PhysRevD 108 034512

PHYSICAL REVIEW D 108, 034512 (2023)
Octet baryon isovector charges from N f = 2 + 1 lattice QCD

Gunnar S. Bali ,1,* Sara Collins ,1,† Simon Heybrock,2 Marius Löffler,1 Rudolf Rödl,1
Wolfgang Söldner ,1,‡ and Simon Weishäupl 1,§
(RQCD Collaboration)
1
Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany
2
European Spallation Source ERIC, Data Management and Software Centre (DMSC),
Box 176, 22100 Lund, Sweden
(Received 15 May 2023; accepted 7 August 2023; published 29 August 2023)
We determine the axial, scalar and tensor isovector charges of the nucleon, sigma and cascade baryons as
well as the difference between the up and down quark masses, mu − md . We employ gauge ensembles with
N f ¼ 2 þ 1 nonperturbatively improved Wilson fermions at six values of the lattice spacing in the range
a ≈ ð0.039–0.098Þ fm, generated by the coordinated lattice simulations (CLS) effort. The pion mass Mπ
ranges from around 430 MeV down to a near physical value of 130 MeV and the linear spatial lattice extent
L varies from 6.5M −1 −1
π to 3.0M π , where LM π ≥ 4 for the majority of the ensembles. This allows us to
perform a controlled interpolation/extrapolation of the charges to the physical mass point in the infinite
volume and continuum limit. Investigating SU(3) flavor symmetry, we find moderate symmetry breaking
effects for the axial charges at the physical quark mass point, while no significant effects are found for the
other charges within current uncertainties.
DOI: 10.1103/PhysRevD.108.034512
I. INTRODUCTION moment of the helicity structure functions g1 ðxÞ is related

to the sum of the individual spins of the quarks within the
A charge of a hadron parameterizes the strength of its
proton. For lattice determinations of the individual quark
interaction at small momentum transfer with a particle that
contributions to its first and third moments, see, e.g.,
couples to this particular charge. For instance, the isovector
Refs. [3–8], respectively. Due to the lack of experimental
axial charge determines the β decay rate of the neutron. At
data on g1 ðxÞ, in particular at small Bjorken-x, and
the same time, this charge corresponds to the difference
difficulties in the flavor separation, usually additional
between the contribution of the spin of the up quarks minus
information is used in determinations of the helicity parton
the spin of the down quarks to the total longitudinal spin of
distribution functions (PDFs) from global fits to exper-
a nucleon in the light front frame that is used in the collinear
imental data [9–13]. In addition to the axial charge gA of the
description of deep inelastic scattering. This intimate
proton, this includes information from hyperon decays, in
connection to spin physics at large virtualities and, more
combination with SU(3) flavor symmetry relations whose
specifically, to the decomposition of the longitudinal proton
validity need to be checked.
spin into contributions of the gluon total angular momen-
In this article we establish the size of the corrections to
tum and the spins and angular momenta for the different
SU(3) flavor symmetry in the axial sector and also for the
quark flavors [1,2] opens up a whole area of intense
scalar and the tensor isovector charges of the octet baryons:
experimental and theoretical research: the first Mellin
in analogy to the connection between axial charges and the
first moments of helicity PDFs, the tensor charges are
*
gunnar.bali@ur.de related to first moments of transversity PDFs. This was
† exploited recently in a global fit by the JAM Collaboration
sara.collins@ur.de
‡
wolfgang.soeldner@ur.de [14,15]. Since no tensor or scalar couplings contribute to
§
simon.weishaeupl@ur.de tree-level Standard Model processes, such interactions may
hint at new physics and it is important to constrain new
Published by the American Physical Society under the terms of interactions (once discovered) using lattice QCD input, see,
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to e.g., Ref. [16] for a detailed discussion. SU(3) flavor
the author(s) and the published article’s title, journal citation, symmetry among the scalar charges is also instrumental
and DOI. Funded by SCOAP3. regarding recent tensions between different determinations
2470-0010=2023=108(3)=034512(47) 034512-1 Published by the American Physical Society

GUNNAR S. BALI et al. PHYS. REV. D 108, 034512 (2023)
of the pion nucleon σ term, see Ref. [17] for a summary of

latest phenomenological and lattice QCD results and, e.g.,
the discussion in Sec. 10 of Ref. [18] about the connection
between Okubo-Zweig-Iizuka (OZI) rule violation,
(approximate) SU(3) flavor symmetry and the value of
the pion nucleon σ term. Finally, the scalar isovector
charges relate the QCD part of the mass splitting between
isospin partners to the difference of the up and down quark
masses.
Assuming SU(3) flavor symmetry, the charges for the
whole baryon octet in a given channel only depend on two
independent parameters. For the proton and the axial FIG. 1. The spin-1=2 baryon octet where S, I 3 and Q ¼ ð1 þ
charge, this relation reads gA ¼ FA þ DA , where in the SÞ=2 þ I 3 label strangeness, isospin and charge, respectively.
massless limit FA and DA correspond to the chiral pertur-
bation theory (ChPT) low energy constants (LECs) F and
D, respectively. Already in the first lattice calculations of between the numbers of strange (s) antiquarks and quarks
the axial charge of the proton [19–21], that were carried out are usually called hyperons. The spin-1=2 baryon octet,
in the quenched approximation, FA and DA have been depicted in Fig. 1, contains the nucleons N ∈ fp; ng,
determined separately. However, in spite of the long history besides the S ¼ −1 hyperons Λ0 and Σ ∈ fΣþ ; Σ0 ; Σ− g
of nucleon structure calculations, SU(3) flavor symmetry and the S ¼ −2 hyperons Ξ ∈ fΞ0 ; Ξ− g (cascades). We
breaking is relatively little explored using lattice QCD: only assume isospin symmetry ml ¼ mu ¼ md, where ml
very few investigations of axial charges of the baryon octet corresponds to the average mass of the physical up (u)
exist to date [22–26] and only one of these includes the and down (d) quarks. In this case, the baryon masses within
scalar and tensor charges [26]. Here we compute these isomultiplets are degenerate and simple relations exist
charges for the light baryon octet. We also predict the between matrix elements that differ in terms of the isospin
difference between the up and down quark masses, the I 3 of the baryons and of the local operator (current).
0
QCD contributions to baryon isospin mass splittings and Baryon charges gJB B are obtained from matrix elements
isospin differences of pion baryon σ terms. of the form
This article is organized as follows. In Sec. II we define 0
the octet baryon charges and some related quantities of hB0 ðp0 ;s0 ÞjūΓJ djBðp;sÞi ¼ gJB B ūB0 ðp0 ;s0 ÞΓJ uB ðp;sÞ ð1Þ
interest. In Sec. III the lattice setup is described, including
the gauge ensembles employed, the computational methods at zero four-momentum transfer q2 ¼ ðp0 − pÞ2 ¼ 0.
used to obtain two- and three-point correlation functions Above, uB ðp; sÞ denotes the Dirac spinor of a baryon B
and the excited state analysis performed to extract the with four momentum p and spin s. We restrict ourselves to
ground state matrix elements of interest. We continue with ΔI 3 ¼ 1 transitions within the baryon octet. In this case
details on the nonperturbative renormalization and order a p0 ¼ p, since in isosymmetric QCD mB0 ¼ mB , and it is
improvement, before explaining our infinite volume, con- sufficient to set p ¼ 0. Rather than using the above I 3 ¼ 1
tinuum limit and quark mass extrapolation strategy. Our currents ūΓJ d (where the vector and axial currents couple
results for the charges in the infinite volume, continuum to the W − boson), it is convenient to define I 3 ¼ 0 isovector
limit at physical quark masses are then presented in Sec. IV. currents,
Subsequently, in Sec. V we discuss SU(3) symmetry
breaking effects, determine the up and down quark mass OJ ðxÞ ¼ ūðxÞΓJ uðxÞ − d̄ðxÞΓJ dðxÞ; ð2Þ
difference from the scalar charge of the Σ baryon, split
isospin breaking effects on the baryon masses into QCD and the corresponding charges gBJ ,
and QED contributions and determine isospin breaking
corrections to the pion baryon σ terms. Throughout this hBðp; sÞjOJ jBðp; sÞi ¼ gBJ ūB ðp; sÞΓJ uB ðp; sÞ; ð3Þ
section we also compare our results to literature values,
before we give a summary and an outlook in Sec. VI. In the which, in the case of isospin symmetry, are trivially related
0
appendices further details regarding the stochastic three- to the gJB B :
point function method are given and additional data tables
and figures are provided. gNJ ≔ gpJ ¼ gpn
J ; ð4Þ
þ
pffiffiffi þ 0
II. OCTET BARYON CHARGES gΣJ ≔ gΣJ ¼ − 2gΣJ Σ ; ð5Þ
All light baryons (i.e., baryons without charm or bottom 0 0 −
quarks) with strangeness S < 0, i.e., with a net difference gΞJ ≔ gΞJ ¼ −gΞJ Ξ : ð6Þ
034512-2
OCTET BARYON ISOVECTOR CHARGES FROM N f ¼ 2 þ 1 … PHYS. REV. D 108, 034512 (2023)
Note that we do not include the Λ baryon here since in utilized to ensure leading Oða2 Þ discretization effects
this case the isovector combination trivially gives zero. are then given. Finally, the strategy for interpolation/
We consider vector (V), axial-vector (A), scalar (S) and extrapolation to the physical point in the infinite volume
tensor (T) operators which are defined through the Dirac and continuum limit is outlined.
matrices ΓJ ¼ γ 4 ; γ i γ 5 ; 1; σ ij for J ∈ fV; A; S; Tg, with
σ μν ¼ 12 ½γ μ ; γ ν , where i; j ∈ f1; 2; 3g and i < j. A. Gauge ensembles
The axial charges in the ms ¼ ml ¼ 0 chiral limit are We employ ensembles generated with N f ¼ 2 þ 1 fla-
important parameters in SU(3) ChPT and enter the expan- vors of nonperturbatively OðaÞ improved Wilson fermions
sion of every baryonic quantity. These couplings can be and a tree-level Symanzik improved gauge action, which
decomposed into two LECs F and D which appear in the were mostly produced within the coordinated lattice sim-
first order meson-baryon Lagrangian for three light quark ulations (CLS) [32] effort. Either periodic or open boun-
flavors (see, e.g., Ref. [27]): dary conditions in time [33] are imposed, where the latter
choice is necessary for ensembles with a < 0.06 fm in
gNA ¼ F þ D; gΣA ¼ 2F; gΞA ¼ F − D: ð7Þ order to avoid freezing of the topological charge and thus to
ensure ergodicity [34]. The hybrid Monte Carlo (HMC)
Due to group theoretical constraints, see, e.g., simulations are stabilized by introducing a twisted mass
Refs. [28,29], such a decomposition also holds for term for the light quarks [35], whereas the strange quark is
ms ¼ ml > 0, for the axial as well as for the other charges. included via the rational hybrid Monte Carlo (RHMC)
We define for m ¼ ms ¼ ml algorithm [36]. The modifications of the target action are
corrected for by applying the appropriate reweighting, see
gNJ ðmÞ ¼ FJ ðmÞ þ DJ ðmÞ; ð8Þ Refs. [17,32,37] for further details.
In total 47 ensembles were analyzed spanning six lattice
gΣJ ðmÞ ¼ 2FJ ðmÞ; ð9Þ spacings a in the range 0.039 fm ≲ a ≲ 0.098 fm, with
pion masses between 430 MeV and 130 MeV (below the
gΞJ ðmÞ ¼ FJ ðmÞ − DJ ðmÞ; ð10Þ physical pion mass), as shown in Fig. 2. The lattice spatial
extent L is kept sufficiently large, where LMπ ≥ 4 for the
where F ¼ FA ð0Þ, D ¼ DA ð0Þ. The vector Ward identity majority of the ensembles. A limited number of smaller
(conserved vector current, CVC relation) implies that gNV ¼ volumes are employed to enable finite volume effects to be
gΞV ¼ FV ¼ 1 and gΣV ¼ 2FV , i.e., in this case the above investigated, with the spatial extent varying across all the
relations also hold for ms ≠ ml, with FV ðmÞ ¼ 1 ensembles in the range 3.0 ≤ LMπ ≤ 6.5. Further details
and DV ðmÞ ¼ 0. are given in Table I. The ensembles lie along three
In this article we determine the charges at many different trajectories in the quark mass plane, as displayed in Fig. 3:
positions in the quark mass plane and investigate SU(3) (i) The symmetric line: the light and strange quark
flavor symmetry breaking, i.e., the extent of violation of masses are degenerate (ml ¼ ms ) and SU(3) flavor
Eqs. (8)–(10). Due to this, other than for J ≠ V where symmetry is exact.
DV =FV ¼ 0, the functions DJ ðmÞ and FJ ðmÞ are not (ii) The tr M ¼ const line: starting at the ml ¼ ms flavor
uniquely determined at the physical point, where symmetric point, the trajectory approaches the
ms ≫ ml . At this quark mass point we will find the physical point holding the trace of the quark mass
approximate ratios DA =FA ≈ ð1.55 − 1.95Þ, DS =FS ≈ matrix (2ml þ ms , i.e., the flavor averaged quark
−ð0.2 − 0.5Þ and DT =FT ≈ 1.5. The first ratio can be mass) constant such that 2M2K þ M2π is close to its
compared to the SU(6) quark model expectation physical value.
DA ðmÞ=FA ðmÞ ¼ 3=2 (see, e.g., Ref. [30]), which is (iii) The ms ¼ const line: the renormalized strange quark
consistent with the large-N c limit [31]. mass is kept near to its physical value [38].
The latter two trajectories intersect close to the physical
point, whereas the symmetric line approaches the SU(3)
III. LATTICE SETUP
chiral limit. In figures where data from different lines are
In this section we present the details of our lattice setup. shown, we will distinguish these, employing the symbol
First, we describe the gauge ensembles employed and the shapes of Fig. 2 (triangle, circle, diamond). The excellent
construction of the correlation functions. The computation coverage of the quark mass plane enables the interpolation/
of the three-point correlation functions via a stochastic extrapolation of the results for the charges to the physical
approach is briefly discussed. Following this, we present point to be tightly constrained. In addition, considering the
the fitting analysis and treatment of excited state contri- wide range of lattice spacings and spatial volumes and the
butions employed to extract the ground state baryon matrix high statistics available for most ensembles, all sources of
elements. The renormalization factors used to match to systematic uncertainty associated with simulating at
the continuum MS scheme and the improvement factors unphysical quark mass, finite lattice spacing and finite
034512-3
FIG. 2. Parameter landscape of the ensembles listed in Table I. The ensembles are grouped according to the three quark mass
trajectories (see the text and Fig. 3): (left) the symmetric line (ml ¼ ms ), (middle) the tr M ¼ const line and (right) the ms ¼ const line.
TABLE I. List of the gauge ensembles analyzed in this work. The rqcd xyz ensembles were generated by the RQCD group using the
BQCD code [39], whereas all other ensembles were created within the CLS effort. Note that for H102 there are two replicas. These have
the same parameters but were generated with slightly different algorithmic setups and, therefore, have to be analyzed separately. N401
and N451 differ in terms of the boundary conditions (bc) imposed in the time direction (open (o) and antiperiodic (p), respectively). The
lattice spacings a are determined in Ref. [17]. In the second to last column t denotes the source-sink separation of the connected three-
point functions. The subscript (superscript) given for each separation indicates the number of measurements performed using the
sequential source (stochastic) method on each configuration (see Secs. III B and III C). The last column gives N cnfg , the number of
configurations analyzed.
Ensemble β a [fm] Trajectory bc N t × N 3s Mπ [MeV] MK [MeV] LM π t=a N cnfg

3 73 ; 93 ; 113 ; 134
A650 3.34 0.098 sym p 48 · 24 371 371 4.43 5062
A653 tr M=sym p 48 · 243 429 429 5.12 73 ; 93 ; 113 ; 134 2525
A654 tr M p 48 · 243 338 459 4.04 723 ; 923 ; 1123 ; 1324 2533
rqcd021 3.4 0.086 sym p 32 · 323 340 340 4.73 82 ; 102 ; 124 ; 144 1541
H101 tr M=sym o 96 · 323 423 423 5.88 82 ; 102 ; 122 ; 142 2000
U103 tr M=sym o 128 · 243 420 420 4.38 81 ; 102 ; 123 ; 144 2470
H102r001 tr M o 96 · 323 354 442 4.92 841 ; 1042 ; 1243 ; 1444 997
H102r002 tr M o 96 · 323 359 444 4.99 841 ; 1042 ; 1243 ; 1444 1000
U102 tr M o 128 · 243 357 445 3.72 81 ; 102 ; 123 ; 144 2210
N101 tr M o 128 · 483 281 467 5.86 81 ; 102 ; 123 ; 144 1457
H105 tr M o 96 · 323 281 468 3.91 821 ; 1022 ; 1223 ; 1424 2038
D101 tr M o 128 · 643 222 476 6.18 81 ; 102 ; 123 ; 144 608
C101 tr M o 96 · 483 222 476 4.63 821 ; 1022 ; 1223 ; 1424 2000
S100 tr M o 128 · 323 214 476 2.98 81 ; 102 ; 123 ; 144 983
D150 tr M=ms p 128 · 643 127 482 3.53 81 ; 102 ; 123 ; 144 603
H107 ms o 96 · 323 368 550 5.12 822 ; 1022 ; 1223 ; 1424 1564
H106 ms o 96 · 323 273 520 3.80 842 ; 1042 ; 1243 ; 1444 1553
C102 ms o 96 · 483 223 504 4.65 842 ; 1042 ; 1243 ; 1444 1500
(Table continued)
034512-4
TABLE I. (Continued)
Ensemble β a [fm] Trajectory bc N t × N 3s Mπ [MeV] MK [MeV] LM π t=a N cnfg

3 94 ; 114 ; 138 ; 168
rqcd030 3.46 0.076 sym p 64 · 32 319 319 3.93 1224
X450 sym p 64 · 483 265 265 4.90 92 ; 112 ; 134 ; 164 400
B450 tr M=sym p 64 · 323 421 421 5.19 93 ; 113 ; 143 ; 164 1612
S400 tr M o 128 · 323 354 445 4.36 921 ; 1122 ; 133 ; 142 ; 1624 2872
N451 tr M p 128 · 483 289 466 5.34 944 ; 1144 ; 134 ; 144 ; 1644 1011
N401 tr M o 128 · 483 287 464 5.30 921 ; 1122 ; 133 ; 142 ; 1624 1086
D450 tr M p 128 · 643 216 480 5.32 924 ; 1124 ; 134 ; 142 ; 1624 621
D452 tr M p 128 · 643 156 488 3.84 94 ; 114 ; 144 ; 164 1000
B452 ms p 64 · 323 352 548 4.34 923 ; 1123 ; 133 ; 142 ; 1624 1944
N450 ms p 128 · 483 287 528 5.30 924 ; 1124 ; 134 ; 142 ; 1624 1132
D451 ms p 128 · 643 219 507 5.39 924 ; 1124 ; 134 ; 142 ; 1624 458
X250 3.55 0.064 sym p 64 · 483 350 350 5.47 112 ; 142 ; 164 ; 194 1493
X251 sym p 64 · 483 268 268 4.19 114 ; 144 ; 168 ; 198 1474
N202 tr M=sym o 128 · 483 414 414 6.47 111 ; 142 ; 162 ; 194 883
N203 tr M o 128 · 483 348 445 5.44 1141 ; 1442 ; 1643 ; 1944 1543
N200 tr M o 128 · 483 286 466 4.47 1141 ; 1442 ; 1643 ; 1944 1712
S201 tr M o 128 · 323 290 471 3.02 111 ; 142 ; 163 ; 194 2092
D200 tr M o 128 · 643 202 484 4.21 1121 ; 1422 ; 1623 ; 1924 2001
E250 tr M=ms p 192 · 963 131 493 4.10 114 ; 144 ; 164 ; 194 490
N204 ms o 128 · 483 353 549 5.52 1122 ; 1422 ; 1623 ; 1924 1500
N201 ms o 128 · 483 287 527 4.49 1122 ; 1422 ; 1623 ; 1924 1522
D201 ms o 128 · 643 200 504 4.17 1141 ; 1442 ; 1643 ; 1944 1078
N300 3.7 0.049 tr M=sym o 128 · 483 425 425 5.15 141 ; 172 ; 212 ; 244 1539
N302 tr M o 128 · 483 348 455 4.21 1421 ; 1722 ; 2123 ; 2424 1383
J303 tr M o 192 · 643 259 479 4.18 1422 ; 1724 ; 2126 ; 2428 998
E300 tr M o 192 · 963 176 496 4.26 1423 ; 1723 ; 2126 ; 2426 1038
N304 ms o 128 · 483 353 558 4.27 1422 ; 1722 ; 2123 ; 2424 1652
J304 ms o 192 · 643 261 527 4.21 1423 ; 1723 ; 2123 ; 2424 1630
J500 3.85 0.039 tr M=sym o 192 · 643 413 413 5.24 171 ; 222 ; 273 ; 324 1837
J501 tr M o 192 · 643 336 448 4.26 1721 ; 2222 ; 2723 ; 3224 1018
volume can be investigated. Our strategy for performing a N α ¼ ϵijk uiα ðujT Cγ 5 dk Þ; ð13Þ
simultaneous continuum, quark mass and infinite volume
extrapolation is given in Sec. III F.
Σα ¼ ϵijk uiα ðsjT Cγ 5 uk Þ; ð14Þ
B. Correlation functions
The baryon octet charges are extracted from two- and Ξα ¼ ϵijk siα ðsjT Cγ 5 uk Þ; ð15Þ
three-point correlation functions of the form
with spin index α, color indices i, j, k and C being the
X
CB2pt ðtÞ ¼ P αβ hBα ðx0 ; tÞB̄β ð0; 0Þi; ð11Þ charge conjugation matrix. The Λ baryon is not considered
þ
x0 here since three-point functions with the currents OJ ¼
X ūΓJ u − d̄ΓJ d vanish in this case. Without loss of generality,
CB3pt ðt; τÞ ¼ P αβ hBα ðx0 ; tÞOJ ðy; τÞB̄β ð0; 0Þi: ð12Þ we place the source space-time position at the origin (0, 0)
x0 ;y and the sink at ðx0 ; tÞ such that the source-sink separation in
time equals t. The current is inserted at ðy; τÞ with
Spin-1=2 baryon states are created (annihilated) using 0 ≤ τ ≤ t.1 The annihilation interpolators are projected
suitable interpolators B̄ (B): for the nucleon, Σ and Ξ, onto zero-momentum via the sums over the spatial sink
we employ interpolators corresponding to the proton, Σþ
and Ξ0 , respectively, Note that in practice we only analyze data with 2a ≤ τ ≤ t−2a.
1
034512-5
FIG. 3. Position of the ensembles in the quark mass plane. The tr M ¼ const line (indicated as a blue line) and ms ¼ const line (green
line) intersect close to the physical point (black cross). The symmetric line (orange line), which approaches the SU(3) chiral limit,
crosses the tr M ¼ const line around M π ¼ 411 MeV.
position, while momentum conservation (and the sum over combination of the current. The two-point functions are
y for the current) means the source is also at rest. constructed in the standard way using point-to-all propa-
We ensure positive parity via the projection operator gators. For the three-point functions either a stochastic
P þ ¼ 12 ð1 þ γ 4 Þ. For the three-point functions, P ¼ P þ for approach (described in the next subsection) or the sequen-
J ¼ V, S and P ¼ iγ i γ 5 P þ for J ¼ A, T. The latter tial source method [41] (on some ensembles in combination
corresponds to taking the difference of the polarizations with the coherent sink technique [42]) is employed. The
(in the i direction). The interpolators are constructed from stochastic approach provides a computationally cost effi-
spatially extended quark fields in order to increase the cient way of evaluating the three-point functions for the
overlap with the ground state of interest and minimize whole of the baryon octet, however, additional noise is
contributions to the correlation functions from excited introduced. The relevant measurements for the nucleon
states. Wuppertal smearing is employed [20], together with (which has the worst signal-to-noise ratio of the octet) have
APE-smeared [40] gauge transporters. The number of already been performed with the sequential source method
smearing iterations is varied with the aim of ensuring that as part of other projects by our group, see, e.g., Ref. [43].
ground state dominance sets in for moderate time separa- We use these data in our analysis and the stochastic
tions. The root mean squared light quark smearing approach for the correlation functions of the Σ and the Ξ
radii range from about 0.6 fm (for M π ≈ 430 MeV) up baryons. Note that along the symmetric line (ml ¼ ms ) the
to about 0.8 fm (for Mπ ≈ 130 MeV), whereas the hyperon three-point functions can be obtained as linear
strange quark radii range from about 0.5 fm (for the combinations of the contractions carried out for the currents
physical strange quark mass) up to about 0.7 fm (for ūΓJ u and d̄ΓJ d within the proton. Therefore, no stochastic
MK ¼ M π ≈ 270 MeV). See Sec. E.1 (and in particular three-point functions are generated in these cases.
Table 15) of Ref. [17] for further details. Figure 4 dem- We typically realize four source-sink separations with
onstrates that, when keeping for similar pion and kaon t=fm ≈ f0.7; 0.8; 1.0; 1.2g in order to investigate excited
masses the smearing radii fixed in physical units, the state contamination and reliably extract the ground state
ground state dominates the two-point functions at similar baryon octet charges. Details of our fitting analysis are
physical times across different lattice spacings. presented in Sec. III D. Multiple measurements are per-
Performing the Wick contractions for the two- and three- formed per configuration, in particular for the larger
point correlation functions leads to the connected quark- source-sink separations to improve the signal, see
line diagrams displayed in Fig. 5. Note that there are no Table I. The source positions are chosen randomly on each
disconnected quark-line diagrams for the three-point func- configuration in order to reduce autocorrelations. On
tions as these cancel when forming the isovector flavor ensembles with open boundary conditions in time only
034512-6
FIG. 5. Quark-line diagrams of the two-point (left) and con-

nected three-point (right) correlation functions where xsrc ¼ ð0;0Þ,
xins ¼ ðy; τÞ and xsnk ¼ ðx0 ; tÞ.
constructing a sequential source [41] which depends on

the baryon sink interpolator (including its temporal position
and momentum). This method has the disadvantage that
one needs to compute a new quark propagator for each
source-sink separation, sink momentum and baryon sink
interpolator. Alternatively, one can estimate the all-to-all
propagator stochastically. This introduces additional noise
on top of the gauge noise, however, the quark-line diagram
can be computed in a very efficient way. The stochastic
approach allows one to factorize the three-point correlation
function into a “spectator” and an “insertion” part which
can be computed and stored independently with all spin
indices and one color index open. This is illustrated in
Fig. 6 and explained in more detail below. The two parts
can be contracted at a later (postprocessing) stage with the
appropriate spin and polarization matrices, such that
arbitrary baryonic interpolators can be realized, making
FIG. 4. Effective masses mBeff ðtÞ ¼ ln½CB2pt ðt− a2Þ=CB2pt ðtþ a2Þ=a
this method ideal for SU(3) flavor symmetry studies.
of the nucleon (top) and Ξ (bottom) determined on ensembles
with M π ≈ 340 MeV and M K ≈ 450 MeV and lattice spacings
ranging from a ¼ 0.098 fm (ensemble A654) down to a ¼
0.039 fm (ensemble J501). The errors of mBeff ðtÞ are expected
to increase in proportion to a−1 but they also vary, e.g., with the
number of effectively independent ensembles analyzed.
the spatial positions are varied and the source and sink time
slices are restricted to the bulk of the lattice (sufficiently
away from the boundaries), where translational symmetry
is effectively restored.
C. Stochastic three-point correlation functions

In the following, we describe the construction of the
connected three-point correlation functions using a computa-
tionally efficient stochastic approach. This method was FIG. 6. Schematic representation of the forward (left) and
introduced for computing meson three-point functions in backward (right, shown in gray) propagating three-point corre-
Ref. [44] and utilized for baryons in Refs. [45,46] and also for lation functions with open spin indices that are computed
mesons in Refs. [47,48]. Similar stochastic approaches have simultaneously with the stochastic approach. The indices corre-
been implemented by other groups, see, e.g., Refs. [49,50]. sponding to the spectator and insertion part from the factorization
in Eq. (21) are color-coded in blue and green, respectively. The
In Fig. 5 the quark-line diagram for the three-point
black solid lines correspond to the standard point-to-all propa-
function contains an all-to-all quark propagator which gators, whereas the green wiggly lines represent stochastic time
connects the current insertion at time τ with the baryon slice-to-all propagators. The temporal positions of the forward/
sink at time t. The all-to-all propagator is too computa- 0;fwdjbwd
backward sink, insertion and source are labeled as x4 , y4
tionally expensive to evaluate exactly. One commonly used and x4 , respectively. The flavor indices are chosen corresponding
approach avoids directly calculating the propagator by to a nucleon three-point function with a ūΓJ u-current insertion.
034512-7
Furthermore, no additional inversions are needed for f. Note that su;i ¼ sd;i since our light quarks are mass-
different sink momenta. degenerate.
As depicted in Fig. 7, we simultaneously compute the Using γ 5 -Hermiticity (γ 5 Df γ 5 ¼ D†f ) and the properties
three-point functions for a baryon propagating (forwards) given in Eqs. (17) and (18), the time slice-to-all propagator
from source time slice x4 to sink time slice x0;fwd 4 and connecting all points of the (forward and backward) sink
0;bwd
propagating (backwards) from x4 to x4 . We start with the time slices x04 to all points of any insertion time slice y4 can
definition of the stochastic source and solution vectors be estimated as
which can be used to construct the time slice-to-all
1 X
propagator (shown as a green wiggly line in Fig. 6). In sto N
0 0
the following i ∈ f1; …; N sto g is the “stochastic index,” we Gf3 ðx 0
; yÞγc0μd ≈ ½γ η ðx0 Þγc0 ½γ 5 sf3 ðyÞμd : ð20Þ
N sto i¼1 5 i
denote spin indices with Greek letters, color indices with
Latin letters (other than f or i) and we use flavor indices
f n ∈ fu; d; sg. We introduce (time partitioned) complex Z2 Combining this time slice-to-all propagator with point-
noise vectors [51,52] to-all propagators for the source position x4 , the baryonic
three-point correlation functions Eq. (12) can be factorized
pffiffiffi 0;fwdjbwd
ðZ2 ⊗ iZ2 Þ= 2 if x4 ¼ x4 ; as visualized in Fig. 6 into a spectator part (S) and an
α
ηi ðxÞa ¼ ð16Þ insertion part (I), leaving all flavor and spin indices open:
0 otherwise;
0 0 0
where the noise vector has support on time slices x0;fwd
4 and C3pt ðp0 ; qjx04 ; y4 ; x4 Þfα1αβ βγ μνγ
f2 f3 f4
0;bwd
x4 . The noise vectors have the properties 1 X sto X
3
N
0 0 0
≈ ðS ðp0 ; x04 ; x4 Þαicαβ βγ
N sto i¼1 c¼1 f1 f2
1 X
N sto
α 1
η ðxÞ ¼ O pffiffiffiffiffiffiffiffi ; ð17Þ × I f3 f4 ðq; y4 ; x4 Þμνγ
N sto i¼1 i a N sto ic Þ: ð21Þ
The spectator and insertion parts are defined as

1 X
N sto
α β 1
η ðxÞ η ðyÞ ¼ δxy δαβ δab þ O pffiffiffiffiffiffiffiffi : ð18Þ
N sto i¼1 i a i b N sto 0 0 0
Sf1 f2 ðp0 ; x04 ; x4 Þαicαβ βγ
X 0
The solution vectors sf;i ðyÞ are defined through the linear ðϵa0 b0 c0 ϵabc Gf1 ðx0 ; xÞaα0 αa Gf2 ðx0 ; xÞbβ0 βb
0
≔
system x0
0
× ½γ 5 ηi ðx0 Þγc0 e−ip ·ðx −xÞ Þ;
0 0
ð22Þ
Df ðx; yÞαβ β α
ab sf;i ðyÞb ¼ ηi ðxÞa ; ð19Þ
where we sum over repeated indices (other than f) and I f3 f4 ðq; y4 ; x4 Þμνγ
ic
X
Df ðx; yÞαβ
ab is the Wilson-Dirac operator for the quark flavor ≔ ½γ 5 sf3 ;i ðyÞμd Gf4 ðy; xÞνγ
þiq·ðy−xÞ
ð23Þ
dc e :
y
Using these building blocks, three-point functions for given

baryon interpolators and currents for any momentum
combination can be constructed. Note that in this article
we restrict ourselves to the case q ¼ p0 ¼ 0. The point-to-
all propagators within the spectator part are smeared at the
source and at the sink, whereas Gf4 is only smeared at the
source. The stochastic source is smeared too, however, this
is carried out after solving Eq. (19). In principle, the
spectator part also depends on f 3 because for f 3 ¼ s
and f 3 ∈ fu; dg different smearing parameters are used.
FIG. 7. Sketch of the source and sink positions of the We ignore the dependence of the spectator part on f 3 since
three-point functions realized using the stochastic approach.
Blue diamonds depict the position of the forward (x0;fwd ) and
here we restrict ourselves to f 3 ; f 4 ∈ fu; dg. For details on
x
the smearing see the previous subsection. Using the same
backward (x0;bwd
x ) sink time slices. Green points correspond to the
source time slices xk4 for k ¼ 0, 1, 2, 3. Each three-point function
set of time slice-to-all propagators, we compute point-to-all
measurement is labeled by the source-sink separation, where propagators for a number of different source positions at
the values given correspond to the setup for the ensembles time slices x4 in-between x0;bwd
4 and x0;fwd
4 which allows us
at β ¼ 3.40. to vary the source-sink distances, see Figs. 6 and 7.
034512-8
FIG. 8. Left: polarized nucleon three-point correlation function propagating in the forward and backward directions for a ūγ y γ z u
current insertion obtained from one stochastic measurement on ensemble N200 (a ¼ 0.064 fm). Note that two different source positions
were needed to obtain the same source-sink separation t ¼ 14a in the two directions, see Fig. 7. Right: forward-backward average of the
stochastic three-point function shown on the left compared to that obtained using the sequential source method (with one source
position, propagating in the forward direction).
The number of stochastic estimates N sto is chosen by X

CB3pt ðt; τ; OJ Þ ¼ −EBn ðt−τÞ eEBm τ ;
balancing the computational cost against the size of the m hnjOJ jmie
ZBn ZB ð25Þ
n;m
stochastic noise introduced. We find that for N sto ≳ 100
the stochastic noise becomes relatively small compared to
the gauge noise and we employ 100 estimates across all the where EBn is the energy of state jni (n ¼ 0; 1; …), created
ensembles. In some channels the signal obtained for the when applying the baryon interpolator B̄ to the vacuum
three-point function, after averaging over the forward and state jΩi and ZBn is the associated overlap factor
backward directions, is comparable to that obtained from ZBn ∝ hnjB̄jΩi. The ground state matrix elements of interest
the traditional sequential source method (for a single
h0jOJ j0i ¼ gB;latt
J can be obtained in the limit of large time
source, computed in the forward direction), as shown in
Fig. 8. Nonetheless, when taking the ratio of the three-point separations from the ratio of the three-point and two-point
function with the two-point function for the fitting analysis, functions
discussed in the next subsection, a significant part of the
gauge noise cancels, while the stochastic noise remains. CB3pt ðt; τ; OJ Þ t;τ→∞ B;latt
RBJ ðt; τÞ ¼ ⟶ gJ : ð26Þ
This results in larger statistical errors in the ratio for the CB2pt ðtÞ
stochastic approach. This is a particular problem in the
vector channel. A more detailed comparison of the two
However, the signal-to-noise ratio of the correlation func-
methods is given in Appendix A.
tions deteriorates exponentially with the time separation
As mentioned above, only flavor conserving currents and
and with current techniques it is not possible to achieve a
zero momentum transfer are considered, however, the data
reasonable signal for separations that are large enough to
to construct three-point functions with flavor changing
ensure ground state dominance. At moderate t and τ, one
currents containing up to one derivative for various differ-
observes significant excited state contributions to the ratio.
ent momenta is also available, enabling an extensive
All states with the same quantum numbers as the baryon
investigation of (generalized) form factors in the future.
interpolator contribute to the sums in Eqs. (24) and (25),
Similarly, meson three-point functions can be constructed
including multi-particle excitations such as Bπ P-wave
by computing the relatively inexpensive meson spectator
and Bππ S-wave scattering states. The spectrum of states
part and (re-)using the insertion part, see Ref. [48] for first
results. becomes increasingly dense as one decreases the pion mass
while keeping the spatial extent of the lattice sufficiently
large, where the lowest lying excitations are multiparticle
D. Fitting and excited state analysis states.
The spectral decompositions of the two- and three-point One possible strategy is to first determine the energies
correlation functions read of the ground state and lowest lying excitations by fitting to
X the two-point function (which is statistically more precise
CB2pt ðtÞ ¼ jZBn j2 e−En t ; ð24Þ
B
than the three-point function) with a suitable func-
n tional form. The energies can then be used in a fit to the
034512-9
three-point function (or the ratio RBJ ) to extract the charge J ∈ fA; S; Tg. As the axial, scalar and tensor chan-
gBJ .2 However, the three-quark baryon interpolators we use nels are the main focus of this study, we only
by design have only a small overlap with the multi-particle consider excluding the vector channel data.
states containing five or more quarks and antiquarks and it (ii) The parametrization: either one (“ES ¼ 1”) or two
is difficult to extract the lower lying excited state spectrum (“ES ¼ 2”) excited states are included in the fits. In
from the two-point function. Nonetheless, multiparticle the latter case, in order to stabilize the fit, we use a
states can significantly contribute to the three-point func- prior for ΔE1 corresponding to the energy gap for
tion if the transition matrix elements hnjOJ j0i are large. the lowest lying multi-particle state. As a cross-
Furthermore, depending on the current, different matrix check we repeat these fits using the average re-
elements, and hence excited state contributions, will sult obtained for ΔE2 in fits 5–8 as a prior and
dominate. In particular, one would expect the axial and leaving ΔE1 as a free parameter (fits 13–20). The
scalar currents to couple to the Bπ P-wave and Bππ S-wave widths of the priors are set to E1 =100 and to E2 =100,
states, respectively, while the tensor and vector currents respectively. In general, the contributions from
may enhance transitions between B and Bππ states when excited state to excited state transitions could not
ππ is in a P-wave. be resolved and the parameters bJ2;4 are set to zero.
The summation method [41] is an alternative approach, We also found that the tensor and vector currents
which involvesP summing the ratio over the operator insertion couple more strongly to the second excited state,
time SBJ ðtÞ ¼ t−τ τ¼τ0 RJ ðt; τÞ, where one can show that the
0 B consistent with the expectations mentioned above,
leading excited state contributions to SBJ ðtÞ only depend on t and the first excited state contributions are omitted
[rather than also on t − τ and τ as for RBJ ðt; τÞ]. However, one for these channels in the ES ¼ 2 fits. Furthermore,
needs a large number of source-sink separations (more than due to the large statistical error of the stochastic
the four values of t that are realized in this study) in order to three-point functions for the Σ and Ξ baryons in the
extract reliable results from this approach. vector channel (see Fig. 9 and the discussion in
These considerations motivate us to extract the charges Appendix A), we are not able to resolve bV1 (and
by fitting to the ratio of correlation functions using a fit analogously bV3 ). For these baryons we also set
form which takes into account contributions from up to two bV1;3 ¼ 0 in all the fits.
excited states, (iii) The fit range: two fit intervals τ ∈ ½δtj ; t − δtj are
used with δt1 ¼ n1 a ≈ 0.15 fm and δt2 ¼ n2 a ≈
RBJ ðt; τÞ ¼ bJ0 þ bJ1 ðe−ΔE1 ðt−τÞ þ e−ΔE1 τ Þ þ bJ2 e−ΔE1 t 0.25 fm.3
A typical fit to the ratios for the cascade baryon is
þ bJ3 ðe−ΔE2 ðt−τÞ þ e−ΔE2 τ Þ þ bJ4 e−ΔE2 t ; ð27Þ
shown in Fig. 9 for ensemble N302 (M π ¼ 348 MeV
and a ¼ 0.049 fm). The variation in the ground state matrix
where ΔEn ¼ EBn − EB0 denotes the energy gap between the
elements extracted from the 20 different fits is shown in
ground state and the nth excited state of baryon B and we Fig. 10, also for the nucleon on the same ensemble. See
have not included transitions between the first and the Appendix C for the analogous plot for the Σ baryon.
second excited state. The amplitude bJ0 ¼ gB;latt J gives the Overall, the results are consistent within errors, however,
charge, while bJ1;3 and bJ2;4 are related to the ground state to some trends in the results can be seen across the different
excited state and excited state to excited state transition ensembles. In the axial channel, in particular the results for
matrix elements, respectively. In practice, even when the fits involving a single excited state (fits 1–4), tend to be
simultaneously fitting to all available source-sink separa- lower than those involving two excited states (fits 5–20).
tions, it is difficult to determine the energy gaps (and The former are, in general, statistically more precise than
amplitudes) for a particular channel J. Similar to the the latter due to the smaller number of parameters in the fit.
strategy pursued in Ref. [53], we simultaneously fit to In order to study the systematics arising from any residual
all four channels J ∈ fV; A; S; Tg for a given baryon. As the excited state contamination in the final results at the physical
same energy gaps are present, the overall number of fit point (in the continuum limit at infinite volume), the
parameters is reduced and the fits are further constrained. extrapolations, detailed in Sec. III F, are performed for
To ensure that the excited state contributions are suffi- the results obtained from fits 1–4 (ES ¼ 1) and fits 5–8
ciently under control, we carry out a variety of different fits, (ES ¼ 2), separately. For each set of fits, 500 samples are
summarized in Table II. We vary drawn from the combined bootstrap distributions of the four
(i) The datasets included in the fit: simultaneous fits
are performed to the data for J ∈ fA; S; T; Vg and 3
Due to nj ≥ 2 and its quantization, δt1 and δt2 depend slightly
on the lattice spacing: δtj ≈ 0.20 fm; 0.29 fm (β ¼ 3.34), δtj ≈
2
Given the precision of the two-point function relative to that 0.17 fm; 0.26 fm (β ¼ 3.40), δtj ≈ 0.15 fm; 0.23 fm (β ¼ 3.46),
of the three-point function, this strategy is very similar to fitting δtj ≈ 0.13 fm; 0.26 fm (β ¼ 3.55), δtj ≈ 0.15 fm; 0.25 fm
CB2pt and CB3pt simultaneously. (β ¼ 3.70), δtj ≈ 0.16 fm; 0.27 fm (β ¼ 3.85).
034512-10
FIG. 9. Unrenormalized ratios RΞJ ðt; τÞ, J ∈ fA; S; T; Vg [defined in Eq. (26)] for the cascade baryon on ensemble N302
(M π ¼ 348 MeV and a ¼ 0.049 fm), where t ≈ f0.7; 0.8; 1.0; 1.2g fm. The gray horizontal lines and bands show the results for
the ground state matrix elements h0jūΓJ u − d̄ΓJ dj0i ¼ gΞ;latt
J , obtained from a simultaneous fit to the ratios for all channels and source-
sink separations using parametrization 7 [see Eq. (27) and Table II]. The data points with τ ∈ ½δt; t − δt, where δt ¼ 2a, are included in
the fit (the faded data points are omitted), which is the maximum fit range possible for our action. The colored curves show the
expectation from the fit for each source-sink separation.
FIG. 10. Results for the four unrenormalized charges of the nucleon (top) and cascade baryon (bottom) obtained from the fits listed in
Table II for ensemble N302 (M π ¼ 348 MeV and a ¼ 0.049 fm). The green (blue) horizontal lines and bands indicate the final results
and errors obtained from the median and 68% confidence level interval of the combined bootstrap distributions determined from the fits
indicated by the green (blue) data points which include one (two) excited state(s). On the right the energy gaps determined in the fits and
those corresponding to the lowest lying multiparticle states are displayed using the same color coding as in Fig. 11.
034512-11
TABLE II. Summary of the fits performed. We vary the

combinations of channels J that are fitted simultaneously as
well as the number of excited states (ES) included in the fit and
the fit interval τ ∈ ½δt; t − δt with δt ∈ fδt1 ; δt2 g, where
δt1 ≈ 0.15 fm, δt2 ≈ 0.25 fm. The last two columns indicate
which parameters in Eq. (27) are constrained by a prior or set
to zero. All other parameters are determined in the fit.
Fit J δt ES Prior Set to zero

1 A, S, T δt1 1 bJ2 , bJ3 , bJ4 , ΔE2
2 A, S, T δt2 1 bJ2 , bJ3 , bJ4 , ΔE2
3 A, S, T, V δt1 1 bJ2 , bJ3 , bJ4 , ΔE2
4 A, S, T, V δt2 1 bJ2 , bJ3 , bJ4 , ΔE2
5 A, S, T δt1 2 ΔE1 bT1 , bJ2 , bJ4
6 A, S, T δt2 2 ΔE1 bT1 , bJ2 , bJ4
7 A, S, T, V δt1 2 ΔE1 bT;V J
1 , b2 , b4
J
8 A, S, T, V δt2 2 ΔE1 T;V

b1 , b2 , bJ4
J
9 A, S, T δt1 2 ΔE1 bJ2 , bJ4 FIG. 11. Results for the first and second excited state energy
10 A, S, T δt2 2 ΔE1 bJ2 , bJ4 gaps of the cascade baryon, ΔE1 (brown data points) and ΔE2
11 A, S, T, V δt1 2 ΔE1 bJ2 , bJ4 (orange data points), respectively, determined on ensembles lying
12 A, S, T, V δt2 2 ΔE1 bJ2 , bJ4 on the tr M ¼ const trajectory with a ¼ 0.064 fm. The pion mass
δt1 ΔE2 decreases from left to right with Mπ ¼ 414 MeV for ensemble
13 A, S, T 2 bT1 , bJ2 , bJ4
N202 and M π ¼ 202 MeV for ensemble D200, see Table I. For
14 A, S, T δt2 2 ΔE2 bT1 , bJ2 , bJ4
each ensemble, the ΔE1 obtained using fits 1–4 of Table II are
15 A, S, T, V δt1 2 ΔE2 bT;V J
1 , b2 , b4
J
shown on the left and the ΔE1 (fixed with a prior to the lowest
16 A, S, T, V δt2 2 ΔE2 T;V
b1 , b2 , bJ4
J
multi-particle energy gap) and ΔE2 resulting from fits 5–8 are
17 A, S, T δt1 2 ΔE2 bJ2 , bJ4 displayed on the right. For comparison, the energy gaps of the
18 A, S, T δt2 2 ΔE2 bJ2 , bJ4 lower lying noninteracting multiparticle states with the quantum
19 A, S, T, V δt1 2 ΔE2 bJ2 , bJ4 numbers of the cascade baryon are shown as horizontal lines,
20 A, S, T, V δt2 2 ΔE2 bJ2 , bJ4 where the momenta utilized for each hadron are indicated in
lattice units.
fit variations. The final result and error, shown as the green two excited states. The ΔE2 energy gaps from the two
and blue bands in Fig. 10, correspond to the median and the excited state fits (with the first excited state fixed with a
68% confidence interval, respectively. Note that we take the prior to the lowest multi-particle level) are consistent with
same 500 bootstrap samples for all the baryons to preserve the next level that is significantly above the first excited
correlations. The final results for all the ensembles are listed state, although for ensemble D200 the errors are too large to
in Tables XVI–XVIII of Appendix B for the nucleon, sigma draw a conclusion. Given that more than one excited state is
and cascade baryons, respectively. contributing significantly, we expect that the latter fits
In terms of the energy gaps extracted, Fig. 10 shows that isolate the ground state contribution more reliably. We
we find consistency across variations in the fit range and remark that within present statistics, two-exponential fits to
whether the vector channel data is included or not. the two-point functions alone give energy gaps aΔE ¼
However, the first excited energy gap ΔE1 obtained 0.390ð37Þ; 0.371ð34Þ; 0.430ð37Þ and 0.312(46) for N202,
from the single excited state fits tends to be higher than N203, N200 and D200, respectively, that are all larger than
the lowest multiparticle level, in particular, as the pion mass
is decreased, suggesting that contributions from higher
excited states are significant. This can be seen in Fig. 11,
where we compare the results for the energy gaps for the TABLE III. Improvement coefficients bJ for J ∈ fA; S; T; Vg
cascade baryon with the lower lying noninteracting Ξπ and from Refs. [54,55].
Ξππ states for four ensembles with a ¼ 0.064 fm and pion
β bA bS bT bV
masses ranging from 414 MeV down to 202 MeV. Note that
the multi-particle levels are modified in a finite volume, 3.34 1.249(16) 1.622(47) 1.471(11) 1.456(11)
although the corresponding energy shifts may be small for 3.4 1.244(16) 1.583(62) 1.4155(48) 1.428(11)
the large volumes realized here. There are a number of 3.46 1.239(15) 1.567(74) 1.367(12) 1.410(13)
3.55 1.232(15) 1.606(98) 1.283(14) 1.388(17)
levels within roughly 500 MeV of the first excited state.
3.7 1.221(13) 1.49(11) 1.125(15) 1.309(22)
Some levels lie close to each other and one would not 3.85 1.211(12) 1.33(16) 0.977(38) 1.247(26)
expect that the difference can be resolved by fits with one or
034512-12
aΔE2 , with the exception of D200, where the two gaps For the renormalization factors, we employ the values
agree within errors. obtained in Ref. [56]. The factors are determined non-
perturbatively in the RI0 -SMOM scheme [57,58] and then
E. Nonperturbative renormalization (for ZS and ZT ) converted to the MS scheme using three-
and improvement loop matching [59–61]. We remark that the techniques for
implementing the Rome-Southampton method were
The isovector lattice charges, gB;latt
J , extracted in the extended in Ref. [56] to ensembles with open boundary
previous subsection need to be matched to the continuum conditions in time. This development enables us to utilize
MS scheme. The renormalized matrix elements suffer ensembles with a < 0.06 fm, where only open boundary
from discretization effects, however, the leading order conditions in time are available due to the need to maintain
effects are reduced to Oða2 Þ when implementing full ergodicity. A number of different methods are employed in
OðaÞ improvement. In the forward limit, in addition to Ref. [56] to determine the renormalization factors. In order
using a nonperturbatively OðaÞ improved fermion action, to assess the systematic uncertainty arising from the
this involves taking mass dependent terms into account. matching in the final results for the charges at the physical
The following multiplicative factors are applied, point in the continuum limit, we make use of two sets of
results, collected in Tables V and VI and referred to as Z1J
gBJ ¼ ZkJ ð1 þ aml bJ þ 3am̄b̃J ÞgB;latt
J þ Oða2 Þ; ð28Þ and Z2J , respectively, in the following. The first set of results
are extracted using the fixed-scale method, where the
for J ∈ fV; A; S; Tg, where ZJ are the renormalization RI0 -SMOM factors are determined at a fixed scale (ignoring
factors and bJ and b̃J are the OðaÞ improvement coef- discretization effects), while the second set are obtained by
ficients. Note that the renormalization factors for the scalar fitting the factors as a function of the scale and the lattice
and tensor currents depend on the scale, ZS;T ¼ ZS;T ðμÞ, spacing, the “fit method.” See Ref. [56] for further details.
where we take μ ¼ 2 GeV. The vector Ward identity lattice In both cases, lattice artefacts are reduced by subtracting
quark mass amq is obtained from the hopping parameter κq the perturbative one-loop expectation. For the axial and
(q ¼ l; s) and the critical hopping parameter κcrit via
amq ¼ ð1=κq − 1=κ crit Þ=2. m̄ ¼ ð2ml þ ms Þ=3 denotes TABLE V. Set of renormalization factors taken from Ref. [56],
the flavor averaged quark mass. The hopping parameters denoted as Z1J in the text. The factors are determined using
for all ensembles used within this work are tabulated in the RI0 -SMOM scheme and the “fixed-scale method” with the
Table XV of Appendix B. For κcrit we utilize the inter- perturbative subtraction of lattice artefacts. For ZA and ZV , the
polation formula [17] values correspond to those listed under Z0A and Z0V , respectively,
which are obtained using renormalization conditions consistent
with the respective Ward identities. The statistical and systematic
1 1 þ 0.28955g2 − 0.1660g6
¼ 8 − 0.402454g2 : ð29Þ errors have been added in quadrature.
κcrit 1 þ 0.22770g2 − 0.2540g4
β ZA S ð2 GeVÞ ZT ð2 GeVÞ
ZMS MS
ZV
The improvement coefficients bJ and b̃J are determined 3.34 0.77610(58) 0.6072(26) 0.8443(35) 0.72690(71)
nonperturbatively in Ref. [54]. We make use of updated 3.4 0.77940(36) 0.6027(25) 0.8560(35) 0.73290(67)
preliminary values, which will appear in a future publica- 3.46 0.78240(32) 0.5985(25) 0.8665(36) 0.73870(71)
tion [55]. These are listed in Tables III and IV, respectively. 3.55 0.78740(22) 0.5930(25) 0.8820(37) 0.74740(82)
Note that no estimates of b̃J are available for β ¼ 3.85. 3.7 0.79560(98) 0.5846(24) 0.9055(42) 0.76150(94)
Considering the size of the statistical errors, the general 3.85 0.8040(13) 0.5764(25) 0.9276(42) 0.77430(76)
reduction of the jb̃J j values with increasing β (and the
decreasing a), at this lattice spacing we set b̃J ¼ 0 for all J.
TABLE VI. Set of renormalization factors denoted as Z2J in the
TABLE IV. Improvement coefficients b̃J for J ∈ fA; S; T; Vg text. These are determined as in Table V but now using the “fit
from Refs. [54,55]. Note that no results are available for method.”
β ¼ 3.85.
β ZA S ð2 GeVÞ
ZMS T ð2 GeVÞ
ZMS ZV
β b̃A b̃S b̃T b̃V
3.34 0.7579(42) 0.6115(93) 0.8321(95) 0.7072(60)
3.34 −0.06ð28Þ −0.24ð55Þ 1.02(16) 1.05(13) 3.4 0.7641(35) 0.6068(86) 0.8462(88) 0.7168(49)
3.4 −0.11ð13Þ −0.36ð23Þ 0.49(17) 0.41(11) 3.46 0.7695(36) 0.6025(79) 0.8585(84) 0.7250(43)
3.46 0.08(11) −0.421ð83Þ 0.115(19) 0.158(28) 3.55 0.7774(36) 0.5968(66) 0.8756(76) 0.7367(37)
3.55 −0.03ð13Þ −0.25ð12Þ 0.000(37) 0.069(42) 3.7 0.7895(32) 0.5880(45) 0.9010(63) 0.7544(30)
3.7 −0.047ð75Þ −0.274ð65Þ −0.0382ð60Þ −0.031ð18Þ 3.85 0.8006(25) 0.5793(35) 0.9243(55) 0.7699(38)
034512-13
TABLE VII. Renormalization factors ZA and ZV obtained from works, see, e.g., Refs. [68–70]. Finite volume effects
the interpolation formulas in Eqs. (C.7) and (C.8) in Ref. [63], appear at Oðp3 Þ with no additional LECs appearing in
denoted as Z3J in the text. the coefficients. Again the signs of the corrections are the
β ZA ZV
opposite to the trend seen in the data and, when included, it
is difficult to resolve the effects of the decuplet baryons. As
3.34 0.7510(11) 0.7154(11) is shown in Sec. IV, the data for all the charges are well
3.4 0.75629(65) 0.72221(65) described when the fit form is restricted to the dominant
3.46 0.76172(39) 0.72898(39) terms, with free coefficients c0 , cπ , cK and cV .
3.55 0.76994(34) 0.73905(35)
We remark that the same set of LECs appear in the Oðp2 Þ
3.7 0.78356(32) 0.75538(33)
3.85 0.79675(45) 0.77089(47) SU(3) ChPT expressions for the three different octet
baryons (for a particular charge). Ideally, one would carry
out a simultaneous fit to the whole baryon octet (taking the
vector currents, we also consider a third set of renormal- correlations between the gBJ determined on the same
ization factors, Z3J , listed in Table VII, that are obtained ensemble into account). However, we obtain very similar
with the chirally rotated Schrödinger functional approach results when fitting the gBJ individually compared to fitting
[62], see Ref. [63]. We emphasize that employing the the results for all the octet baryons simultaneously. For
different sets of renormalization factors should lead to simplicity, we choose to do the former, such that the
consistent results for the charges in the continuum limit. coefficients cX for the different baryons are independent of
one another.
F. Extrapolation strategy Lattice spacing effects also need to be taken into account
and we add both mass independent and mass dependent
In the final step of the analysis the renormalized charges terms to the continuum fit ansatz to give
gBJ determined at unphysical quark masses and finite lattice
spacing and spatial volume are extrapolated to the physical
point in the continuum and infinite volume limits. We gBJ ðM π ; MK ; L; aÞ ¼ gBJ ðMπ ; MK ; L; 0Þ
employ a similar strategy to the one outlined in Ref. [64] þ ca a2 þ c̄a M̄2 a2 þ δca δM2 a2
and choose continuum fit functions of the form
þ ca;3 a3 ; ð31Þ
gBJ ðM π ; MK ; L; a ¼ 0Þ
e−LMπ where M̄2 ¼ ð2M2K þ M2π Þ=3 and δM2 ¼ M2K − M2π .
¼ c0 þ cπ M 2π þ cK M2K þ cV M 2π pffiffiffiffiffiffiffiffiffiffi ; ð30Þ
LMπ The meson masses areprescaledffiffiffiffiffiffi with the Wilson flow
scale t0 [71], Mπ;K ¼ 8t0 M π;K to form dimensionless
to parameterize the quark mass and finite volume depend- combinations. This rescaling is required to implement full
ence, where L is the spatial lattice extent and the coef- OðaÞ improvement (along with employing a fermion action
ficients cX , X ∈ f0; π; K; Vg are understood to depend on and isovector currents that are nonperturbatively OðaÞ
the baryon B and the current J. The leading order improved) when simulating at fixed bare lattice coupling
coefficients c0 give the charges in the SU(3) chiral limit, instead of at fixed lattice spacing, see Sec. 4.1 of Ref. [17]
which can be expressed in terms of two LECs, e.g., F and for a detailed discussion of this issue. The values of t0 =a2
D, for the axial charges, see Eq. (7). and the pion and kaon masses in lattice units for our
Equation (30) is a phenomenological fit form based on set of ensembles are given in Table XV of Appendix B.
the SU(3) ChPT expressions for the axial charge. It contains We translate between different lattice spacings using t⋆0 ,
the expected Oðp2 Þ terms for the quark mass dependence the value of t0 along the symmetric line where 12t0 M2π ¼
and the dominant finite volume corrections. The Oðp3 Þ pffiffiffiffiffiffiffi
1.110 [72], i.e., a ¼ a= 8t⋆0 . The values, determined in
expressions for gBA [65–67] contain log terms with coef-
Ref. [17], are listed in Table VIII. Note that we include a
ficients completely determined by the LECs F and D. In an
earlier study of the axial charges on the ms ¼ ml subset of term that is cubic in the lattice spacing in the fit form,
the ensembles used here [64], we found that including these however, this term is only utilized in the analysis of the
terms did not provide a satisfactory description of the data. vector charge, for which we have the most precise data.
When terms arising from loop corrections that contain
decuplet baryons are taken into account, additional LECs TABLE VIII. Values for t⋆0 =a2 at each β-value as determined in
enter that are difficult to resolve. If the coefficient of the log Ref. [17].
term is left as a free parameter, one finds that the coefficient
has the opposite sign to the ChPT expectation without β 3.34 3.4 3.46 3.55 3.7 3.85
decuplet loops. We made similar observations in this study t⋆0 2.204(6) 2.888(8) 3.686(11) 5.157(15) 8.617(22) 13.988(34)
a2
and this is also consistent with the findings of previous
034512-14
To obtain results at the physical quark mass point, we IV. EXTRAPOLATIONS TO THE CONTINUUM,
make use of the scale setting parameter INFINITE VOLUME, PHYSICAL QUARK
MASS LIMIT
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð20Þ
8t0;phys ¼ 0.4098ð25Þ fm; ð32Þ We present the extrapolations to the physical point in the
continuum and infinite volume limits of the isovector
vector (V), axial (A), scalar (S) and tensor (T) charges
determined in Ref. [17] and take the isospin corrected pion
for the nucleon (N), sigma (Σ) and cascade (Ξ) octet
and kaon masses quoted in the FLAG 16 review [73] to
baryons.
define the physical point in the quark mass plane,
Mphys ¼ 134.8ð3Þ MeV; ð33Þ A. Vector charges

π
The isovector vector charges for the nucleon, cascade
and sigma baryons are gNV ¼ gΞV ¼ 1 and gΣV ¼ 2, up to
Mphys
K ¼ 494.2ð3Þ MeV: ð34Þ second order isospin breaking corrections [74]. These
values also apply to our isospin symmetric lattice results
In practice, we choose to fit to the bare lattice charges in the continuum limit for any quark mass combination and
gB;latt
J rather than the renormalized ones as this enables us to volume. A determination of the vector charges provides an
include the uncertainties of the renormalization and important cross-check of our analysis methods and allows
improvement factors (which are the same for all ensembles us to demonstrate that all systematics are under control.
at fixed β) consistently. Therefore, our final fit form reads To start with, we display the ratios of the hyperon
charges over the nucleon charge in Fig. 12. The renorm-
gBJ ðMπ ; MK ; L; aÞ alization factors drop out in the ratio and lattice spacing
gB;latt
J ¼ ; ð35Þ effects are expected to cancel to some extent. As one can
ZkJ ðβÞð1 þ aml bJ ðβÞ þ 3am̄b̃J ðβÞÞ
see, the results align very well with the expected values.
For the individual charges, we perform a continuum
where the dependence of the factors on the β-value is made extrapolation of the data using the fit form
explicit and the superscript k of ZkJ refers to the different
determinations of the renormalization factors that we
gV ¼ c0 þ ca a2 þ c̄a M̄2 a2 þ δca δM2 a2 þ ca;3 a3 : ð36Þ
consider, k ¼ 1, 2, 3 for J ∈ fA; Vg and k ¼ 1, 2 for
J ∈ fS; Tg (see Tables V–VII in the previous subsection).
Note that there is no dependence on the pion or kaon mass
We introduce a separate parameter for ZkJ , bJ and b̃J for
nor on the spatial volume in the continuum limit. M2 and
each β-value and add corresponding “prior” terms to the χ 2
δM2 represent the flavor average and difference of the
function. The statistical uncertainties of these quantities are
kaon and pion masses squared, rescaled with the scale
incorporated by generating pseudobootstrap distributions.
The systematic uncertainty in the determination of the
charges at the physical point is investigated by varying the
fit model and by employing different cuts on the ensembles
that enter the fits. For the latter we consider
(1) no cut: including all the available data points,
denoted as dataset 0, DS (0),
(2) pion mass cut: excluding all ensembles with
Mπ > 400 MeV, DSðM<400 π
MeV Þ,
(3) pion mass cut: excluding all ensembles with

Mπ > 300 MeV, DSðM<300 π
MeV
Þ,
(4) a lattice spacing cut: excluding the coarsest lattice
spacing, i.e., the ensembles with a ≈ 0.098 fm,
DSða<0.1 fm Þ, and
(5) a volume cut: excluding all ensembles with
LMπ < 4, DSðLM>4 π Þ. FIG. 12. Ratio of the hyperon (B ¼ Σ; Ξ) vector charges over
In some cases, more than one cut is applied, e.g., cut 2 and the nucleon charge, gBV =gNV , as a function of the rescaled pion
4, with the dataset denoted DSðM<400 π
MeV <0.1 fm
;a Þ, etc. mass squared (8t0 M 2π ¼ M2π ). The data were extracted using two
Our final results are obtained by carrying out the averaging excited states in the fitting analysis, see Sec. III D, and not
procedure described in Appendix B of Ref. [64] which corrected for lattice spacing or volume effects. Circles (dia-
gives an average and error that incorporates both the monds) correspond to the tr M ¼ const (ms ¼ const) trajectories,
statistical and systematic uncertainties. the triangles to the ms ¼ ml line.
034512-15
pffiffiffiffiffiffi
parameter t0, while the lattice spacing a ¼ a= 8t0 . See the
previous section for further details of the extrapolation
procedure. We implement full OðaÞ improvement and
leading discretization effects are quadratic in the lattice
spacing. However, the data for the nucleon vector charge
are statistically very precise and higher order effects can be
resolved. This motivates the addition of the cubic term in
Eq. (36). The data for gΣV and gΞV are less precise as they are
determined employing the stochastic approach outlined in
Sec. III C which introduces additional noise, see Appendix A
for further discussion.
The data are well described by Eq. (36), as demonstrated
by the fit, shown in Fig. 13, for gNV which has a goodness of FIG. 14. Results for the nucleon vector charge gNV in the
fit of χ 2 =N dof ¼ 0.92. The data are extracted using two continuum limit at the physical point obtained using Z3V (see
excited states in the fitting analysis (see Sec. III D) and we Table VII) and five different parametrizations applied to dataset
employ the most precise determination of the renormaliza- DSðM <400
π
MeV
Þ (fits to Eq. (31) with different coefficients set to
tion factors (Z3V , see Table VII). A cut of M π < 400 MeV is zero labeled 1, …, 5, see the text) and DSðM <400 π
MeV ; a<0.1 fm Þ
imposed on the ensembles entering the fit, however, fits (6, …, 10). See Sec. III F for the definitions of the data sets. The
including all data points are also performed, as detailed data were extracted including two excited states in the fitting
below. When the data are corrected for the discretization analysis, see Sec. III D. The model average is shown as the red
effects according to the fit, we see consistency with gNV ¼ 1, data point and the green horizontal line and band. On the right the
model averaged distribution is displayed as a histogram where
for all pion and kaon masses. Using the fit to shift the data
also the median and the 68% confidence level interval, which
points to the physical point, we observe that the lattice form the final result, are indicated (green lines). The top panel
shows the weights (gray points) assigned to the individual fits,
with the corresponding χ 2 =N dof values given above.
spacing dependence is moderate but statistically signifi-

cant, with a 3–4% deviation from the continuum value at
the coarsest lattice spacing (lower panel of Fig. 13).
In order to investigate the uncertainty arising from
the choice of parametrization and the importance of
the different terms, we repeat the extrapolations employing
five different parametrizations (listed in terms of the
coefficients of the terms entering the fit): ð1; fc0 ; ca gÞ,
ð2; fc0 ; ca ; δca gÞ, ð3; fc0 ; ca ; ca;3 gÞ, ð4; fc0 ; ca ; ca;3 ; δca gÞ
and ð5; fc0 ; ca ; ca;3 ; c̄a ; δca gÞ.4 Regarding the lattice spac-
ing dependence, the mass independent term ca is always
included as the other terms are formally at a higher order.
These five fits are performed on two datasets. The first
set contains ensembles with Mπ < 400 MeV [dataset
DSðM<400π
MeV
Þ], while in the second set the ensembles
with the coarsest lattice spacing are also excluded
FIG. 13. Continuum limit extrapolation of the nucleon iso- (DSðM<400π
MeV <0.1 fm
;a Þ, þ5 is added to the fit number).
vector vector charge gNV for a five parameter fit [Eq. (36)] using See the end of Sec. III F for the definitions of the datasets.
the renormalization factors Z3V (see Table VII) and imposing the The results for gNV , displayed in Fig. 14, show that the
cut Mπ < 400 MeV. The data were extracted including two cubic term and at least one mass dependent term are needed
excited states in the fitting analysis, see Sec. III D. The upper to obtain a reasonable description of the data in terms of the
panel shows the data points corrected for discretization effects
according to the fit. They are consistent with gNV ¼ 1. The bottom
χ 2 =N dof . Two of the fit forms with large χ 2 =N dof values
panel shows the lattice spacing dependence at the physical point. (corresponding to 1, 2, 6, 7, with negligible weight in the
The blue lines and gray bands indicate the expectations from the
fit. For better visibility, the data point for ensemble D452, which We also investigated the possibility of residual OðaÞ effects,
4
has a relatively large error (see Table XVI), is not displayed. in spite of the nonperturbative improvement of the current and the
Circles (diamonds) correspond to the tr M ¼ const (ms ¼ const) action. Indeed, the coefficients of additional terms ∝ a were
trajectories, the triangles to the ms ¼ ml line. found to be consistent with zero.
034512-16
TABLE IX. Results for gBV , B ∈ fN; Σ; Ξg, obtained with three within a combined statistical and systematic uncertainty of
different sets of renormalization factors. The errors include the about 1‰.
statistical and all the systematic uncertainties. The above analysis is also performed utilizing the sets
of renormalization factors Z1V and Z2V , determined via the
Renormalization gNV gΣV gΞV
RI0 -SMOM scheme [56]. The results for the nucleon vector
Z1V (Table V) 0.9975ð20Þ
ð22Þ
2.012ð16Þ
ð26Þ
1.012ð11Þ
ð13Þ
charge are compared in Fig. 15. The uncertainties on these
Z2V (Table VI) 0.9945ð41Þ
ð66Þ
2.014ð16Þ
ð36Þ
1.008ð8Þ
ð19Þ factors are larger, in particular for Z2V , than those of set Z3V ,
which is derived using the chirally rotated Schrödinger
Z3V (Table VII) 1.0012ð11Þ
ð12Þ
2.021ð27Þ
ð21Þ
1.015ð11Þ
ð10Þ
functional approach [63]. This translates into larger errors
for gNV for those fits. The lattice spacing dependence is
model averaging procedure) give values that are incon- somewhat different: the first quadratic mass dependent term
sistent with the continuum expectation. The results are in Eq. (36) and the cubic term can no longer be fully
stable under the removal of the coarsest ensembles. resolved and also parametrization ð2; fc0 ; ca ; δca gÞ gives a
Performing the model averaging procedure, the final χ 2 =N dof ¼ 1.00 (0.95) when employing Z1V (Z2V ).
result for the nucleon, given in the last row of the first The systematic uncertainty of the results due to residual
column of Table IX, agrees with the expectation gNV ¼ 1 excited state contamination and the range of pion masses
FIG. 15. Results for the nucleon vector charge gNV as in Fig. 14 but now also including those obtained employing Z1V and Z2V .
FIG. 16. Overview of the results for the vector charges gBV , B ∈ fN; Σ; Ξg, obtained from different datasets. These are labeled by the
number of excited states used in the fitting analysis (ES ¼ 1, 2), the pion mass cut imposed (denoted A or B) and the set of
renormalization factors employed (ZkV , k ¼ 1, 2, 3). Each data point represents a model averaged result. The label A indicates that 15 fits
[5 fit variations applied to three datasets, DS(0), DSðM π<400 MeV Þ and DSða<0.1 fm Þ] are averaged, while the results labeled with B are
based on the set of 10 fits utilized in Fig. 14 (5 fit variations applied to two datasets, DSðMπ<400 MeV Þ and DS[M <400
π
MeV , a<0.1 fm )]. See
k
Sec. III F for the definitions of the datasets. The final results for each ZV (filled squares) are listed in Table IX.
034512-17
employed in the extrapolations is also considered. employing the parametrization in Eq. (31) [with the
Figure 16 shows the model averaged results discussed so continuum form in Eq. (30)]. The discretization effects
far, displayed as filled squares, and also those obtained are found to be fairly mild and we are not able to resolve the
using several other sets of fits. These are labeled in terms of quadratic mass dependent terms or a cubic term. These
the number of excited states (one or two) included in the terms are omitted throughout. As already mentioned in
fitting analysis, the cuts imposed on the pion mass (A or B) Sec. III F we are also not able to resolve any higher order
and the renormalization factors utilized. For the results ChPT terms in the continuum parametrization.
from pion mass cut A, 15 fits enter the model average, the A five parameter fit, with free coefficients
five different parametrizations are applied to three datasets fc0 ; cπ ; cK ; cV ; ca g, describes the data well, as demon-
DS(0), DSðM<400 π
MeV
Þ and DSða<0.1 fm Þ. Note that the first strated in Fig. 17 for the nucleon (with χ 2 =N dof ¼ 0.86) and
and third dataset include ensembles with pion masses up to
430 MeV. For mass cut B, datasets DSðM <400 π
MeV
Þ and DS
<400 MeV <0.1 fm
[M π ,a )] are used, giving 10 fits in total. The
results only depend on the choice of renormalization
factors, suggesting that the systematic uncertainties due
to excited state contamination and the cut made on the pion
mass are very small.
Repeating the whole procedure for the sigma and the
cascade baryons gives vector charges which are also con-
sistent with the expected values to within 1.5σ, as shown in
Fig. 16 (see Figs. 41 and 42 in Appendix C for the individual
fits for mass cut B). The statistical noise introduced by the
stochastic approach dominates, leading to much less precise
values and very little variation between the results for the
different hyperon data sets. We take the values obtained from
the datasets (2, B, ZkV ), listed in Table IX, as our estimates of
the vector charges as these datasets give the most reliable
determinations of the charges across the different channels
(as discussed in the following subsections).
Overall, the results demonstrate that the systematics
arising from excited state contamination, renormalization
and finite lattice spacing are under control in our analysis in
this channel (to within an error of 1‰ for the nucleon).
B. Axial charges
In the following we present our results for the
nucleon, sigma and cascade isovector axial charges gBA , FIG. 17. Simultaneous quark mass, continuum and finite vol-
B ∈ fN; Σ; Ξg. The nucleon axial charge is very precisely ume extrapolation of the nucleon isovector axial charge gNA
extracted on ensembles with Mπ < 400 MeV using two excited
measured in experiment, λ ¼ gNA =gNV ¼ 1.2754ð13Þ [75],
states in the fitting analysis (see Sec. III D) and renormalization
and serves as another benchmark quantity when assessing factors Z3A (see Table VII). A five parameter fit form is employed,
the size of the systematics of the final results. Note, see the text. (Top) Pion mass dependence of gNA , where the data
however, that possible differences of up to 2%, due to points are corrected, using the fit, for finite volume and discre-
radiative corrections, between λ computed in QCD and an tization effects and shifted (depending on the ensemble) to kaon
effective λ measured in experiment have been discussed masses corresponding to the tr M ¼ const and ms ¼ const tra-
recently [76,77]. Lattice determinations of gNA are known to jectories. The fit is shown as a gray band with the three trajectories
be sensitive to excited state contributions, finite volume distinguished by blue (tr M ¼ const, circles), green (ms ¼ const,
effects and other systematics. Whereas there is a long diamonds) and orange (ms ¼ ml , triangles) lines, respectively.
history of lattice QCD calculations of gNA , see, e.g., the The vertical dashed line indicates the physical point. (Middle)
Lattice spacing dependence at the physical point in the infinite
FLAG 21 review [78], there are very few lattice compu- volume limit. (Bottom) Finite volume dependence at the physical
tations of hyperon axial charges [22–26] and only few point in the continuum limit. The dashed blue line (band) indicates
phenomenological estimates exist from measurements of the infinite volume result. For better visibility, the data points for
semileptonic hyperon decay rates. ensembles D150, E250, and D452, which have relatively large
We carry out simultaneous continuum, quark mass and errors (see Table XVI), are not displayed. The black cross at the
finite volume fits to the individual baryon charges physical point indicates the experimental value [75].
034512-18
FIG. 18. The same as Fig. 17 for the isovector axial charges gBA of the sigma baryon (left) and the cascade baryon (right). For better
visibility, the data points for ensembles D452 and D451, which have relatively large errors (see Table XVIII), are not displayed for the
cascade baryon. Compared to the nucleon, for the analysis of the hyperon charges a reduced set of ensembles is employed, see
Tables XVII and XVIII for a complete list of ensembles.
Fig. 18 for the sigma and cascade baryons (with χ 2 =N dof ¼ The results of the eight fits and their model averages for
0.85 and 1.25, respectively). The data are extracted using the three different determinations of the renormalization
two excited states (ES ¼ 2) in the fitting analysis (see factors are shown in Fig. 19 for the nucleon and in Fig. 43
Sec. III D) and renormalized with factors Z3A (that are the of Appendix C for the hyperon axial charges. In all cases,
most precise of the three determinations considered, see we find consistent results across the different fits and choice
Table VII). For the cascade baryon, with two strange of renormalization factor suggesting that the statistical
quarks, the data on the three quark mass trajectories errors dominate. The additional lattice spacing term is
(tr M ¼ const, ms ¼ const and ml ¼ ms ) are clearly delin- not really resolved with the goodness of fit only changing
eated, however, note the different scale on the right of slightly, while the errors on the coefficients increase. For
Fig. 18. The availability of ensembles on two trajectories the nucleon and sigma baryon, all fits have a χ 2 =N dof < 1
which intersect at the physical point helps to constrain the and are given a similar weight in the model average, while
physical value of the axial charge. In terms of the finite for the cascade baryon, the cut M<300 π
MeV
is needed to
volume effects, only the nucleon shows a significant achieve a goodness of fit around 1 and these fits have the
dependence on the spatial extent. The quark mass depend- highest weight factors.
ence is also pronounced in this case. In order to further explore the systematics, additional
As in the vector case, we quantify the systematics datasets are considered. We assess the sensitivity of the
associated with the extraction of the charges at the physical results to excited state contributions by performing extrap-
point (in the continuum and infinite volume limits) by olations of the data extracted using only one excited state
varying the parametrization and the set of ensembles that (“ES=1”) in the fitting analysis. In addition, as only the
are included in the fit. We consider two fit forms Oðp2 Þ ChPT terms are included in the continuum para-
ð1; fc0 ; cπ ; cK ; cV ; ca gÞ and ð2; fc0 ; cπ ; cK ; cV ; ca ; δca g) metrization, we test the description of the quark mass
and four datasets, DS(M<400 π
MeV
), DSðM <300 π
MeV
Þ, dependence by performing 10 fits, involving the two
DSðM<400
π
MeV <0.1 fm
; a Þ and DSðM <400 MeV
π ; LM >4
π Þ, see parametrization variations above, applied to five datasets,
Sec. III F for their definitions. DS(0), DSðM<400π
MeV
Þ, DSðM <300
π
MeV
Þ, DSða<0.1 fm Þ and
034512-19
FIG. 19. The same as Fig. 15 for the nucleon axial charge gNA . The eight fits correspond to two fit variations, see the text, applied to four
datasets, DSðM <400
π
MeV
Þ, DSðM π<300 MeV Þ, DSðM<400
π
MeV <0.1 fm
;a Þ and DSðM <400π
MeV
π Þ. The data are extracted using two
; LM >4
excited states in the fitting analysis, see Sec. III D.
DSðLM>4 π Þ. The first, fourth and fifth datasets include As detailed in Sec. III D, more than one excited state is
ensembles with pion masses up to 430 MeV. contributing significantly to the ratio of three-point over
The results for the axial charges from model averaging two-point correlation functions and including two excited
the 10 fits (denoted A) employing the 5 datasets and also states in the fitting analysis enables the ground state matrix
from the 8 fits (denoted B) using the 4 datasets given above, element to be isolated more reliably. Considering the pion
for the ES ¼ 1 and ES ¼ 2 data and the different renorm- mass cuts, to be conservative we take the results of the
alization factors are displayed in Fig. 20. Very little model averages of the B datasets (where all the ensembles
variation is seen in the results in terms of the range of have M π < 400 MeV) as only the dominant mass depen-
pion masses included and, as before, the renormalization dent terms are included in the continuum parametrization.
factors employed, suggesting the associated systematics are Our estimates, corresponding to the (ES ¼ 2, B, ZkA ) results
accounted for within the combined statistical and system- in Fig. 20, are listed in Table X.
atic error (which includes the uncertainty due to lattice
spacing and finite volume effects). However, the results are
C. Scalar charges
sensitive to the number of excited states included in the
fitting analysis. This is only a significant effect for the As there is no isovector scalar current interaction at tree-
nucleon, for which the ES ¼ 1 results lie around 2.5σ level in the Standard Model, the scalar charges cannot be
below experiment. Similar underestimates of gNA have been measured directly in experiment. However, the conserved
observed in many earlier lattice studies [78]. vector current (CVC) relation can be used to estimate the
FIG. 20. The same as Fig. 16 for the nucleon, sigma and cascade axial charges. The label A indicates that 10 fits enter the model
average corresponding to two fit variations, see the text, applied to 5 datasets, DS(0), DSðM <400
π
MeV
Þ, DSðM π<300 MeV Þ, DSða<0.1 fm Þ and
DSðLM π Þ. For the data points labeled with B, the results of the 8 fits employed in Fig. 19 are averaged. For the nucleon, the FLAG 21
>4
average for N f ¼ 2 þ 1 [53,79] and the experimental value [75] are indicated (black diamonds).
034512-20
TABLE X. Results for gBA, B ∈ fN; Σ; Ξg, obtained with three

different sets of renormalization factors. The errors include the
statistical and all the systematic uncertainties.
Renormalization gNA gΣA gΞA

Z1A (Table V) 1.299ð29Þ
ð28Þ
0.885ð42Þ
ð30Þ
−0.269ð13Þ
ð14Þ
Z2A (Table VI) 1.295ð29Þ

ð28Þ
0.882ð42Þ
ð30Þ
−0.269ð12Þ
ð14Þ
Z3A (Table VII) 1.284ð27Þ

ð28Þ
0.875ð39Þ
ð30Þ
−0.267ð12Þ
ð13Þ
charges from determinations of the up and down quark

mass difference, δm ¼ mu − md , and the QCD contribution
to baryon mass isospin splittings, e.g., between the mass of
the proton and the neutron, ΔmQCD N
N , (for gS see Eq. (55)
below). Reference [80] finds gS ¼ 1.02ð11Þ employing
N
lattice estimates for δm and an average of lattice and

phenomenological values for ΔmQCD N , which is consistent
with the FLAG 21 [78] N f ¼ 2 þ 1 result of gNS ¼
1.13ð14Þ [53]. Estimates can also be made of the isovector
scalar charges of the other octet baryons, see the discussion
in Sec. VA. Conversely, direct determinations of the scalar
charges can be used to predict δm , as presented in Sec. V C.
So far, there has been only one previous study of the
hyperon scalar charges [26].
For the extrapolation of the scalar charges and the FIG. 21. The same as Fig. 17 for the nucleon scalar charge gNS .
extraction of the value at the physical point, we follow The factors Z1S are used for the matching (see Table V). For
the same procedures as for the axial channel, presented in orientation, the FLAG 21 result for N f ¼ 2 þ 1 [53] is indicated
the previous subsection. The five parameter fit (with (black diamond) at the physical point. For better visibility, the
coefficients fc0 ; cπ ; cK ; cV ; ca g) can again account for data points for ensembles D150, E250, and D452, which have
the observed quark mass, lattice spacing and volume relatively large errors (see Table XVI), are not displayed.
dependence as illustrated in Fig. 21 for the nucleon (with
χ 2 =N dof ¼ 0.56) and Fig. 46 of Appendix C for the sigma scalar charges for all the model averages performed are
and cascade baryons (with χ 2 =N dof ¼ 0.97 and 1.14, compiled in Fig. 22. There are no significant variations in
respectively). The data are extracted using two excited the results obtained using the different renormalization
states in the fitting analysis. For both hyperons, the quark factors and datasets (A or B). For the nucleon, there is also
mass and lattice spacing effects can be resolved, in contrast agreement between the values for the data extracted
to the nucleon, while for all baryons the dependence on the including one (ES ¼ 1) or two (ES ¼ 2) excited states in
spatial volume is marginal. When investigating the sys- the fitting analysis and consistency with the current FLAG
tematics in the estimates of the charges at the physical 21 result. For the sigma baryon, and to a lesser extent for
point, we perform model averages of the results of (A): 8 the cascade baryon, there is a tension between the ES ¼ 1
fits from the two fit variations (as for the axial case) and the and ES ¼ 2 determinations. As discussed previously, the
four datasets, DS(0), DSðM<400 MeV Þ, DSða<0.1 fm Þ and
π (“ES ¼ 2,” B, ZkS ) values are considered the most reliable.
DSðLMπ Þ, (B): 6 fits from the two fit variations to the
>4
These are listed in Table XI.
three datasets DSðM<400 π
MeV
Þ, DSðM <400
π
MeV <0.1 fm
;a Þ
and DS(M<400 π
MeV
, LM >4
π ). Note that a cut on the pion
mass Mπ < 300 MeV is not considered. The scalar matrix
elements are generally less precise than the axial ones and TABLE XI. Results for gBS, B ∈ fN; Σ; Ξg, obtained with two
utilizing such a reduced dataset leads to instabilities in the different sets of renormalization factors. The errors include the
extrapolation and spurious values of the coefficients.
For illustration, the values from the individual fits and Renormalization gNS gΣS gΞS
the model averages over the B datasets for the two different
determinations of the renormalization factors are given in Z1S (Table V) ð14Þ
1.11ð16Þ
ð22Þ
3.98ð24Þ
ð11Þ
2.57ð11Þ
Fig. 44 in Appendix C. The results are consistent across the Z2S (Table VI) ð14Þ
1.12ð17Þ
ð23Þ
4.00ð24Þ
ð11Þ
2.57ð11Þ
different fits, although the weights vary. The values of the
034512-21
FIG. 22. The same as Fig. 16 for the nucleon, sigma and cascade scalar charges. The label A indicates that 8 fits enter the model
average corresponding to two fit variations, see the text, applied to 4 datasets, DS(0), DSðM π<400 MeV Þ, DSða<0.1 fm Þ and DSðLM >4π Þ. For
the data points labeled with B, the two fit variations are performed on 3 datasets, DSðM <400 π
MeV Þ, DSðM <400 MeV ; a<0.1 fm Þ and
π
DSðM <400 MeV ; LM >4 Þ, giving a total of 6 fits for the average. For the nucleon, the FLAG 21 result for N ¼ 2 þ 1 [53] is also shown
π π f
(black diamond).
D. Tensor charges fits have very large uncertainties, unless some assumptions
In the isosymmetric limit, the nucleon tensor charge is are made. In fact, in some analyses, the fit is constrained to
equal to the first moment of the nucleon isovector trans- reproduce the lattice results for the isovector charge,
versity parton distribution function. Due to the lack of see Refs. [14,15]. The FLAG 21 review [78] gives as
experimental data, estimates of gNT from phenomenological the N f ¼ 2 þ 1 value for the nucleon tensor charge the
result of Ref. [53], gNT ¼ 0.965ð61Þ, whereas, as far as we
know, there is only one previous study of the hyperon
tensor charges [26].
The extraction of the octet baryon tensor charges at the
physical point follows the analysis of the axial charges in
Sec. IV B. In particular, the parametrizations employed
and the datasets considered are the same. Figure 23
displays a typical example of an extrapolation for the
nucleon tensor charge for a five parameter fit with a
χ 2 =N dof ¼ 0.63. See Fig. 47 in Appendix C for the
analogous figures for the sigma and cascade baryons.
The variation of the fits with the parametrization and the
datasets utilized and the corresponding model averages,
for the datasets with pion mass cut B (see Sec. IV B), are
shown in Fig. 45.
An overview of the model averaged results for all
variations of the input data is given in Fig. 24. The
agreement between the different determinations suggests
the systematics associated with the extrapolation are under
control. Although the results utilizing data extracted with
two excited states (ES ¼ 2) in the fitting analysis are
consistently above or below those extracted from the
ES ¼ 1 data, considering the size of the errors of the
model averages (which combine the statistical and system-
atic uncertainties), the differences are not significant. Our
estimates for the tensor charges, corresponding to the
FIG. 23. The same as Fig. 17 for the nucleon tensor charge gNT . (ES ¼ 2, B, ZkT ) values, are listed in Table XII.
The factors Z1T are used for the matching (see Table V). For
orientation, the FLAG 21 result for N f ¼ 2 þ 1 [53] is indicated V. DISCUSSION OF THE RESULTS
(black diamond) at the physical point. For better visibility, the
data points for ensembles D150 and D452, which have relatively Our values for the vector, axial, scalar and tensor charges
large errors (see Table XVI), are not displayed. of the nucleon, sigma and cascade baryons are given in
034512-22
FIG. 24. The same as Fig. 16 for the nucleon, sigma and cascade tensor charges. For the nucleon, the FLAG 21 result for N f ¼ 2 þ 1
[53] is also shown (black diamond).
Tables IX–XII, respectively. In each case, we take the most dynamical fermions employed, including charm quarks in
precise value as our final result, i.e., the one obtained using the sea is not expected to lead to a discernible effect.
Z3V and Z3A for the vector and axial channels, respectively, Regarding the hyperon axial charges, far fewer works
and Z1S and Z1T for the scalar and the tensor. In the following exist. Lin et al. [22,92] performed the first study, utiliz-
we compare with previous determinations of the charges ing N f ¼ 2 þ 1 ensembles with pion masses ranging
taken from the literature and discuss the SU(3) flavor between 350 MeV and 750 MeV and a single lattice
symmetry breaking effects in the different channels. We use spacing of 0.12 fm. After an extrapolation to the physical
the conserved vector current relation and our result for the pion mass they obtain gΣA ¼ 0.900ð42Þstat ð54Þsys and gΞA ¼
scalar charge of the sigma baryon to determine the up −0.277ð15Þstat ð19Þsys , where estimates of finite volume and
and down quark mass difference. We compute the QCD discretization effects are included in the systematic uncer-
contributions to baryon isospin mass splittings and evaluate tainty. Note that we have multiplied their result for gΣA by
the isospin breaking effects on the pion baryon σ terms. a factor of two to match our normalization convention.
In Refs. [24,93] ETMC determined all octet and decuplet
A. Individual charges
We first consider the axial charges. Our final values read
ð28Þ ð30Þ ð13Þ

gNA ¼ 1.284ð27Þ ; gΣA ¼ 0.875ð39Þ ; gΞA ¼ −0.267ð12Þ : ð37Þ
The result for the nucleon compares favorably with the

experimental value gNA =gNV ¼ 1.2754ð13Þ [75] and the
FLAG 21 [78] average for N f ¼ 2 þ 1, gNA ¼ 1.248ð23Þ.
The latter is based on the determinations in Refs. [53,79].
All sources of systematic uncertainty must be reasonably
under control to be included in the FLAG average and a
number of more recent studies incorporate continuum,
quark mass and finite volume extrapolations. A compilation
of results for gNA is displayed in Fig. 25. Although the
determinations are separated in terms of the number of
TABLE XII. Results for gBT, B ∈ fN; Σ; Ξg, obtained with two
different sets of renormalization factors. The errors include the
FIG. 25. Compilation of recent lattice determinations of the
Renormalization gNT gΣT gΞT nucleon axial charge gNA with N f ¼ 2 þ 1 [26,53,79,81–86] and
N f ¼ 2 þ 1 þ 1 [68,69,87–91] dynamical fermions. Values with
Z1T (Table V) 0.984ð29Þ
ð19Þ
0.798ð21Þ
ð15Þ
−0.1872ð41Þ
ð59Þ
filled symbols were obtained via a chiral, continuum and finite
Z2T (Table VI) 0.979ð27Þ
ð19Þ
0.793ð21Þ
ð17Þ
−0.1872ð42Þ
ð59Þ
volume extrapolation. The vertical black line gives the exper-
imental result [75].
034512-23
(i.e., nucleon, hyperon and Δ) axial couplings employing

N f ¼ 2 þ 1 þ 1 ensembles with pion masses between
210 MeV and 430 MeV and two lattice spacings
a ∈ f0.065 fm; 0.082 fmg. Using a simple linear ansatz
for the quark mass extrapolation, they quote gΣA ¼
0.7629ð218Þstat and gΞA ¼ −0.2479ð87Þstat , where the errors
are purely statistical.
More recently, Savanur et al. [25] extracted the axial
charges on N f ¼ 2 þ 1 þ 1 ensembles with three different
lattice spacings a ∈ f0.06 fm; 0.09 fm; 0.12 fmg, pion
masses between 135 MeV and 310 MeV and volumes in
the range 3.3 ≤ LMπ ≤ 5.5. The ratios gΣA =gNA and gΞA =gNA
are extrapolated taking the quark mass dependence and
lattice spacing and finite volume effects into account. The
experimental value of gNA is then used to obtain gΣA ¼
0.891ð11Þstat ð13Þsys (again multiplied by a factor of two to
meet our conventions) and gΞA ¼ −0.2703ð47Þstat ð13Þsys .
Finally, QCDSF-UKQCD-CSSM presented results for
the isovector axial, scalar and tensor charges in
Ref. [26]. They employ N f ¼ 2 þ 1 ensembles lying on
a tr M ¼ const trajectory with pion masses ranging FIG. 26. Comparison of lattice determinations [22,24,25] of the
between 220 MeV and 470 MeV and five different values hyperon axial charges for the Σ and Ξ baryons normalized to the
of the lattice spacing in the range ð0.052–0.082Þ fm. The nucleon axial charge. Some of the results from Lin et al. at
Feynman-Hellmann theorem is used to calculate the baryon heavier pion masses are not shown. The gray vertical dashed line
matrix elements. Performing an extrapolation to the physi- indicates the physical pion mass point. Only our results from the
cal mass point including lattice spacing and finite volume ms ¼ const trajectory are shown. The gray bands indicate the
effects, they find gΣA ¼ 0.876ð26Þstat ð09Þsys and gΞA ¼ (continuum limit, infinite volume) quark mass behavior accord-
ing to the fits displayed in Figs. 17 and 18. Note that the data
−0.206ð22Þstat ð19Þsys .
points are not corrected for finite volume or discretization effects.
We also mention the earlier studies of Erkol et al. [23] All data are converted to our phase and normalization conven-
(N f ¼ 2), utilizing pion masses above 500 MeV, and tions, Eqs. (1)–(6). The Lin et al. and ETMC results are obtained
QCDSF-UKQCD (N f ¼ 2 þ 1) carried out at a single by taking the ratio of the individual charges and employing error
lattice spacing [94]. propagation.
In Fig. 26 we compare the ratios of the hyperon axial
charges to the nucleon axial charge, gBA =gNA , from
Refs. [22,24,25], obtained on individual ensembles to
our results. A comparison of the charges themselves cannot
be made since, as mentioned above, Savanur et al. only
present results for the ratio. As the strange quark mass is
held approximately constant in these works, only our
results from the ms ¼ const trajectory are displayed.
Similarly, the QCDSF-UKQCD-CSSM values are omitted
as the ensembles utilized lie on a tr M ¼ const trajectory.
We observe reasonable agreement between the data. Note
that our continuum, infinite volume limit result (the gray
band in the figure) for gΣA =gNA lies slightly below the central
values of most of our ms ¼ const data points.
The individual hyperon axial charges at the physical point
are shown in Fig. 27, along with a number of phenomeno-
logical determinations employing a variety of quark models
FIG. 27. Comparison of our results for the axial charges gΣA and
[95–97], the chiral soliton model [98] and SU(3) covariant
gΞA (blue symbols and error bands) with other lattice determi-
baryon ChPT [67]. Within errors, the lattice results are nations [22,24–26] and phenomenological estimates [67,95–98].
consistent apart from the rather low value for gΣA from ETMC Values with filled symbols were obtained via a chiral, continuum
[24] and the rather high value for gΞA from QCDSF-UKQCD- and finite volume extrapolation. All results are converted to our
CSSM [26]. The phenomenological estimates for gΣA are in phase and normalization conventions, Eqs. (1)–(6).
034512-24
reasonable agreement with our value, while there is a large

spread in the expectations for gΞA.
We remark that, in analogy to the CVC relation
(discussed in Sec. V C below), the axial Ward identity,
∂μ ðūγ μ γ 5 dÞ ¼ iðmd þ mu Þūγ 5 d, connects the axial and
pseudoscalar charges,
mB B
gBP ¼ g ; ð38Þ
ml A
where mB and ml correspond to the baryon and the light

quark mass, respectively. This relation was employed in
Ref. [80] to determine the pseudoscalar charge of the
nucleon, which is defined as the pseudoscalar form factor in
the forward limit. Taking the baryon masses of isosym-
metric QCD from Table 14 of Ref. [17] and the isospin
averaged light quark mass ml ¼ 3.381ð40Þ MeV in the
N f ¼ 4 flavor MS scheme at μ ¼ 2 GeV from the FLAG FIG. 28. As in Fig. 25 for the nucleon scalar charge gNS
with N f ¼ 2 þ 1 [26,53,81,83–86,106,107] and N f ¼ 2 þ 1 þ 1
21 review [78], we find
[69,87,89,91] dynamical fermions. González-Alonso et al. esti-
ð9Þ ð11Þ mate the scalar charge via the conserved vector current (CVC)
gNP;N f ¼4 ¼ 356ð9Þ ; gΣP;N f ¼4 ¼ 308ð14Þ ; relation [80].
ð5Þ
gΞP;N f ¼4 ¼ −104ð5Þ : ð39Þ
splitting [108] and their QCD contributions to the baryon
Turning to the scalar charges, our final results in the three mass splittings [109], we obtain
flavor MS scheme at μ ¼ 2 GeV read5

gNS ¼ 1.05ð13Þ; gΣS ¼ 3.35ð19Þ; gΞS ¼ 2.29ð15Þ; ð41Þ
gNS ¼ 1.11ð16Þ ; gΣS ¼ 3.98ð24Þ ; gΞS ¼ 2.57ð11Þ : ð40Þ
which agree with our results to within two standard

For the nucleon, our result for gNS agrees with the FLAG 21
deviations. Note that a smaller value for jδm j (see
value gNS ¼ 1.13ð14Þ for N f ¼ 2 þ 1 [78] (taken from
Sec. V C) would uniformly increase these charges.
Ref. [53]) and more recent lattice determinations, see Regarding the tensor charges we find in the N f ¼ 3 MS
Fig. 28. There is only one previous lattice determination scheme at μ ¼ 2 GeV
of the hyperon scalar couplings by QCDSF-UKQCD-
CSSM [26], who obtain gΣS ¼ 2.80ð24Þstat ð05Þsys and
gΞS ¼ 1.59ð11Þstat ð04Þsys . These values are much smaller ð19Þ ð15Þ ð59Þ
gNT ¼ 0.984ð29Þ ; gΣT ¼ 0.798ð21Þ ; gΞT ¼ −0.1872ð41Þ : ð42Þ
than ours.
One can also employ the CVC relation and estimates of
the QCD contribution to the isospin mass splittings and Since the anomalous dimension of the tensor bilinear is
the light quark mass difference to determine the scalar smaller than for the scalar case, we would expect no
charges. For a detailed discussion see Sec. V C below. statistically relevant difference between the N f ¼ 3 and
Reference [80] obtains gNS ¼ 1.02ð11Þ assuming ΔmQCD N ¼ N f ¼ 4 schemes at μ ¼ 2 GeV. The nucleon charge agrees
mQCD
p − mQCD
n ¼ −2.58ð18Þ MeV and the quark mass with the FLAG 21 [78] value of gNT ¼ 0.965ð61Þ [53] for
difference δm ¼ mu − md ¼ −2.52ð19Þ MeV. Similarly, N f ¼ 2 þ 1 and other recent lattice studies. These are shown
using the results by BMWc on the light quark mass in Fig. 29 along with determinations from phenomenology.
The large uncertainties of the latter reflect the lack of
5
Using Version 3 of RunDec [99], we compute the conversion experimental data. In particular, in Refs. [14,15] the JAM
factor from N f ¼ 3 to N f ¼ 4: 1.00082ð2ÞΛ ð1Þpert ð56Þmc ¼ collaboration constrain the first Mellin moment of the
1.0008ð6Þ. The errors reflect the uncertainty of the Λ-parameter isovector combination of the transverse parton distribution
[100], the difference between 5-loop running [101,102]/4-loop functions to reproduce a lattice result for gNT. QCDSF-
decoupling [103–105] and 4-loop running/3-loop decoupling and
a 200 MeV uncertainty in the charm quark on-shell mass, UKQCD-CSSM also determined the hyperon tensor charges
respectively: at μ ¼ 2 GeV there is no noteworthy difference 26]]. Their results gΣT ¼ 0.805ð15Þstat ð02Þsys and gΞT ¼
between N f ¼ 3 and N f ¼ 4 MS pseudo(scalar) charges. −0.1952ð74Þstat ð10Þsys are in good agreement with ours.
034512-25
FIG. 30. The ratios 2gBA =gΣA − 1 for B ∈ fN; Ξg as a function of

the pion mass squared. The latter are rescaled with the Wilson
flow scale t0 . The red diamonds are the continuum and infinite
volume limit results at the physical point (indicated by the dashed
vertical line) obtained from our extrapolations of the individual
charges. The yellow bands depict the corresponding ms ¼ ml ¼ 0
predictions D=F. Circles (diamonds) correspond to the
tr M ¼ const. (ms ¼ const.) trajectories, the triangles to the
ms ¼ ml line. The data are not corrected for lattice spacing or
volume effects.
FIG. 29. As in Fig. 25 for the nucleon tensor charge gNT
with N f ¼ 2 þ 1 [26,53,81,83–86,106,110] and N f ¼ 2þ1þ1
[69,87,89,91,111,112] dynamical fermions. Recent pheno- Using Eqs. (8)–(10), we obtain for m ¼ ms ¼ ml
menological estimates are also displayed for comparison
[14,15,113–117]. Values with filled symbols were obtained via 2gBJ ðmÞ D ðmÞ
a quark mass, continuum and finite volume extrapolation. In −1¼ J ; ð43Þ
addition, the filled ETM (2022) [112] point is obtained from a gΣJ ðmÞ FJ ðmÞ
continuum limit extrapolation of results determined on three
physical point ensembles with large spatial volumes. where “þ” and “−” corresponds to B ¼ N and B ¼ Ξ,
respectively. Figure 30 shows these combinations for the
axial charges, as functions of the squared pion mass,
B. SU(3) flavor symmetry breaking compared to the chiral, continuum limit expectations
On the SU(3) flavor symmetric line, i.e., for D=F (yellow bands) determined from our global fit
m ¼ ml ¼ ms , the baryon charges gBJ ðmÞ can be decom- (see Fig. 18 and Table XIII). The chiral limit value agrees
ð27Þ
posed into two functions, FJ ðmÞ and DJ ðmÞ, see Sec. II and with our earlier result D=F ¼ 1.641ð44Þ [64] (model aver-
Eqs. (8)–(10). For the axial charges gBA , the values of these aging recomputed for D=F instead of F=D) within 1.5
functions in the SU(3) chiral limit correspond to the LECs standard errors. The data shown in the figure are not
F ¼ FA ð0Þ and D ¼ DA ð0Þ. We will not consider the corrected for volume or lattice spacing effects. Note that the
vector channel (J ¼ V) here since FV ðmÞ ¼ 1 and
DV ðmÞ ¼ 0, which holds even at ms ≠ ml , due to charge
TABLE XIII. The combinations 2gBJ =gΣJ − 1 of Eq. (43) with
conservation.
B ∈ fN; Ξg at the physical point, in the continuum and infinite
Estimates of baryon structure observables often rely on volume limit. These are computed from the individual charges in
SU(3) flavor symmetry arguments, however, it is not Tables X (for Z3A ), XI and XII (for Z1J ). The last row gives the
known a priori to what extent this symmetry is broken combination DJ ð0Þ=FJ ð0Þ in the chiral, continuum and infinite
for ms ≠ ml and, in particular, at the physical point. Since volume limit (yellow bands in Figs. 30–32), computed from the
within this analysis, we only determined three isovector individual charges: DJ =FJ ¼ ðgNJ − gΞJ Þ=gΣJ .
charges (B ∈ fN; Σ; Ξg) for each channel (J ∈ fA; S; Tg),
we cannot follow the systematic approach to investigate B 2gBA =gΣA − 1 2gBS =gΣS − 1 2gBT =gΣT − 1
SU(3) flavor symmetry breaking of matrix elements pro- ð12Þ ð67Þ
N 1.93ð15Þ ð76Þ
−0.441ð89Þ 1.467ð99Þ
posed in Ref. [29]. Nevertheless, constructing appropriate ð91Þ
ratios from the individual charges will provide us with Ξ ð37Þ
−1.609ð40Þ 0.288ð96Þ ð17Þ
−1.469ð16Þ
ð11Þ ð54Þ
estimates of the flavor symmetry breaking effects for each DJ ð0Þ=FJ ð0Þ 1.79ð8Þ ð46Þ
−0.416ð49Þ 1.530ð56Þ
channel.
034512-26
renormalization factors and improvement coefficients and, carry out the same comparison for the tensor charges.
possibly, other systematics cancel from Eq. (43). For the In this case, within errors of a few percent, no flavor
ratio of the Ξ over the Σ axial charge we see no significant symmetry violation is seen. Moreover, DT ðmÞ=FT ðmÞ ¼
difference between the physical point value and that DT ð0Þ=FT ð0Þ within errors.
obtained for the same average quark mass at the flavor In order to quantify the symmetry breaking effect
symmetric point. The symmetry breaking effect of the between matrix elements involving the current J as a
combination involving gNA =gΣA can be attributed to the pion function of the quark mass splitting ms − ml , we define
mass dependence of gNA , see Fig. 17. The red symbols at the
physical point (dashed vertical line) correspond to our gΞJ þ gNJ − gΣJ
δJSUð3Þ ¼ ; ð44Þ
continuum, infinite volume limit extrapolated results, listed gΞJ þ gNJ þ gΣJ
in Table XIII for the combinations Eq. (43).
In Fig. 31 the combinations Eq. (43) are shown for the where for ms ¼ ml, δJSUð3Þ ¼ ð2FJ −2FJ Þ=ð2FJ þ2FJ Þ ¼ 0,
isovector scalar charges. These are compared to our SU(3)
see Eqs. (8)–(10). Also from these ratios some of the
chiral limit extrapolated results (yellow bands) and the
systematics as well as the renormalization factors and
continuum, infinite volume limit results at the physical
improvement terms will cancel. We define a dimensionless
point (red diamonds). We find no statistically significant
SU(3) breaking parameter x ¼ ðM2K −M 2π Þ=ð2M2K þM 2π Þ ∼
symmetry breaking in this case. However, the statistical
ms −ml and assume a polynomial dependence:
errors are larger than for the axial case and also FS > FA .
Therefore, we cannot exclude symmetry breaking of a X
similar size as for the axial charges, in particular, in the ratio δJSUð3Þ ¼ aJn xn : ð45Þ
of the Ξ over the Σ baryon charge. Finally, in Fig. 32 we n>0
The data for δASUð3Þ ðxÞ depicted in Fig. 33 become more and
more positive as the physical point (vertical dashed line) is
approached. This observation agrees with findings from
earlier studies [22–26]. We fit to data for which the average
quark mass is kept constant (blue circles). However, there is
no significant difference between these and the ms ≈ const
points (black squares). Both linear and quadratic fits in x
(aAn ¼ 0 for n ≠ 1 and aAn ¼ 0 for n ≠ 2, respectively) give
FIG. 31. The same as Fig. 30 for the scalar channel. The yellow
bands depict the ms ¼ ml ¼ 0 predictions DS ð0Þ=FS ð0Þ ob-
tained from the extrapolations of the individual charges.
FIG. 33. The SU(3) symmetry breaking ratio δASUð3Þ ðxÞ

[Eq. (45)] for the axial charges as a function of x ¼ ðM 2K − M2π Þ=
ð2M 2K þ M 2π Þ. Circles (diamonds) correspond to the tr M ¼ const
(ms ¼ const) trajectories, the triangle to ms ¼ ml . The gray
(green) band shows the result from a linear (quadratic) one
parameter fit including only the blue data points that correspond
to the ensembles on the tr M ¼ const line. Black data points
correspond to ensembles on the ms ¼ const trajectory. The
vertical gray dashed line indicates the physical point. The red
diamond corresponds to the result derived from the values for the
FIG. 32. The same as Fig. 30 for the tensor channel. individual charges, see Sec. VA.
034512-27
adequate descriptions of the data and agree with our C. The up and down quark mass difference
continuum, infinite volume limit extrapolated physical Our results on the scalar charges, in particular, gΣS , enable
point result (red diamond) us to determine the quark mass splitting δm ¼ mu − md .
While we simulate the isosymmetric theory, in Nature
ð23Þ this symmetry is broken. The extent of isospin symmetry
δASUð3Þ ¼ 0.075ð27Þ ; ð46Þ breaking is determined by two small parameters, δm =ΛQCD
and the fine structure constant αQED , which are similar in
derived from the values for the individual charges. Effects size. The vector Ward identity relates δm to the QCD
of this sign and magnitude were also reported previously. contributions to baryon mass splittings within an isomul-
ETMC [24] find gNA þ gΞA − gΣA ¼ 0.147ð24Þ, whereas tiplet. In particular, to leading order in δm =ΛQCD and αQED ,
Savanur and Lin [25] quote ðgNA þgΞA −gΣA Þ=gNA ¼ 0.087ð15Þ. the difference between the Σþ and Σ− baryon masses is a
For J ≠ A no statistically significant effects were pure QCD effect from which, with our knowledge of gΣS , we
observed. Nevertheless, for completeness we carry out can extract δm without additional assumptions.
the same analysis for J ¼ S and J ¼ T, see Fig. 34. Our We consider isospin multiplets of baryons BQ ∈
continuum, infinite volume limit extrapolated physical fN Q ; ΣQ ; ΞQ g with electric charges Q ¼ I 3 þ 12 ð1 þ SÞ ∈
point results f0; 1g (N þ ¼ p, N 0 ¼ n) and define the mass differences
ΔmBQþ1 ¼ mBQþ1 − mBQ . Note that for the Σ there are two
differences [75],
ð37Þ ð16Þ
δSSUð3Þ ¼ −0.040ð41Þ ; δTSUð3Þ ¼ −0.001ð23Þ ð47Þ
ΔmΣþ ¼ mΣþ − mΣ0 ¼ −3.27ð7Þ MeV; ð48Þ
provide upper limits on the relative size of SU(3) flavor
violation at the physical point. ΔmΣ0 ¼ mΣ0 − mΣ− ¼ −4.81ð4Þ MeV: ð49Þ
The other splittings read [75]
ΔmΞ ¼ −6.85ð21Þ MeV; ΔmN ≈ −1.293 MeV: ð50Þ
The mass differences can be split into QCD (∼δm ) and QED
(∼αQED ΛQCD ) contributions:
ΔmB ¼ ΔmQCD
B þ ΔmQED
B : ð51Þ
The splitting depends on the scale, the renormalization

scheme and the matching conventions between QCD and
QCD þ QED. The Cottingham formula [118] relates the
leading QED contribution to hadron masses to the total
electric charge squared times a function of the unpolarized
Compton forward-amplitude, i.e., to leading order in
αQED the electric contribution to charge-neutral hadron
masses should vanish (as was suggested in the massless
limit by Dashen [119]). Moreover, for δm ¼ 0 this implies
that the leading QED contributions to the masses of
the Σþ and Σ− baryons are the same. Therefore, up to
OðαQED ; δm =ΛQCD Þ · δm terms,
1
ΔmQCD
Σ ¼ ðmΣþ − mΣ− Þ
2
¼ −4.04ð4Þ MeV; ð52Þ
1
ΔmQED
Σþ
¼ ðmΣþ þ mΣ− Þ − mΣ0
2
FIG. 34. The same as Fig. 33 for the scalar (top) and tensor
(bottom) charges. ¼ 0.77ð5Þ MeV ¼ −ΔmQED
Σ0
: ð53Þ
034512-28
From the Ademollo-Gatto theorem [74] we know that the TABLE XIV. Comparison of the light quark mass difference.
leading isospin breaking effects on the vector charges
gNV ¼ gΞV ¼ 1 and gΣV ¼ 2 are quadratic functions of Nf δm /MeV
δm =ΛQCD and αQED , whereas the scalar charges gBS are RM123 [121] 2þ1þ1 −2.38ð18Þ
subject to linear corrections in αQED and δm =ΛQCD . FNAL-MILCa [122] 2þ1þ1 ð9Þ
−2.55ð7Þ
The Lorentz decomposition of the on-shell QCD matrix MILCa [123] 2þ1þ1 ð11Þ
−2.57ð6Þ
element for the isovector vector current between baryons BMWc [108] 2þ1 −2.41ð12Þ
B0 ¼ BQþ1 and B ¼ BQ (that differ by ΔI 3 ¼ 1 in their This work 2þ1 −2.03ð12Þ
isospin) gives [see Eq. (1)] a
FNAL-MILC [122] and MILC [123] only quote the ratios
ð131Þ ð157Þ
B0 B
i∂μ hB0 ðp0 Þjd̄γ μ ujBðpÞi ¼ gV i∂μ ūB0 ðp0 Þγ μ uB ðpÞ mu =md ¼ 0.4556ð93Þ and mu =md ¼ 0.4529ð82Þ , respectively
0
(all errors added in quadrature). Using the FLAG 21 [78]
¼ gVB B ΔmQCD
B ½1 þ Oðδm =ΛQCD Þ; average ml ¼ 3.410ð43Þ, we combine these results to form
δm ¼ 2ml ðmu =md − 1Þ=ðmu =md þ 1Þ and compute the error
ð54Þ by error propagation.
where the leading correction is due to q0 ¼ p00 − p0 ¼

ΔmQCDB ¼ jqj. In the last step we used the equations of We expect jOðδm =ΛQCD ; αQED Þj ≲ 1% corrections from
motion. Combining this with the vector Ward identity higher order effects to this result, which we can neglect
i∂μ d̄γ μ u ¼ ðmu − md Þd̄u gives at the present level of accuracy.
We can compare our value of δm with results from the
0
gVB B ΔmQCD
B
0
¼ gSB B ðmu − md Þ ð55Þ literature in Table XIV. This includes the N f ¼ 2 þ 1 result
of BMWc [108] and the N f ¼ 2 þ 1 þ 1 continuum limit
as the QCD contribution to the mass difference, with results of RM123 [121], FNAL-MILC [122] and MILC
corrections that are suppressed by powers of the symmetry [123]. In the latter two cases we convert results for mu =md
breaking parameters. Note that the normalization conven- into δm as described in the table caption. We see a tension
tion of the charges gBJ defined in Eq. (3), between the previous determinations and our result on the
pffiffiffi two to three σ level.
þ Σ0 0 −
gpn
J ¼ gJ ;
N
gΣJ ¼ −gΣJ = 2; gΞJ Ξ ¼ −gΞJ ; We remark that all the previous results utilize the
dependence of the pion and kaon masses on the quark
cancels in the above equation so that we can replace masses and the electromagnetic coupling. We consider our
0
gJB B ↦ gBJ to obtain method of determining the quark mass splitting from the
scalar coupling gΣS and the mass difference between the Σþ
gBV and the Σ− baryons as more direct. In Ref. [124] ΔmQCD ¼
δm ¼ mu − md ¼ ΔmQCD ; ð56Þ N
gBS B −1.87ð16Þ MeV (which agrees within errors with lattice
determinations, including ours, see below) is determined
which we refer to as the CVC relation.6 from experimental input. A larger (negative) QCD differ-
Using our physical point, continuum and infinite volume ence would require a larger QED contribution to the proton
ð22Þ mass. As discussed above, the QED contribution to the mass
limit result gΣS ¼ 3.98ð24Þ , assuming gΣV ¼ 2 and applying
of the Σþ baryon is 0.77(5) MeV (similar in size to ΔmQED ¼
Eq. (56) for the Σ baryon, we obtain in the N f ¼ 3 MS N
0.58ð16Þ MeV [124]) and it would be surprising if this
scheme at μ ¼ 2 GeV
increased when replacing a strange quark by a down quark.
ð12Þ Assuming ΔmQCD ¼ −1.87ð16Þ MeV and a value δm ≈
mu − md ¼ −2.03ð12Þ MeV: ð57Þ N
−2.50ð10Þ MeV as suggested by Refs. [108,121–123]
would require a coupling gNS ¼ 0.75ð7Þ to satisfy the
0
6
Note that also the relations between gSB B and gBS receive CVC relation Eq. (56). This in turn is hard to reconcile
OðαQED Þ corrections. Therefore terms ∝ ml αQED , ∝ δm αQED and with the majority of lattice results compiled in Fig. 28. With
∝ δ2m =ΛQCD can be added to Eq. (56). Since ml is similar in size a lower value for jδm j (and/or a larger jΔmQCD N j) this
to δm , we can neglect the first of these terms too, whose inconsistency disappears.
appearance is related to the mixing in QCD þ QED of ml and
δm under renormalization. Using the MS scheme at μ ¼ 2 GeV
corresponds to the suggestion of Ref. [120], however, for quark D. QCD and QED isospin breaking effects
masses ml ∼ δm this additional scale-dependence can be ne- on the baryon masses
glected with good accuracy, as pointed out above. In addition,
there are small Oðα2QED ΛQCD Þ terms due to the QED contribu- We proceed to compute the QED contributions to the
tions to the β- and γ-functions, which are also of higher order. proton and Ξ− masses, ΔmQED
N and −ΔmQED
Ξ :
034512-29
ΔmQED
N ¼ ΔmN − gNS ðmu − md Þ by BMWc [108] from simulations of QCD with quenched
QED, differs by 2.2 standard deviations from our result in
2gNS
¼ ΔmN − ΔmQCD
Σ ; ð58Þ Eq. (57). Also other lattice results on the QCD contribution
gΣS to the mass-splittings (summarized in Fig. 35), obtained at a
single lattice spacing from Endres et al. [125] using QEDTL
2gΞS and QEDM , Brantley et al. [126] and CSSM-QCDSF-
ΔmQED
Ξ ¼ ΔmΞ − ΔmQCD
Σ : ð59Þ
gΣS UKQCD [127] agree within errors.
We mention the possibility of an enhancement of the
This gives (higher order) δ2m =ΛQCD correction to the Σ0 mass due to the
possibility of mixing with the Λ0 , which, however, appears
ð31Þ
ΔmQED
N ¼ 0.97ð36Þ MeV; to be a very small effect [128]. A positive contribution to the
ð31Þ Σ0 mass would increase ΔmQED Σ but leave ΔmQCD Σ [and
ΔmQCD
N ¼ −2.26ð36Þ MeV; ð60Þ
therefore the quark mass difference Eq. (57)] invariant.
The electromagnetic contributions to the p, Σ and Ξ−
ΔmQED
Σ ¼ 0.77ð05Þ MeV; masses are all similar to 1 MeV, with an enhancement for
ΔmQCD ¼ −4.04ð04Þ MeV; ð61Þ the heavier, more compact cascade baryon. Recently,
Σ
combining the Cottingham formula [118] with experimen-
ð37Þ tal input from elastic scattering and parton distribution
ΔmQED
Ξ ¼ −1.65ð39Þ MeV;
functions, the value ΔmQED N ¼ 0.58ð16Þ MeV was deter-
ð42Þ
ΔmQCD
Ξ ¼ −5.20ð44Þ MeV: ð62Þ mined in Ref. [124]. While within errors our result Eq. (60)
agrees with this value, the number obtained in Ref. [124] is
For completeness we included the values for the Σ baryons more inline with the suggested ordering ΔmQED N <
that we determined from the experimental masses alone, ΔmQED
Σ < −Δm QED
Ξ . Combining their value with our deter-
without lattice input. The above mass splittings agree with ð28Þ
mination of gNS gives δm ¼ −1.69ð26Þ MeV, somewhat
the BMWc [109] continuum limit results from simulations
smaller in modulus than our result Eq. (57) and certainly
of QCD plus QED, see Fig. 35 (errors added in quadrature).
in tension with, e.g., δm ¼ −2.41ð12Þ MeV [108].
Nevertheless, as mentioned above, the value of δm , reported
We find that the effect of md > mu on the Ξ and Σ mass
splittings is much bigger than for the nucleon since this is
proportional to gBS =gBV and gNS < gΣS =2 < gΞS . This hierarchy
is due to gNS ≈ FS þ DS , gΣS =2 ≈ FS and gΞS ≈ FS − DS with
FS > 0 and DS < 0. Interestingly, the pion baryon σ terms
σ πB ¼ σ uB þ σ dB that encode the up plus down quark mass
contribution to the baryon masses exhibit the opposite
ordering [17], σ πN > σ πΣ > σ πΞ .
E. Isospin breaking effects on the

pion baryon σ terms
Having determined the quark mass differences, we can
also compute the leading isospin violating corrections to
the pion baryon σ terms σ πB ¼ σ uB þ σ dB . One can either
work with matrix elements [129], using the identity
δm
mu ūu þ md d̄d ¼ ml ðūu þ d̄dÞ þ ðūu − d̄dÞ; ð63Þ
2
or one can start from the Feynman-Hellmann theorem
FIG. 35. Comparison of the QCD contributions to isospin mass hBjq̄qjBi ∂m
splittings [75,109,124–127]. Note that in our normalization σ qB ¼ mq ¼ mq B : ð64Þ
mΣþ − mΣ− ≈ 2ΔmQCD . Values with filled symbols were obtained
hBjBi ∂mq
Σ
via a quark mass, continuum and finite volume extrapolation.
Endres et al. [125] only quote values for ΔmQED from which we Writing mp ¼ mN þ ΔmQCD N =2 þ ΔmN
QED
and mn ¼
N
compute ΔmN QCD
employing the experimental proton-neutron mN − ΔmQCD N =2, where Δm QCD
N ¼ δ g
m S
N N
=g V , and realizing
mass splitting. that the dependence of the QED contributions on the quark
034512-30

masses is of higher order in the isospin breaking, we obtain gΣA ¼ 0.875ð39Þ ; gΣS ¼ 3.98ð24Þ ; gΣT ¼ 0.798ð21Þ ;
at linear order
gΞA ¼ −0.267ð12Þ ; gΞS ¼ 2.57ð11Þ ; gΞT ¼ −0.1872ð41Þ :
∂mp ∂mp 1
σ πp ¼ mu þ md ¼ σ πN þ ΔmQCD
N ; ð65Þ
∂mu ∂md 2 A comparison with previous works is presented in
Sec. VA. We quantify SU(3) symmetry breaking effects
∂mn ∂m 1 for the axial charge at the physical point in terms of the
σ πn ¼ mu þ md n ¼ σ πN − ΔmQCD
N : ð66Þ combination
∂mu ∂md 2
gΞA þ gNA − gΣA ð23Þ
The same can be carried out for the Σ , Ξ0 and Ξ− baryons. δASUð3Þ ¼ ¼ 0.075ð27Þ ;
Using the results for the σ terms of the isosymmetric theory gΞA þ gNA þ gΣA
of Ref. [17], we obtain
see Fig. 33. In particular the axial charge of the nucleon
ð4.7Þ ð4.7Þ
deviates from its value in the SU(3) chiral limit, as can be
σ πp ¼ 42.8ð4.7Þ MeV; σ πn ¼ 45.0ð4.7Þ MeV; ð67Þ seen in Fig. 30 and Table XIII. No significant symmetry
breaking is observed for the other charges within current
ð3.8Þ ð3.8Þ precision, see Figs. 31, 32, and 34.
σ πΣþ ¼ 21.9ð6.1Þ MeV; σ πΣ− ¼ 29.9ð6.1Þ MeV; ð68Þ
To cross-check the analysis methods, the vector charges
are determined and the expected values, gNV ¼ gΞV ¼ 1 and
ð4.5Þ
σ πΞ0 ¼ 8.6ð6.4Þ MeV;
ð4.5Þ
σ πΞ− ¼ 13.8ð6.4Þ MeV; ð69Þ gΣV ¼ 2, are reproduced reasonably well:
whereas the pion σ term for the Σ0 is not affected at linear gNV ¼ 1.0012ð11Þ ; gΣV ¼ 2.021ð27Þ ; gΞV ¼ 1.015ð11Þ :
ð3.8Þ
order: σ πΣ0 ≈ σ πΣ ¼ 25.9ð6.1Þ MeV. We refrain from further
Furthermore, we exploit the conserved vector current
decomposing the pion baryon σ terms into the individual relation to predict the quark mass difference
up and down quark contributions. However, this can
easily be accomplished [129]. It is worth noting that mu − md ¼ −2.03ð12Þ MeV
ðσ πΞ− − σ πΞ0 Þ=σ πΞ ≫ ðσ πn − σ πp Þ=σ πN , in spite of the same
isospin difference. from the scalar charge of the Σ baryon. We utilize this to
decompose isospin mass splittings between the baryons
VI. SUMMARY AND OUTLOOK into QCD and QED contributions [see Eqs. (60)–(62)] and
to predict the leading isospin corrections to the pion baryon
We determined the axial, scalar and tensor isovector σ terms [see Eqs. (67)–(69)].
charges of the nucleon, sigma and cascade baryons using A computationally efficient stochastic approach was
N f ¼ 2 þ 1 lattice QCD simulations. The analysis is based employed in the analysis, which allows for the simulta-
on 47 gauge ensembles, spanning a range of pion masses neous evaluation of the three-point correlation functions of
from 430 MeV down to a near physical value of 130 MeV all baryons with a variety of current insertions and
across six different lattice spacings between a ≈ 0.039 fm momentum combinations. This work is a first step toward
and a ≈ 0.098 fm and linear spatial lattice extents determining hyperon decay form factors which are relevant
3.0M−1 −1
π ≤ L ≤ 6.5M π . The availability of ensembles lying for the study of CP violation [130]. A complementary
on three trajectories in the quark mass plane enables SU(3) study of the baryon octet σ terms on the same dataset as
flavor symmetry breaking to be explored systematically used here is already ongoing [131].
and the quark mass dependence of the charges to be tightly
constrained. Simultaneous extrapolations to the physical ACKNOWLEDGMENTS
point in the continuum and infinite volume limit are
performed. Systematic errors are assessed by imposing We thank all our coordinated lattice simulations (CLS)
cuts on the pion mass, the lattice spacing and the volume as colleagues for discussions and the joint production of the
well as using different sets of renormalization factors. Our gauge ensembles used. Moreover, we thank Benjamin
results (in the MS scheme at μ ¼ 2 GeV) for the nucleon Gläßle and Piotr Korcyl for their contributions regarding
charges are the (stochastic) three-point function code. S. C. and S. W.
received support through the German Research Foundation
ð28Þ ð14Þ ð19Þ (DFG) Grant CO 758/1-1. The work of G. B. was funded in
gNA ¼ 1.284ð27Þ ; gNS ¼ 1.11ð16Þ ; gNT ¼ 0.984ð29Þ : part by the German Federal Ministry of Education and
Research (BMBF) Grant No. 05P18WRFP1. Additional
For the hyperon charges we find support from the European Union’s Horizon 2020
034512-31
research and innovation programme under the Marie positions of the point-to-all propagators. To form the three-
Skłodowska-Curie Grant Agreement No. 813942 (ITN point functions for the nucleon an additional N sto ¼ 100
EuroPLEx) and Grant Agreement No. 824093 light stochastic solves (for the time slice-to-all propagators
(STRONG 2020) is gratefully acknowledged, as well as connecting the sink and current time slices, the wiggly line
initial stage funding through the German Research in Fig. 6) are performed. This setup provides 8 measure-
Foundation (DFG) collaborative research centre SFB/ ments of the nucleon three-point function (as shown in
TRR-55. The authors gratefully acknowledge the Gauss Fig. 7, with the source-sink separations t=a ¼ 11, 14, 16,
Centre for Supercomputing (GCS) for providing computing 19) and includes all polarizations (and the unpolarized
time through the John von Neumann Institute for case) as well as a range of sink momenta (almost) for free.
Computing (NIC) on the supercomputer JUWELS [132] In principle, decuplet baryon three-point functions can also
and in particular on the Booster partition of the supercom- be constructed at the analysis stage. This setup is evaluated
puter JURECA [133] at Jülich Supercomputing Centre twice on each configuration leading to a total of 296
(JSC). GCS is the alliance of the three national super- inversions. Similarly, an additional ð4 × 12 þ 100Þ × 2
computing centres HLRS (Universität Stuttgart), JSC strange solves are performed in order to form the three-
(Forschungszentrum Jülich), and LRZ (Bayerische point functions for all the (octet and decuplet) hyperons,
Akademie der Wissenschaften), funded by the BMBF including strangeness changing currents that we did not
and the German State Ministries for Research of Baden- consider here.
Württemberg (MWK), Bayern (StMWFK) and Nordrhein- In the sequential source setup, we compute the three-
Westfalen (MIWF). Additional simulations were carried point function for the nucleon at rest, again for source-sink
out on the QPACE 3 Xeon Phi cluster of SFB/TRR-55 and separations t=a ¼ 11, 14, 16 and 19. Ten measurements are
the Regensburg Athene 2 Cluster. The authors also thank carried out per configuration (corresponding to 1, 2, 3 and 4
the JSC for their support and for providing services and measurements for each t, respectively), where in each case
computing time on the HDF Cloud cluster [134] at JSC, the two light quark flavors (u and d) of the current and the
funded via the Helmholtz Data Federation (HDF) pro- four possible polarizations of the nucleon require 2 × 4
gramme. Most of the ensembles were generated using sequential sources to be constructed. This amounts to
openQCD [35] within the CLS effort. A few additional performing ð4 þ 10 × 2 × 4Þ × 12 ¼ 1008 light solves.
ensembles were generated employing the BQCD-code [39] The additional 4 × 12 inversions refer to the point-to-all
on the QPACE supercomputer of SFB/TRR-55. For the propagators for 4 different source positions that connect the
computation of hadronic two- and three-point functions we source to the sink (and the current). This is three-times the
used a modified version of the Chroma [135] software cost of the stochastic approach (for the nucleon three-point
package along with the LibHadronAnalysis library and the functions), which realizes a range of sink momenta.
multigrid solver implementation of Refs. [136,137] (see The ratios of the three-point over two-point functions for
also Ref. [138]) as well as the IDFLS solver [139] of the nucleon obtained from the two different approaches are
openQCD. We used Matplotlib [140] to create the figures. compared in Fig. 36. A significant part of the gauge noise
cancels in the ratio, while the (additional) stochastic noise
remains. For our setup, this leads to larger statistical errors
APPENDIX A: FURTHER DETAILS OF THE
for the stochastic data compared to the sequential source
THREE-POINT FUNCTION MEASUREMENTS
results. This difference can clearly be seen for the ratio in
1. Comparison of the stochastic the vector channel, for which the gauge noise is minimal,
and sequential source methods however, the difference is less pronounced for the other
We computed the connected three-point functions for all charges. For the sigma and cascade baryons we generally
the octet baryons utilizing the computationally efficient find a good statistical signal in the ratios employing the
stochastic approach outlined in Sec. III C. This approach stochastic approach, see Fig. 37, although, also in this case
introduces additional stochastic noise on top of the gauge the ratios for the hyperon vector charges suffer from large
noise. In the analysis presented, for the nucleon, we make errors.
use of statistically more precise three-point correlation In the case of a large-scale analysis effort including high
function measurements determined via the sequential statistics, as presented in this article, the disk space required
source method as part of other projects. In the following, to store the stochastic three-point function data is signifi-
we compare the computational costs of the stochastic and cant. The individual spectator (S) and insertion (I) parts as
the sequential source methods and the results for the ratios defined in Eqs. (22) and (23), respectively, are stored
of the three-point over two-point functions for the nucleon. with all indices open. In general this amounts to N ¼
As a typical example, we consider the measurements N½S þ N½I complex double precision floating point num-
performed on ensemble N200 (M π ¼ 286 MeV and bers for each gauge field configuration where
a ¼ 0.064 fm). For our setup, illustrated in Figs. 6 and
7, a total of 4 × 12 solves are needed for the 4 source N½S ¼ N SF · N p0 · N snk · N src · N sto · N c · N 5s ; ðA1Þ
034512-32
FIG. 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.
034512-33
FIG. 37. Unrenormalized ratio of three-point over two-point functions for the Σ (left) and Ξ (right) scalar charge on ensemble N200.
N½I ¼ N IF · N q · N ∂μ · N τ · N src · N sto · N c · N 2s : ðA2Þ
Here N S=I
F denotes the number of flavor combinations for
the spectator and insertion parts (typically 4 and 2,
respectively), N p gives the number of momentum combi-
nations for a maximum momentum jpj (with p either being
the sink momentum p0 or the momentum transfer q),
N src=snk corresponds to the number of source (4) and sink
(2) positions, N c ¼ 3 and N s ¼ 4 are the dimensions of
color and spin space and N τ is the number of current
insertion time slices, usually the distance between the two
sink time slices, see Figs. 6 and 7. N ∂μ refers to the number
of derivatives included in the current insertion. We consider
all currents including up to one derivative (N ∂μ ¼ 1 þ 4),
FIG. 38. Nucleon three-point function for the scalar channel
although only the currents without derivatives are presented
with a source-sink separation t=a ¼ 16 for a single source position
in this work. This adds up to a file size of the order of GBs on ensemble D200. The average and standard error is computed
for a single gauge field configuration and disk space usage including and excluding one particular gauge configuration.
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.
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 π ¼ 202 MeV and a ¼ 0.064 fm). The three-point func-
tion for the scalar charge with source-sink separation t=a ¼
16 for a single source position is displayed. In this case one FIG. 39. The numbers of configurations separated by a distance
outlier is identified (according to the criterion given below) jOwi − Ôj=ΔO from the central value Ô for the three-point
and one sees a substantial change in the configuration function data of Fig. 38 on the insertion time slice τ=a ¼ 15.
average and a reduction in the standard deviation if this The blue dashed line indicates the cutoff K ¼ 100, which is
measurement (on a particular configuration) is excluded. exceeded by a single measurement. Note the cut and different
The Mainz group [141] reported similar outliers when scales of the ordinate. Due to this representation a few histogram
determining the nucleon σ terms on a subset of the CLS entries are not shown.
034512-34
gauge ensembles employed here. We remark that such where

outliers do not seem to occur in the distributions of the two-
point function measurements. To overcome this obstacle we Owup þ Owlow Owup − Owlow
Ô ¼ ; ΔO ¼ ; ðA5Þ
constrain the analyzed data and discard “outlier” configu- 2 2
rations on which measurements give contributions that are
very far away from the expected central value. As discussed setting the cutoff K to a large value (K ¼ 100). In Fig. 39
in Sec. III A we employ the appropriate reweighting factors we show the distribution of the measurements for the
where wi ¼ wli wsi is the product of the light and strange nucleon three-point function for a scalar current (at a single
reweighting factors, determined on each configuration i, source position and current insertion time) on the ensemble
see in particular the discussion in Appendix G. 2 of D200 discussed above (see Fig. 38). The outlier is more
Ref. [17]. In order to have a robust estimate for the variance than 400 ΔO away from Ô. Considering all the three-point
of the reweighted data function measurements across the different ensembles, we
uniformly set the cutoff in Eq. (A4) to K ¼ 100 and remove
wi Oi all configurations from the analysis on which at least one
Owi ¼ ; ðA3Þ measurement satisfies this criterion.
w
P
where w ¼ N −1 cnfg i wi , we determine the lower and upper APPENDIX B: ADDITIONAL TABLES
boundary values Owlow and Owup of the central 68% interval
In Table XV we include further details on the gauge
of this distribution. We then remove configurations i with ensembles. In Tables XVI–XVIII the results for the
unrenormalized charges for the three octet baryons are
jOwi − Ôj > KΔO; ðA4Þ collected.
TABLE XV. The gauge ensembles utilized in this work: the light and strange hopping parameters κ l and κ s , the gradient flow scale
parameter t0 =a2 and the pion (M π ) and kaon masses (M K ) [17].
Ensemble κl κs t0 =a2 aM π aM K
A653 0.1365716 0.1365716 2.1729(50) 0.21235(94) 0.21235(94)
A650 0.136600 0.136600 2.2878(72) 0.1833(13) 0.1833(13)
A654 0.136750 0.136216193 2.1950(77) 0.1669(11) 0.22714(91)
H101 0.13675962 0.13675962 2.8545(81) 0.18283(57) 0.18283(57)
U103 0.13675962 0.13675962 2.8815(57) 0.18133(61) 0.18133(61)
H107 0.13694566590798 0.136203165143476 2.7193(76) 0.15913(73) 0.23745(53)
H102r002 0.136865 0.136549339 2.8792(90) 0.15490(92) 0.19193(77)
U102 0.136865 0.136549339 2.8932(63) 0.15444(84) 0.19235(61)
H102r001 0.136865 0.136549339 2.8840(89) 0.15311(98) 0.19089(78)
rqcd021 0.136813 0.136813 3.032(15) 0.14694(88) 0.14694(88)
H105 0.136970 0.13634079 2.8917(65) 0.1213(14) 0.20233(64)
N101 0.136970 0.13634079 2.8948(39) 0.12132(58) 0.20156(30)
H106 0.137015570024 0.136148704478 2.8227(68) 0.1180(21) 0.22471(67)
C102 0.13705084580022 0.13612906255557 2.8682(47) 0.09644(77) 0.21783(36)
C101 0.137030 0.136222041 2.9176(38) 0.09586(64) 0.20561(33)
D101 0.137030 0.136222041 2.910(10) 0.0958(11) 0.20572(45)
S100 0.137030 0.136222041 2.9212(91) 0.0924(31) 0.20551(57)
D150 0.137088 0.13610755 2.9476(30) 0.05497(79) 0.20834(17)
B450 0.136890 0.136890 3.663(11) 0.16095(49) 0.16095(49)
S400 0.136984 0.136702387 3.6919(74) 0.13535(42) 0.17031(38)
B452 0.1370455 0.136378044 3.5286(66) 0.13471(47) 0.20972(34)
rqcd030 0.1369587 0.1369587 3.914(15) 0.12202(68) 0.12202(68)
N451 0.1370616 0.1365480771 3.6822(46) 0.11067(32) 0.17828(20)
N401 0.1370616 0.1365480771 3.6844(52) 0.10984(57) 0.17759(37)
N450 0.1370986 0.136352601 3.5920(42) 0.10965(31) 0.20176(18)
X450 0.136994 0.136994 3.9935(92) 0.10142(62) 0.10142(62)
D451 0.137140 0.136337761 3.6684(36) 0.08370(31) 0.19385(15)
D450 0.137126 0.136420428639937 3.7076(75) 0.08255(41) 0.18354(12)
D452 0.137163675 0.136345904546 3.7251(37) 0.05961(50) 0.18647(13)
(Table continued)
034512-35
TABLE XV. (Continued)
Ensemble κl κs t0 =a2 aM π aM K
N202 0.137000 0.137000 5.165(14) 0.13388(35) 0.13388(35)
N204 0.137112 0.136575049 4.9473(79) 0.11423(33) 0.17734(29)
X250 0.137050 0.137050 5.283(28) 0.11319(39) 0.11319(39)
N203 0.137080 0.136840284 5.1465(63) 0.11245(30) 0.14399(24)
S201 0.137140 0.13672086 5.1638(91) 0.09379(47) 0.15220(37)
N201 0.13715968 0.136561319 5.0427(75) 0.09268(31) 0.17040(22)
N200 0.137140 0.13672086 5.1600(71) 0.09236(29) 0.15061(24)
X251 0.137100 0.137100 5.483(26) 0.08678(40) 0.08678(40)
D200 0.137200 0.136601748 5.1793(39) 0.06540(33) 0.15652(15)
D201 0.1372067 0.136546844 5.1378(66) 0.06472(42) 0.16302(18)
E250 0.137232867 0.136536633 5.2027(41) 0.04227(23) 0.159370(61)
N300 0.137000 0.137000 8.576(21) 0.10642(38) 0.10642(38)
N304 0.137079325093654 0.136665430105663 8.322(20) 0.08840(33) 0.13960(31)
N302 0.137064 0.1368721791358 8.539(19) 0.08701(41) 0.11370(36)
J304 0.137130 0.1366569203 8.497(12) 0.06538(18) 0.13181(14)
J303 0.137123 0.1367546608 8.615(14) 0.06481(19) 0.11975(16)
E300 0.137163 0.1366751636177327 8.6241(74) 0.04402(20) 0.12397(15)
J500 0.136852 0.136852 14.013(34) 0.08116(34) 0.08116(34)
J501 0.1369032 0.136749715 13.928(39) 0.06589(26) 0.08798(23)
TABLE XVI. Results for the unrenormalized nucleon charges gN;latt

J for J ∈ fA; S; T; Vg. #ES labels the number of excited states used
to determine the ground state matrix element, see the discussion in Sec. III D.
1 2
#ES
Ensemble gN;latt
A gN;latt
S gN;latt
T gN;latt
V gN;latt
A gN;latt
S gN;latt
T gN;latt
V
A653 1.563(17) 1.449(60) 1.273(13) 1.4140(24) 1.589(18) 1.43(10) 1.279(13) 1.4136(21)

A650 1.554(13) 1.468(68) 1.236(10) 1.4173(35) 1.581(18) 1.41(13) 1.243(12) 1.4171(32)
A654 1.545(22) 1.62(21) 1.216(30) 1.4353(85) 1.538(33) 1.76(34) 1.211(19) 1.4365(87)
H101 1.5662(88) 1.760(55) 1.221(13) 1.39160(46) 1.584(16) 1.795(88) 1.224(13) 1.39152(46)
U103 1.495(17) 1.61(11) 1.219(23) 1.39001(77) 1.511(26) 1.68(13) 1.227(18) 1.38990(66)
H107 1.630(37) 1.46(13) 1.248(19) 1.40067(98) 1.667(29) 1.34(17) 1.260(20) 1.40051(77)
H102r002 1.596(21) 1.70(15) 1.221(22) 1.3986(11) 1.616(29) 1.84(21) 1.231(21) 1.39817(87)
U102a 1.449(41) 1.42(17) 1.180(56) 1.3988(22) 1.448(41) 1.32(24) 1.194(31) 1.3982(11)
H102r001 1.582(15) 1.67(10) 1.230(21) 1.39706(67) 1.598(32) 1.78(19) 1.235(21) 1.39698(62)
rqcd021 1.546(16) 1.67(12) 1.188(22) 1.383(13) 1.567(30) 1.60(23) 1.190(22) 1.383(12)
H105 1.533(29) 1.44(20) 1.194(37) 1.4077(18) 1.524(51) 1.29(44) 1.199(32) 1.4074(15)
N101a 1.623(19) 1.662(87) 1.229(16) 1.40416(63) 1.670(32) 1.64(21) 1.236(18) 1.40411(53)
H106 1.589(34) 1.30(27) 1.223(35) 1.4084(20) 1.597(64) 1.20(50) 1.231(37) 1.4081(17)
C102 1.699(36) 1.70(32) 1.184(25) 1.4083(13) 1.755(58) 1.74(54) 1.198(29) 1.4083(12)
C101 1.675(37) 1.67(17) 1.214(17) 1.40908(85) 1.758(44) 1.76(40) 1.232(18) 1.40872(69)
D101a 1.647(51) 1.74(34) 1.222(40) 1.4091(18) 1.700(80) 2.09(77) 1.236(36) 1.4088(11)
S100a 1.86(98) 1.6(1.2) 1.27(21) 1.4091(49) 1.75(16) 1.6(1.5) 1.257(57) 1.4072(25)
D150a 1.49(21) 2.8(2.9) 0.88(23) 1.430(13) 1.39(31) 5.0(4.0) 0.92(14) 1.430(15)
B450 1.549(12) 1.642(72) 1.228(13) 1.3729(31) 1.587(18) 1.61(13) 1.233(14) 1.3724(27)
S400 1.525(13) 1.645(90) 1.191(22) 1.37682(65) 1.523(28) 1.63(17) 1.193(20) 1.37683(59)
B452 1.555(15) 1.49(10) 1.232(17) 1.3781(38) 1.568(30) 1.39(24) 1.235(19) 1.3781(36)
rqcd030 1.510(14) 1.531(92) 1.178(15) 1.3769(33) 1.553(27) 1.47(20) 1.182(16) 1.3769(30)
N451 1.595(14) 1.603(89) 1.212(15) 1.3828(35) 1.636(31) 1.49(21) 1.217(18) 1.3828(30)
N401 1.584(26) 2.00(27) 1.166(37) 1.3836(11) 1.603(51) 2.27(46) 1.180(31) 1.38328(91)
N450 1.598(13) 1.67(11) 1.208(17) 1.3840(29) 1.625(31) 1.58(26) 1.210(19) 1.3838(29)
X450 1.558(23) 1.99(18) 1.166(23) 1.3884(78) 1.606(51) 2.15(37) 1.174(24) 1.3873(69)
(Table continued)
034512-36
TABLE XVI. (Continued)
1 2
#ES
Ensemble gN;latt
A gN;latt
S gN;latt
T gN;latt
V gN;latt
A gN;latt
S gN;latt
T gN;latt
V
D451 1.583(25) 1.42(25) 1.175(36) 1.3807(78) 1.554(80) 0.88(77) 1.176(38) 1.379(10)

D450 1.623(25) 1.46(30) 1.201(22) 1.3761(90) 1.672(66) 1.04(77) 1.207(27) 1.3769(84)
D452b 1.54(21) 0.1(4.5) 0.95(19) 1.07(42) 1.53(32) −1.0ð5.1Þ 1.00(13) 1.13(22)
N202 1.537(12) 1.926(84) 1.170(17) 1.34793(29) 1.556(20) 2.03(11) 1.179(16) 1.34780(28)
N204 1.570(13) 1.694(91) 1.200(14) 1.35344(39) 1.587(25) 1.80(19) 1.205(15) 1.35335(37)
X250 1.5232(98) 1.814(80) 1.163(14) 1.3513(25) 1.546(19) 1.88(15) 1.167(14) 1.3512(23)
N203 1.532(10) 1.748(68) 1.176(16) 1.35169(27) 1.545(26) 1.69(15) 1.176(16) 1.35169(27)
S201a 1.436(25) 1.49(21) 1.155(41) 1.35576(78) 1.52(12) 0.82(98) 1.160(41) 1.3555(10)
N201 1.568(18) 1.67(16) 1.158(21) 1.35653(50) 1.594(36) 1.54(35) 1.162(23) 1.35649(49)
N200 1.565(21) 1.47(15) 1.174(19) 1.35533(37) 1.607(40) 1.33(32) 1.179(22) 1.35530(34)
X251 1.532(25) 1.94(16) 1.135(14) 1.3568(56) 1.604(25) 2.18(26) 1.149(14) 1.3558(41)
D200 1.582(24) 1.94(30) 1.136(32) 1.35952(43) 1.617(61) 2.37(72) 1.140(32) 1.35997(77)
D201 1.562(35) 1.60(41) 1.135(48) 1.35922(82) 1.597(91) 1.82(98) 1.141(45) 1.35922(67)
E250a 1.66(14) 1.8(1.2) 1.134(77) 1.353(28) 2.00(20) 3.2(3.8) 1.175(55) 1.358(13)
N300 1.479(12) 1.813(95) 1.149(20) 1.31543(19) 1.485(22) 1.83(14) 1.152(19) 1.31540(16)
N304 1.495(23) 1.64(18) 1.137(28) 1.31926(32) 1.501(36) 1.52(31) 1.140(27) 1.31924(31)
N302 1.498(19) 1.82(18) 1.097(29) 1.31893(29) 1.523(35) 1.83(30) 1.103(27) 1.31889(27)
J304 1.529(19) 1.77(22) 1.107(22) 1.32310(36) 1.567(39) 1.67(51) 1.113(23) 1.32305(37)
J303 1.518(17) 1.51(14) 1.096(29) 1.32206(21) 1.526(56) 1.16(53) 1.094(36) 1.32198(25)
E300 1.557(37) 1.67(36) 1.086(25) 1.3135(88) 1.646(61) 1.44(92) 1.100(33) 1.312(11)
J500 1.451(11) 1.858(92) 1.108(18) 1.29090(12) 1.453(18) 1.91(11) 1.114(13) 1.290855(87)
J501 1.484(37) 2.13(30) 1.072(59) 1.29352(16) 1.499(38) 2.10(28) 1.088(29) 1.29349(19)
a
Ensemble only enters the analysis of the nucleon charges since no data for the hyperon charges are available.
b
The nucleon three-point functions are computed with the “stochastic” approach. (For all the other ensembles the sequential source
method was used.)
TABLE XVII. Results for the unrenormalized hyperon charges gJΣ;latt for J ∈ fA; S; T; Vg. #ES labels the number of excited states used
to determine the ground state matrix element, see the discussion in Sec. III D. The three-point functions are computed employing the
“stochastic” approach.
1 2
#ES
Ensemble gAΣ;latt gΣ;latt
S gTΣ;latt gVΣ;latt gAΣ;latt gΣ;latt
S gΣ;latt
T gVΣ;latt
A653 1.180(10) 4.40(14) 1.018(14) 2.8354(32) 1.189(16) 4.45(14) 1.023(12) 2.8347(27)
A650 1.1764(90) 4.46(12) 0.9968(95) 2.8381(39) 1.190(16) 4.49(15) 1.000(10) 2.8380(37)
A654 1.173(12) 4.01(16) 1.028(13) 2.852(12) 1.204(37) 4.11(30) 1.028(16) 2.850(14)
H101 1.1877(81) 5.03(12) 0.982(10) 2.78711(66) 1.198(13) 5.12(14) 0.987(10) 2.78686(65)
U103 1.137(13) 4.69(18) 0.983(16) 2.78535(89) 1.147(21) 4.75(18) 0.989(16) 2.78507(80)
H107 1.207(18) 4.25(19) 1.030(12) 2.823(15) 1.232(28) 4.35(27) 1.036(13) 2.820(11)
H102r002 1.177(13) 4.67(17) 0.988(13) 2.807(14) 1.199(25) 4.77(22) 0.992(14) 2.805(12)
H102r001 1.157(14) 4.73(19) 0.973(16) 2.788(11) 1.146(27) 4.79(28) 0.978(17) 2.788(11)
rqcd021 1.149(15) 5.27(24) 0.958(19) 2.790(14) 1.145(27) 5.35(33) 0.962(19) 2.790(13)
H105 1.147(22) 4.83(44) 0.958(23) 2.812(20) 1.162(40) 5.25(55) 0.969(21) 2.809(16)
H106 1.139(15) 4.24(20) 0.968(16) 2.796(15) 1.149(38) 4.27(48) 0.970(19) 2.796(15)
C102 1.140(19) 4.32(29) 0.950(27) 2.791(23) 1.133(45) 3.94(76) 0.952(22) 2.792(18)
C101 1.174(21) 4.58(43) 0.957(22) 2.837(21) 1.240(51) 4.80(94) 0.963(21) 2.834(17)
B450 1.1743(76) 4.80(14) 0.992(11) 2.7464(30) 1.195(15) 4.90(18) 0.997(11) 2.7460(28)
S400 1.159(16) 5.25(29) 0.945(17) 2.751(13) 1.180(23) 5.43(27) 0.962(16) 2.752(11)
B452 1.147(13) 4.33(18) 0.990(16) 2.764(13) 1.163(28) 4.45(33) 0.994(16) 2.763(12)
rqcd030 1.140(12) 5.20(22) 0.944(16) 2.7576(35) 1.157(22) 5.41(33) 0.950(15) 2.7573(33)
N451 1.168(15) 4.90(34) 0.976(14) 2.795(18) 1.193(30) 5.43(38) 0.988(11) 2.789(12)
N401 1.130(52) 5.22(68) 0.933(50) 2.781(42) 1.100(67) 5.23(82) 0.949(33) 2.776(31)
(Table continued)
034512-37
TABLE XVII. (Continued)
1 2
#ES
Ensemble gAΣ;latt gΣ;latt
S gTΣ;latt gVΣ;latt gAΣ;latt gΣ;latt
S gΣ;latt
T gVΣ;latt
N450 1.148(18) 4.11(17) 0.994(20) 2.738(24) 1.154(43) 3.54(65) 0.992(17) 2.737(21)
X450 1.168(23) 6.05(38) 0.924(25) 2.7700(84) 1.195(49) 6.38(58) 0.933(25) 2.7690(79)
D451 1.143(53) 4.79(95) 0.955(38) 2.67(10) 1.18(15) 3.8(3.0) 0.948(45) 2.624(56)
D450 1.129(63) 4.9(1.1) 0.938(52) 2.721(77) 1.09(11) 3.9(2.1) 0.944(42) 2.730(48)
D452 1.115(52) 4.8(1.1) 0.923(52) 2.789(28) 1.07(14) 5.3(2.6) 0.935(44) 2.787(29)
N202 1.166(10) 5.68(20) 0.939(12) 2.69777(45) 1.182(16) 5.85(15) 0.955(13) 2.69721(41)
N204 1.1465(91) 4.30(12) 0.980(11) 2.686(11) 1.160(26) 4.22(30) 0.979(12) 2.686(12)
X250 1.1549(88) 5.85(20) 0.928(12) 2.7051(32) 1.167(16) 6.12(21) 0.940(12) 2.7046(27)
N203 1.156(12) 5.60(26) 0.941(10) 2.7246(96) 1.182(19) 6.05(22) 0.9623(87) 2.7184(69)
N201 1.150(17) 5.22(43) 0.953(18) 2.737(17) 1.163(42) 5.70(53) 0.965(17) 2.733(14)
N200 1.119(13) 5.03(21) 0.947(13) 2.721(11) 1.116(26) 5.37(36) 0.954(12) 2.720(10)
X251 1.145(16) 6.59(33) 0.903(13) 2.7165(69) 1.195(24) 7.06(37) 0.916(14) 2.7145(55)
D200 1.082(48) 6.4(1.4) 0.841(59) 2.774(26) 1.058(82) 7.7(1.8) 0.867(38) 2.777(29)
D201 1.125(34) 5.7(1.3) 0.924(37) 2.730(31) 1.133(62) 6.8(1.2) 0.944(24) 2.726(19)
N300 1.121(12) 5.52(19) 0.918(17) 2.63208(25) 1.126(19) 5.59(18) 0.925(16) 2.63197(23)
N304 1.106(18) 4.84(31) 0.925(21) 2.642(17) 1.130(30) 4.87(45) 0.928(20) 2.641(15)
N302 1.086(26) 5.41(35) 0.864(34) 2.668(25) 1.118(42) 5.38(49) 0.868(26) 2.666(21)
J304 1.091(29) 5.95(65) 0.864(30) 2.646(25) 1.149(54) 6.61(74) 0.879(23) 2.645(18)
J303 1.100(19) 5.23(59) 0.914(22) 2.671(31) 1.198(61) 5.88(96) 0.927(17) 2.669(26)
E300 1.047(93) 7.9(3.8) 0.853(57) 2.728(87) 1.04(12) 9.1(3.1) 0.884(41) 2.705(55)
J500 1.100(10) 5.78(20) 0.887(13) 2.58250(15) 1.106(15) 5.82(16) 0.894(11) 2.58241(12)
J501 1.053(35) 6.01(86) 0.857(46) 2.629(22) 1.038(54) 6.26(66) 0.878(22) 2.628(19)
TABLE XVIII. Results for the unrenormalized hyperon charges gΞ;latt J for J ∈ fA; S; T; Vg. #ES labels the number of excited states
used to determine the ground state matrix element, see the discussion in Sec. III D. The three-point functions are computed employing
the “stochastic” approach.
1 2
#ES
Ensemble gΞ;latt
A gΞ;latt
S gΞ;latt
T gΞ;latt
V gΞ;latt
A gΞ;latt
S gΞ;latt
T gΞ;latt
V
A653 −0.399ð16Þ 3.13(18) −0.2495ð54Þ 1.4229(30) −0.401ð11Þ 3.019(89) −0.2534ð38Þ 1.4206(21)

A650 −0.3849ð96Þ 3.13(11) −0.2375ð43Þ 1.4213(28) −0.3905ð95Þ 3.109(92) −0.2423ð43Þ 1.4211(20)
A654 −0.3528ð58Þ 2.65(12) −0.2407ð47Þ 1.4344(67) −0.354ð12Þ 2.63(17) −0.2408ð59Þ 1.4349(67)
H101 −0.3810ð53Þ 3.331(91) −0.2342ð41Þ 1.39556(41) −0.3854ð82Þ 3.349(93) −0.2371ð48Þ 1.39528(40)
U103 −0.3580ð67Þ 3.09(11) −0.2389ð48Þ 1.39547(47) −0.362ð12Þ 3.05(14) −0.2380ð43Þ 1.39563(47)
H107 −0.3602ð71Þ 2.89(16) −0.2413ð41Þ 1.4165(82) −0.3653ð77Þ 2.93(12) −0.2453ð38Þ 1.4126(64)
H102r002 −0.3707ð54Þ 3.122(91) −0.2336ð34Þ 1.4045(52) −0.3708ð92Þ 3.16(12) −0.2357ð42Þ 1.4038(48)
H102r001 −0.3671ð52Þ 3.12(10) −0.2362ð35Þ 1.3987(55) −0.3683ð99Þ 3.16(14) −0.2383ð45Þ 1.3987(52)
rqcd021 −0.418ð17Þ 3.93(22) −0.2211ð82Þ 1.415(15) −0.425ð16Þ 3.81(21) −0.2275ð71Þ 1.409(10)
H105 −0.370ð11Þ 3.25(26) −0.2214ð68Þ 1.4050(86) −0.372ð11Þ 3.24(22) −0.2261ð72Þ 1.4047(73)
H106 −0.3448ð45Þ 2.605(83) −0.2347ð27Þ 1.4052(42) −0.3553ð84Þ 2.56(22) −0.2359ð39Þ 1.4049(46)
C102 −0.3431ð42Þ 2.672(96) −0.2296ð41Þ 1.4007(52) −0.341ð13Þ 2.52(28) −0.2286ð50Þ 1.4000(57)
C101 −0.373ð23Þ 3.15(38) −0.2398ð58Þ 1.423(13) −0.387ð14Þ 3.19(35) −0.2436ð33Þ 1.4177(99)
B450 −0.3822ð73Þ 3.321(81) −0.2322ð46Þ 1.3737(22) −0.3914ð98Þ 3.289(96) −0.2352ð56Þ 1.3737(19)
S400 −0.3560ð57Þ 3.20(12) −0.2332ð41Þ 1.3774(56) −0.3577ð91Þ 3.28(13) −0.2373ð50Þ 1.3778(51)
B452 −0.3484ð52Þ 2.873(96) −0.2309ð35Þ 1.3669(53) −0.3523ð85Þ 2.90(13) −0.2333ð45Þ 1.3680(57)
rqcd030 −0.384ð11Þ 3.93(22) −0.2249ð65Þ 1.3819(34) −0.394ð14Þ 3.96(25) −0.2313ð67Þ 1.3807(28)
N451 −0.3693ð57Þ 3.55(15) −0.2183ð42Þ 1.3969(68) −0.3720ð94Þ 3.74(17) −0.2248ð46Þ 1.3948(55)
N401 −0.372ð13Þ 3.17(18) −0.2296ð58Þ 1.389(13) −0.389ð16Þ 3.07(32) −0.2325ð68Þ 1.388(12)
N450 −0.3489ð63Þ 2.92(14) −0.2294ð42Þ 1.3764(77) −0.351ð14Þ 2.87(27) −0.2297ð59Þ 1.3759(78)
X450 −0.403ð18Þ 4.29(31) −0.2314ð88Þ 1.3809(53) −0.414ð23Þ 4.39(34) −0.2369ð74Þ 1.3810(45)
D451 −0.308ð19Þ 2.70(25) −0.226ð19Þ 1.361(35) −0.286ð36Þ 1.87(98) −0.225ð13Þ 1.351(24)
(Table continued)
034512-38
TABLE XVIII. (Continued)
1 2
#ES
Ensemble gΞ;latt
A gΞ;latt
S gΞ;latt
T gΞ;latt
V gΞ;latt
A gΞ;latt
S gΞ;latt
T gΞ;latt
V
D450 −0.365ð21Þ 3.79(88) −0.2334ð98Þ 1.391(18) −0.372ð29Þ 4.27(62) −0.2377ð53Þ 1.391(11)

D452 −0.320ð13Þ 3.75(66) −0.2213ð80Þ 1.399(13) −0.292ð33Þ 4.41(95) −0.2268ð84Þ 1.397(11)
N202 −0.3758ð48Þ 3.78(10) −0.2202ð41Þ 1.34982(24) −0.3759ð89Þ 3.817(99) −0.2244ð50Þ 1.34959(23)
N204 −0.3365ð44Þ 2.916(92) −0.2262ð35Þ 1.3484(47) −0.326ð11Þ 2.91(15) −0.2252ð46Þ 1.3479(51)
X250 −0.3753ð54Þ 4.18(14) −0.2195ð43Þ 1.3541(24) −0.3782ð80Þ 4.26(14) −0.2257ð46Þ 1.3536(19)
N203 −0.3612ð42Þ 3.652(95) −0.2178ð30Þ 1.3549(53) −0.3649ð75Þ 3.74(12) −0.2220ð36Þ 1.3542(47)
N201 −0.350ð11Þ 3.44(32) −0.2222ð49Þ 1.377(12) −0.355ð10Þ 3.56(20) −0.2285ð47Þ 1.3714(79)
N200 −0.3515ð42Þ 3.58(13) −0.2147ð38Þ 1.3665(49) −0.3542ð78Þ 3.73(16) −0.2184ð42Þ 1.3653(43)
X251 −0.3942ð89Þ 4.68(18) −0.2226ð60Þ 1.3600(46) −0.409ð11Þ 4.84(21) −0.2295ð43Þ 1.3576(40)
D200 −0.347ð16Þ 4.2(1.0) −0.2127ð76Þ 1.3738(89) −0.350ð15Þ 4.69(49) −0.2237ð51Þ 1.3707(70)
D201 −0.3393ð78Þ 3.38(26) −0.2082ð60Þ 1.3647(83) −0.344ð17Þ 3.47(51) −0.2100ð74Þ 1.3645(84)
N300 −0.3596ð58Þ 3.74(10) −0.2246ð45Þ 1.31662(16) −0.359ð10Þ 3.78(11) −0.2279ð49Þ 1.31649(12)
N304 −0.3307ð51Þ 3.12(11) −0.2196ð43Þ 1.3182(66) −0.325ð10Þ 3.09(17) −0.2169ð42Þ 1.3175(70)
N302 −0.361ð11Þ 3.77(19) −0.2136ð60Þ 1.341(11) −0.373ð15Þ 3.72(22) −0.2170ð66Þ 1.3383(89)
J304 −0.3366ð73Þ 3.45(22) −0.2090ð50Þ 1.3286(83) −0.346ð11Þ 3.63(28) −0.2143ð58Þ 1.3277(67)
J303 −0.3416ð75Þ 3.66(23) −0.2133ð48Þ 1.3306(94) −0.358ð15Þ 3.88(34) −0.2208ð59Þ 1.3298(81)
E300 −0.334ð17Þ 3.75(49) −0.194ð13Þ 1.342(14) −0.340ð31Þ 3.5(1.0) −0.195ð14Þ 1.344(17)
J500 −0.3508ð46Þ 3.89(11) −0.2174ð36Þ 1.291575(73) −0.3454ð91Þ 3.93(11) −0.2198ð38Þ 1.291515(71)
J501 −0.353ð10Þ 4.14(25) −0.2055ð62Þ 1.314(11) −0.356ð13Þ 4.17(21) −0.2071ð55Þ 1.3135(98)
APPENDIX C: ADDITIONAL FIGURES

We provide additional figures (Figs. 40–47).
FIG. 40. The same as Fig. 10 for the sigma baryon.
034512-39
FIG. 41. The same as Fig. 13 for the isovector vector charges gBV 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 XVII and XVIII), are not displayed. See Tables XVII
and XVIII for the set of ensembles used.
FIG. 42. The same as Fig. 15 for the vector charge gBV of the sigma (right) and cascade (left) baryon.
FIG. 43. The same as Fig. 19 for the axial charge gBA of the sigma (right) and cascade (left) baryon.
034512-40
FIG. 44. The same as Fig. 19 for the scalar charge gBS 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 datasets, DSðM π<400 MeV Þ, DSðM π<400 MeV ; a<0.1 fm Þ and
DSðM <400 π Þ. The data are extracted using two excited states in the fitting analysis, see Sec. III D. Only two different sets of
MeV
π ; LM >4
renormalization factors are used.
FIG. 45. The same as Fig. 19 for the tensor charge gBT 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 D. Only two different sets of renormalization
factors are used.
034512-41
FIG. 46. The same as Fig. 21 for the isovector scalar charges gBS 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 XVII), is not displayed for the sigma baryon.
See Tables XVII and XVIII for the set of ensembles used.
FIG. 47. The same as Fig. 23 for the isovector tensor charges gBT 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 XVIII), is not displayed for the cascade baryon.
See Tables XVII and XVIII for the set of ensembles used.
034512-42
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Adaptive aggregation based domain decomposition arXiv:2303.08741.
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PHYSICAL REVIEW D 108, 034512 (2023)

Octet baryon isovector charges from N f = 2 + 1 lattice QCD

(Received 15 May 2023; accepted 7 August 2023; published 29 August 2023)

I. INTRODUCTION moment of the helicity structure functions g1 ðxÞ is related

2470-0010=2023=108(3)=034512(47) 034512-1 Published by the American Physical Society

of the pion nucleon σ term, see Ref. [17] for a summary of

Ensemble β a [fm] Trajectory bc N t × N 3s Mπ [MeV] MK [MeV] LM π t=a N cnfg

Ensemble β a [fm] Trajectory bc N t × N 3s Mπ [MeV] MK [MeV] LM π t=a N cnfg

FIG. 5. Quark-line diagrams of the two-point (left) and con-

constructing a sequential source [41] which depends on

C. Stochastic three-point correlation functions

 The spectator and insertion parts are defined as

Using these building blocks, three-point functions for given

The number of stochastic estimates N sto is chosen by X

TABLE II. Summary of the fits performed. We vary the

Fit J δt ES Prior Set to zero

8 A, S, T, V δt2 2 ΔE1 T;V

Mphys ¼ 134.8ð3Þ MeV; ð33Þ A. Vector charges

(3) pion mass cut: excluding all ensembles with

spacing dependence is moderate but statistically signifi-

TABLE X. Results for gBA, B ∈ fN; Σ; Ξg, obtained with three

Renormalization gNA gΣA gΞA

Z2A (Table VI) 1.295ð29Þ

Z3A (Table VII) 1.284ð27Þ

charges from determinations of the up and down quark

lattice estimates for δm and an average of lattice and

ð28Þ ð30Þ ð13Þ

The result for the nucleon compares favorably with the

(i.e., nucleon, hyperon and Δ) axial couplings employing

reasonable agreement with our value, while there is a large

where mB and ml correspond to the baryon and the light

ð14Þ ð22Þ ð11Þ

which agree with our results to within two standard

FIG. 30. The ratios 2gBA =gΣA − 1 for B ∈ fN; Ξg as a function of

FIG. 33. The SU(3) symmetry breaking ratio δASUð3Þ ðxÞ

The other splittings read [75]

ΔmΞ ¼ −6.85ð21Þ MeV; ΔmN ≈ −1.293 MeV: ð50Þ

The splitting depends on the scale, the renormalization

where the leading correction is due to q0 ¼ p00 − p0 ¼

E. Isospin breaking effects on the

ð30Þ ð22Þ ð15Þ

N½I ¼ N IF · N q · N ∂μ · N τ · N src · N sto · N c · N 2s : ðA2Þ

gauge ensembles employed here. We remark that such where

TABLE XV. (Continued)

TABLE XVI. Results for the unrenormalized nucleon charges gN;latt

A653 1.563(17) 1.449(60) 1.273(13) 1.4140(24) 1.589(18) 1.43(10) 1.279(13) 1.4136(21)

TABLE XVI. (Continued)

D451 1.583(25) 1.42(25) 1.175(36) 1.3807(78) 1.554(80) 0.88(77) 1.176(38) 1.379(10)

TABLE XVII. (Continued)

A653 −0.399ð16Þ 3.13(18) −0.2495ð54Þ 1.4229(30) −0.401ð11Þ 3.019(89) −0.2534ð38Þ 1.4206(21)

TABLE XVIII. (Continued)

D450 −0.365ð21Þ 3.79(88) −0.2334ð98Þ 1.391(18) −0.372ð29Þ 4.27(62) −0.2377ð53Þ 1.391(11)

APPENDIX C: ADDITIONAL FIGURES

FIG. 40. The same as Fig. 10 for the sigma baryon.

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The spectator and insertion parts are defined as