MILC111FNAL-MILC [119] and MILC [120] only quote the ratios $m_{u}/m_{d}=0.4556^{(131)}_{(93)}$ and $m_{u}/m_{d}=0.4529^{(157)}_{(82)}$, respectively (all errors added in quadrature). Using the FLAG 21 [77] average $m_{\ell}=3.410(43)$, we combine these results to form $\delta_{m}=2m_{\ell}(m_{u}/m_{d}-1)/(m_{u}/m_{d}+1)$ and compute the error by error propagation. [120] | 2+1+1 | $-2.57^{(11)}_{(6)}$
BMWc [113] | 2+1 | $-2.41(12)$
This work | 2+1 | $-2.03(12)$

We can compare our value of $\delta_{m}$ with results from the literature in
Table 14. This includes the $N_{f}=2+1$ result of BMWc [113] and the
$N_{f}=2+1+1$ continuum limit results of RM123 [118], FNAL-MILC [119] and MILC
[120]. In the latter two cases we convert results for $m_{u}/m_{d}$ into
$\delta_{m}$ as described in the table caption. We see a tension between the
previous determinations and our result on the two to three $\sigma$ level.

We remark that all the previous results utilize the dependence of the pion and
kaon masses on the quark masses and the electromagnetic coupling. We consider
our method of determining the quark mass splitting from the scalar coupling
$g^{\Sigma}_{S}$ and the mass difference between the $\Sigma^{+}$ and the
$\Sigma^{-}$ baryons as more direct. In Ref. [121] $\Delta
m_{N}^{\text{QCD}}=-1.87(16)\,\text{MeV}$ (which agrees within errors with
lattice determinations, including ours, see below) is determined from
experimental input. A larger (negative) QCD difference would require a larger
QED contribution to the proton mass. As discussed above, the QED contribution
to the mass of the $\Sigma^{+}$ baryon is $0.77(5)\,\text{MeV}$ (similar in
size to $\Delta m_{N}^{\text{QED}}=0.58(16)\,\text{MeV}$ [121]) and it would
be surprising if this increased when replacing a strange quark by a down
quark. Assuming $\Delta m_{N}^{\text{QCD}}=-1.87(16)\,\text{MeV}$ and a value
$\delta_{m}\approx-2.50(10)\,\text{MeV}$ as suggested by Refs. [113, 118, 119,
120] would require a coupling $g_{S}^{N}=0.75(7)$ to satisfy the CVC relation
Eq. (60). This in turn is hard to reconcile with the majority of lattice
results compiled in Fig. 28. With a lower value for $|\delta_{m}|$ (and/or a
larger $|\Delta m_{N}^{\text{QCD}}|$) this inconsistency disappears.

###  V.4 QCD and QED isospin breaking effects on the baryon masses

We proceed to compute the QED contributions to the proton and $\Xi^{-}$
masses, $\Delta m_{N}^{\text{QED}}$ and $-\Delta m_{\Xi}^{\text{QED}}$:

$\displaystyle\Delta m^{\text{QED}}_{N}$ $\displaystyle=\Delta
m_{N}-g_{S}^{N}(m_{u}-m_{d})$ $\displaystyle=\Delta
m_{N}-\frac{2g_{S}^{N}}{g_{S}^{\Sigma}}\Delta m_{\Sigma}^{\text{QCD}},$ (62)
$\displaystyle\Delta m^{\text{QED}}_{\Xi}$ $\displaystyle=\Delta
m_{\Xi}-\frac{2g_{S}^{\Xi}}{g_{S}^{\Sigma}}\Delta m_{\Sigma}^{\text{QCD}}.$
(63)

This gives

$\displaystyle\Delta m_{N}^{\text{QED}}$
$\displaystyle=0.97^{(31)}_{(36)}\,\text{MeV},$
$\displaystyle\\!\\!\\!\\!\Delta m_{N}^{\text{QCD}}$
$\displaystyle=-2.26^{(31)}_{(36)}\,\text{MeV},$ (64) $\displaystyle\Delta
m_{\Sigma}^{\text{QED}}$ $\displaystyle=0.77(05)\,\text{MeV},$
$\displaystyle\\!\\!\\!\\!\Delta m_{\Sigma}^{\text{QCD}}$
$\displaystyle=-4.04(04)\,\text{MeV},$ (65) $\displaystyle\Delta
m_{\Xi}^{\text{QED}}$ $\displaystyle=-1.65^{(37)}_{(39)}\,\text{MeV},$
$\displaystyle\\!\\!\\!\\!\Delta m_{\Xi}^{\text{QCD}}$
$\displaystyle=-5.20^{(42)}_{(44)}\,\text{MeV}.$ (66)

For completeness we included the values for the $\Sigma$ baryons that we
determined from the experimental masses alone, without lattice input. The
above mass splittings agree with the BMWc [114] continuum limit results from
simulations of QCD plus QED, see Fig. 35 (errors added in quadrature).
Nevertheless, as mentioned above, the value of $\delta_{m}$, reported by BMWc
[113] from simulations of QCD with quenched QED, differs by 2.2 standard
deviations from our result in Eq. (61). Also other lattice results on the QCD
contribution to the mass-splittings (summarized in Fig. 35), obtained at a
single lattice spacing from Endres et al. [122] using $\text{QED}_{\text{TL}}$
and $\text{QED}_{\text{M}}$, Brantley et al. [123] and CSSM-QCDSF-UKQCD [124]
agree within errors.

Figure 35: Comparison of the QCD contributions to isospin mass splittings.
Note that in our normalization $m_{\Sigma^{+}}-m_{\Sigma^{-}}\approx 2\Delta
m_{\Sigma}^{\text{QCD}}$.

We mention the possibility of an enhancement of the (higher order)
$\delta_{m}^{2}/\Lambda_{\text{QCD}}$ correction to the $\Sigma^{0}$ mass due
to the possibility of mixing with the $\Lambda^{0}$, which, however, appears
to be a very small effect [125]. A positive contribution to the $\Sigma^{0}$
mass would increase $\Delta m_{\Sigma}^{\text{QED}}$ but leave $\Delta
m_{\Sigma}^{\text{QCD}}$ (and therefore the quark mass difference Eq. (61))
invariant.

The electromagnetic contributions to the $p$, $\Sigma^{\pm}$ and $\Xi^{-}$
masses are all similar to $1\,\text{MeV}$, with an enhancement for the
heavier, more compact cascade baryon. Recently, combining the Cottingham
formula [115] with experimental input from elastic scattering and parton
distribution functions, the value $\Delta
m_{N}^{\text{QED}}=0.58(16)\,\text{MeV}$ was determined in Ref. [121]. While
within errors our result Eq. (64) agrees with this value, the number obtained
in Ref. [121] is more inline with the suggested ordering $\Delta
m_{N}^{\text{QED}}<\Delta m_{\Sigma}^{\text{QED}}<-\Delta
m_{\Xi}^{\text{QED}}$. Combining their value with our determination of
$g_{S}^{N}$ gives $\delta_{m}=-1.69^{(53)}_{(51)}\text{MeV}$, in agreement
with our result Eq. (61) but also in tension with, e.g.,
$\delta_{m}=-2.41(12)\,\text{MeV}$ [113].

We find that the effect of $m_{d}>m_{u}$ on the $\Xi$ and $\Sigma$ mass
splittings is much bigger than for the nucleon since this is proportional to
$g_{S}^{B}/g_{V}^{B}$ and $g_{S}^{N}<g_{S}^{\Sigma}/2<g_{S}^{\Xi}$. This
hierarchy is due to $g_{S}^{N}\approx F_{S}+D_{S}$, $g_{S}^{\Sigma}/2\approx
F_{S}$ and $g_{S}^{\Xi}\approx F_{S}-D_{S}$ with $F_{S}>0$ and $D_{S}<0$.
Interestingly, the pion baryon $\sigma$ terms $\sigma_{\pi
B}=\sigma_{uB}+\sigma_{dB}$ that encode the up plus down quark mass
contribution to the baryon masses exhibit the opposite ordering [17],
$\sigma_{\pi N}>\sigma_{\pi\Sigma}>\sigma_{\pi\Xi}$.

###  V.5 Isospin breaking effects on the pion baryon $\sigma$ terms

Having determined the quark mass differences, we can also compute the leading
isospin violating corrections to the pion baryon $\sigma$ terms $\sigma_{\pi
B}=\sigma_{uB}+\sigma_{dB}$. One can either work with matrix elements [126],
using the identity

$m_{u}\bar{u}u+m_{d}\bar{d}d=m_{\ell}\left(\bar{u}u+\bar{d}{d}\right)+\frac{\delta_{m}}{2}\left(\bar{u}u-\bar{d}d\right),$
(67)

or one can start from the Feynman-Hellmann theorem

$\sigma_{qB}=m_{q}\frac{\langle B|\bar{q}q|B\rangle}{\langle
B|B\rangle}=m_{q}\frac{\partial m_{B}}{\partial m_{q}}.$ (68)

Writing $m_{p}=m_{N}+\Delta m_{N}^{\text{QCD}}/2+\Delta m_{N}^{\text{QED}}$
and $m_{n}=m_{N}-\Delta m_{N}^{\text{QCD}}/2$, where $\Delta
m_{N}^{\text{QCD}}=\delta_{m}g_{S}^{N}/g_{V}^{N}$, and realizing that the
dependence of the QED contributions on the quark masses is of higher order in
the isospin breaking, we obtain at linear order

$\displaystyle\sigma_{\pi p}$ $\displaystyle=m_{u}\frac{\partial
m_{p}}{\partial m_{u}}+m_{d}\frac{\partial m_{p}}{\partial m_{d}}=\sigma_{\pi
N}+\frac{1}{2}\Delta m_{N}^{\text{QCD}},$ (69) $\displaystyle\sigma_{\pi n}$
$\displaystyle=m_{u}\frac{\partial m_{n}}{\partial m_{u}}+m_{d}\frac{\partial
m_{n}}{\partial m_{d}}=\sigma_{\pi N}-\frac{1}{2}\Delta m_{N}^{\text{QCD}}.$
(70)

The same can be carried out for the $\Sigma^{\pm}$, $\Xi^{0}$ and $\Xi^{-}$
baryons. Using the results for the $\sigma$ terms of the isosymmetric theory
of Ref. [17], we obtain

$\displaystyle\sigma_{\pi p}$
$\displaystyle=42.8^{(4.7)}_{(4.7)}\,\text{MeV},$ $\displaystyle\sigma_{\pi
n}$ $\displaystyle=45.0^{(4.7)}_{(4.7)}\,\text{MeV},$ (71)
$\displaystyle\sigma_{\pi\Sigma^{+}}$
$\displaystyle=21.9^{(3.8)}_{(6.1)}\,\text{MeV},$
$\displaystyle\sigma_{\pi\Sigma^{-}}$
$\displaystyle=29.9^{(3.8)}_{(6.1)}\,\text{MeV},$ (72)
$\displaystyle\sigma_{\pi\Xi^{0}}$
$\displaystyle=8.6^{(4.5)}_{(6.4)}\,\text{MeV},$
$\displaystyle\sigma_{\pi\Xi^{-}}$
$\displaystyle=13.8^{(4.5)}_{(6.4)}\,\text{MeV},$ (73)

whereas the pion $\sigma$ term for the $\Sigma^{0}$ is not affected at linear
order:
$\sigma_{\pi\Sigma^{0}}\approx\sigma_{\pi\Sigma}=25.9^{(3.8)}_{(6.1)}\,\text{MeV}$.
We refrain from further decomposing the pion baryon $\sigma$ terms into the
individual up and down quark contributions. However, this can easily be
accomplished [126]. It is worth noting that
$(\sigma_{\pi\Xi^{-}}-\sigma_{\pi\Xi^{0}})/\sigma_{\pi\Xi}\gg(\sigma_{\pi
n}-\sigma_{\pi p})/\sigma_{\pi N}$, in spite of the same isospin difference.

##  VI Summary and outlook

We determined the axial, scalar and tensor isovector charges of the nucleon,
sigma and cascade baryons using $N_{f}=2+1$ lattice QCD simulations. The
analysis is based on 47 gauge ensembles, spanning a range of pion masses from
$430\,\text{MeV}$ down to a near physical value of $130\,\text{MeV}$ across
six different lattice spacings between $a\approx 0.039\,\text{fm}$ and
$a\approx 0.098\,\text{fm}$ and linear spatial lattice extents
$3.0\,M_{\pi}^{-1}\leq L\leq 6.5\,M_{\pi}^{-1}$. The availability of ensembles
lying on three trajectories in the quark mass plane enables SU(3) flavour
symmetry breaking to be explored systematically and the quark mass dependence
of the charges to be tightly constrained. Simultaneous extrapolations to the
physical point in the continuum and infinite volume limit are performed.
Systematic errors are assessed by imposing cuts on the pion mass, the lattice
spacing and the volume as well as using different sets of renormalization
factors. Our results (in the $\overline{\text{MS}}$ scheme at
$\mu=2\,\text{GeV}$) for the nucleon charges are

$\displaystyle g_{A}^{N}$ $\displaystyle=1.284^{(28)}_{(27)},$ $\displaystyle
g_{S}^{N}$ $\displaystyle=1.11^{(14)}_{(16)},$ $\displaystyle g_{T}^{N}$
$\displaystyle=0.984^{(19)}_{(29)}.$

For the hyperon charges we find

$\displaystyle g_{A}^{\Sigma}$ $\displaystyle=0.875^{(30)}_{(39)},$
$\displaystyle g_{S}^{\Sigma}$ $\displaystyle=3.98^{(22)}_{(24)},$
$\displaystyle g_{T}^{\Sigma}$ $\displaystyle=0.798^{(15)}_{(21)},$
$\displaystyle g_{A}^{\Xi}$ $\displaystyle=-0.267^{(13)}_{(12)},$
$\displaystyle g_{S}^{\Xi}$ $\displaystyle=2.57^{(11)}_{(11)},$ $\displaystyle
g_{T}^{\Xi}$ $\displaystyle=-0.1872^{(59)}_{(41)}.$

A comparison with previous works is presented in Sec. V.1. We quantify SU(3)
symmetry breaking effects for the axial charge at the physical point in terms
of the combination

$\delta^{A}_{\text{SU(3)}}=\frac{g_{A}^{\Xi}+g_{A}^{N}-g_{A}^{\Sigma}}{g_{A}^{\Xi}+g_{A}^{N}+g_{A}^{\Sigma}}=0.075^{(23)}_{(27)},$

see Fig. 33. In particular the axial charge of the nucleon deviates from its
value in the SU(3) chiral limit, as can be seen in Fig. 30. No significant
symmetry breaking is observed for the other charges within current precision,
see Figs. 31, 32 and III.1.

To cross-check the analysis methods, the vector charges are determined and the
expected values, $g_{V}^{N}=g_{V}^{\Xi}=1$ and $g_{V}^{\Sigma}=2$, are
reproduced reasonably well:

$\displaystyle g_{V}^{N}$ $\displaystyle=1.0012^{(12)}_{(11)},$ $\displaystyle
g_{V}^{\Sigma}$ $\displaystyle=2.021^{(21)}_{(27)},$ $\displaystyle
g_{V}^{\Xi}$ $\displaystyle=1.015^{(10)}_{(11)}.$

Furthermore, we exploit the conserved vector current relation to predict the
quark mass difference

$m_{u}-m_{d}=-2.03(12)\,\text{MeV}$

from the scalar charge of the $\Sigma$ baryon. We utilize this to decompose
isospin mass splittings between the baryons into QCD and QED contributions
(see Eqs. (64)–(66)) and to predict the leading isospin corrections to the
pion baryon $\sigma$ terms (see Eqs. (71)–(73)).

A computationally efficient stochastic approach was employed in the analysis,
which allows for the simultaneous evaluation of the three-point correlation
functions of all baryons with a variety of current insertions and momentum
combinations. This work is a first step towards determining hyperon decay form
factors which are relevant for the study of CP violation [127]. A
complementary study of the baryon octet $\sigma$ terms on the same data set as
used here is already ongoing [128].

###### Acknowledgements.

We thank all our Coordinated Lattice Simulations (CLS) colleagues for
discussions and the joint production of the gauge ensembles used. Moreover, we
thank Benjamin Gläßle and Piotr Korcyl for their contributions regarding the
(stochastic) three-point function code. S.C. and S.W. received support through
the German Research Foundation (DFG) grant CO 758/1-1. The work of G.B. was
funded in part by the German Federal Ministry of Education and Research (BMBF)
grant no. 05P18WRFP1. Additional support from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement no. 813942 (ITN EuroPLEx) and grant agreement no. 824093 (STRONG
2020) is gratefully acknowledged, as well as initial stage funding through the
German Research Foundation (DFG) collaborative research centre SFB/TRR-55. The
authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS) for
providing computing time through the John von Neumann Institute for Computing
(NIC) on the supercomputer JUWELS [129] and in particular on the Booster
partition of the supercomputer JURECA [130] at Jülich Supercomputing Centre
(JSC). GCS is the alliance of the three national supercomputing centres HLRS
(Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ (Bayerische
Akademie der Wissenschaften), funded by the BMBF and the German State
Ministries for Research of Baden-Württemberg (MWK), Bayern (StMWFK) and
Nordrhein-Westfalen (MIWF). Additional simulations were carried out on the
QPACE 3 Xeon Phi cluster of SFB/TRR-55 and the Regensburg Athene 2 Cluster.
The authors also thank the JSC for their support and for providing services
and computing time on the HDF Cloud cluster [131] at JSC, funded via the
Helmholtz Data Federation (HDF) programme. Most of the ensembles were
generated using openQCD [34] within the CLS effort. A few additional ensembles
were generated employing the BQCD-code [38] on the QPACE supercomputer of
SFB/TRR-55. For the computation of hadronic two- and three-point functions we
used a modified version of the Chroma [132] software package along with the
LibHadronAnalysis library and the multigrid solver implementation of Refs.
[133, 134] (see also ref. [135]) as well as the IDFLS solver [136] of openQCD.
We used Matplotlib [137] to create the figures.

##  Appendix A Further details of the three-point function measurements

###  A.1 Comparison of the stochastic and sequential source methods

We computed the connected three-point functions for all the octet baryons
utilizing the computationally efficient stochastic approach outlined in Sec.
III.3. This approach introduces additional stochastic noise on top of the
gauge noise. In the analysis presented, for the nucleon, we make use of
statistically more precise three-point correlation function measurements
determined via the sequential source method as part of other projects. In the
following, we compare the computational costs of the stochastic and the
sequential source methods and the results for the ratios of the three-point
over two-point functions for the nucleon.

As a typical example, we consider the measurements performed on ensemble N200
($M_{\pi}=286$ MeV and $a=0.064$ fm). For our set-up, illustrated in Figs. 6
and 7, a total of $4\times 12$ solves are needed for the 4 source positions of
the point-to-all propagators. To form the three-point functions for the
nucleon an additional $N_{\text{sto}}=100$ light stochastic solves (for the
timeslice-to-all propagators connecting the sink and current timeslices, the
wiggly line in Fig. 6) are performed. This set-up provides 8 measurements of
the nucleon three-point function (as shown in Fig. 7, with the source-sink
separations $t/a=11$, $14$, $16$, $19$) and includes all polarizations (and
the unpolarized case) as well as a range of sink momenta (almost) for free. In
principle, decuplet baryon three-point functions can also be constructed at
the analysis stage. This set-up is evaluated twice on each configuration
leading to a total of 296 inversions. Similarly, an additional $(4\times
12+100)\times 2$ strange solves are performed in order to form the three-point
functions for all the (octet and decuplet) hyperons, including strangeness
changing currents that we did not consider here.

In the sequential source set-up, we compute the three-point function for the
nucleon at rest, again for source-sink separations $t/a=11$, $14$, $16$ and
$19$. Ten measurements are carried out per configuration (corresponding to 1,
2, 3 and 4 measurements for each $t$, respectively), where in each case the
two light quark flavours ($u$ and $d$) of the current and the four possible
polarizations of the nucleon require $2\times 4$ sequential sources to be
constructed. This amounts to performing $(4+10\times 2\times 4)\times 12=1008$
light solves. The additional $4\times 12$ inversions refer to the point-to-all
propagators for 4 different source positions that connect the source to the
sink (and the current). This is three-times the cost of the stochastic
approach (for the nucleon three-point functions), which realizes a range of
sink momenta.

The ratios of the three-point over two-point functions for the nucleon
obtained from the two different approaches are compared in Fig. III.1. A
significant part of the gauge noise cancels in the ratio, while the
(additional) stochastic noise remains. For our set-up, this leads to larger
statistical errors for the stochastic data compared to the sequential source
results. This difference can clearly be seen for the ratio in the vector
channel, for which the gauge noise is minimal, however, the difference is less
pronounced for the other charges. For the sigma and cascade baryons we
generally find a good statistical signal in the ratios employing the
stochastic approach, see Fig. 37, although, also in this case the ratios for
the hyperon vector charges suffer from large errors.

In the case of a large-scale analysis effort including high statistics, as
presented in this article, the disk space required to store the stochastic
three-point function data is significant. The individual spectator ($S$) and
insertion ($I$) parts as defined in Eqs. (25) and (26), respectively, are
stored with all indices open. In general this amounts to $N=N[S]+N[I]$ complex
double precision floating point numbers for each gauge field configuration
where

$\displaystyle N[S]$ $\displaystyle=N_{F}^{S}\cdot
N_{\mathbf{p}^{\prime}}\cdot N_{\text{snk}}\cdot N_{\text{src}}\cdot
N_{\text{sto}}\cdot N_{c}\cdot N_{s}^{5}\ ,$ (74) $\displaystyle N[I]$
$\displaystyle=N_{F}^{I}\cdot N_{\mathbf{q}}\cdot N_{\partial_{\mu}}\cdot
N_{\tau}\cdot N_{\text{src}}\cdot N_{\text{sto}}\cdot N_{c}\cdot N_{s}^{2}\ .$
(75)

Here $N_{F}^{S/I}$ denotes the number of flavour combinations for the
spectator and insertion parts (typically 4 and 2, respectively),
$N_{\mathbf{p}}$ gives the number of momentum combinations for a maximum
momentum $|\mathbf{p}|$ (with $\mathbf{p}$ either being the sink momentum
$\mathbf{p}^{\prime}$ or the momentum transfer $\mathbf{q}$),
$N_{\text{src}/\text{snk}}$ corresponds to the number of source (4) and sink
(2) positions, $N_{c}=3$ and $N_{s}=4$ are the dimensions of colour and spin
space and $N_{\tau}$ is the number of current insertion timeslices, usually
the distance between the two sink timeslices, see Figs. 6 and 7.
$N_{\partial_{\mu}}$ refers to the number of derivatives included in the
current insertion. We consider all currents including up to one derivative
($N_{\partial_{\mu}}=1+4$), although only the currents without derivatives are
presented in this work. This adds up to a file size of the order of GBs for a
single gauge field configuration and disk space usage of the order of TBs for
a typical CLS gauge ensemble. Storing the data with all indices open allows
for a very flexible analysis. Octet or decuplet baryon three-point functions
can be constructed from the spectator and insertion parts for different
polarizations, current insertions as well as for a large number of momentum
combinations.

Figure 36: Unrenormalized ratio of three-point over two-point functions for
the nucleon vector, axial, scalar and tensor charge (from top to bottom) on
ensemble N200. The left hand side shows the data from the sequential source
method from 1,2,3 and 4 measurements (for increasing values of $t$) compared
to the same measurements obtained from the stochastic approach with four
measurements for each value of $t$ on the right hand side.

Figure 37: Unrenormalized ratio of three-point over two-point functions for
the $\Sigma$ (left) and $\Xi$ (right) scalar charge on ensemble N200.

###  A.2 Treatment of outliers

When analysing the three-point functions on some of the ensembles, we observe
a small number of three-point function results that are by many orders of
magnitude larger than the rest. These outliers, whose origin currently remains
unexplained, would have a significant impact on the analysis, as illustrated
in Fig. 38 for ensemble D200 ($M_{\pi}=202$ MeV and $a=0.064$ fm). The three-
point function for the scalar charge with source-sink separation $t/a=16$ for
a single source position is displayed. In this case one outlier is identified
(according to the criterion given below) and one sees a substantial change in
the configuration average and a reduction in the standard deviation if this
measurement (on a particular configuration) is excluded. The Mainz group [138]
reported similar outliers when determining the nucleon $\sigma$ terms on a
subset of the CLS gauge ensembles employed here. We remark that such outliers
do not seem to occur in the distributions of the two-point function
measurements.

Figure 38: Nucleon three-point function for the scalar channel with a source-
sink separation $t/a=16$ for a single source position on ensemble D200. The
average and standard error is computed including and excluding one particular
gauge configuration.

To overcome this obstacle we constrain the analysed data and discard ‘outlier’
configurations on which measurements give contributions that are very far away
from the expected central value. As discussed in Sec. III.1 we employ the
appropriate reweighting factors where $w_{i}=w_{i}^{\ell}w_{i}^{s}$ is the
product of the light and strange reweighting factors, determined on each
configuration $i$, see in particular the discussion in Appendix G.2 of Ref.
[17]. In order to have a robust estimate for the variance of the reweighted
data

$\displaystyle O_{i}^{w}=\frac{w_{i}O_{i}}{w},$ (76)

where $w=N_{\text{cnfg}}^{-1}\sum_{i}w_{i}$, we determine the lower and upper
boundary values $O^{w}_{\text{low}}$ and $O^{w}_{\text{up}}$ of the central
68% interval of this distribution. We then remove configurations $i$ with

$\displaystyle|O_{i}^{w}-\widehat{O}|>K\,\Delta O,$ (77)

where

$\displaystyle\widehat{O}$
$\displaystyle=\frac{O^{w}_{\text{up}}+O^{w}_{\text{low}}}{2},$
$\displaystyle\Delta O$
$\displaystyle=\frac{O^{w}_{\text{up}}-O^{w}_{\text{low}}}{2},$ (78)

setting the cutoff $K$ to a large value ($K=100$). In Fig. 39 we show the
distribution of the measurements for the nucleon three-point function for a
scalar current (at a single source position and current insertion time) on the
ensemble D200 discussed above (see Fig. 38). The outlier is more than
$400\,\Delta O$ away from $\widehat{O}$. Considering all the three-point
function measurements across the different ensembles, we uniformly set the
cutoff in Eq. (77) to $K=100$ and remove all configurations from the analysis
on which at least one measurement satisfies this criterion.

Figure 39: The numbers of configurations separated by a distance
$|O_{i}^{w}-\widehat{O}|/\Delta O$ from the central value $\widehat{O}$ for
the three-point function data of Fig. 38 on the insertion timeslice
$\tau/a=15$. The blue dashed line indicates the cutoff $K=100$, which is
exceeded by a single measurement. Note the cut and different scales of the
ordinate. Due to this representation a few histogram entries are not shown.

##  Appendix B Additional tables

In Table B we include further details on the gauge ensembles. In Tables B–B
the results for the unrenormalized charges for the three octet baryons are
collected.

Table 15: The gauge ensembles utilized in this work: the light and strange
hopping parameters $\kappa_{\ell}$ and $\kappa_{s}$, the gradient flow scale
parameter $t_{0}/a^{2}$ and the pion ($M_{\pi}$) and kaon masses ($M_{K}$)
[17]. Table 16: Results for the unrenormalized nucleon charges
$g_{J}^{N,\text{latt}}$ for $J\in\\{A,S,T,V\\}$. #ES labels the number of
excited states used to determine the ground state matrix element, see the
discussion in Sec. III.4.  11footnotetext: Ensemble only enters the analysis
of the nucleon charges since no data for the hyperon charges are
available.22footnotetext: The nucleon three-point functions are computed with
the “stochastic” approach. (For all the other ensembles the sequential source
method was used.) Table 17: Results for the unrenormalized hyperon charges
$g_{J}^{\Sigma,\text{latt}}$ for $J\in\\{A,S,T,V\\}$. #ES labels the number of
excited states used to determine the ground state matrix element, see the
discussion in Sec. III.4. The three-point functions are computed employing the
“stochastic” approach.  Table 18: Results for the unrenormalized hyperon
charges $g_{J}^{\Xi,\text{latt}}$ for $J\in\\{A,S,T,V\\}$. #ES labels the
number of excited states used to determine the ground state matrix element,
see the discussion in Sec. III.4. The three-point functions are computed
employing the “stochastic” approach.

##  Appendix C Additional figures

We provide additional figures (Figs. 40–47).

Figure 40: The same as Fig. III.1 for the sigma baryon.

Figure 41: The same as Fig. 13 for the isovector vector charges $g_{V}^{B}$ of
the sigma baryon (left) and the cascade baryon (right). For better visibility,
the data points for ensemble D451, which have large errors (see Tables B and
B), are not displayed. See Tables B and B for the set of ensembles used.

Figure 42: The same as Fig. 15 for the vector charge $g_{V}^{B}$ of the sigma
(right) and cascade (left) baryon.

Figure 43: The same as Fig. 19 for the axial charge $g_{B}^{B}$ of the sigma
(right) and cascade (left) baryon.

Figure 44: The same as Fig. 19 for the scalar charge $g_{S}^{B}$ of the
nucleon (top, left), sigma (top, right) and cascade (bottom) baryons. The six
fits correspond to two fit variations, see the text, applied to three data
sets, DS($M_{\pi}^{\footnotesize{<400\,\text{MeV}}}$),
DS($M_{\pi}^{\footnotesize{<400\,\text{MeV}}}$,
$a^{\footnotesize{<0.1\,\text{fm}}}$) and
DS($M_{\pi}^{\footnotesize{<400\,\text{MeV}}}$,
$LM_{\pi}^{\footnotesize{>4}}$). The data are extracted using two excited
states in the fitting analysis, see Sec. III.4. Only two different sets of
renormalization factors are used.

Figure 45: The same as Fig. 19 for the tensor charge $g_{T}^{B}$ for the
nucleon (top, left), sigma (top, right) and cascade (bottom) baryons. The data
are extracted using two excited states in the fitting analysis, see Sec.
III.4. Only two different sets of renormalization factors are used.

Figure 46: The same as Fig. 21 for the isovector scalar charges $g_{S}^{B}$ of
the sigma baryon (left) and the cascade baryon (right). For better visibility,
the data point for ensemble E300, which has a relatively large error (see
Table B), is not displayed for the sigma baryon. See Tables B and B for the
set of ensembles used.

Figure 47: The same as Fig. 23 for the isovector tensor charges $g_{T}^{B}$ of
the sigma baryon (left) and the cascade baryon (right). For better visibility,
the data point for ensemble E300, which has a relatively large error (see
Table B), is not displayed for the cascade baryon. See Tables B and B for the
set of ensembles used.

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