0.268 | 0.255 | -15.142 | 4.189 | 0.852 | 0.169
0.100 | 2285.0 | 2251.3 | 0.0233 | 1.188 | 7.367 | 0.201 | 0.225 | -14.907 | 4.014 | 0.850 | 0.208
0.100 | 2325.0 | 2267.9 | 0.0229 | 0.464 | 4.258 | 0.150 | 0.231 | -12.881 | 2.418 | 0.662 | 0.364
0.100 | 2365.0 | 2284.4 | 0.0225 | 3.874 | 4.411 | 0.287 | 0.739 | -10.969 | 2.221 | 0.630 | 0.435
0.100 | 2405.0 | 2300.8 | 0.0222 | -3.542 |  4.432 |  0.404 |  1.238 | -11.716 | 2.166 |  0.681 |  0.430
---|---|---|---|---|---|---|---|---|---|---|---
0.100 | 2445.0 | 2317.0 | 0.0218 | -2.553 | 4.665 | 0.329 | 1.321 | -10.259 | 2.300 | 0.588 | 0.452
0.100 | 2485.0 | 2333.2 | 0.0214 | -0.625 | 5.832 | 0.631 | 1.708 | -8.202 | 3.259 | 0.638 | 0.364
0.100 | 2525.0 | 2349.2 | 0.0211 | -4.785 | 5.523 | 0.537 | 1.282 | -12.208 | 3.150 | 0.733 | 0.456
0.100 | 2565.0 | 2365.1 | 0.0208 | -5.088 | 5.431 | 0.310 | 1.000 | -8.565 | 2.863 | 0.495 | 0.953
0.100 | 2605.0 | 2380.9 | 0.0205 | -4.249 | 6.307 | 0.211 | 0.465 | -7.268 | 3.132 | 0.448 | 1.061
0.100 | 2645.0 | 2396.6 | 0.0201 | -1.454 | 6.765 | 0.106 | 0.354 | -12.333 | 3.318 | 0.730 | 0.448
0.100 | 2685.0 | 2412.2 | 0.0198 | -3.477 | 7.573 | 0.214 | 0.650 | -13.003 | 3.659 | 0.759 | 0.300
0.100 | 2725.0 | 2427.7 | 0.0196 | -8.275 | 8.178 | 0.403 | 0.726 | -13.838 | 4.016 | 0.831 | 0.316
0.100 | 2765.0 | 2443.2 | 0.0193 | -5.867 | 7.309 | 0.298 | 0.926 | -15.363 | 4.422 | 0.968 | 1.116
0.100 | 2805.0 | 2458.5 | 0.0190 | -2.355 | 7.693 | 0.229 | 0.668 | -15.399 | 5.141 | 1.072 | 1.665
0.100 | 2845.0 | 2473.7 | 0.0187 | -5.013 | 6.881 | 0.306 | 0.212 | -12.836 | 4.092 | 0.743 | 0.682
0.100 | 2885.0 | 2488.8 | 0.0185 | -4.007 | 7.609 | 0.178 | 1.216 | -15.019 | 4.365 | 0.880 | 1.561
0.150 | 227.5 | 1075.8 | 0.3514 | 3.537 | 6.103 | 9.555 | 8.491 | -61.256 | 10.170 | 11.998 | 11.833
0.150 | 270.0 | 1112.2 | 0.2961 | 12.933 | 5.666 | 2.324 | 5.877 | -50.296 | 8.559 | 3.450 | 9.452
0.150 | 310.0 | 1145.5 | 0.2579 | 25.440 | 4.679 | 2.025 | 7.919 | -94.374 | 6.515 | 3.322 | 9.936
0.150 | 350.0 | 1177.8 | 0.2284 | 10.614 | 4.508 | 3.682 | 6.815 | -143.747 | 5.921 | 4.929 | 8.771
0.150 | 390.0 | 1209.2 | 0.2050 | 4.516 | 4.762 | 5.869 | 6.254 | -172.890 | 5.986 | 7.373 | 7.870
0.150 | 430.0 | 1239.9 | 0.1859 | 5.435 | 5.039 | 4.319 | 4.480 | -131.759 | 6.205 | 5.353 | 5.498
0.150 | 470.0 | 1269.8 | 0.1701 | 9.906 | 4.742 | 3.185 | 1.466 | -84.240 | 5.769 | 4.149 | 1.659
0.150 | 510.0 | 1299.0 | 0.1567 | 8.116 | 4.447 | 2.427 | 0.564 | -49.114 | 5.182 | 3.235 | 0.789
0.150 | 550.0 | 1327.6 | 0.1453 | 7.676 | 3.913 | 1.581 | 0.962 | -28.701 | 4.284 | 2.042 | 1.501
0.150 | 590.0 | 1355.5 | 0.1355 | 8.485 | 3.482 | 1.581 | 0.894 | -20.635 | 3.634 | 1.832 | 1.424
0.150 | 630.0 | 1383.0 | 0.1269 | 8.687 | 3.196 | 1.663 | 0.799 | -18.807 | 3.235 | 2.089 | 1.208
0.150 | 670.0 | 1409.8 | 0.1193 | 8.416 | 3.141 | 1.160 | 1.107 | -16.606 | 3.148 | 1.882 | 1.194
0.150 | 710.0 | 1436.2 | 0.1126 | 6.033 | 3.297 | 0.852 | 0.856 | -14.093 | 3.397 | 1.168 | 1.285
0.150 | 750.0 | 1462.1 | 0.1066 | 7.997 | 3.653 |  0.433 |  0.693 | -10.516 |  3.717 |  0.663 |  1.011
---|---|---|---|---|---|---|---|---|---|---|---
0.150 | 790.0 | 1487.6 | 0.1012 | 9.738 | 4.130 | 0.866 | 1.523 | -8.556 | 3.763 | 1.072 | 1.499
0.150 | 830.0 | 1512.6 | 0.0963 | 2.912 | 4.650 | 1.160 | 2.293 | -3.873 | 3.830 | 0.970 | 2.076
0.150 | 870.0 | 1537.2 | 0.0919 | 3.913 | 4.955 | 1.031 | 1.372 | -5.819 | 4.151 | 1.259 | 1.293
0.150 | 910.0 | 1561.4 | 0.0878 | 1.937 | 5.032 | 0.322 | 1.756 | -2.842 | 4.165 | 0.739 | 2.246
0.150 | 950.0 | 1585.3 | 0.0841 | 0.456 | 4.813 | 0.523 | 2.733 | -6.978 | 3.791 | 0.905 | 1.895
0.150 | 990.0 | 1608.8 | 0.0807 | 14.691 | 8.089 | 1.680 | 1.313 | -16.938 | 5.876 | 1.031 | 1.181
0.150 | 1030.0 | 1631.9 | 0.0776 | 3.467 | 4.850 | 0.282 | 1.390 | -6.773 | 3.744 | 0.312 | 0.817
0.150 | 1070.0 | 1654.8 | 0.0747 | 1.026 | 4.520 | 0.336 | 3.396 | -12.652 | 3.690 | 0.675 | 1.371
0.150 | 1110.0 | 1677.3 | 0.0720 | -9.527 | 4.579 | 0.439 | 0.637 | -8.883 | 3.715 | 0.701 | 0.726
0.150 | 1150.0 | 1699.5 | 0.0695 | -2.496 | 4.546 | 0.339 | 1.135 | -10.505 | 3.456 | 0.620 | 0.495
0.150 | 1190.0 | 1721.5 | 0.0672 | 2.371 | 4.692 | 0.317 | 2.825 | -13.350 | 3.472 | 0.673 | 1.156
0.150 | 1230.0 | 1743.1 | 0.0650 | -1.243 | 4.488 | 0.380 | 1.791 | -13.722 | 3.419 | 1.102 | 0.724
0.150 | 1270.0 | 1764.5 | 0.0629 | 6.832 | 5.027 | 0.665 | 1.436 | -16.786 | 3.863 | 0.914 | 1.377
0.150 | 1310.0 | 1785.7 | 0.0610 | 10.278 | 6.246 | 1.002 | 1.441 | -17.229 | 4.181 | 0.953 | 0.635
0.150 | 1350.0 | 1806.6 | 0.0592 | -5.703 | 6.405 | 0.230 | 1.623 | -6.508 | 4.207 | 0.473 | 0.739
0.150 | 1390.0 | 1827.2 | 0.0575 | -3.262 | 6.414 | 0.275 | 0.385 | -6.172 | 4.219 | 0.424 | 0.589
0.150 | 1430.0 | 1847.7 | 0.0559 | -1.099 | 6.157 | 0.284 | 0.697 | -12.787 | 4.285 | 0.806 | 0.952
0.150 | 1470.0 | 1867.8 | 0.0544 | -4.573 | 7.312 | 0.282 | 0.403 | -0.691 | 4.506 | 0.127 | 0.559
0.150 | 1510.0 | 1887.8 | 0.0529 | -1.400 | 7.457 | 0.157 | 0.385 | -7.804 | 4.531 | 0.507 | 0.342
0.150 | 1550.0 | 1907.6 | 0.0516 | -13.139 | 6.321 | 0.736 | 0.198 | -20.054 | 4.098 | 1.409 | 0.202
0.150 | 1590.0 | 1927.2 | 0.0503 | 2.114 | 6.761 | 0.262 | 0.327 | -9.427 | 4.236 | 0.565 | 0.139
0.150 | 1630.0 | 1946.6 | 0.0490 | -2.797 | 6.089 | 0.247 | 0.319 | -7.940 | 3.750 | 0.528 | 0.207
0.150 | 1670.0 | 1965.8 | 0.0479 | 9.262 | 6.393 | 0.764 | 0.528 | -17.845 | 3.881 | 1.020 | 0.214
0.150 | 1710.0 | 1984.8 | 0.0467 | 3.543 | 4.445 | 0.773 | 2.983 | -12.193 | 2.476 | 0.661 | 0.843
0.150 | 1750.0 | 2003.6 | 0.0457 | -10.549 | 4.368 | 0.358 | 1.964 | -12.722 | 2.418 | 0.789 | 0.593
0.150 | 1790.0 | 2022.2 | 0.0447 | -1.536 | 4.307 |  0.243 |  0.227 | -14.143 |  2.386 |  0.781 |  0.312
---|---|---|---|---|---|---|---|---|---|---|---
0.150 | 1830.0 | 2040.7 | 0.0437 | 3.487 | 3.931 | 0.384 | 0.864 | -14.068 | 2.187 | 0.743 | 0.288
0.150 | 1870.0 | 2059.0 | 0.0427 | 1.752 | 4.319 | 0.330 | 0.782 | -13.281 | 2.276 | 0.725 | 0.529
0.150 | 1910.0 | 2077.2 | 0.0419 | -1.936 | 4.157 | 0.259 | 1.197 | -11.872 | 2.187 | 0.679 | 0.343
0.150 | 1950.0 | 2095.1 | 0.0410 | 3.002 | 4.098 | 0.427 | 1.225 | -10.719 | 2.254 | 0.594 | 0.517
0.150 | 1990.0 | 2113.0 | 0.0402 | -5.769 | 4.182 | 0.278 | 0.932 | -8.489 | 2.467 | 0.518 | 0.451
0.150 | 2030.0 | 2130.7 | 0.0394 | 0.327 | 4.132 | 0.156 | 0.364 | -12.323 | 3.232 | 0.679 | 0.221
0.150 | 2070.0 | 2148.2 | 0.0386 | -0.294 | 4.412 | 0.145 | 0.286 | -12.112 | 3.424 | 0.675 | 0.137
0.150 | 2110.0 | 2165.6 | 0.0379 | -0.836 | 4.747 | 0.136 | 0.261 | -12.090 | 3.619 | 0.679 | 0.139
0.150 | 2150.0 | 2182.9 | 0.0372 | -1.796 | 6.129 | 0.163 | 0.305 | -11.343 | 3.367 | 0.646 | 0.148
0.150 | 2190.0 | 2200.0 | 0.0365 | 7.273 | 6.624 | 0.496 | 0.202 | -11.858 | 3.544 | 0.630 | 0.145
0.150 | 2230.0 | 2217.0 | 0.0358 | 0.350 | 6.497 | 0.130 | 0.355 | -10.022 | 3.422 | 0.565 | 0.160
0.150 | 2270.0 | 2233.9 | 0.0352 | -2.228 | 5.215 | 0.119 | 0.253 | -11.921 | 3.702 | 0.679 | 0.179
0.150 | 2310.0 | 2250.6 | 0.0346 | -1.973 | 5.038 | 0.124 | 0.367 | -12.010 | 3.503 | 0.729 | 0.237
0.150 | 2350.0 | 2267.2 | 0.0340 | -2.392 | 4.845 | 0.146 | 0.486 | -11.864 | 3.313 | 0.873 | 0.550
0.150 | 2390.0 | 2283.7 | 0.0334 | 6.938 | 5.780 | 0.460 | 1.499 | -9.429 | 2.599 | 0.548 | 0.737
0.150 | 2430.0 | 2300.1 | 0.0329 | -7.159 | 5.875 | 0.716 | 2.450 | -9.369 | 2.500 | 0.612 | 0.803
0.150 | 2470.0 | 2316.3 | 0.0324 | -7.662 | 5.669 | 0.419 | 2.665 | -8.457 | 2.455 | 0.526 | 0.718
0.150 | 2510.0 | 2332.5 | 0.0318 | -11.704 | 5.834 | 0.413 | 3.805 | -6.864 | 2.889 | 0.452 | 0.703
0.150 | 2550.0 | 2348.5 | 0.0313 | -12.251 | 5.803 | 0.497 | 3.442 | -8.156 | 2.970 | 0.566 | 0.695
0.150 | 2590.0 | 2364.4 | 0.0309 | -2.773 | 6.027 | 0.516 | 2.832 | -5.731 | 2.812 | 0.359 | 1.226
0.150 | 2630.0 | 2380.3 | 0.0304 | -1.879 | 6.743 | 0.276 | 1.265 | -6.109 | 3.010 | 0.386 | 1.347
0.150 | 2670.0 | 2396.0 | 0.0299 | 0.356 | 6.534 | 0.110 | 0.490 | -11.304 | 2.891 | 0.655 | 0.434
0.150 | 2710.0 | 2411.6 | 0.0295 | -0.429 | 6.377 | 0.185 | 0.460 | -11.109 | 2.753 | 0.643 | 0.483
0.150 | 2750.0 | 2427.1 | 0.0291 | -4.804 | 7.336 | 0.277 | 4.534 | -9.701 | 2.939 | 0.590 | 0.729
0.150 | 2790.0 | 2442.5 | 0.0287 | 0.726 | 6.995 | 0.418 | 2.757 | -12.856 | 3.135 | 0.705 | 2.397
0.150 | 2830.0 | 2457.8 | 0.0282 | 6.860 | 6.432 |  0.509 |  0.653 | -13.259 |  3.183 |  0.774 |  2.246
---|---|---|---|---|---|---|---|---|---|---|---
0.150 | 2870.0 | 2473.1 | 0.0279 | 1.415 | 5.449 | 0.366 | 0.752 | -10.079 | 2.562 | 0.629 | 0.588
0.150 | 2910.0 | 2488.2 | 0.0275 | -3.242 | 5.784 | 0.087 | 1.329 | -7.754 | 2.637 | 0.496 | 0.747
0.200 | 252.5 | 1074.3 | 0.4221 | 13.790 | 3.318 | 10.709 | 7.560 | -38.694 | 6.467 | 14.450 | 11.419
0.200 | 295.0 | 1110.8 | 0.3613 | 9.440 | 3.819 | 1.772 | 3.899 | -67.255 | 6.461 | 2.727 | 7.448
0.200 | 335.0 | 1144.1 | 0.3181 | 6.040 | 3.367 | 1.457 | 4.517 | -93.382 | 5.187 | 2.519 | 7.215
0.200 | 375.0 | 1176.5 | 0.2842 | 2.222 | 3.248 | 2.957 | 5.386 | -135.555 | 4.669 | 4.306 | 7.699
0.200 | 415.0 | 1207.9 | 0.2568 | 1.626 | 3.529 | 3.853 | 5.221 | -150.585 | 4.786 | 5.154 | 6.992
0.200 | 455.0 | 1238.6 | 0.2342 | 6.285 | 3.749 | 3.791 | 3.749 | -132.563 | 4.892 | 4.953 | 4.900
0.200 | 495.0 | 1268.6 | 0.2153 | 12.098 | 3.308 | 2.779 | 1.806 | -83.584 | 4.189 | 3.587 | 2.171
0.200 | 535.0 | 1297.8 | 0.1992 | 7.023 | 4.035 | 1.809 | 1.064 | -48.229 | 4.965 | 2.323 | 1.226
0.200 | 575.0 | 1326.4 | 0.1854 | 6.935 | 5.100 | 1.208 | 0.692 | -30.576 | 6.097 | 1.489 | 0.859
0.200 | 615.0 | 1354.4 | 0.1733 | 7.341 | 4.967 | 1.082 | 0.486 | -24.222 | 5.800 | 1.276 | 0.729
0.200 | 655.0 | 1381.8 | 0.1627 | 4.639 | 4.694 | 0.865 | 1.129 | -18.166 | 5.372 | 1.073 | 1.064
0.200 | 695.0 | 1408.7 | 0.1534 | 3.931 | 4.665 | 0.465 | 1.293 | -9.967 | 5.240 | 0.648 | 1.100
0.200 | 735.0 | 1435.1 | 0.1450 | 3.131 | 4.572 | 0.514 | 0.634 | -13.385 | 5.003 | 0.670 | 0.769
0.200 | 775.0 | 1461.0 | 0.1375 | 8.223 | 4.833 | 0.548 | 0.294 | -11.448 | 4.895 | 0.505 | 0.480
0.200 | 815.0 | 1486.5 | 0.1308 | 5.739 | 4.748 | 0.399 | 0.370 | -6.351 | 4.691 | 0.421 | 0.376
0.200 | 855.0 | 1511.6 | 0.1247 | -6.748 | 4.703 | 0.476 | 0.971 | 0.020 | 4.560 | 0.362 | 0.646
0.200 | 895.0 | 1536.2 | 0.1191 | -2.512 | 4.395 | 0.310 | 1.484 | -2.180 | 4.243 | 0.223 | 0.887
0.200 | 935.0 | 1560.4 | 0.1140 | 1.714 | 3.861 | 0.168 | 0.450 | -5.796 | 3.710 | 0.240 | 0.312
0.200 | 975.0 | 1584.3 | 0.1093 | -2.916 | 3.869 | 0.225 | 0.645 | -3.320 | 3.575 | 0.233 | 0.430
0.200 | 1015.0 | 1607.8 | 0.1050 | 0.873 | 3.603 | 0.197 | 0.730 | -2.289 | 3.234 | 0.138 | 0.436
0.200 | 1055.0 | 1631.0 | 0.1010 | 3.411 | 3.369 | 0.260 | 0.408 | -6.416 | 2.977 | 0.303 | 0.442
0.200 | 1095.0 | 1653.8 | 0.0973 | 0.917 | 3.160 | 0.313 | 0.612 | -8.377 | 2.874 | 0.424 | 0.417
0.200 | 1135.0 | 1676.4 | 0.0939 | -5.114 | 3.164 | 0.227 | 0.571 | -6.155 | 2.851 | 0.423 | 0.278
0.200 | 1175.0 | 1698.6 | 0.0907 | -1.376 | 3.306 |  0.289 |  0.562 | -8.777 |  2.679 |  0.469 |  0.309
---|---|---|---|---|---|---|---|---|---|---|---
0.200 | 1215.0 | 1720.6 | 0.0877 | -0.408 | 3.322 | 0.192 | 0.583 | -7.995 | 2.632 | 0.402 | 0.504
0.200 | 1255.0 | 1742.2 | 0.0849 | -2.934 | 3.316 | 0.269 | 0.652 | -10.711 | 2.603 | 0.594 | 0.281
0.200 | 1295.0 | 1763.7 | 0.0823 | 6.539 | 3.535 | 0.628 | 0.385 | -15.119 | 2.762 | 0.751 | 0.232
0.200 | 1335.0 | 1784.8 | 0.0798 | 4.969 | 4.095 | 0.587 | 0.663 | -14.784 | 2.905 | 0.781 | 0.283
0.200 | 1375.0 | 1805.7 | 0.0775 | 8.241 | 4.053 | 0.696 | 0.874 | -17.368 | 3.007 | 0.870 | 1.131
0.200 | 1415.0 | 1826.4 | 0.0753 | 0.926 | 3.820 | 0.250 | 0.803 | -10.907 | 2.816 | 0.584 | 1.000
0.200 | 1455.0 | 1846.8 | 0.0733 | 3.559 | 3.600 | 0.372 | 0.670 | -12.090 | 2.645 | 0.596 | 0.728
0.200 | 1495.0 | 1867.0 | 0.0713 | 0.530 | 3.654 | 0.316 | 0.729 | -15.627 | 2.638 | 0.827 | 0.344
0.200 | 1535.0 | 1887.0 | 0.0694 | -4.291 | 3.955 | 0.254 | 0.595 | -10.485 | 2.692 | 0.618 | 0.387
0.200 | 1575.0 | 1906.8 | 0.0677 | 3.464 | 4.031 | 0.439 | 0.637 | -10.308 | 2.640 | 0.540 | 0.265
0.200 | 1615.0 | 1926.4 | 0.0660 | 8.330 | 3.991 | 0.567 | 0.499 | -10.810 | 2.576 | 0.524 | 0.186
0.200 | 1655.0 | 1945.8 | 0.0644 | 3.729 | 4.024 | 0.344 | 0.748 | -10.536 | 2.561 | 0.545 | 0.262
0.200 | 1695.0 | 1965.0 | 0.0629 | 6.435 | 4.111 | 0.493 | 1.400 | -12.606 | 2.557 | 0.635 | 0.403
0.200 | 1735.0 | 1984.0 | 0.0614 | -0.882 | 4.177 | 0.696 | 2.282 | -9.990 | 2.533 | 0.611 | 0.659
0.200 | 1775.0 | 2002.8 | 0.0600 | -8.126 | 4.105 | 0.369 | 1.713 | -8.385 | 2.429 | 0.567 | 0.547
0.200 | 1815.0 | 2021.5 | 0.0587 | 0.684 | 4.115 | 0.317 | 0.175 | -11.252 | 2.415 | 0.613 | 0.357
0.200 | 1855.0 | 2039.9 | 0.0575 | 4.958 | 4.019 | 0.557 | 0.802 | -12.005 | 2.325 | 0.633 | 0.385
0.200 | 1895.0 | 2058.2 | 0.0562 | 3.502 | 3.838 | 0.377 | 1.037 | -13.349 | 2.130 | 0.708 | 0.566
0.200 | 1935.0 | 2076.4 | 0.0551 | 0.067 | 3.829 | 0.229 | 0.877 | -11.955 | 2.073 | 0.659 | 0.348
0.200 | 1975.0 | 2094.4 | 0.0540 | 1.574 | 3.852 | 0.309 | 0.829 | -12.275 | 2.045 | 0.673 | 0.556
0.200 | 2015.0 | 2112.2 | 0.0529 | -3.076 | 3.872 | 0.163 | 0.606 | -9.834 | 2.166 | 0.562 | 0.458
0.200 | 2055.0 | 2129.9 | 0.0519 | -0.166 | 3.756 | 0.144 | 0.447 | -10.504 | 2.661 | 0.580 | 0.251
0.200 | 2095.0 | 2147.5 | 0.0509 | 0.521 | 3.764 | 0.190 | 0.628 | -12.039 | 2.622 | 0.655 | 0.304
0.200 | 2135.0 | 2164.9 | 0.0499 | -0.129 | 3.769 | 0.174 | 0.766 | -12.558 | 2.555 | 0.687 | 0.374
0.200 | 2175.0 | 2182.2 | 0.0490 | -2.033 | 4.404 | 0.130 | 0.701 | -10.399 | 2.292 | 0.585 | 0.248
0.200 | 2215.0 | 2199.3 | 0.0481 | 3.207 | 4.469 |  0.320 |  0.843 | -10.564 |  2.364 |  0.558 |  0.220
---|---|---|---|---|---|---|---|---|---|---|---
0.200 | 2255.0 | 2216.3 | 0.0473 | -1.724 | 4.238 | 0.166 | 1.127 | -8.273 | 2.255 | 0.465 | 0.276
0.200 | 2295.0 | 2233.2 | 0.0464 | -4.587 | 3.704 | 0.186 | 0.530 | -8.809 | 2.296 | 0.508 | 0.191
0.200 | 2335.0 | 2249.9 | 0.0456 | -0.694 | 3.681 | 0.140 | 0.571 | -8.788 | 2.071 | 0.722 | 0.258
0.200 | 2375.0 | 2266.5 | 0.0449 | -1.545 | 4.021 | 0.248 | 0.394 | -7.649 | 2.309 | 1.340 | 0.537
0.200 | 2415.0 | 2283.0 | 0.0441 | 1.542 | 4.321 | 0.191 | 0.822 | -10.287 | 2.297 | 0.533 | 0.464
0.200 | 2455.0 | 2299.4 | 0.0434 | 0.391 | 5.027 | 0.110 | 0.113 | -8.194 | 2.753 | 0.445 | 0.151
0.200 | 2495.0 | 2315.7 | 0.0427 | 3.748 | 4.711 | 0.295 | 0.214 | -9.663 | 2.251 | 0.516 | 0.150
0.200 | 2535.0 | 2331.8 | 0.0420 | -3.462 | 5.315 | 0.176 | 0.098 | -8.667 | 2.675 | 0.512 | 0.156
0.200 | 2575.0 | 2347.9 | 0.0414 | -4.257 | 5.050 | 0.185 | 0.200 | -8.722 | 2.502 | 0.549 | 0.173
0.200 | 2615.0 | 2363.8 | 0.0408 | 8.723 | 4.601 | 0.583 | 0.728 | -8.317 | 2.103 | 0.429 | 0.275
0.200 | 2655.0 | 2379.6 | 0.0401 | 3.937 | 5.560 | 0.269 | 0.943 | -9.021 | 2.691 | 0.494 | 0.336
0.240 | 277.5 | 1077.6 | 0.4609 | 14.650 | 4.723 | 4.956 | 2.176 | -58.830 | 9.270 | 11.761 | 6.767
0.240 | 320.0 | 1113.9 | 0.3997 | 12.332 | 5.068 | 3.379 | 2.528 | -70.081 | 8.721 | 6.884 | 5.335
0.240 | 360.0 | 1147.1 | 0.3553 | 7.192 | 3.694 | 2.159 | 2.890 | -88.335 | 5.967 | 3.823 | 4.975
0.240 | 400.0 | 1179.4 | 0.3197 | 2.021 | 3.783 | 3.169 | 3.019 | -120.832 | 5.717 | 4.758 | 4.737
0.240 | 440.0 | 1210.8 | 0.2907 | 4.501 | 3.909 | 3.769 | 3.340 | -137.414 | 5.594 | 5.342 | 4.805
0.240 | 480.0 | 1241.4 | 0.2664 | 6.774 | 3.906 | 3.661 | 3.144 | -128.423 | 5.340 | 4.992 | 4.292
0.240 | 520.0 | 1271.3 | 0.2460 | 14.674 | 3.778 | 2.474 | 1.586 | -80.133 | 4.966 | 3.240 | 2.040
0.240 | 560.0 | 1300.5 | 0.2284 | 7.420 | 4.538 | 1.556 | 0.935 | -49.234 | 5.786 | 1.970 | 1.101
0.240 | 600.0 | 1329.0 | 0.2132 | 7.277 | 5.292 | 1.062 | 0.343 | -31.454 | 6.538 | 1.279 | 0.407
0.240 | 640.0 | 1357.0 | 0.1998 | 7.134 | 4.791 | 0.877 | 0.302 | -25.574 | 5.755 | 1.026 | 0.413
0.240 | 680.0 | 1384.3 | 0.1881 | 5.395 | 4.392 | 0.649 | 0.732 | -19.033 | 5.145 | 0.738 | 0.730
0.240 | 720.0 | 1411.2 | 0.1776 | 5.324 | 4.210 | 0.398 | 0.796 | -8.635 | 4.816 | 0.359 | 0.743
0.240 | 760.0 | 1437.5 | 0.1683 | 5.075 | 3.738 | 0.433 | 0.466 | -9.629 | 4.143 | 0.443 | 0.651
0.240 | 800.0 | 1463.4 | 0.1599 | 11.109 | 3.709 | 0.816 | 0.378 | -9.196 | 3.833 | 0.620 | 0.465
0.240 | 840.0 | 1488.8 | 0.1523 | 8.491 | 3.343 |  0.508 |  0.572 | -3.113 |  3.381 |  0.434 |  0.539
---|---|---|---|---|---|---|---|---|---|---|---
0.240 | 880.0 | 1513.8 | 0.1453 | -1.176 | 3.047 | 0.373 | 1.171 | -0.789 | 3.030 | 0.312 | 0.940
0.240 | 920.0 | 1538.4 | 0.1390 | 0.851 | 2.695 | 0.282 | 0.745 | -6.161 | 2.672 | 0.281 | 0.580
0.240 | 960.0 | 1562.6 | 0.1332 | 0.765 | 2.565 | 0.229 | 0.344 | -7.973 | 2.506 | 0.365 | 0.296
0.240 | 1000.0 | 1586.5 | 0.1279 | 2.504 | 2.841 | 0.333 | 0.620 | -4.932 | 2.610 | 0.266 | 0.657
0.240 | 1040.0 | 1610.0 | 0.1230 | 7.956 | 3.699 | 0.488 | 0.925 | -5.591 | 3.240 | 0.376 | 0.993
0.240 | 1080.0 | 1633.1 | 0.1184 | 8.457 | 3.690 | 0.610 | 1.337 | -10.510 | 3.174 | 0.581 | 0.804
0.240 | 1120.0 | 1655.9 | 0.1142 | 7.155 | 3.552 | 0.714 | 1.814 | -8.929 | 2.983 | 0.686 | 0.918
0.240 | 1160.0 | 1678.4 | 0.1103 | -2.486 | 3.677 | 0.271 | 1.572 | -4.796 | 3.045 | 0.353 | 0.780
0.240 | 1200.0 | 1700.7 | 0.1066 | 3.012 | 4.048 | 0.286 | 0.734 | -8.299 | 3.277 | 0.380 | 0.531
0.240 | 1240.0 | 1722.6 | 0.1031 | -0.472 | 4.318 | 0.124 | 0.329 | -4.527 | 3.370 | 0.244 | 0.461
0.240 | 1280.0 | 1744.2 | 0.0999 | -2.042 | 4.252 | 0.242 | 0.593 | -8.851 | 3.259 | 0.499 | 0.351
0.240 | 1320.0 | 1765.6 | 0.0969 | 5.330 | 4.242 | 0.478 | 0.522 | -11.534 | 3.199 | 0.539 | 0.244
0.240 | 1360.0 | 1786.7 | 0.0940 | 1.181 | 4.249 | 0.345 | 0.447 | -12.226 | 3.152 | 0.633 | 0.312
0.240 | 1400.0 | 1807.6 | 0.0914 | 10.185 | 4.394 | 0.741 | 1.519 | -13.575 | 3.332 | 0.628 | 0.599
0.240 | 1440.0 | 1828.3 | 0.0888 | 1.459 | 4.292 | 0.188 | 0.128 | -7.033 | 3.187 | 0.365 | 0.150
0.240 | 1480.0 | 1848.7 | 0.0864 | 2.829 | 4.054 | 0.323 | 0.300 | -8.625 | 2.967 | 0.439 | 0.308
0.240 | 1520.0 | 1868.9 | 0.0841 | -1.767 | 3.848 | 0.250 | 0.831 | -13.948 | 2.755 | 0.781 | 1.197
0.240 | 1560.0 | 1888.8 | 0.0820 | -4.030 | 4.057 | 0.270 | 0.714 | -9.832 | 2.685 | 0.603 | 0.357
0.240 | 1600.0 | 1908.6 | 0.0799 | 5.364 | 3.852 | 0.389 | 1.677 | -9.721 | 2.455 | 0.478 | 1.420
0.240 | 1640.0 | 1928.2 | 0.0780 | 6.050 | 3.600 | 0.506 | 1.492 | -10.500 | 2.293 | 0.543 | 0.503
0.240 | 1680.0 | 1947.5 | 0.0761 | 3.945 | 3.470 | 0.354 | 0.943 | -10.827 | 2.190 | 0.613 | 0.329
0.240 | 1720.0 | 1966.7 | 0.0744 | 3.671 | 3.498 | 0.356 | 1.263 | -10.053 | 2.174 | 0.515 | 0.443
0.240 | 1760.0 | 1985.7 | 0.0727 | -0.841 | 3.164 | 0.523 | 1.685 | -8.908 | 1.963 | 0.560 | 0.538
0.240 | 1800.0 | 2004.5 | 0.0711 | -2.307 | 3.086 | 0.324 | 1.077 | -8.801 | 1.749 | 0.512 | 0.415
0.240 | 1840.0 | 2023.2 | 0.0695 | 3.159 | 3.330 | 0.376 | 0.155 | -10.280 | 2.061 | 0.531 | 0.255

0.240 | 1880.0 | 2041.6 | 0.0680 | 2.450 | 3.275 |  0.338 |  0.899 | -10.236 |  2.031 |  0.549 |  0.319
---|---|---|---|---|---|---|---|---|---|---|---
0.240 | 1920.0 | 2059.9 | 0.0666 | 0.830 | 3.171 | 0.200 | 0.731 | -9.732 | 1.944 | 0.521 | 0.482
0.240 | 1960.0 | 2078.1 | 0.0653 | 3.365 | 3.127 | 0.333 | 0.535 | -11.821 | 1.916 | 0.607 | 0.291
0.240 | 2000.0 | 2096.1 | 0.0639 | 0.102 | 2.949 | 0.204 | 0.575 | -11.153 | 1.617 | 0.600 | 0.391
0.240 | 2040.0 | 2113.9 | 0.0627 | -1.216 | 2.907 | 0.134 | 0.449 | -9.477 | 1.765 | 0.520 | 0.254
0.240 | 2080.0 | 2131.6 | 0.0615 | 1.361 | 3.078 | 0.192 | 0.110 | -8.447 | 1.934 | 0.440 | 0.203
0.240 | 2120.0 | 2149.1 | 0.0603 | 3.651 | 3.059 | 0.350 | 0.350 | -13.124 | 1.902 | 0.668 | 0.527
0.240 | 2160.0 | 2166.5 | 0.0592 | 1.622 | 3.051 | 0.244 | 0.600 | -12.831 | 1.743 | 0.668 | 0.624
0.240 | 2200.0 | 2183.8 | 0.0581 | -0.408 | 3.305 | 0.128 | 0.444 | -10.021 | 1.960 | 0.535 | 0.308
0.240 | 2240.0 | 2200.9 | 0.0571 | -1.516 | 3.507 | 0.153 | 0.317 | -8.893 | 2.203 | 0.486 | 0.206
0.240 | 2280.0 | 2217.8 | 0.0561 | -5.569 | 3.579 | 0.245 | 0.334 | -6.705 | 2.221 | 0.405 | 0.214
0.240 | 2320.0 | 2234.7 | 0.0551 | -4.849 | 3.695 | 0.278 | 0.720 | -7.365 | 2.189 | 0.426 | 0.198

Table 4: Table of $\sigma_{\mathrm{LT}}(\nu,Q^{2})$ and
$\sigma_{\mathrm{TT}}(\nu,Q^{2})$ on 3He. From left to right: Four-momentum
transfer squared; energy transfer $\nu$; Invariant mass $W$; Bjorken scaling
variable $x$; Cross-section, statistical uncertainty, uncorrelated systematic
uncertainty, and correlated uncertainty for $\sigma_{\mathrm{LT}}$ and
$\sigma_{\mathrm{TT}}$, respectively.


Assume $a\in V^{\prime}\setminus\text{\rm Leaf }(T)$. We choose $a$ as the
root and denote the partial order of $T$, if by $\succeq$. Let $n$ be the
degree of $a$ and $a_{1},a_{2},\ldots a_{n}$ be the neighbors of $a$. We
define trees $T_{j}=(V_{j},E_{j},W_{j})$ and $V_{j}^{\prime}\subset V_{j}$,
for $j=1,2,\ldots n$ by

$V_{j}=\\{x\in V:x\succeq
a_{j}\\}\cup\\{a\\},E_{j}=E\cap[V_{j}]^{2},W_{j}=W|_{E_{j}}\text{ and
}V_{j}^{\prime}=V^{\prime}\cap V_{j}$

Then the cardinality of $V_{j}^{\prime}\setminus\text{\rm Leaf }(T_{j})$ is
strictly smaller than the cardinality of $V^{\prime}\setminus\text{\rm Leaf
}(T)$, and we can apply the inductive hypothesis to obtain for each
$j=1,2\ldots n$ a tree
$T^{\prime}_{j}=(V_{j}^{\prime},E_{j}^{\prime},W^{\prime}_{j})$ satisfying
(44).

Thus since the sets $V_{j}^{\prime}\setminus\\{a\\}$, are pair wise disjoint
and $a\in V_{j}$, for $j=1,2\ldots,n$
$T^{\prime}=(V^{\prime},E^{\prime},W^{\prime})$, with
$E^{\prime}=\cup_{j=1}^{n}E^{\prime}_{1}$,
$W^{\prime}:E^{\prime}\to(0,\infty),e\mapsto W_{j}^{\prime}(e)$, if $e\in
E^{\prime}_{j}$ is also a tree. Since every path from an element in $V_{i}$ to
and element in $V_{j}$ has to contain $a$ it follows that
$T^{\prime}=(V^{\prime},E^{\prime},W^{\prime})$ satisfies (44). ∎

Using Theorem 6.2 and Corollary 6.5 we obtain the following results

######  Corollary 6.7.

Assume that $(M,d)$ embeds $D$-stochastically into trees, then $(M,d)$ embeds
bijectively $8D$-stochastically int trees.

Thus there are $n$, so that for $i=1,2,\ldots$ there is a set
$E_{i}\subset[M]^{2}$, a metric $d_{i}$, and numbers $p_{i}\in(0,1]$, so that
$\sum_{i=1}^{n}p_{i}=1$, $T_{i}=(M,E_{i})$ is a tree, and $d_{i}$ a geodesic
distance on $M$ with respect to $T_{i}$ and so that for all $x,y\in M$

(49) $\displaystyle d(x,y)\leq d_{i}(x,y),\text{ for $i=1,2,\ldots,n$,}$ (50)
$\displaystyle\sum_{j=1}^{n}p_{i}d_{i}(x,y)\leq Dd(x,y).$

Moreover we can assume that for $i=1,2,\ldots n$ actually $d_{i}$ is the
geodesic metric generated by the weight function

$w_{i}:E_{i}\to[0,\infty),\quad\text{ with }w_{i}(e)=d(u,v)\text{ if
}e=\\{u,v\\}\in E_{i}.$

###### Proof.

The existence of $n$ sets $E_{i}\subset[M]^{2}$, a metrics $d_{i}$, and
numbers $p_{i}\in(0,1]$, for $i=1,2,\ldots,n$ follows from Theorem 6.2 and
Corollary 6.5. To see the moreover part, let us denote for $i=1,2,\ldots,n$
the geodesic distance on $M$ generated by the above defined weight function
$w_{i}$ by $\tilde{d}_{i}$. From the triangle inequality, it follows that
$\tilde{d}_{i}(x,y)\geq d(x,y)$. We note that from (49) it follows that for
every $i=1,2,\ldots,n$, any $e=\\{u,v\\}\in E_{i}$, we have that $d(u,v)\leq
d_{i}(u,v)$, and thus $d(x,y)\leq\tilde{d}_{i}(x,y)\leq d_{i}(x,y)$, for any
$x,y\in M$, which implies (50). ∎

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