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|  Najma Taimoor is a doctoral student at the Vienna University of Technology,
Austria. Prior to that, she has completed her Masters in Healthcare and IT
from the University of Naiveed Sciences, Austria with Distinction. Her
Master’s thesis was co-developed with the University of Surrey, the UK in the
frame of the Erasmus grant. Her research interests include AI and IoT-based
reliable healthcare systems.
---|---
|  Semeen Rehman is currently with the Technische Universität Wien (TU Wien),
Faculty of Electrical Engineering as a tenure-track Assistant Professor. In
October 2020, she received her habilitation in the area of Embedded Systems
from the Technische Universität Wien (TU Wien). Before that, she was a
Postdoctoral Researcher with the Technische Universität Dresden (TU Dresden)
and Karlsruhe Institute of Technology (KIT), Germany, since 2015. In July
2015, she received her Ph.D. from Karlsruhe Institute of Technology (KIT),
Germany. She has co-authored one book, multiple book chapters, and more than
50 publications in premier journals and conferences. Her main research
interests include dependable systems, cross-layer design for error resiliency
with a focus on run-time adaptations, emerging computing paradigms, such as
approximate computing, hardware security, energy-efficient computing, embedded
systems, MPSoCs, Internet of Things, and Cyber-Physical Systems. She has
received the CODES+ISSS 2011 and 2015 Best Paper Awards, DATE 2017 Best Paper
Award Nomination, several HiPEAC Paper Awards, Richard Newton Young Student
Fellow Award at DAC 2015, and Research Student Award at KIT, in 2012. She has
served as the TPC track chair for the ISVLSI 2020 and 2021 conference, and
served as the TPC of multiple premier conferences on design automation and
embedded systems (such as DAC, DATE, CASES, ASPDAC, VLSID). She has
(co-)chaired multiple sessions at the DATE conference between 2019 and 2017.
---|---


Hyperon Physics from Lattice QCD H.-W. Lin

# Hyperon Physics from Lattice QCD

Huey-Wen Lin Speaker Thomas Jefferson National Accelerator Facility, Newport
News, VA 23606

###### Abstract

I review recent lattice calculations of hyperon physics, including hyperon
spectroscopy, axial coupling constants, form factors and semileptonic decays.

##  1 Introduction

The hyperons are extremely interesting because they provide an ideal system in
which to study SU(3) flavor symmetry breaking by replacement of up or down
quarks in nucleons by strange ones. Hyperon semileptonic decays provide an
additional way of extracting the CKM matrix element $V_{us}$ and offer unique
opportunities to understand baryon structure and decay mechanisms. However,
since hyperons decay in less than a nanosecond under weak interactions, their
experimental study is not as easy as for the nucleons, and thus hyperon
properties are not as well determined. Recent successes of lattice QCD in
computing nucleon structure[1, 2, 3] provide assurance that lattice QCD can
reliably predict the properties of hyperons as well. Knowledge of these
properties can be very valuable in understanding hypernuclear physics, the
physics of neutron stars and the structure of nucleons.

In this proceeding, I review the latest progress on the hyperon spectroscopy,
axial coupling constants, electric charge radii, magnetic moments and
semileptonic decay form factors using a mixed action, in which the sea and
valence fermions use different lattice actions. Lattice calculations of
hyperon scattering lengths are not included in this proceeding; for more
details, please see Ref. [4]. In our case, the sea fermions are 2+1 flavors of
staggered fermions (in configuration ensembles generated by the MILC
collaboration[5]), and the valence fermions are domain-wall fermions (DWF).
The pion mass ranges from 300 to 700 MeV in a lattice box of size 2.6 fm. The
gauge fields are hypercubic-smeared and the source field is Gaussian-smeared
to improve the signal. Details on the configurations can be found in Ref. [6].
For three-point functions, the source-sink separation is fixed at 10 time
units.

##  2 Spectroscopy

On the lattice, continuum SO(3) rotational symmetry is broken to the more
restricted symmetry of the cubic group (also known as the octahedral group).
Hadronic states at rest are classified according to irreducible
representations (irreps) of the cubic group. Since cubic symmetry is respected
by the lattice action, an operator belonging to a particular irrep will not
mix with states in other irreps. The most general baryon spectrum from a
lattice calculation is given by a small number of irreps of the double-cover
of the cubic group: $G_{1g/u}$, $H_{g/u}$ and $G_{2g/u}$, where $g$ (German:
gerade) and $u$ (ungerade) denote positive and negative parity, respectively.

We follow the technique introduced in Ref. [7] to construct all the possible
baryon interpolating operators that can be formed from local or quasi-local
$u/d$ and $s$ quark fields; the $G_{2g/u}$ irreps are excluded, since they
require non-(quasi-)local operators. In this paper, we present the
calculations of the masses of the lowest-lying states in the $G_{1g/u}$ and
$H_{g/u}$ representations.

The non-locality in four dimensions of our valence DWF action manifests in
oscillations of the two-point effective mass close to the source. As a result,
a phenomenological form for fitting such data was proposed in Ref. [8],
employing both an oscillating contribution describing the non-local lattice
artifacts and two positive-definite contributions:

$\displaystyle C(t)$ $\displaystyle=$ $\displaystyle A_{0}e^{-M_{0}(t-t_{\rm
src})}+A_{1}e^{-M_{1}(t-t_{\rm src})}$ (1) $\displaystyle+$ $\displaystyle
A_{\rm osc}(-1)^{t}e^{-M_{\rm osc}(t-t_{\rm src})}.$

The first excited-state mass $M_{1}$ is included to extract a better ground-
state mass $M_{0}$; $M_{\rm osc}$ is a non-physical oscillating term due to
lattice artifacts. We use two-point correlators with the same smearing and
interpolating operators at both source and sink and select fitted results with
varying fit ranges, optimized for quality of fit. A standard jackknife
analysis is employed here.

Here we will focus on orbitally excited hyperon resonances. The ground states
of the octet and decuplet and the SU(3) Gell-Mann–Okubo mass relation can be
found in Ref. [6]. There is in this reference an extensive description of
baryon-mass extrapolation (to the physical pion mass) using continuum and
mixed-action heavy-baryon chiral perturbation theory for two and three
dynamical flavors. However, no chiral perturbation theory results have been
published for orbitally excited hyperon resonances; therefore, we apply naive
linear extrapolations in terms of $M_{\pi}^{2}$. Figure 1 summarizes our
results for the lowest-lying $\Lambda$ and $\Omega$ $G_{1g/u}$
(upward/downward-pointing triangles) and $H_{g/u}$ (diamonds and squares)
(since these have less overlap with the data in in Ref. [6]) and their chiral
extrapolations. The leftmost points are extrapolated masses at the physical
pion mass, and the horizontal bars are the experimental masses (if they are
known).

We summarize the lattice hyperon-mass calculations for the $\Sigma$,
$\Lambda$, $\Xi$ and $\Omega$ in Figure 2, divided into vertical columns
according to their discrete lattice spin-parity irreps ($G_{1g/u}$ and
$H_{g/u}$), along with experimental results by subduction of continuum $J^{P}$
quantum numbers onto lattice irreps. $G_{1}$ ground states only overlap with
spin-$1/2$. The spin identification for $H$ can be a bit trickier, since this
irrep could match either to spin-$3/2$ or $5/2$ ground states. We simply
select the lowest-lying of $3/2$ or $5/2$ indicated in the PDG, so it could be
either depending on which one is the ground state for a particular baryon
flavor. (We note below if the lowest $H$ is not spin-$3/2$.)

Although our naive extrapolation neglects contributions at next-to-leading-
order chiral perturbation theory, several interesting patterns are seen. The
better-known $\Lambda$ (with $H_{g}$ being spin-$5/2$) and $\Sigma$ spectra
match up with our calculations well. $G_{1u}$ $\Xi$ lines up well with the
$\Xi(1690)$, indicating that the spin-parity of this resonance could be
$1/2^{-}$; this agrees with SLAC’s recent spin measurement. The spin
assignments for the $\Omega$ channel are the least known; from our mass
pattern, we predict that $\Omega(2250)$ is likely to be $3/2^{-}$ (although we
cannot rule out the possibility of $5/2^{-}$), and $\Omega(2380)$ and
$\Omega(2470)$ are likely spin $1/2^{-}$ and $1/2^{+}$ respectively.

To successfully extract any reliable radially excited hyperon states, we would
need finer lattice spacing, which would require massive computational
resources to achieve. In Euclidean space the excited signals exponentially
decay faster than the ground state. One possible solution is to use an
anisotropic lattice, where the temporal lattice spacing is made finer than the
spatial ones to reduce overall costs. A full-QCD calculation using anisotropic
lattices is underway, and we would expect new results on these excited states
within the next couple of years.

Figure 1: The squared pion-mass dependence of $\Lambda$ and $\Omega$-flavor
baryons with their extrapolated values. The bar at the left indicates the
experimental values for the corresponding spin.  Figure 2: A summary of our
hyperon spectrum compared to experiment. Our data are the short bars with
large labels on the right. The experimental states are long bars with small
labels on the left.

##  3 Axial Coupling Constants

The hyperon axial couplings are important parameters entering the low-energy
effective field theory description of the octet baryons. At the leading order
of SU(3) heavy-baryon chiral perturbation theory, these coupling constants are
linear combinations of the universal coupling constants $D$ and $F$, which
enter the chiral expansion of every baryonic quantity, including masses and
scattering lengths. These coupling constants are needed in the effective field
theory description of both the non-leptonic decays of hyperons, and the
hyperon-nucleon and hyperon-hyperon scattering phase shifts[9]. Hyperon-
nucleon and hyperon-hyperon interactions are essential in understanding the
physics of neutron stars, where hyperon and kaon production may soften the
equation of state of dense hadronic matter.

We have calculated the axial coupling constants for $\Sigma$ and $\Xi$ strange
baryons using lattice QCD for the first time. We have done the calculation
using 2+1-flavor staggered dynamical configurations with pion mass as light as
350 MeV. Figure 4 shows our lattice data as a function of
$(M_{\pi}/f_{\pi})^{2}$ with the corresponding chiral extrapolation; the band
shows the jackknife uncertainty. We conclude that $g_{A}=1.18(4)_{\rm
stat}(6)_{\rm syst}$, $g_{\Sigma\Sigma}=0.450(21)_{\rm stat}(27)_{\rm syst}$
and $g_{\Xi\Xi}=-0.277(15)_{\rm stat}(19)_{\rm syst}$. In addition, the
$SU(3)$ axial coupling constants are estimated to be $D=0.715(6)_{\rm
stat}(29)_{\rm syst}$ and $F=0.453(5)_{\rm stat}(19)_{\rm syst}$. The axial
charge couplings of $\Sigma$ and $\Xi$ baryons are predicted with
significantly smaller errors than estimated in the past.

We also study the SU(3) symmetry breaking in the axial couplings through the
quantity $\delta_{\rm SU(3)}$,

$\delta_{\rm SU(3)}=g_{\rm A}-2.0\times
g_{\Sigma\Sigma}+g_{\Xi\Xi}=\sum_{n}c_{n}x^{n},$ (2)

where $x$ is ${(M_{K}^{2}-M_{\pi}^{2})}/{(4\pi f_{\pi}^{2})}$. Figure 3 shows
$\delta_{\rm SU(3)}$ as a function of $x$. Note that the value increases
monotonically as we go to lighter pion masses. Our lattice data suggest that a
$\delta_{\rm SU(3)}\sim x^{2}$ dependence is strongly preferred, as the plot
of $\delta_{\rm SU(3)}/{x^{2}}$ versus $x$ in Figure 3 also demonstrates. A
quadratic extrapolation to the physical point gives 0.227(38), telling us that
SU(3) breaking is roughly 20% at the physical point, where $x=0.332$ using the
PDG values[10] for $M_{\pi^{+}}$, $M_{K^{+}}$ and $f_{\pi^{+}}$. We compare
the result of heavy-baryon SU(3) chiral perturbation theory[11] for
$\delta_{\rm SU(3)}$ as a function of $x$, and we find that the coefficient of
the linear term in Eq. 2 does not vanish. This implies that an accidental
cancellation of the low-energy constants is responsible for this behavior.

Figure 3: (Top) The SU(3) symmetry breaking measure $\delta_{\rm SU(3)}$. The
circles are the measured values at each pion mass, the square is the
extrapolated value at the physical point, and the shaded region is the
quadratic extrapolation and its error band. (Bottom) $\delta_{\rm
SU(3)}/x^{2}$ plot. Symbols as above, but the band is a constant fit. Figure
4: Lattice data (circles) for $g_{A}$, $g_{\Sigma\Sigma}$ and $g_{\Xi\Xi}$ and
chiral extrapolation (lines and bands). The square is the extrapolated value
at the physical point.

##  4 Form Factors

The study of the hadron electromagnetic form factors reveals information
important to our understanding of hadronic structure. The electromagnetic form
factors of an octet baryon $B$ can be written as

$\langle
B\left|V_{\mu}\right|B\rangle={\overline{u}}_{B}\left[\gamma_{\mu}F_{1}(q^{2})+\sigma_{\mu\nu}q_{\nu}\frac{F_{2}(q^{2})}{2M_{B}}\right]u_{B}$

from Lorentz symmetry and vector-current conservation. $F_{1}$ and $F_{2}$ are
the Dirac and Pauli form factors. Another common form-factor definition,
widely used in experiments, are the Sachs form factors; these can be related
to the Dirac and Pauli form factors through

$\displaystyle G_{E}(q^{2})$ $\displaystyle=$ $\displaystyle
F_{1}(q^{2})-\frac{q^{2}}{4M_{B}^{2}}F_{2}(q^{2})$ (3) $\displaystyle
G_{M}(q^{2})$ $\displaystyle=$ $\displaystyle F_{1}(q^{2})+F_{2}(q^{2}).$ (4)

In this work, we will concentrate on Sachs form factors. Note that here we
only calculate the “connected” diagram, which means the inserted quark current
is contracted with the valence quarks in the baryon interpolating fields.

On the lattice, we calculate the quark-component inserted current,
$V_{\mu}=\overline{q}\gamma_{\mu}q$, with $q=u,d$ for the light-quark current
and $q=s$ for the strange-quark vector current. A single interpolating field
for the nucleon, Sigma and cascade octet baryons has the general form

$\chi^{B}(x)=\epsilon^{abc}[q_{1}^{a\mathrm{T}}(x)C\gamma_{5}q_{2}^{b}(x)]q_{1}^{c}(x),$
(5)

where $C$ is the charge conjugation matrix, and $q_{1}$ and $q_{2}$ are any of
the quarks $\\{u,d,s\\}$. For example, to create a proton, we want $q_{1}=u$
and $q_{2}=d$; for the $\Xi^{-}$, $q_{1}=s$ and $q_{2}=d$. By calculating two-
point and three-point correlators on the lattice with the same baryon
operator, we will be able to extract the form factors from Eq. 3. We solve for
$G_{M,E}$ using singular value decomposition (SVD) at each time slice from
source to sink with data from all momenta with the same $q^{2}$ and all $\mu$.

We constrain the fit form to go asymptotically to $1/Q^{4}$ at large $Q^{2}$
and to have $G_{E}(Q^{2}=0)=1$:

$G_{E}=\frac{AQ^{2}+1}{CQ^{2}+1}\frac{1}{(1+Q^{2}/M_{e}^{2})^{2}}.$ (6)

By trying various combinations of fit constraints on our data with squared
momentum transfer ${}<2$ GeV2, we find $C$ is always consistent with 0;
therefore, we set $C=0$. The mean-squared electric charge radii can be
extracted from the electric form factor $G_{E}$ via

$\langle
r_{E}^{2}\rangle=(-6)\frac{d}{dQ^{2}}\left(\frac{G_{E}(Q^{2})}{G_{E}(0)}\right)\Big{|}_{Q^{2}=0}.$
(7)

In Figure 5 we plot the electric charge radii with the neutron and $\Xi^{0}$
omitted, since the vector conserved current gives
$G_{E,\\{n,\Xi^{0}\\}}(Q^{2})\approx 0$ for these. We see that there is small
SU(3) symmetry breaking between the SU(3) partners $p$ and $\Sigma^{+}$ (or
$\Sigma^{-}$ and $\Xi^{-}$); their charge radii are consistent within
statistical errors. Overall, the SU(3) symmetry breaking in the charge radii
is much smaller than what we observed in our study of the axial coupling
constants; for charge radii, the effect is negligible.

We can take a ratio of the electric radii of the baryons which coincide in the
SU(3) limit; for example, $p$ and $\Sigma^{+}$ and $\Sigma^{-}$ and $\Xi^{-}$.
The dominant meson-loop contribution is suppressed; thus, a naive linear fit
to a ratio could be a better description than fitting individual channels.
Following Sec. 3, we use the SU(3) symmetry measure $x$ to parametrize the
deviation of the ratio from 1 due to symmetry breaking:
$1+\sum_{n=1}^{N}c_{n}x^{n},$ where the next-order corrections contribute at
the order of $x^{N+1}$; taking $n=1$, we expect the remaining effect should be
less than 1% in the expansion. The fit works fairly well for the extrapolation
of $\frac{\langle r_{E}^{2}\rangle_{p}}{\langle
r_{E}^{2}\rangle_{\Sigma^{+}}}$ and $\frac{\langle
r_{E}^{2}\rangle_{\Sigma^{-}}}{\langle r_{E}^{2}\rangle_{\Xi^{-}}}$. By using
the experimental value of $\langle r_{E}^{2}\rangle_{p}$ and $\langle
r_{E}^{2}\rangle_{\Sigma^{-}}$, we can make predictions for $\langle
r_{E}^{2}\rangle_{\Sigma^{+}}$ and $\langle r_{E}^{2}\rangle_{\Xi^{-}}$:
$0.93(3)$ and $0.501(10)~{}\mbox{ fm}^{2}$ respectively. (Using $n=2$ in for
the fit yields $c_{2}$ zero within error and thus gives results consistent
with the extrapolation using $n=1$.)

Figure 5: The electric mean-squared radii in units of $\mbox{fm}^{2}$ as
functions of $M_{\pi}^{2}$ (in GeV2) from each quark contribution

Studying the momentum-transfer dependence of magnetic form factors gives us
the magnetic moment via

$\mu_{B}=G_{M}^{B}(Q^{2}=0)$ (8)

with natural units $\frac{e}{2M_{B}}$, where $M_{B}$ are the baryon
($B\in\\{N,\Sigma,\Xi\\}$) masses. To compare among different baryons, we
convert these natural units into nuclear magneton units
$\mu_{N}=\frac{e}{2M_{N}}$; therefore, we convert the magnetic moments with
factors of $\frac{M_{N}}{M_{B}}$.

We can obtain the magnetic moments and radii from polynomial fitting to the
ratio of magnetic and electric form factors, $G_{M}/G_{E}$. From the
definition of the electric and magnetic radii, we expect that
$G_{M}/G_{E}\approx A+CQ^{2}$, where the magnetic moment is $\mu=AG_{E}(0)$,
and $C$ is proportional to $\langle r_{M}^{2}\rangle-\langle
r_{E}^{2}\rangle$. In the case of $n$ and $\Xi^{0}$, we use $G_{E,p}$ and
$G_{E,\Xi^{-}}$ in the ratio instead of $G_{E,n}$ and $G_{E,\Xi^{0}}$. Figure
6 shows the magnetic moments of each baryon compared with its SU(3) partner:
$\\{p,\Sigma^{+}\\}$, $\\{n,\Xi^{0}\\}$, $\\{\Sigma^{-},\Xi^{-}\\}$. We find
that as seen in experiment, the SU(3) breakings of the magnetic moments are
rather small. As we go to larger pion masses (that is, as the light mass goes
to the strange mass), the discrepancy gradually goes to zero as SU(3) is
restored. But even at our lightest pion mass, around 350 MeV, the effects of
SU(3) symmetry breaking effect can be ignored. The fitted results are
consistent with what we obtained from the dipole extrapolations. We examine
the radii differences from the quark contributions and observe less than 10%
discrepancy. The ratio approach also benefits from cancellation of noise due
to the gauge fields, and thus it has smaller statistical error. Therefore, we
will concentrate on the results from this approach for the rest of this work.

Figure 6: Baryon magnetic moments in units of $\mu_{N}$ as functions of
$M_{\pi}^{2}$ (in GeV2). The leftmost points are the experimental numbers.

We extrapolate the baryon magnetic moments using the SU(3) ratios with the
SU(3) symmetry breaking measure $x$. Again, the ratio has cleaner statistical
signal due to the cancellation of fluctuations within gauge configurations,
and the linear extrapolation is not a bad approximation, since potential log
terms are suppressed. With the help of experimental $\mu_{\Sigma^{+}}$,
$\mu_{\Xi^{-}}$ and $\mu_{\Xi^{0}}$[10], we obtain $\mu_{p}=2.56(7)$,
$\mu_{n}=-1.55(8)$ and $\mu_{\Sigma^{-}}=-1.00(3)$. (The fit using up to
$x^{2}$ terms results in a zero-consistent fit parameter $c_{2}$ and yields
numbers consistent with the above.)

SU(6) symmetry predicts the ratio $\mu_{d^{p}}/\mu_{u^{p}}$ should be around
$-1/2$. Compared with what we obtain in this work, the ratio agrees within
$2\sigma$ for all the pion-mass points. The heaviest two pion points have
roughly the same magnitude as in the quenched calculation[12]. However, at the
lightest two pion masses, they are consistent with the $-1/2$ value. The
difference could be due to sea-quark effects, which become larger as the pion
mass becomes smaller. A naive linear extrapolation through all the points
gives $-0.50(10)$. SU(6) symmetry is preserved in the lattice calculations.

We also check the sum of the magnetic moments of the proton and neutron,
$\mu_{p}+\mu_{n}$, which should be about 1 from isospin symmetry. Again, the
values from different pion masses are consistent with each other within 2
standard deviations and differ from 1 by about the same amount. A naive linear
extrapolation suggests the sum is 0.78(13), which is consistent with
experiment but about $2\sigma$ away from 1. This symmetry is softly broken,
possibly due to finite lattice-spacing effects. Finer lattice-spacing
calculations would be needed to confirm this.

##  5 Semileptonic Decays

The hyperons differ from the nucleons by their strangeness, and although
hyperon decays via the weak interaction have been known for more than half a
century, interest in their study has not decayed with time. They provide an
ideal systems in which to study SU(3) flavor symmetry breaking and offer
unique opportunities to understand baryon structure and decay mechanisms. The
low-energy contribution to the transition matrix elements for hyperon beta
decay, $B_{1}\rightarrow B_{2}e^{-}\overline{\nu}$ can be written in general
form as

$\displaystyle{\cal
M}=\frac{G_{s}}{\sqrt{2}}\overline{u}_{B_{2}}(O_{\alpha}^{\rm
V}+O_{\alpha}^{\rm
A}){u}_{B_{1}}\overline{u}_{e}\gamma^{\alpha}(1+\gamma_{5})v_{\nu}.$ (9)

From Lorentz symmetry, we expect the matrix element composed of any two
spin-$1/2$ nucleon states, $B_{1}$ and $B_{2}$, to have the form

$O_{\alpha}^{V}=f_{1}(q^{2})\gamma^{\alpha}+\frac{f_{2}(q^{2})}{M_{B_{1}}}\sigma_{\alpha\beta}q^{\beta}+\frac{f_{3}(q^{2})}{M_{B_{1}}}q_{\alpha}$
$O_{\alpha}^{A}=\left(g_{1}(q^{2})\gamma^{\alpha}+\frac{g_{2}(q^{2})}{M_{B_{1}}}\sigma_{\alpha\beta}q^{\beta}+\frac{g_{3}(q^{2})}{M_{B_{1}}}q_{\alpha}\right)\gamma_{5}$

with transfer momentum $q=p_{B_{2}}-p_{B_{1}}$ and $V,A$ indicating the vector
and axial currents respectively. $f_{1}$ and $g_{1}$ are the vector and axial
form factors, and $f_{2}$ and $g_{2}$ are the weak magnetic and induced-
pseudoscalar form factors. They are non-zero even with $B_{1}=B_{2}$. SU(3)
flavor symmetry breaking accounts for the non-vanishing induced scalar and
weak electric form factors, $f_{3}$ and $g_{2}$ respectively. Due to the
difficulty in disentangling experimental form factor contributions, they tend
to be set to zero. We will be able to determine these form factors using
theoretical technique and will find them to be non-negligible.

So far, there are only two “quenched” (where the fermion masses in the sea
sector are infinitely heavy) lattice calculations of hyperon beta decay, and
they are in different channels, $\Sigma\rightarrow n$ and
$\Xi^{0}\rightarrow\Sigma^{+}$. Guadagnoli et al.[13] calculated the matrix
element $\Sigma\rightarrow n$ with all of the pion masses larger than 700 MeV.
Sasaki et al.[14] used lighter pion masses in the range 530–650 MeV and DWF to
look at the $\Xi^{0}$ decay channel. They extrapolate the vector form factor
$f_{1}$ using the parameter $\delta=(M_{B_{2}}-M_{B_{1}})/M_{B_{2}}$. In this
work, we calculate both decay channels and remove the quenched approximation,
which often causes notoriously large systematic error. Our fermion sea sector
contains degenerate up and down quarks plus the strange. We use pion masses as
light as 350 MeV, which ameliorates some of the uncertainty in the
extrapolation to the physical pion mass. To condense our work for this
proceeding, we will concentrate on the results from $\Sigma\rightarrow n$.

To obtain $V_{us}$, we need to extrapolate $f_{1}$ to zero momentum-transfer
(we cannot calculate this point directly due to the discrete values of
momentum accessible in a finite volume) and the physical pion and kaon masses.
Fortunately, $f_{1}$ is protected by the Ademollo-Gatto (AG) theorem such that
there is no first-order SU(3) breaking. Therefore, the quantity deviates from
its SU(3) value by the order of the symmetry breaking term of
$O({H^{\prime}}^{2})$, where $H^{\prime}$ is the SU(3) symmetry breaking
Hamiltonian; the natural candidate an observable to track this breaking is the
mass splitting between the kaon and pion. Combining with momentum
extrapolation (using a dipole form in this case), we use a single simultaneous
fit:

$f_{1}(q^{2})=\frac{1+\left(M_{K}^{2}-M_{\pi}^{2}\right)^{2}\left(A_{1}+A_{2}\left(M_{K}^{2}+M_{\pi}^{2}\right)\right)}{\left(1-\frac{q^{2}}{M_{0}+M_{1}\left(M_{K}^{2}+M_{\pi}^{2}\right)}\right)^{2}}.$
(10)

Figure 7 shows the result from simultaneously fitting over all $q^{2}$ and
mass combinations for the $\Sigma^{-}\rightarrow n$ decay. The $z$-direction
indicates $f_{1}$, while the $x$\- and $y$-axes indicate mass and transfer
momentum. The surface is the fit using Eq. 10 with color to indicate different
masses. The columns are the data and the momentum points from different pion
masses line up in bands. Our preliminary result for $f_{1}$ is $-0.95(3)$
(which is consistent with the quenched result[13]: $f_{1}=-0.988(29)_{\rm
stat}$.)

Figure 7: Simultaneous extrapolation in $q^{2}$ and mass

The axial form factor $g_{1}$ is not protected by the AG theorem.
Experimentally, one is interested in its ratio with the vector form factor,
$g_{1}/f_{1}$, at zero momentum transfer. Here we adopt naive linear mass
combinations $(M_{K}^{2}+M_{\pi}^{2})$ and $(M_{K}^{2}-M_{\pi}^{2})$ and
momentum dependence to extrapolate and find $g_{1}(0)/f_{1}(0)=-0.336(52)$,
which is consistent with the experimental value of $-0.340(17)$. Similar
extrapolations are applied to other form factor ratios, such as
$f_{2}(0)/f_{1}(0)=-1.28(19)$, which is consistent with Cabibbo model value of
$-1.297$[15].

Figure 8: The momentum dependence of the form factor ratios $f_{3}/f_{1}$
(top) and $g_{2}/f_{1}$ (bottom) with various pion masses labeled by
triangles, squares, downward triangles and diamonds from light to heavy; the
band indicates the extrapolation at the physical mass

Finally, we turn our discussion toward the SU(3)-vanishing weak-electric
$g_{2}$ and induced-scalar $f_{3}$ form factors. Figure 8 presents the
momentum dependence of the ratios $g_{2}(q^{2})/f_{1}(q^{2})$ and
$f_{3}(q^{2})/f_{1}(q^{2})$ for sea-pion masses ranging 350–700 MeV. The band
indicates our mass extrapolation in terms of a naive linear dependence of
$M_{K}^{2}-M_{\pi}^{2}$ and $M_{K}^{2}+M_{\pi}^{2}$. We find
$f_{3}(0)/f_{1}(0)=-0.17(11)$ and $g_{2}(0)/f_{1}(0)=-0.29(20)$, which are 1.5
standard deviations from zero. Experimentally, only a combination of axial
form factors $\left|g_{1}(0)/f_{1}(0)-0.133g_{2}(0)/f_{1}(0)\right|$ is
determined. They find 0.327(7)(19), which is consistent with our result,
$0.297(60)$.

##  6 Summary and Outlook

The lattice formulation is a powerful method for calculating nonperturbative
quantities used in fundamental tests of QCD. The results presented here
demonstrate clear progress toward the long-term goals of determining the
hyperon spectrum, structure and decays from first principles. Ongoing efforts
from lattice community using improved operators and algorithms (to improve the
noise-to-signal ratios), finer lattice spacings (to reduce systematic error
due to discretization), lighter pion masses (to reduce uncertainty introduced
by chiral extrapolation) on dynamical sea quarks will continue to better
approximate the physical world and provide observables that elude experiments.

## Acknowledgements

HWL thanks collaborators Kostas Orginos, David Richards, Colin Morningstar and
Saul Cohen for useful discussions. These calculations were performed using the
Chroma software suite[16] on clusters at Jefferson Laboratory using time
awarded under the SciDAC Initiative. Authored by Jefferson Science Associates,
LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Government retains
a non-exclusive, paid-up, irrevocable, world-wide license to publish or
reproduce this manuscript for U.S. Government purposes.

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