# Nucleon structure with two flavors of dynamical domain-wall fermions

Huey-Wen Lin<EMAIL_ADDRESS>Jefferson Laboratory, 12000 Jefferson Avenue,
Newport News, VA 23606  RIKEN-BNL Research Center, Brookhaven National
Laboratory, Upton, NY 11973  Tom Blum<EMAIL_ADDRESS>Physics
Department, University of Connecticut, Storrs, CT 06269-3046  RIKEN-BNL
Research Center, Brookhaven National Laboratory, Upton, NY 11973  Shigemi Ohta
<EMAIL_ADDRESS>Institute of Particle and Nuclear Studies, KEK, Tsukuba,
305-0801, Japan  RIKEN-BNL Research Center, Brookhaven National Laboratory,
Upton, NY 11973  Physics Department, Sokendai Graduate U. Adv. Studies,
Hayama, Kanagawa 240-0193, Japan  Shoichi Sasaki<EMAIL_ADDRESS>Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113,
Japan  Takeshi Yamazaki<EMAIL_ADDRESS>Physics Department,
University of Connecticut, Storrs, CT 06269-3046  RIKEN-BNL Research Center,
Brookhaven National Laboratory, Upton, NY 11973

(Feb. 4, 2008)

###### Abstract

We present a numerical lattice quantum chromodynamics calculation of isovector
form factors and the first few moments of the isovector structure functions of
the nucleon. The calculation employs two degenerate dynamical flavors of
domain-wall fermions, resulting in good control of chiral symmetry breaking.
Non-perturbative renormalization of the relevant quark currents is performed
where necessary. The DBW2 gauge action is used to further improve the chiral
behavior while maintaining a reasonable physical lattice volume. The inverse
lattice spacing, $a^{-1}$, is approximately 1.7 GeV. Degenerate up and down
dynamical quark masses of approximately 1, 3/4 and 1/2 times the strange quark
mass are used. The physical volume of the lattice is about $(1.9\mbox{
fm})^{3}$. The ratio of the isovector vector to axial charges, $g_{A}/g_{V}$,
tends to a somewhat lower value than the experimental value as the quark mass
is reduced toward the physical point. Momentum-transfer dependences of the
isovector vector, axial, induced tensor and induced pseudoscalar form factors
are calculated. The Goldberger-Treiman relation holds at low momentum transfer
and yields an estimation of the pion-nucleon coupling, $g_{\pi NN}=15.5(1.4)$,
where the quoted error is only statistical. We find that the flavor non-
singlet quark momentum fraction $\langle x\rangle_{u-d}$ and quark helicity
fraction $\langle x\rangle_{\Delta u-\Delta d}$ overshoot their experimental
values after linear chiral extrapolation. We discuss possible systematic
errors for this discrepancy. An estimate for transversity, $\langle
1\rangle_{\delta u-\delta d}=0.93(6)$ in $\overline{\rm MS}$ at 2 GeV is
obtained and a twist-3 polarized moment, $d_{1}$, appears small, suggesting
that the Wandzura-Wilczek relation holds approximately. We discuss in detail
the systematic errors in the calculation, with particular attention paid to
finite volume, excited state contamination, and chiral extrapolations.

###### pacs:

11.15.Ha, 11.30.Rd, 12.38.Aw, 12.38.-t 12.38.Gc

††preprint: RBRC-675,KEK-TH-1150

##  I Introduction

In this paper we report numerical lattice quantum chromodynamics (QCD)
calculations of the isovector form factors and moments of the isovector
structure functions of the nucleon on gauge field configurations with two
degenerate flavors of dynamical domain-wall fermions (DWF) Kaplan (1992);
Shamir (1993); Furman and Shamir (1995) generated by the RIKEN-BNL-Columbia
(RBC) Collaboration.

Four isovector form factors parameterize neutron $\beta$ decay: the vector and
induced tensor form factors from the vector current,

$\langle
p|V^{+}_{\mu}(x)|n\rangle=\bar{u}_{p}\left[\gamma_{\mu}g_{V}(q^{2})-q_{\lambda}\sigma_{\lambda\mu}{g_{T}(q^{2})}\right]u_{n}e^{iq\cdot
x},$ (1)

and the axial and induced pseudoscalar form factors from the axial current,

$\langle
p|A^{+}_{\mu}(x)|n\rangle=\bar{u}_{p}\left[\gamma_{\mu}\gamma_{5}g_{A}(q^{2})+iq_{\mu}\gamma_{5}{g_{P}(q^{2})}\right]u_{n}e^{iq\cdot
x},$ (2)

which are given here in the Euclidean metric convention. Thus, $q^{2}$ as used
in this paper stands for Euclidean four-momentum squared, and corresponds to
the time-like momentum squared as $q_{M}^{2}=-q^{2}<0$ in Minkowski space. We
also note a slight change in the sign convention of the pseudoscalar form
factor from our earlier publications, such as Ref. Sasaki et al. (2003). Here
$q=p_{n}-p_{p}$ is the momentum transfer between the proton ($p$) and neutron
($n$). In the limit $|{\vec{q}}|\rightarrow 0$, the momentum transfer should
be small because the mass difference of the neutron and proton is only about
1.3 MeV. This makes the limit $q^{2}\rightarrow 0$, where the vector and axial
form factors dominate, a good approximation. Their values in this limit are
called the vector and axial charges of the nucleon: $g_{V}=g_{V}(q^{2}=0)$ and
$g_{A}=g_{A}(q^{2}=0)$. Experimentally, $g_{V}=\cos\theta_{C}$ (with the
Cabibbo mixing angle $\theta_{C}$), and $g_{A}=1.2695(29)\times g_{V}$ Yao et
al. (2006).

These form factors are calculable on the lattice Liu et al. (1995); Gockeler
et al. (2005); Alexandrou et al. (2006); Hagler et al. (2007); Alexandrou et
al. (2007); Gockeler et al. (2007a); Yamazaki and Ohta (2007); Sasaki and
Yamazaki (2007) but quite often are prohibitively complicated if one uses
conventional staggered or Wilson fermions. The staggered fermions, with their
complicated flavor/taste structure, make even definition of baryon operators
difficult. The Wilson fermions make the necessary current renormalization
complicated due to large explicit violation of chiral symmetry. The domain-
wall fermions (DWF) Kaplan (1992, 1993); Shamir (1993); Furman and Shamir
(1995), with their exponential suppression of chiral symmetry breaking, make
such renormalizations easy. In particular, currents that are connected by
chiral transformation such as vector and axial ones should share the same
renormalization. Thus, the ratio $g_{A}/g_{V}$ is naturally renormalized in
DWF lattice calculations Sasaki et al. (2003).

Earlier (mostly quenched) lattice calculations Fukugita et al. (1995); Liu et
al. (1994); Gockeler et al. (1996) of this ratio gave significant
underestimations by up to 20%. In a quenched calculation with DWF we found
much of this deficit comes from the small lattice volumes employed in those
earlier studies Sasaki et al. (2003). With the sufficiently large lattice
volume of about $(2.4\mbox{ fm})^{3}$ used in our quenched calculation with
DWF and the improved DBW2 gauge action Takaishi (1996); de Forcrand et al.
(2000), we observed that the ratio does not depend strongly on the quark mass
and obtained a value of $1.212\pm 0.027({\rm stat})\pm 0.024({\rm norm})$ in
the chiral limit. The value is almost consistent with experiment, and combined
with the very weak dependence on the quark mass, does not require any chiral
extrapolation but a linear one in quark mass. Thus an obvious question arises:
do these observations hold when the quarks are treated dynamically?

There has been recent interest in the pseudoscalar form factor,
$g_{P}(q^{2})$, due to the MuCap Collaboration’s new high-precision experiment
studying ordinary muon capture (OMC) by protons, $\mu^{-}p\to\nu_{\mu}n$
Andreev et al. (2007). The OMC experiments determine the induced pseudoscalar
coupling $g_{P}=m_{\mu}g_{P}(q^{2})$ at $q^{2}=0.88m_{\mu}^{2}$, where
$m_{\mu}$ is the muon mass. The new experiment yields $g_{P}=7.3\pm 1.1$,
which is consistent with the value predicted by chiral perturbation theory,
$g_{P}^{\rm ChPT}=8.26\pm 0.16$ Bernard et al. (2002), but much smaller than
the last world average for OMC, $(g_{P}^{\rm OMC})_{\rm ave}=10.5\pm 1.8$
given in Ref. Gorringe and Fearing (2004), and the value from a single
experiment of radiative muon capture (RMC), $\mu^{-}p\to\nu_{\mu}n\gamma$,
$g_{P}^{\rm RMC}=12.4\pm 1.0$ Jonkmans et al. (1996). Recently, a quenched DWF
calculation Sasaki and Yamazaki (2007) reported a result, $g_{P}=8.15\pm
0.54({\rm stat})\pm 0.16({\rm norm})$, consistent with the MuCap experiment.
It is interesting to see what value is obtained in a $n_{f}=2$ dynamical DWF
calculation.

In this paper we also report on this form factor, $g_{P}(q^{2})$. It is the
induced part of Eq. (2), directly related to the pion-nucleon coupling,
$g_{\pi NN}$, which should satisfy $\displaystyle m_{N}g_{A}=F_{\pi}g_{{\pi
NN}};$ at finite momentum transfer,

$2m_{N}g_{A}(q^{2})-q^{2}{g_{P}(q^{2})}=\frac{2g_{{\pi
NN}}F_{\pi}m_{\pi}^{2}}{q^{2}+m_{\pi}^{2}}$ (3)

with a more traditional convention of $F_{\pi}=f_{\pi}/\sqrt{2}\sim 92$ MeV.
These should hold up to residual dependence on the momentum transfer $q^{2}$;
the pion-nucleon coupling at high momentum transfer may significantly differ
from its value near zero momentum transfer. On the lattice there may also
arise some corrections from the finite lattice volume.

Since we study the dependence of these form factors at relatively low momentum
transfer, we can try to extract corresponding mean square radii, $\langle
r^{2}\rangle_{{V,T,A}}$, defined by $\displaystyle
g_{V,T,A}(q^{2})/g_{V,T,A}(0)\sim 1-\frac{1}{6}q^{2}\langle
r^{2}\rangle_{{V,T,A}}+\cdots$.

The structure functions are measured in deep inelastic scattering of leptons
from nucleons Breidenbach et al. (1969); Friedman (1991); Kendall (1991);
Taylor (1991); Gluck et al. (1996); Gehrmann and Stirling (1996); Lai et al.
(1997); Adams et al. (1997); Adeva et al. (1998); Gluck et al. (1998);
Ackerstaff et al. (1999); Martin et al. (2002), the cross section of which is
factorized in terms of leptonic and hadronic tensors:

$\displaystyle
k_{0}^{\prime}\frac{d\sigma}{d^{3}k^{\prime}}=\frac{2M}{s-M^{2}}\frac{\alpha^{2}}{(q^{2})^{2}}l^{\mu\nu}W_{\mu\nu}.$
(4)

The leptonic tensor is known to be

$l_{\mu\nu}(k,k^{\prime})=2\left(k_{\mu}k^{\prime}_{\nu}+k^{\prime}_{\mu}k_{\nu}-\frac{1}{2}Q^{2}g_{\mu\nu}\right).$
(5)

Hence, the cross section provides us with structure information about the
target nucleon through the hadronic tensor, $W_{\mu\nu}$, which is decomposed
into symmetric unpolarized and antisymmetric polarized parts:

$W^{\\{\mu\nu\\}}(x,Q^{2})=\left(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\right){F_{1}(x,Q^{2})}+\left(P^{\mu}-\frac{\nu}{q^{2}}q^{\mu}\right)\left(P^{\nu}-\frac{\nu}{q^{2}}q^{\nu}\right)\frac{F_{2}(x,Q^{2})}{\nu}$
(6)
$W^{[\mu\nu]}(x,Q^{2})=i\epsilon^{\mu\nu\rho\sigma}q_{\rho}\left(\frac{S_{\sigma}}{\nu}({g_{1}(x,Q^{2})}\right.+\left.{g_{2}(x,Q^{2})})-\frac{q\cdot
SP_{\sigma}}{\nu^{2}}{g_{2}(x,Q^{2})}\right).$ (7)

$\nu=q\cdot P$, $S^{2}=-M^{2}$, and $x=Q^{2}/2\nu$, and $Q^{2}=|q^{2}|$. The
unpolarized structure functions are $F_{1}(x,Q^{2})$ and $F_{2}(x,Q^{2})$, and
the polarized, $g_{1}(x,Q^{2})$ and $g_{2}(x,Q^{2})$. Their moments are
described in terms of Wilson’s operator product expansion:

$\displaystyle 2\int_{0}^{1}dx\,x^{n-1}{F_{1}(x,Q^{2})}$ $\displaystyle=$
$\displaystyle\sum_{q=u,d}c^{(q)}_{1,n}(\mu^{2}/Q^{2},g(\mu))\>\langle
x^{n}\rangle_{q}(\mu)+{O(1/Q^{2})},$
$\displaystyle\int_{0}^{1}dx\,x^{n-2}{F_{2}(x,Q^{2})}$ $\displaystyle=$
$\displaystyle\sum_{f=u,d}c^{(q)}_{2,n}(\mu^{2}/Q^{2},g(\mu))\>\langle
x^{n}\rangle_{q}(\mu)+{O(1/Q^{2})},$ $\displaystyle
2\int_{0}^{1}dx\,x^{n}{g_{1}(x,Q^{2})}$ $\displaystyle=$
$\displaystyle\sum_{q=u,d}e^{(q)}_{1,n}(\mu^{2}/Q^{2},g(\mu))\>\langle
x^{n}\rangle_{\Delta q}(\mu)+{O(1/Q^{2})},$ $\displaystyle
2\int_{0}^{1}dx\,x^{n}{g_{2}(x,Q^{2})}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{n}{n+1}\sum_{q=u,d}\left[e^{q}_{2,n}(\mu^{2}/Q^{2},g(\mu))\>d_{n}^{q}(\mu)-2e^{q}_{1,n}(\mu^{2}/Q^{2},g(\mu))\>\langle
x^{n}\rangle_{\Delta q}(\mu)\right]+{O(1/Q^{2})},$ (8)

where $c_{1}$, $c_{2}$, $e_{1}$, and $e_{2}$ are perturbatively known Wilson
coefficients and ${\langle x^{n}\rangle_{q}(\mu)}$, ${\langle
x^{n}\rangle_{\Delta q}(\mu)}$ and $d_{n}(\mu)$ are calculable on the lattice
as forward nucleon matrix elements of certain local operators.

Again, the conventional staggered or Wilson fermions would complicate such
lattice calculations for the same reasons as discussed for the form factors.
The DWF calculations are simpler because of easier renormalizations due to
good chiral symmetry. In particular the first moments $\langle x\rangle_{u-d}$
(quark momentum fraction) and $\langle x\rangle_{\Delta u-\Delta d}$ (quark
helicity fraction) share a common renormalization, and so their ratio is
naturally renormalized in DWF calculations Orginos et al. (2006).

During the last few years lattice computations have provided many interesting
results for these structure function moments Gockeler et al. (1996, 1999a);
Detmold et al. (2001); Dolgov et al. (2002); Gockeler et al. (2002); Orginos
(2002); Hagler et al. (2003); Ohta and Orginos (2004, 2005); Sasaki et al.
(2003); LHPC et al. (2004); Gockeler et al. (2004a); Khan et al. (2004);
Gockeler et al. (2004b); Orginos et al. (2006); Edwards et al. (2006);
Gockeler et al. (2007b); Lin and Ohta (2006); Orginos (2006); Schroers (2007);
Hagler et al. (2007); Lin (2007a, b); Yamazaki and Ohta (2007); Hagler (2007),
in both quenched and full QCD. These calculations provide first-principles
values for the moments of structure functions at leading twist. One of the
major unresolved issues in these previous calculations is significant
overestimation of the moments $\langle x\rangle_{u-d}$ and $\langle
x\rangle_{\Delta u-\Delta d}$ compared with results from fits to the
experimental data Gockeler et al. (1996); Dolgov et al. (2002); Gockeler et
al. (2004b); Orginos et al. (2006). In our quenched calculation Orginos et al.
(2006) of these quantities we encountered similar overestimations.
Interestingly, however, the ratio of these quantities, $\langle
x\rangle_{u-d}/\langle x\rangle_{\Delta u-\Delta d}$ showed very weak
dependence on the quark mass and agreed well with the experimental ratio in
the chiral limit. Again an obvious question is whether this behavior survives
when the quarks are treated dynamically.

We address these questions with a lattice QCD calculation using two degenerate
dynamical flavors of domain-wall fermions and DBW2 rectangle-improved gauge
action Aoki et al. (2005). Three bare sea quark masses of 0.02, 0.03 and 0.04
in lattice units (corresponding to pion masses of about 0.5, 0.6 and 0.7 GeV)
are used with about 200 gauge configurations each. The lattice cutoff is
$a^{-1}\sim 1.7\mbox{ GeV}$, and the spatial volume is about $(1.9\mbox{
fm})^{3}$. The lattice cutoff is sufficiently high to allow us to take
advantage of various benefits of DWF such as good chiral and flavor symmetries
in performing fully-nonperturbative renormalization for our nucleon
observables. Unlike our past quenched calculations Sasaki et al. (2003);
Orginos et al. (2006) which did not show significant dependence on quark mass
in most of the observables, we sometimes observe significant deviations at the
lightest quark mass from the heavier quark mass results. The pion mass times
lattice extent, $m_{\pi}L$, is slightly less than 5 at the lightest quark
mass, which may be problematic for calculations involving a large hadron such
as the nucleon. We must therefore perform quark-mass extrapolation on a case-
by-case basis and with great caution.

The rest of the paper is organized as follows. In Section II, we first briefly
summarize the numerical ensembles used in the calculation. Then we discuss the
choice of nucleon source and sink and the operators used to calculate the form
factors and moments of the structure functions. Finally we briefly summarize
the numerical nonperturbative renormalization methods. In Section III we begin
with discussion of the form factors at zero momentum transfer. We show the
conservation of the isovector vector current is under good control and then
give an evaluation of the nucleon axial charge. Then we discuss momentum
dependence of various form factors. In particular the Dirac mean-squared
charge radius is extracted. The Goldberger-Treiman relation is shown to hold
and provides an estimate of the pion-nucleon coupling. Finally, we describe
our calculations of some low moments of the structure functions. We summarize
our conclusions and future plans in Section IV.

##  II Numerical Method

We use DWF lattice ensembles generated by the RBC Collaboration Aoki et al.
(2005). These were generated with two degenerate flavors of dynamical quarks
described by the domain-wall fermion Kaplan (1992, 1993); Shamir (1993);
Furman and Shamir (1995) action and gluons described by the doubly-blocked
Wilson (DBW2) gauge action Takaishi (1996); de Forcrand et al. (2000). There
are three such ensembles with sea quark masses of $m_{\rm sea}=0.02$, 0.03,
and 0.04 in lattice units, which respectively correspond to about 1/2, 3/4 and
1 times the strange quark mass, or degenerate pseudoscalar meson masses of
about 500, 600 and 700 MeV. The inverse lattice spacing is about 1.7 GeV, set
from the $\rho$-meson mass, yielding a physical volume of $(1.9\mbox{
fm})^{3}$.

Table 1:  Summary of simulation parameters and the numbers of configurations. The pion mass, the nucleon mass and the pion decay constant ($F_{\pi}$) are also tabulated.  $m_{f}$ | 0.02 | 0.03 | 0.04
---|---|---|---
$t_{\rm snk}-t_{\rm src}$ | 12 | 10 | 10 | 10
$t_{\rm src}$ | $0,9,18$ | $0,15$ | $0,15$ | $0,15$
$r_{\rm gauss}$ | 8 | 4.35 | 4.35 | 4.35
# of conf. | 185 each | 220 each | 220 each | 220 each
$m_{\pi}$ (GeV)Aoki et al. (2005) | 0.493(5) | 0.607(4) | 0.695(4)
$m_{N}$ (GeV)Aoki et al. (2005) | 1.28(2) | 1.43(2) | 1.55(2)
$F_{\pi}$ (GeV)Aoki et al. (2005) | 0.1141(7) | 0.1232(7) | 0.1329(7)

We study nucleon matrix elements using quark propagators obtained with
Gaussian gauge-invariant sources with radius ($r_{\rm gauss}$) of 4.35 for all
three quark masses and additionally with Gaussian smearing radius of 8 for the
lightest quark mass. The former parameter was previously used for the neutron
dipole moments calculation on the $m_{\rm sea}=0.03$ and 0.04 ensembles
Berruto et al. (2006). The latter is a better size for overlap with the ground
state of the nucleon. For details of our source definition see Ref. Berruto et
al. (2006). We use only unitary valence quark mass values, $m_{\rm
valence}=m_{\rm sea}=m_{f}$, when computing quark propagators. We use two
different sequential sources generated with source-sink locations, $[t_{\rm
src},t_{\rm sink}]=[0,10]$ and [15, 25], where the source-sink separation
$t_{\rm sep}$ is fixed as 10 lattice units, on a given gauge configuration for
all three quark masses with the smaller $r_{\rm gauss}$. As for the larger
$r_{\rm gauss}$, we use a different choice of source-sink separation, $t_{\rm
sep}=12$. The longer separation causes the larger statistical noise.
Therefore, we use three sources placed at $[t_{\rm src},t_{\rm sink}]=[0,12]$,
[9, 21] and [18, 30] to increase the statistics. Later, we will discuss the
possibility of excited-state contaminations in our calculations by comparing
results obtained from two sets of parameters, $\\{t_{\rm sep},r_{\rm
gauss}\\}$, on the lightest quark ensemble. The parameter values and the
number of configurations used from each ensemble are summarized in Table 1. We
also compile some basic physics results from Ref. Aoki et al. (2005) in the
same table.

We define our nucleon two-point functions with a nucleon interpolation field
$\chi$ and smearing parameters $A$ and $B$ as follows:

$\langle\chi_{A}(t,{\vec{p}})\overline{\chi}_{B}(0,-{\vec{p}})\rangle=\sum_{s}\langle
0|\chi_{A}|p,s\rangle\langle p,s|\overline{\chi}_{B}|0\rangle
e^{-E(p)t}+\cdot\cdot\cdot=\frac{E({\vec{p}})\gamma^{t}-i\vec{\gamma}\cdot\vec{p}+m_{N}}{2E}\sqrt{z_{A}(p)z_{B}(p)}\;e^{-E({\vec{p}})t}+\cdot\cdot\cdot$
(9)

with normalized states defined as

$\displaystyle\langle 0|\chi_{A}|p,s\rangle$ $\displaystyle=$
$\displaystyle\sqrt{z_{A}({\vec{p}})}u_{s}(\vec{p}),$ (10)

and spinors satisfying

$\displaystyle\sum_{s}u_{s}(\vec{p})\bar{u}_{s}(\vec{p})$ $\displaystyle=$
$\displaystyle E(\vec{p})\gamma^{t}-i\vec{\gamma}\cdot\vec{p}+m_{N},$ (11)

where $E({\vec{p}})=\sqrt{m_{N}^{2}+{\vec{p}}^{2}}$. The ellipsis in Eq. (9)
denotes excited-state contributions, which can be ignored in the case of $t\gg
1$.

We often use the projector $\displaystyle{\cal P}_{+}=\frac{1+\gamma_{t}}{2}$
to the positive-parity states,

$\Gamma^{(2)}_{AB}(t_{\rm snk},t_{\rm src};\vec{p})=\frac{1}{4}{\rm Tr}({\cal
P}_{+}\langle\chi_{A}(t_{\rm snk},{\vec{p}})\overline{\chi}_{B}(t_{\rm
src},-{\vec{p}})\rangle)=\frac{E({\vec{p}})+m_{N}}{2E}\sqrt{z_{A}({\vec{p}})z_{B}({\vec{p}})}\;e^{-E({\vec{p}})(t_{\rm
snk}-t_{\rm src})}+\cdot\cdot\cdot.$ (12)

These two-point functions provide appropriate normalization factors when we
extract matrix elements from three-point correlation functions, as well as
estimates for the nucleon mass.

Now we define the three-point functions:

$\displaystyle G_{\mu,{AB}}(t_{\rm src},t,t_{\rm snk})$ $\displaystyle=$
$\displaystyle\langle\chi_{A}(t_{\rm snk},\vec{p}_{\rm
snk})\,J_{\mu}(t,\vec{q})\,\overline{\chi}_{B}(t_{\rm src},\vec{p}_{\rm
src})\rangle$ (13) $\displaystyle=$
$\displaystyle\frac{E^{\prime}\gamma^{t}+i\vec{\gamma}\cdot\vec{p}^{\prime}+m_{N}}{2E^{\prime}}\cdot\Sigma_{\mu}\cdot\frac{E\gamma^{t}-i\vec{\gamma}\cdot\vec{p}+m_{N}}{2E}\,\sqrt{z_{A}({\vec{p}}^{\prime})z_{B}({\vec{p}})}\;e^{-E^{\prime}(t_{\rm
snk}-t)}e^{-E(t-t_{\rm src})}+\cdots,$

where ${\vec{q}}={\vec{p}}^{\prime}-{\vec{p}}$, ${\vec{p}}={\vec{p}}_{\rm
src}$, ${\vec{p}}^{\prime}={\vec{p}}_{\rm snk}$, $E=E({\vec{p}}_{\rm src})$
and $E^{\prime}=E(\vec{p}_{\rm snk})$. The operator $\Sigma_{\mu}$ is
appropriately selected for each observable of interest discussed in the
following subsections.

We calculate the nucleon isovector form factors and some low moments of
structure functions, namely the isovector vector ($g_{V}$), induced tensor
($g_{T}$), isovector axial ($g_{A}$), and induced pseudoscalar ($g_{P}$), form
factors at both zero and finite momentum transfer, and structure function
moments corresponding to the momentum fraction $\langle x\rangle_{u-d}$,
helicity fraction $\langle x\rangle_{\Delta u-\Delta d}$, a twist-3 moment
$d_{1}$, and transversity $\langle 1\rangle_{\delta u-\delta d}$. The
respective choice of the lattice operators for these observables are the same
as in earlier RBC reports Sasaki et al. (2002, 2003); Orginos et al. (2006);
Sasaki and Yamazaki (2007) on quenched calculations, and are briefly
summarized in what follows.

Before we move on to discuss the operator choices for our observables of
interest, we would like to briefly explain our different choice of source-sink
separation, $t_{\rm sep}$, and Gaussian smearing radius, $r_{\rm gauss}$, for
the lightest quark mass ensemble (see Lin and Ohta (2006); Ohta and Orginos
(2005); Orginos (2003) for the earlier calculations). We started our
calculation by setting $t_{\rm sep}=10$ time units and the Gaussian source
smearing radius to 4.35, parameters previously used for the neutron dipole
moments calculation on the $m_{f}=0.03$ and 0.04 ensemblesBerruto et al.
(2006). Although the new results of the Gaussian source in the heaver quark-
mass region agree with the heavier quark-mass points of the box source, for
the same calculation with $m_{f}=0.02$, we find a significant decrease in the
axial charge compared to our previous box-source calculation, which used a
separation of 12 time units as shown in Fig. 1. This discrepancy prompted us
to examine whether there is a systematic error due to excited-state
contamination in the new results.

The effective masses plotted in Fig. 2 show that the Gaussian source with
smearing radius 4.35 has a ground state plateau that begins at $t-t_{\rm
src}=7$ or 8 (top figure), while the box source appears to plateau sooner
(bottom figure). Thus one may suspect that excited-state contamination
accounts for the effect observed in Fig. 1. Since the major difference between
the two calculations is the source type and separation, we extend the source-
sink separation to $t_{\rm sep}=12$ for the Gaussian source (the same as in
the previous box-source calculation) and also increase its radius to 8 in
order to reduce excited-state contamination. Indeed, the Gaussian source with
radius 8 provides similar quality of plateau in the effective mass plot, as
shown in the middle panel of Fig. 2. However, as shown in Fig. 1, the
resulting $g_{A}/g_{V}$ with the second parameter set $\\{t_{\rm sep},r_{\rm
gauss}\\}=\\{12,8\\}$ is consistent with that of the first one $\\{10,4.35\\}$
rather than the previous result of the box source. So, the discrepancy between
the results is probably not caused by excited-state contamination.

As we mentioned before, to reduce statistical fluctuations, we use multiple
sources in the present studies. This is another difference from the previous
calculation, where only a single source was utilized. To make this point
clear, we show the dependence of bare $g_{V}$ and $g_{A}$ on the location of
the current insertion $t$ for each choice of the source location in Fig. 3. In
the top panels, all calculations of the bare $g_{V}$ show clear plateaus
between source and sink locations. Although the slight dependence on the
source location may be observed among the three sources in the $\\{t_{\rm
sep},r_{\rm gauss}\\}=\\{12,8\\}$ case, all the values of these plateaus agree
with one another within the statistical errors. We see no evidence of excited-
state contamination in the bare $g_{V}$.

On the other hand, for the bare $g_{A}$ (bottom panels), both the results
obtained from box and Gaussian sources with $t_{\rm sep}=12$ exhibit larger
fluctuations and less clear plateaus, while two consistent plateaus clearly
appear in the cases of $t_{\rm sep}=10$. This indicates that the larger
separation causes the larger fluctuations, since the absolute values of three-
point functions are exponentially suppressed as a function of $t_{\rm sep}$.
Indeed, the $t_{\rm sep}=12$ Gaussian source results obtained from a single
source-sink location $[t_{\rm src},t_{\rm sink}]=[9,21]$ agree with those of
the box source due to the large statistical errors, while the average from all
three sources for $\\{t_{\rm sep},r_{\rm gauss}\\}=\\{12,8\\}$ provides good
agreement with the results of $\\{t_{\rm sep},r_{\rm gauss}\\}=\\{10,4.35\\}$.

In this context, the lower value of the bare $g_{A}$ obtained from the present
studies is more statistically significant, since the box source results have
been obtained only for a single source. Although a systematic error from
excited-state contaminations in the result of $\\{t_{\rm sep},r_{\rm
gauss}\\}=\\{10,4.35\\}$ might be hidden, owing to the large fluctuations in
that of $\\{t_{\rm sep},r_{\rm gauss}\\}=\\{12,8\\}$, we may choose to live
with the former for the final result. This is mainly because the finite
momentum calculation suffers much from such large fluctuations. We obtain
results for the form factors only from the case of $t_{\rm sep}=10$.

Figure 1: Renormalized isovector axial charge $g_{A}/g_{V}$ calculated with box Ohta and Orginos (2005) and Gaussian smeared sources with various source-sink separations ($t_{\rm sep}$). The value from the Gaussian source with $t_{\rm sep}=10$ and smearing radius 4.35 at the lightest quark mass of $m_{f}=0.02$ deviates significantly from the previous box-source calculation. In that case the box size is chosen for better overlap with the ground state of the nucleon and the longer source-sink separation ($t_{\rm sep}=12$) is adopted to avoid possible excited-state contaminations. This discrepancy is not resolved by using a Gaussian source of an radius 8 and the same source-sink separation. As discussed in the text, this is likely caused by the larger statistical fluctuation in the longer source-sink separation.  |  |
---|---|---
Figure 2: Nucleon effective mass plots from two-point functions. Gaussian source with smearing radius, $r_{\rm gauss}=4.35$ (left), 8 (center), and box source (right) from the calculation in Ref. Ohta and Orginos (2005). |  |
---|---|---
|  |
Figure 3: Comparison of the bare isovector vector charge $g_{V}$ (top) and
axial charge $g_{A}$ (bottom) from source-sink separation $t_{\rm sep}=10$
Gaussian source (left panels), $t_{\rm sep}=12$ Gaussian source (middle
panels) and $t_{\rm sep}=12$ box source (right panels).

###  II.1 Form factors

For numerical convenience, we redefine the form factors as dimensionless
quantities. Thus, the isovector vector current in Eq. (1) is rewritten as

$\langle
p|V^{+}_{\mu}(0)|n\rangle=\bar{u}_{p}\left[\gamma_{\mu}G_{V}(q^{2})-q_{\lambda}\sigma_{\lambda\mu}\frac{G_{T}(q^{2})}{2m_{N}}\right]u_{n}$
(14)

in terms of the dimensionless vector $G_{V}$ and induced-tensor $G_{T}$ form
factors. Here the $\Sigma_{\mu}$ in Eq. (13) is
$\gamma_{\mu}G_{V}(q^{2})-q_{\lambda}\sigma_{\lambda\mu}\frac{G_{T}(q^{2})}{2m_{N}}$.

Likewise the isovector axial current in Eq. (1) is rewritten as

$\langle
p|A^{+}_{\mu}(0)|n\rangle=\bar{u}_{p}\left[\gamma_{\mu}\gamma_{5}G_{A}(q^{2})+iq_{\mu}\gamma_{5}\frac{G_{P}(q^{2})}{2m_{N}}\right]u_{n}$
(15)

with the dimensionless axial $G_{A}$ and induced-pseudoscalar $G_{P}$ form
factors. Note that the latter is normalized with twice the nucleon mass
$2m_{N}$ unlike in some muon-capture literature where it is normalized with
the muon mass $m_{\mu}$. This is for numerical convenience in this paper,
where the nucleon mass calculated on the lattice is heavier than its physical
value. Here the $\Sigma_{\mu}$ in Eq. (13) is
$\gamma_{\mu}\gamma_{5}G_{A}(q^{2})+iq_{\mu}\gamma_{5}\frac{G_{P}(q^{2})}{2m_{N}}$.
(In the above, $g_{V,A}(q^{2})\equiv G_{V,A}(q^{2})$.)

The right-hand sides of the above two equations have the most general form
consistent with Lorentz covariance. The momentum transfer $q=p_{n}-p_{p}$
becomes very small in the forward limit, because of the small mass difference
between the neutron and proton.

We use two projection operators to help us extract the momentum dependence of
the form factor:

${\Gamma^{(3),{\cal P}}_{\mu,{AB}}(t_{\rm snk},t;p_{\rm src},p_{\rm
snk})}=\frac{1}{4}{\rm Tr}({\cal P}G_{\mu,{AB}}(t_{\rm snk},t;p_{\rm
src},p_{\rm snk})),$ (16)

where we choose ${\cal P}={\cal P}_{A_{z}}={\cal P}_{+}\gamma_{5}\gamma_{z}$
for both axial and vector currents and ${\cal P}={\cal P}_{+}$ for vector
current. We combine three- and two-point functions to remove time dependence
and redundant source or sink normalization ($z$) factors Hagler et al. (2003):

$R_{j_{\mu}}=\frac{\Gamma^{(3),{\cal P}}_{\mu,GG}(t_{\rm src},t,t_{\rm
snk},\vec{p}_{\rm src},\vec{p}_{\rm snk})}{\Gamma^{(2)}_{GG}(t_{\rm
src},t_{\rm snk},\vec{p}_{\rm
snk})}\times\left(\frac{\Gamma^{(2)}_{LG}(t,t_{\rm snk},\vec{p}_{\rm
src})\Gamma^{(2)}_{GG}(t_{\rm src},t,\vec{p}_{\rm
snk})\Gamma^{(2)}_{LG}(t_{\rm src},t_{\rm snk},\vec{p}_{\rm
snk})}{\Gamma^{(2)}_{LG}(t,t_{\rm snk},\vec{p}_{\rm
snk})\Gamma^{(2)}_{GG}(t_{\rm src},t,\vec{p}_{\rm
src})\Gamma^{(2)}_{LG}(t_{\rm src},t_{\rm snk},\vec{p}_{\rm
src})}\right)^{\frac{1}{2}}$ (17)

where $L$ denotes a local (point) source or sink and $G$ stands for a
Gaussian-smeared one. In this work, we fix the sink momentum to zero.
Therefore, in the axial-current case, Eq. (17) gives

$\displaystyle{\cal R}_{\rm A_{i}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2E(\vec{p})(E(\vec{p})+m_{N})}}$ (18)
$\displaystyle\times$
$\displaystyle\left[\delta_{iz}(E(\vec{p})+m_{N})G_{A}-\frac{p_{z}p_{i}}{2m_{N}}G_{P}\right]$

with $i\in\\{x,y,z\\}$ for corresponding insertion of operator ${A_{i}}$. In
the vector current case, we use the projection operator $P_{+}$ on $V_{4}$ and
$P_{A_{z}}$ on $V_{1}$ and $V_{2}$ to extract $G_{V}$ and induced $G_{T}$; the
necessary equations are respectively

$\displaystyle\sqrt{\frac{E(\vec{p})+m_{N}}{2E(\vec{p})}}\left(G_{V}-\frac{E(\vec{p})-m_{N}}{2m_{N}}G_{T}\right),$
(19)
$\displaystyle\frac{-ip_{y}}{\sqrt{2E(\vec{p})(E(\vec{p})+m_{N})}}(G_{V}+G_{T}),$
(20)
$\displaystyle\frac{+ip_{x}}{\sqrt{2E(\vec{p})(E(\vec{p})+m_{N})}}(G_{V}+G_{T}).$
(21)

One can in principle solve for $G_{A}(q^{2})$, $G_{P}(q^{2})$, $G_{V}(q^{2})$,
and $G_{T}(q^{2})$ with overconstrained data for each $q^{2}$.

###  II.2 Moments of structure functions

Of the moments of the structure functions summarized in Eq. (I), we calculate
those which do not require finite momentum transfer: the quark momentum
fraction $\langle x\rangle_{q}$, helicity fraction $\langle x\rangle_{\Delta
q}$, transversity $\langle 1\rangle_{\delta q}$ and the twist-3 moment
$d_{1}$. Of these we calculate the isovector contribution for the former
three: $q=u-d$. This simplifies their renormalization as the flavor-singlet
contribution drops out. Thus we fully nonperturbatively renormalize these
three so the results are compared with the experiments under an assumption of
good isospin symmetry. On the other hand we do not renormalize the twist-3
moment: here our interest is to look at its individual up and down components
to see if the Wandzura-Wilczek relation Wandzura and Wilczek (1977) holds. Our
choice of operators ($\Sigma_{\mu}$ in Eq. (13)) for the three-point functions
follow an earlier RBC paper Orginos et al. (2006) and are summarized in Table
2.

Table 2:  Operators used in our structure-function moments calculations,
including the notation for the operator, the explicit operator form, the
hypercubic group representation, the correlator ratios and the projection
operators used in the numerical nonperturbative renormalization in Eq. (26).
quark momentum fraction $\langle x\rangle_{q}$
---
$\Sigma_{\mu}$ | $\displaystyle{\cal O}^{q}_{44}=\overline{q}\left[\gamma_{4}\stackrel{{\scriptstyle\leftrightarrow}}{{D_{4}}}-\frac{1}{3}\sum_{k}\gamma_{k}\stackrel{{\scriptstyle\leftrightarrow}}{{D_{k}}}\right]q$
hypercubic group rep. | ${\bf 3}^{+}_{1}$
correlator ratio | $\displaystyle R_{\langle x\rangle_{q}}=\frac{C^{\Gamma,{\cal O}^{q}_{44}}_{\rm 3pt}}{C_{\rm 2pt}}=m_{N}\langle x\rangle_{q}$
NPR projection | ${{\cal P}^{q}_{44}}^{-1}=\gamma_{4}p_{4}-\frac{1}{3}\sum_{i=1,3}\gamma_{i}p_{i}$
quark helicity fraction $\langle x\rangle_{\Delta q}$
$\Sigma_{\mu}$ | $\displaystyle{\cal O}^{5q}_{\\{34\\}}=i\overline{q}\gamma_{5}\left[\gamma_{3}\stackrel{{\scriptstyle\leftrightarrow}}{{D_{4}}}+\gamma_{4}\stackrel{{\scriptstyle\leftrightarrow}}{{D_{3}}}\right]q$
hypercubic group rep. | ${\bf 6}^{-}_{3}$
correlator ratio | $\displaystyle R_{\langle x\rangle_{\Delta q}}=\frac{C^{\Gamma,{\cal O}^{5q}_{\\{34\\}}}_{\rm 3pt}}{C_{\rm 2pt}}=m_{N}\langle x\rangle_{\Delta q}$
NPR projection | ${{\cal P}^{5q}_{34}}^{-1}=i\gamma_{5}\left(\gamma_{3}p_{4}+\gamma_{4}p_{3}\right)$
transversity $\langle 1\rangle_{\delta q}$
$\Sigma_{\mu}$ | $\displaystyle{\cal O}^{\sigma q}_{34}=\overline{q}\gamma_{5}\sigma_{34}q$
hypercubic group rep. | ${\bf 6}^{+}_{1}$
correlator ratio | $\displaystyle R_{\langle 1\rangle_{\delta q}}=\frac{C^{\Gamma,{\cal O}^{\sigma q}_{\\{34\\}}}_{\rm 3pt}}{C_{\rm 2pt}}=\langle 1\rangle_{\delta q}$
NPR projection | ${{\cal P}^{\sigma q}_{34}}^{-1}=\gamma_{5}\sigma_{34}$
twist-3 matrix element $d_{1}$
$\Sigma_{\mu}$ | $\displaystyle{\cal O}^{5q}_{[34]}=i\overline{q}\gamma_{5}\left[\gamma_{3}\stackrel{{\scriptstyle\leftrightarrow}}{{D_{4}}}-\gamma_{4}\stackrel{{\scriptstyle\leftrightarrow}}{{D_{3}}}\right]q$
hypercubic group rep. | ${\bf 6}^{+}_{1}$
correlator ratio | $\displaystyle R_{d_{1}}=\frac{C^{\Gamma,{\cal O}^{5q}_{[34]}}_{\rm 3pt}}{C_{\rm 2pt}}=d_{1}$
NPR projection | ${{\cal P}^{5q}_{[34]}}^{-1}=i\gamma_{5}\left(\gamma_{3}p_{4}-\gamma_{4}p_{3}\right)$

###  II.3 Nonperturbative renormalization

In order to compare our calculation with the experimental values, we need to
establish the proper connection to the continuum through renormalization.
Fortunately, the well-preserved chiral and flavor symmetries of the domain-
wall fermions (DWF) make this task much easier than in the cases of more
conventional fermions Blum et al. (2003, 2002); Dawson (2003); Aoki et al.
(2004).

For the form factors, the chiral symmetry of DWF assures the isovector vector
and axial local currents which are used in the present calculation share a
common renormalization: $Z_{V}=Z_{A}$, up to higher order discretization
errors, $O(a^{2})$. Further, since the vector current is conserved, the vector
renormalization is calculated from the vector charge, $g_{V}$, as
$Z_{V}=1/g_{V}$. These will be demonstrated in the numerical results section.
Note the induced tensor and pseudoscalar form factors share this common
renormalization as well and do not require any additional calculation for
renormalization.

For structure function moments, we follow Ref. Martinelli et al. (1995) and
implement the regularization-independent momentum-subtraction (RI/MOM) scheme
to nonperturbatively renormalize the operators. This was first applied to
structure functions by Gimenez et al. Gimenez et al. (1998). The procedure may
be described as follows. First, the Fourier transform of the Green function
with our operator of interest is constructed:

$\displaystyle
G_{O_{\Gamma}}(p;a)=\sum_{x,y}e^{-ip\cdot(x-y)}\langle\psi(x){O_{\Gamma}}(0)\overline{\psi}(y)\rangle,$
(22)

where $O_{\Gamma}$ is one of ${\cal O}^{q}_{44}$, ${\cal O}^{\sigma q}_{34}$,
${\cal O}^{5q}_{\\{34\\}}$ or ${\cal O}^{5q}_{[34]}$ for the first-moment
operator renormalization. In the lattice calculation, we need to produce
Fourier-transformed point-source $S(pa;0)$ and point-split–source
$D_{\mu}S(pa;0)$ propagators,

$\displaystyle S(pa;0)$ $\displaystyle=$ $\displaystyle\sum_{x}e^{-ip\cdot
x}S(x;0)$ (23) $\displaystyle D_{\mu}S(pa;0)$ $\displaystyle=$
$\displaystyle\sum_{x}\frac{1}{2}e^{-ip\cdot
x}\left[S(x;-\hat{\mu})U_{\mu}(-\hat{\mu})\right.$ (24) $\displaystyle-$
$\displaystyle\left.S(x;\hat{\mu})U_{\mu}^{\dagger}(0)\right],$

to construct the Green function in Eq. (22). In this work, we use momenta
$(p_{x},p_{y},p_{z},p_{t})$ ranging from $(0,0,0,0)$ to $(2,2,3,5)$ in units
of $2\pi/L$.

The next step is to truncate the external legs from the Green function:

$\displaystyle\Lambda_{O_{\Gamma}}(p;a)=S(p;a)^{-1}G_{O_{\Gamma}}(p;p^{\prime};a)S(p^{\prime};a)^{-1},$
(25)

where $Z^{\rm RI}$ is obtained after the projection

$\displaystyle Z_{O_{\Gamma}}(\mu;a)^{-1}Z_{q}(\mu;a)=\frac{1}{12}{{\rm
Tr}\left(\Lambda_{O_{\Gamma}}(p;a)P_{\Gamma}\right)}|_{p^{2}=\mu^{2}}.$ (26)

The projection operators for various structure function operators are listed
in Table 2. Note that $\mu$ must fall inside the renormalization window
$\Lambda_{\rm QCD}\ll\mu\ll 1/a$.

The following steps allow us to convert the renormalization constants into the
continuum $\overline{\rm MS}$ scheme at 2 GeV. See Appendix .1 for specific
details on the strong-coupling scheme-matching coefficients, RGI running
equation and anomalous dimensions for various operators.

  1. 1.

Obtain $Z^{\rm RI}(\mu)$:
The ratio of $Z_{O_{\Gamma}}(\mu;a)/Z_{q}(\mu;a)$ to $Z_{A}/Z_{q}(\mu;a)$ is
computed and yields $Z_{O_{\Gamma}}(\mu;a)/Z_{A}$. Each of the factors in the
ratio is first exptrapolated to the chiral limit, $m_{f}=-m_{\rm res}$, at
fixed momentum. Using $Z_{V}=1/g_{V}$, we can determine
$Z_{O_{\Gamma}}(\mu;a)$ in Eq. (26), the renormalization constant in the RI
scheme, which we denote as $Z^{\rm RI}$. (See the lightly-filled circles in
Fig. 4.)

Figure 4: Renormalization constants in the chiral limit. The lightly-filled
circles are the renormalization constants in RI-MOM scheme, and the diamonds
are $\overline{\rm MS}$-scheme at 2 GeV. The fits (solid lines) are used to
remove residual $(ap)^{2}$ artifacts.

  2. 2.

Convert to $\overline{\rm MS}$ scheme:
We are interested in continuum quantities, mostly calculated in the
$\overline{\rm MS}$ scheme. The conversion factors between RI and
$\overline{\rm MS}$ schemes for the operators discussed here have been
calculated in Refs. Gockeler et al. (1999b); Floratos et al. (1977). To get
$Z^{\rm\overline{MS}}(\mu)$, we use the three-loop $\alpha_{s}(\mu)$ running
coupling constants defined in Ref. Blum et al. (2002) with
$\Lambda(n_{f}=2)=300$ MeVIzubuchi (2004).

  3. 3.

Remove $(ap)^{2}$ lattice artifacts:
To remove ${\cal O}((ap)^{2})$ errors that might mimic continuum scale
dependence, we first divide out the continuum running factor. The resulting
renormalization factors should be scale-independent, so any remaining
$(ap)^{2}$ dependence must be a lattice artifact. Hence, $Z^{\overline{\rm
MS},{\rm RGI}}$ can be obtained from a fit to the form $f=A(pa)^{2}+B$. (See
Fig. 4.)

  4. 4.

Running to 2 GeV:
Finally, we use the RGI formula to obtain $Z^{\rm\overline{MS}}(2\mbox{
GeV})$.

Table 3 summarizes the renormalization factors in $\overline{\rm MS}$ scheme
at 2 GeV for each operator.

Table 3:  Summary of renormalization factors in $\overline{\rm MS}$ scheme at 2 GeV in the chiral limit. The number of configurations is about 100.  $m_{f}$ | ${\cal O}^{q}_{44}$ | ${\cal O}^{5q}_{34}$ | ${\cal O}^{\sigma q}_{34}$
---|---|---|---
$-m_{\rm res}$ | 1.28(7) | 1.23(7) | 0.776(14)

These renormalization constants will be applied in Section III.3.

##  III Numerical Results

###  III.1 Vector and axial charge

The isovector vector and axial charges, $g_{V}$ and $g_{A}$, are defined as
the zero-momentum-transfer limits of the corresponding form factors,
$G_{V}(q^{2})$ and $G_{A}(q^{2})$. Because of chiral symmetry, a Takahashi-
Ward identity ensures that the two currents, which are related by chiral
transformation, share a common renormalization: $Z_{A}=Z_{V}$ up to a lattice
discretization error of $O(a^{2},m_{f}^{2}\,a^{2})$. Since the vector current
is conserved, its renormalization is easily obtained as the inverse of the
vector charge $g_{V}=G_{V}(q^{2}=0)$. Thus, by calculating the ratio of the
three-point functions for $g_{A}/g_{V}$, we get the renormalized axial charge,
$(g_{A})^{\rm ren}$Sasaki et al. (2003).

Let us first discuss the vector charge. We note, however, that since the
vector charge is conserved, it can not provide a transition matrix element
from one state to another, in particular from any excited state to the ground
state. Thus the precocious plateau does not necessarily mean that excited
state contamination is absent in the signal after one time unit. As mentioned
earlier, to avoid excited-state contributions at the lightest quark mass, we
have lengthened the source-sink separation to 12 time units and used a larger
smearing radius for the Gaussian-smeared source and sink. From the fitting
ranges of 3–9 for $t_{\rm sep}=12$ and 3–7 for $t_{\rm sep}=10$ (see also Fig.
5),

Figure 5: Bare vector charge as a function of the current insertion time.
Results are averaged over all sources. The top figure depicts $t_{\rm sep}=12$
calculation at the lightest quark mass, $m_{f}=0.02$. The other three figures
are obtained from $t_{\rm sep}=10$ calculations at all three quark masses (in
order of increasing mass from top to bottom). The fit ranges are shown by
horizontal lines.

we obtain the plot in Fig. 6.

Figure 6: The isovector vector-charge $(g_{V})^{\rm bare}$ as a function of
the pion mass squared.

This figure shows that larger fluctuations appear in the case of the longer
time separation. By linearly extrapolating three points of $t_{\rm sep}=10$ to
zero quark mass, we obtain an estimate of $g_{V}=1.308(9)$ and
$Z_{V}=0.764(5)$. The resulting value agrees reasonably well with the value of
$Z_{A}=0.75734(55)$ found in Ref. Aoki et al. (2005), which was determined
from the ratio of two-point meson correlation functions. Thus we conclude that
excited state contamination in our value of $g_{V}$ is negligible.

In Fig. 7

Figure 7: The bare axial charge as in Fig. 5

we present the raw results for the bare axial charge. After taking an average
of all sources, the a plateau emerges, even in the case of $t_{\rm sep}=12$
(top figure), albeit with larger statistical errors. As can be seen from this
figure, the plateaus settle in after about three time slices for both $t_{\rm
sep}$ cases at the lightest quark mass. This indicates that excited-state
contamination is negligible compared to the statistical error. Corresponding
renormalized values are obtained from the ratio of correlation functions, and
are presented in Fig. 8.

Figure 8: The axial-to-vector charge ratio as in Fig. 5

As expected, we also find that plateaus settle in after about three time
slices for all cases. By using the fitting ranges shown in this figure, we
obtain the values in Table 4

Table 4: Bare isovector vector and axial charges, $g_{V}^{\rm bare}$ and $g_{A}^{\rm bare}$ and their ratio $(g_{A}/g_{V})^{\rm bare}$ which is by-definition renormalized, extracted from data summarized in Figs. 5, 7 and 8. $m_{f}$ | $\\{t_{\rm sep},r_{\rm gauss}\\}$ | $g_{V}^{\rm bare}$ | $g_{A}^{\rm bare}$ | $(g_{A}/g_{V})^{\rm ren}$
---|---|---|---|---
0.02 | {12, 8} | 1.28(3) | 1.28(12) | 1.01(10)
0.02 | {10, 4.35} | 1.299(6) | 1.27(4) | 0.97(4)
0.03 | {10, 4.35} | 1.291(3) | 1.53(3) | 1.19(3)
0.04 | {10, 4.35} | 1.2865(13) | 1.51(2) | 1.171(16)

and Fig. 9.

Figure 9: The axial-to-vector charge ratio as a function of the pion mass
squared. The lightest mass points with either $t_{\rm sep}=10$ or $t_{\rm
sep}=12$ deviate significantly from the heavier mass points. Omitting the
lightest mass point, a constrained linear fit is performed.

We observe that our axial charge value at the lightest quark mass deviates
significantly from the heavier quark mass points. This deviation causes the
linear extrapolation to the physical point to deviate from experiment by more
than five standard deviations. (The obtained value is given as
$g_{A}/g_{V}=0.89(6)$ with a poor value of $\chi^{2}/{\rm dof}=11.3$) Omitting
the lightest mass point, a constrained linear fit gives $g_{A}/g_{V}=1.23(12)$
at the physical point, in good agreement with experiment. As mentioned in
Section II, our physical volume is about $(1.9\mbox{ fm})^{3}$. In light of
our previous quenched calculation Sasaki et al. (2003) where we saw that a
small volume resulted in a small axial charge, one may worry that a similar
finite-volume effect occurs here.

Since our quenched calculation Sasaki et al. (2003) was published, a way to
interpret the finite-volume effect has been proposed by Beane and Savage Beane
and Savage (2004) within the small scale expansion (SSE) scheme Hemmert et al.
(1998), which is one possible extension of HBChPT with explicit $\Delta$
degrees of freedom. The SSE scheme can provide milder quark-mass dependence
for the axial charge, which seems to be consistent with lattice results, while
either leading order (LO) or next-to-leading order (NLO) HBChPT shows strong
quark-mass dependence in the vicinity of the chiral limit Procura et al.
(2007). There is a caveat that we have to arbitrarily fix at least one of the
four parameters in SSE. However, if one considers only the finite-volume
correction $\delta g_{A}$ (which is $g_{A}(L)-g_{A}(L=\infty)$) within this
model, $\delta g_{A}$ depends only on two phenomenological parameters, the
$N$-$\Delta$ coupling ($c_{A}$) and $\Delta$-$\Delta$ coupling ($g_{1}$) 111
The correspondence between the SSE notation and the parameters used in Ref.
Beane and Savage (2004) can be found in Ref. Khan et al. (2006). Here, we
prefer to use the original SSE notation. , which can be barely fixed.
Furthermore, if the $N$-$\Delta$ coupling is set to zero, the Beane-Savage
formula for the finite-volume correction to $g_{A}$ reduces to that of HBChPT
at leading order. In the original paper Beane and Savage (2004), $\delta
g_{A}$ is predicted to be positive. This remains true in the case of
$c_{A}=0$, where SSE reduces to LO HBChPT. In this sense, this formula fails
to account for the negative $\delta g_{A}$ observed in the quenched DWF
calculation.

In Ref. Khan et al. (2006), a finite-volume study of $g_{A}$ has been done
with $n_{f}=2$ dynamical clover simulations. They also observe negative
$\delta g_{A}$ in both quenched and dynamical simulations. Although their
lattice data points are outside of the range of applicability of SSE (or LO
HBChPT), they can fit their data using the SSE formula for $g_{A}$ with the
finite-volume corrections, which should be negative. They insisted that the
parameters adopted in the Beane-Savage paper do not fully reflect decoupling
constraints, which are guaranteed to reduce SSE to LO HBChPT when $c_{A}=0$.
Indeed, their adopted parameter set yields $\delta g_{A}<0$.

Apart from the concept of SSE, this suggests that the sign of $\delta g_{A}$
predicted from SSE is very sensitive to adopted parameters. We find that the
choice of $c_{A}$ strongly affects the sign of $\delta g_{A}$, as shown in
Fig. 10.

Figure 10: Finite volume correction of the axial charge $g_{A}$ is predicted
within ${\cal O}(\varepsilon^{3})$ SSE. The sign of $\delta g_{A}$ is very
sensitive to the choice of $c_{A}$.

We fix the chiral limit value of the axial charge $g_{A}$, the pion decay
constant $F_{\pi}$ and the $N$-$\Delta$ mass splitting $\Delta_{0}$ to their
physical values and set $g_{1}=3.0$ as in Ref Beane and Savage (2004). We also
set $L=1.9$ fm. We vary the remaining parameter of the $N$-$\Delta$ coupling
$c_{A}$ from 0 to 1.5. Here, we recall that $c_{A}=1.5$ is adopted in Ref.
Khan et al. (2006), while $c_{A}=1$ in Ref. Beane and Savage (2004). The
predicted correction at our simulation point, less than 5%, is negligible.
Furthermore, the sign of $\delta g_{A}$ seems to be positive, against our
expectation. Therefore, the finite-volume correction from SSE can not account
for our data either qualitatively or quantitatively. Again, it is likely that
the heavy quark masses in our simulation do not allow use of such formulae. We
will come back to this finite-volume question when we discuss the momentum
dependence of the form factors and quark momentum and helicity fractions.

###  III.2 Momentum dependence of form factors

As described in the previous subsection, data sets with $\\{t_{\rm sep},r_{\rm
gauss}\\}=\\{10,4.35\\}$ do not seem to suffer from excited-state
contamination. Therefore, we focus on those data sets for analysis of the form
factors at finite momentum transfer in this subsection.

####  III.2.1 Vector current

Under the present assumption of exact isospin symmetry that ignores strange
quark contribution, a straight-forward exercise in SU(2) Lie algebra leads to
the following well known relations between the weak vector-current matrix
elements in neutron beta decay and the isovector part of the nucleon
electromagnetic current matrix elements:

$\langle p|\bar{u}\gamma_{\alpha}d|n\rangle=\langle
p|\bar{u}\gamma_{\alpha}u-\bar{d}\gamma_{\alpha}d|p\rangle=\langle
p|j_{\alpha}^{\rm em}|p\rangle-\langle n|j_{\alpha}^{\rm em}|n\rangle$ (27)

where $j_{\alpha}^{\rm
em}=\frac{2}{3}\bar{u}\gamma_{\alpha}u-\frac{1}{3}\bar{d}\gamma_{\alpha}d$.
This relates the weak vector and induced tensor form factors with the
isovector part of the nucleon electromagnetic form factors:

$\displaystyle F_{1}^{p}(q^{2})-F_{1}^{n}(q^{2})$ $\displaystyle=$
$\displaystyle G_{V}(q^{2})$ (28) $\displaystyle F_{2}^{\it
p}(q^{2})-F_{2}^{\it n}(q^{2})$ $\displaystyle=$ $\displaystyle G_{T}(q^{2}).$
(29)

These are respectively are called the Dirac and Pauli form factors. They are
related to the more conventional Sachs electric $G_{E}$ and magnetic $G_{M}$
form factors:

$\displaystyle G_{E}^{N}(q^{2})$ $\displaystyle=$ $\displaystyle
F_{1}^{N}(q^{2})-\frac{q^{2}}{4m^{2}_{N}}F_{2}^{N}(q^{2})$ (30) $\displaystyle
G_{M}^{N}(q^{2})$ $\displaystyle=$ $\displaystyle
F_{1}^{N}(q^{2})+F_{2}^{N}(q^{2})$ (31)

where $N$ represents $p$ (proton) or $n$ (neutron). The isovector ones
$G_{E,M}^{v}$ are defined by $G_{E,M}^{v}=G_{E,M}^{p}-G_{E,M}^{n}$.

As such we can compare our lattice results computed at finite momentum
transfer with the experiments. The easiest quantity for such a comparison is
the charge radius, which is conventionally defined as $\langle
r^{2}\rangle=-6{dF_{1}}/{dq^{2}}$. The proton mean-squared charge radius is
known experimentally Yao et al. (2006) as $0.7656(119)\mbox{ fm}^{2}$ and the
neutron, -0.1161(22), resulting in an estimate of $0.636(12)\mbox{ fm}^{2}$
for the mean-squared “Dirac” radius of the nucleon. (For details, see Appendix
A of Ref Sasaki and Yamazaki (2007)).

The raw results from the lattice are presented in Fig. 11.

Figure 11: Bare Dirac isovector form factor as a function of the current
insertion time at the heaviest pion mass ($m_{f}=0.04$). The horizontal lines
represent the average value (solid lines) and their one standard deviations
(dashed lines) in the fitted range.

From these we extract the values of the Dirac isovector form factor,
normalized by the value at $q^{2}=0$, as listed in Table 5.

Table 5: Momentum transfer dependence of the isovector form factors. $n^{2}$ refers to the momenta carried by the vertex in units of $(2\pi/L)^{2}$. | $m_{f}$ | $n^{2}=0$ | $n^{2}=1$ | $n^{2}=2$ | $n^{2}=3$ | $n^{2}=4$
---|---|---|---|---|---|---
| 0.02 | 1.299(6) | 0.86(3) | 0.68(3) | 0.63(7) | 0.51(17)
$G_{V}$ | 0.03 | 1.291(3) | 0.882(18) | 0.73(3) | 0.62(4) | 0.48(5)
| 0.04 | 1.2865(13) | 0.930(12) | 0.725(17) | 0.61(3) | 0.58(5)
| 0.02 | N/A | 2.13(15) | 1.53(11) | 1.28(18) | 1.1(4)
$G_{T}$ | 0.03 | N/A | 2.67(13) | 2.05(10) | 1.57(12) | 1.11(13)
| 0.04 | N/A | 2.81(9) | 2.04(7) | 1.69(8) | 1.18(13)
| 0.02 | 1.27(4) | 1.09(4) | 0.92(5) | 0.87(9) | 0.58(19)
$G_{A}$ | 0.03 | 1.53(3) | 1.16(2) | 0.97(4) | 0.87(5) | 0.69(8)
| 0.04 | 1.51(2) | 1.21(2) | 0.99(3) | 0.88(4) | 0.82(8)
| 0.02 | N/A | 9.3(8) | 4.9(5) | 3.3(5) | 1.8(7)
$G_{P}$ | 0.03 | N/A | 9.5(6) | 6.1(4) | 4.2(4) | 2.6(4)
| 0.04 | N/A | 9.7(5) | 6.1(4) | 4.4(3) | 3.1(4)

In Fig. 12,

Figure 12: Momentum-transfer dependence of vector form factors. A dashed curve
corresponds to the dipole form with the empirical value of the dipole mass
$M_{V}=0.857(8)$ GeV, which is evaluated from the electric charge and
magnetization radii of the proton and neutron.

we plot all three $m_{\rm f}$ results for $G_{V}(q^{2})$ as a function of
Euclidean four-momentum squared together with the dipole form of
$(1+q^{2}/M^{2})^{-2}$ with the empirical value of the isovector Dirac dipole
mass $M=0.857(8)$ GeV, which is evaluated from the electric charge and
magnetization radii of the proton and neutron. Different symbols represent the
values obtained from different quark masses. As can be seen, there is no large
quark-mass dependence. Our calculation points are located far from the
empirical curve.

The slope of the form factor at $q^{2}=0$ determines the mean-squared radius,
which can be related to the corresponding dipole mass $M$ as $\langle
r^{2}\rangle=12/M^{2}$. To extract the mean-squared radius from our data, we
simply adopt the dipole form for fitting three lower $q^{2}$ points including
the $q^{2}=0$ value. The fitted values of the Dirac dipole mass $M_{V}$ and
corresponding mean-squared Dirac radii $\langle r_{V}^{2}\rangle$ are listed
in Table 6.

Table 6: Dirac mean-squared charge radius obtained from fitting to the conventional dipole form.  $m_{f}$ |  $M_{V}$ (GeV) | $\langle r_{V}^{2}\rangle({\rm fm}^{2})$
---|---|---
0.02 | 1.40(5) | 0.239(19)
0.03 | 1.49(4) | 0.209(12)
0.04 | 1.57(3) | 0.190(8)
Expt. | 0.857(8) | 0.636(12)

We show the quark-mass dependence of measured mean square Dirac radius in Fig.
13.

Figure 13: The mean-squared Dirac radii $\langle r_{V}^{2}\rangle$ from
$G_{V}(q^{2})$. A simple linear extrapolation with respect to the pion mass
squared underestimates the experimental point (asterisk). A dotted curve is
the predicted quark-mass dependence of mean-squared Dirac radius from LO
HBChPT in the vicinity of the physical point.

Although it is difficult to extrapolate the lattice estimates at the three
heavy pion masses to the physical pion mass, the mild $m_{f}$ dependence in
our observed range of $m_{\pi}^{2}$ allows us to perform a simple linear
extrapolation. Then we obtain the mean-squared radius of the vector form
factor at the physical point: $\langle r_{V}^{2}\rangle=0.281(32)\mbox{
fm}^{2}$, which is much smaller than the experimental value of
$0.636(12)\mbox{ fm}^{2}$.

However, it is well known that this particular quantity, as well as the pion
charge radius, has a logarithmic divergence in heavy baryon chiral
perturbation theory (HBChPT) as one approaches the chiral limit Beg and Zepeda
(1972). The dotted curve plotted in Fig. 13, is the expected quark-mass
dependence of $\langle r_{V}^{2}\rangle$ in the vicinity of the physical point
within leading order (LO) HBChPT. (See Appendix .2.2 for more details.) As can
be seen, the very steep $m_{\pi}^{2}$ dependence, which is associated with the
logarithmic divergence, near the physical point is predicted by HBChPT at
leading order. This may account for the smaller values of our measured mean-
squared radius, which are calculated in the heavy quark mass region
($m_{\pi}>0.49$ GeV).

Thus, we may expect that there is non-linear behavior in terms of
$m_{\pi}^{2}$ in the vicinity of the physical point. This expectation may look
like somewhat contradictory to what we argue for $g_{A}$ with HBChPT and SSE.
However, we recall that the logarithmic divergence can not be cured by higher-
order corrections, whereas the strong quark-mass dependence of $g_{A}$, which
simultaneously implies slower convergence of the chiral expansion, should be
modified by higher-order corrections. Indeed, both the one-loop effective
field theory approach with explicit $\Delta$ degrees of freedom (SSE) Procura
et al. (2007) and two loop HBChPT calculation Bernard and Meissner (2006)
present flat quark-mass dependence at least down to $m_{\pi}\sim 0.3$ GeV. To
observe this, we have to at least push the pion mass down to 0.3 GeV, which is
approximately the location of the intersection between the LO HBChPT curve and
the extrapolated line obtained from our three data points in Fig. 13. We leave
this issue to future calculations.

Next, we extract the induced-tensor (Pauli) form factor, $G_{T}$, from the
vector current; the raw data is listed in Table 5, and an example of
$m_{f}=0.04$ at various transfer momenta is shown in Fig. 14.

Figure 14: The bare Pauli isovector form factor as in Fig. 11

This corresponds to the combination $F_{2}^{p}-F_{2}^{n}$ of the nucleon
electromagnetic form factors. The renormalized Pauli form factor can be
obtained from $G_{T}^{\rm ren}(q^{2})=G_{T}(q^{2})/G_{V}(0)$ with vector
current renormalization $Z_{V}=1/G_{V}(0)$. Fig. 15

Figure 15:  Momentum-transfer dependence of the induced-tensor form factors.
The dashed curve corresponds to the dipole form with the experimental value of
the dipole mass $M_{T}=0.778(23)$ GeV and nucleon magnetic moment
$F_{2}(0)=3.70589$.

shows a comparison of our data and the dipole form $G_{T}^{\rm
ren}(0)/(1+q^{2}/M_{T}^{2})^{2}$ with the experimental values of $G_{T}^{\rm
ren}(0)=F_{2}(0)=3.70589$ and $M_{T}=0.778(23)$ GeV, as described in Appendix
A of Ref Sasaki and Yamazaki (2007). The heaviest two pion mass points are
almost degenerate, as was the case with the Dirac form factor. However, the
data from lightest pion mass here is closer to the experimental values. Again,
this could either be interpreted as a trend toward the experimental values
with decreasing pion mass or merely finite-volume effects.

Furthermore, we fit our $G_{T}^{\rm ren}(q^{2})$ to the dipole form to extract
the Pauli mean squared radius, which is related to the corresponding dipole
mass $M_{T}$ by $\langle r_{T}^{2}\rangle=12/M_{T}^{2}$. In contrast to the
dipole fit on $G_{V}(q^{2})$, it is a two-parameter fit since we do not have
data on the value of $G_{T}^{\rm ren}(q^{2})$ at $q^{2}=0$ without the $q^{2}$
extrapolation. Here, the value of the $G_{T}^{\rm ren}(0)$ is associated with
the difference between the proton and neutron magnetic moments
$\mu_{p}-\mu_{n}=1+F_{2}(0)$. The fitted results are summarized in Table 7.
Here we perform the dipole fit with the two lower $q^{2}$ points.

Table 7: Pauli mean-squared charge radius obtained from fitting to the conventional dipole form and the extrapolated value of the ratio $G_{M}^{v}(q^{2})/G_{E}^{v}(q^{2})$ at $q^{2}=0$ with the linear $q^{2}$ fitting form.  $m_{f}$ | $F_{2}(0)$ | $M_{T}({\rm GeV})$ | $\langle r_{T}^{2}\rangle({\rm fm}^{2})$ | $G^{v}_{M}(0)/G^{v}_{E}(0)$
---|---|---|---|---
0.02 | 2.57(37) | 1.29(16) | 0.28(7) | 3.69(41)
0.03 | 2.89(22) | 1.53(12) | 0.20(3) | 4.23(26)
0.04 | 3.32(17) | 1.36(6) | 0.25(2) | 4.24(17)
Expt. | 3.70589 | 0.778(23) | 0.773(32) | 4.70589

In Fig. 16, we plot values of $\langle r_{T}^{2}\rangle$ and $F_{2}(0)$ as a
functions of pion mass squared

Figure 16:  The mean-squared Pauli radius $\langle r_{T}^{2}\rangle$ (top) and
the value of $G_{T}^{\rm ren}(0)=F_{2}(0)$, which is associated with the
difference between the proton and neutron magnetic moments,
$\mu_{p}-\mu_{n}=1+F_{2}(0)$, (bottom) and as a function of pion mass squared.
A simple linear extrapolation with respect to the pion mass squared
underestimates the experimental point (asterisk).

along with a naive linear extrapolation to the physical pion mass. In the
upper panel of Fig. 16, we linearly extrapolate through all three points,
finding $\langle r_{T}^{2}\rangle=0.17(11)\mbox{ fm}^{2}$. This is much
smaller than the experimental measurement, $0.773(32)\mbox{ fm}^{2}$, and
similar to what we observed for the Dirac mean-squared radius. Lighter pions
in future measurements would be desirable to see whether there is an increase
in radius as we approach the physical pion mass.

In the lower panel of Fig. 16, although the value measured at the heaviest
point is close to the experimental value, the extrapolated value of $F_{2}(0)$
at the physical point also tends to somewhat underestimate the experimental
value. However, we recall that the value of $F_{2}(0)$ is highly dependent on
our adopted fitting form. For example, a monopole fit yields a larger
$F_{2}(0)$ which lies closer to the experimental value. Therefore, our
estimation of $F_{2}(0)$ should carry a large systematic uncertainty due to
the $q^{2}$ extrapolation.

We also have an alternative way to evaluate $F_{2}(0)=\mu_{p}-\mu_{n}-1$. The
ratio of the isovector magnetic form factor $G^{v}_{M}(q^{2})$ and the
isovector electric form factor $G^{v}_{E}(q^{2})$ provides the difference
between the proton and neutron magnetic moments in the forward limit,
$\mu_{p}-\mu_{n}=G^{v}_{M}(0)/G_{E}^{v}(0)$, which is related to $1+F_{2}(0)$.
Experimentally, it is known that this ratio shows no $q^{2}$ dependence at low
$q^{2}$ since both form factors are well fitted by the dipole form with the
comparable dipole masses. Therefore, this ratio may have milder $q^{2}$
dependence than $G_{T}^{\rm ren}(q^{2})$ Sasaki and Yamazaki (2007). Some RBC
results on related ratios for the heavier quark masses were reported in Ref.
Berruto et al. (2006).

In Fig. 17, we show the $q^{2}$ dependence of the ratio
$G^{v}_{M}(q^{2})/G_{E}^{v}(q^{2})$, which clearly exhibits mild dependence.
We therefore use a simple linear fitting form with respect to $q^{2}$ for an
alternative evaluation of the value $\mu_{p}-\mu_{n}$. In Fig. 18, we plot
values of $\mu_{p}-\mu_{n}$, from two determinations, $1+F_{2}(0)$ and
$G_{M}(0)/G_{E}(0)$, as functions of pion mass squared. Both determinations
are roughly consistent with each other within the statistical error. Weak
$m_{\pi}^{2}$ dependence is observed; the heavier two points in the case of
$G_{M}(0)/G_{E}(0)$ are the same within error, while slight downward
$m_{\pi}^{2}$ dependence appears in the results of $1+F_{2}(0)$. The simple
linear extrapolation of $G_{M}(0)/G_{E}(0)$ yields $\mu_{p}-\mu_{n}=3.4(7)$,
which gives the better agreement with the experimental value. We may quote
this value for our final value of $\mu_{p}-\mu_{n}$, since the latter approach
appears to have smaller systematic error in the $q^{2}$-extrapolation.

Figure 17: The $q^{2}$ dependence of the ratio
$G^{v}_{M}(q^{2})/G^{v}_{E}(q^{2})$. A simple linear extrapolation with
respect to $q^{2}$ is utilized for an alternative evaluation of
$\mu_{p}-\mu_{n}$ thanks to its mild $q^{2}$ dependence. Figure 18: Comparison
of two determinations of $\mu_{p}-\mu_{n}$. There is no appreciable
$m_{\pi}^{2}$ dependence for the results from $G^{v}_{M}(0)/G^{v}_{E}(0)$. A
simple linear extrapolation is applied to them. The resulting extrapolated
value at the physical point shows the better agreement with the experimental
value (asterisk).

####  III.2.2 Axial-vector current

In Fig. 19

Figure 19: The bare axial form factor as in Fig. 11

we present the raw data for the axial form factor at finite momentum transfer.
These allow us to extract the momentum-transfer dependence of the form factor
as listed in Table 5 and plotted in the top of Fig. 20.

Figure 20:  (Top) Momentum-transfer dependence of vector form factors. The
dashed curve corresponds to the dipole form with the experimental value of the
dipole mass $M_{A}=1.026(21)$ GeV, evaluated from pion electroproduction
experiments.
(Bottom) Mean-square axial radii $\langle r^{2}_{A}\rangle$ from
$G_{A}(q^{2})$. The shaded regions are naive linear extrapolations (with
errors) of our data points to the physical pion mass. An unexpected reduction
of the measured $\langle r^{2}_{A}\rangle$ at the lightest quark mass is
observed.

The axial form factor is phenomenologically fitted with the dipole form, at
least at low $q^{2}$; so are the Dirac and Pauli form factors Bernard et al.
(2002). The dashed curve in the top of Fig. 20 shows the dipole form with an
experimental value of the axial dipole mass $M_{A}=1.026(21)$ GeV Bernard et
al. (2002). Our lattice data lie above the experimental curve. Notice that the
likely finite volume effect observed earlier for the axial charge leads to the
non-monotonic behavior of the form factor with quark mass, and leads to the
downward curvature of the mean-square axial radius.

To extract the mean-squared radius of the axial vector form factor, we perform
the dipole fit to the three lowest $q^{2}$ points and convert the axial dipole
mass $M_{A}$ into the mean-squared axial radius $\langle r_{A}^{2}\rangle$.
The obtained values are listed in Table 8 and the bottom of Fig. 20.

Table 8: Dipole fits for momentum-transfer dependence of the axial form factor. $m_{f}$ |  $M_{A}$ (GeV) |  $\langle r_{A}^{2}\rangle$ $({\rm fm}^{2})$
---|---|---
0.02 | 2.26(25) | 0.091(20)
0.03 | 1.72(7) | 0.158(11)
0.04 | 1.90(7) | 0.129(9)
Expt. | 1.026(21) | 0.444(19)

We also plot the mean-squared axial radius as a function of the pion mass
squared. As can be seen, the value of $\langle r_{A}^{2}\rangle$ at the
lightest pion mass shows a large reduction beyond statistical fluctuations.
This large reduction in the axial radius suggests finite-volume effects could
be significant at the lightest quark mass.

There is renewed interest in the pseudoscalar form factor, $G_{P}(q^{2})$, due
to the recent MuCap Collaboration Andreev et al. (2007) high-precision
experiment studying ordinary muon capture by protons,
$\mu^{-}p\rightarrow\nu_{\mu}n$, and because of improved electroweak radiative
correction calculationsCzarnecki et al. (2007) that allow precise extraction
of the form factor from these experiments. Using the new MuCap results, a
value of $g_{P}=\frac{m_{\mu}}{2m_{N}}G_{P}(0.88m_{\mu}^{2})=7.3\pm 1.1$ is
foundAndreev et al. (2007); Czarnecki et al. (2007) which, using PCAC and
chiral perturbation theory, is now in good agreement with QCD, $g_{P}=8.26\pm
0.12$ Bernard et al. (2002). Earlier, a TRIUMF group obtained a value of
$12.4\pm 1.0$ Jonkmans et al. (1996). Clark et al. Clark et al. (2006) got
$g_{P}=10.6\pm 1.1$ after reanalyzing the TRIUMF data. Including the new MuCap
result, the “world average” is $8.7\pm 1.0$ Czarnecki et al. (2007).

The induced pseudoscalar form factor $G_{P}$ is obtained from Eq. (18), and is
shown in Fig. 21.

Figure 21: The bare pseudoscalar form factor as in Fig. 11

This allows us to extract its momentum dependence which is listed in Table 5
and plotted in Fig. 22.

Figure 22: Momentum-transfer dependence of the induced pseudoscalar form
factor compared with experiments.

As is noted in the previous section, we use a dimensionless definition for
this, as well as the other three form factors, and use twice the nucleon mass
$2m_{N}$ estimated for each ensemble of configurations. This is different from
some experimental analyses where the muon mass is used for normalization.
Momentum-transfer dependence of the pseudoscalar form factor $G_{P}(q^{2})$
has so far only been studied by one pion electroproduction experiment at
Saclay in 1993 Choi et al. (1993) with range below $0.2\mbox{ GeV}^{2}$.

Fig. 22 shows the $q^{2}$ dependence of the ratio of the induced pseudoscalar
to axial form factors, $G_{P}(q^{2})/G_{A}(q^{2})$. Though our lattice momenta
transfer are much higher than experimentally explored values, they trend
upward toward the experiments.

Another topic regarding the induced pseudoscalar form factor is its relation
with the axial form factor through the PCAC relation and pion-pole dominance
(PPD) model Braun et al. (2006):

$G^{\rm PPD}_{P}(q^{2})=\frac{4m_{N}^{2}G_{A}(q^{2})}{q^{2}+m_{\pi}^{2}}.$
(32)

To see how the pion-pole behavior is preserved in measured $G_{P}(q^{2})$, we
consider the following ratio:

$\alpha_{{}_{\rm PPD}}=\frac{G_{P}(q^{2})}{G^{\rm PPD}_{P}(q^{2})},$ (33)

which is inspired by the above PCAC prediction. If the measured $G_{P}(q^{2})$
has exactly the same form described in Eq. (32), this ratio yields unity in
the entire $q^{2}$ region.

In Fig. 23,

Figure 23: The ratio of $G_{P}(q^{2})$ and $G_{P}^{\rm PPD}(q^{2})$ as a
function of four-momentum squared $q^{2}$. Deviation of the ratio from unity
indicated by dashed line is the deviation from the PPD model. However, the
deviation is rather modest. At least, the $q^{2}$ dependence of the PPD model
approximately accounts for that of measured $G_{P}(q^{2})$.

we plot the above ratio as a function of four-momentum squared $q^{2}$. There
is no appreciable $q^{2}$ dependence. Thus, the $q^{2}$ dependence of the PPD
model approximately accounts for that of measured $G_{P}(q^{2})$. Four
different $q^{2}$ points of the ratio $\alpha_{{}_{\rm PPD}}$ reveal a $q^{2}$
independent plateau within the statistical errors. We simply take the weighted
average of $\alpha_{{}_{\rm PPD}}$ within four measured $q^{2}$ points, then
plot them against the pion mass squared. As shown in Fig. 24,

Figure 24: The quenching factor $\alpha_{{}_{\rm PPD}}$ is plotted as a
function of the pion mass squared.

the average values of $\alpha_{{}_{\rm PPD}}$ gradually approach unity as the
pion mass decreases. A simple linear extrapolation of the quenching factor
$\alpha_{{}_{\rm PPD}}$ to the chiral limit yields 0.85(8), which is only
about 2 standard deviations away from the theoretical expectation. One can
obtain $g_{P}$ with input from $\alpha_{{}_{\rm PPD}}$:

$g_{P}=\alpha_{{}_{\rm
PPD}}\frac{g_{A}}{(1+q_{0}^{2}/M_{A}^{2})^{2}}\frac{2m_{N}m_{\mu}}{(m_{\pi}^{2}+q_{0}^{2})},$
(34)

where $q_{0}^{2}=0.88m_{\mu}^{2}$ with $m_{\mu}=0.10568$ GeV and
$m_{\pi}=0.13957$ GeV. $M_{A}$ and $g_{A}$ are 1.38(16) GeV and 1.23(12)
respectively from naive linear fit using two heavier pion mass points. We
found

$g_{P}=7.68\pm 1.03$ (35)

with statistical error evaluated by jackknife analysis. This is consistent
with the “world average” value of $8.7\pm 1.0$ Czarnecki et al. (2007).

The pseudoscalar form factor $G_{P}$ is related to the pion-nucleon coupling,
$g_{\pi NN}$, through the Goldberger-Treiman relation Goldberger and Treiman
(1958):

$2m_{N}G_{A}(q^{2})-q^{2}\frac{G_{P}(q^{2})}{2m_{N}}=\frac{2g_{{\pi
NN}}F_{\pi}m_{\pi}^{2}}{q^{2}+m_{\pi}^{2}}.$ (36)

At zero momentum transfer, the relation reduces to the following form:

$m_{N}g_{A}=F_{\pi}g_{\pi NN}.$ (37)

Testing the relation experimentally in this zero-momentum form is a little
tricky because the value of the pion-nucleon coupling, $g_{\pi NN}$, is not
known at zero momentum transfer, but at the pion pole. Worse, the pion-pole
value of the coupling varies from analysis to analysis. A conventional
partial-wave analysis of the pion-nucleon elastic scattering data to 2 GeV
provides $13.75\pm 0.15$ Arndt et al. (1994) while a more recent reanalysis in
Ref. Bugg (2004) gives $g_{\pi NN}=13.169\pm 0.057$. Substituting the latter,
we find there is a slight discrepancy from the Goldberger-Treiman relation at
zero momentum transfer, $\Delta_{\rm GT}$, Coon and Scadron (1981) defined as

$\Delta_{\rm GT}=1-{m_{N}g_{A}}/{F_{\pi}g_{\pi NN}}$ (38)

which is $(2.259\pm 0.591)\%$ if we use the mean nucleon mass $m_{N}=938.9$
MeV.

Testing the relation at finite momenta transfer is more difficult, as the
pion-nucleon coupling is even more poorly known from experiment. Thus it is
useful to extract the pion-nucleon coupling, $g_{{\pi NN}}$, from our lattice
data:

$g_{\pi NN,{\rm lat}}=\frac{m_{N,{\rm lat}}g_{A,{\rm lat}}}{F_{\pi,{\rm
lat}}}.$ (39)

See Fig. 25

Figure 25: Lattice estimate for the pion-nucleon coupling, $g_{\pi NN}^{\rm
lat}$, from the Goldberger-Treiman relation as defined in Eq. (39). Omitting
the lightest mass point, a constrained linear fit is performed.

for $g_{\pi NN,{\rm lat}}$ results. A linear fit is performed without the
lightest mass point to give an estimate of $g_{\pi NN}=15.5\pm 1.4$ at the
physical point. This should be compared with the experimental estimates at the
pion pole such as $13.75\pm 0.15$ or $13.169\pm 0.057$ in the above.

Now let us examine the Goldberger-Treiman relation by looking at the
discrepancy $\Delta_{\rm GT}(q^{2})$ defined by the lattice quantities as
follows:

$1-\frac{q^{2}+m_{\pi,{\rm lat}}^{2}}{2m_{N,{\rm lat}}}\frac{4m_{N,{\rm
lat}}^{2}G_{A,{\rm lat}}(q^{2})-q^{2}G_{P,{\rm lat}}(q^{2})}{2g_{{\pi NN,{\rm
lat}}}F_{\pi,{\rm lat}}m_{\pi,{\rm lat}}^{2}};$ (40)

see Fig. 26

Figure 26: Momentum-transfer dependence of the Goldberger-Treiman discrepancy
as defined in Eq. (40).

for the value of $\Delta_{\rm GT}(q^{2})$. Note again that we are using the
pion-nucleon coupling value determined at zero-momentum transfer, and cannot
account for its variation at finite momentum transfer. From our data for the
two heavier quark mass values, we observe the discrepancy is strongly
dependent on the momentum transfer beyond about $q^{2}\approx 0.5~{}\mbox{
GeV}^{2}$. This suggests the pion-nucleon coupling at such high momentum
transfer values is very much different from the low momentum-transfer region.
Note our pion mass squared is roughly $m_{\pi}^{2}$ = 0.25, 0.36 and
$0.49\mbox{ GeV}^{2}$, or below the lowest $q^{2}$ in the plot, and the
physical pion mass is even lower. Thus our data also suggest the Goldberger-
Treiman relation holds at low momentum transfer, $q^{2}\leq 0.5\mbox{
GeV}^{2}$. In that region the lightest quark mass result is in broad agreement
with the heavier masses, albeit with large statistical error. It is desirable
to investigate this further at lower momentum transfer.

###  III.3 Momentum and helicity fractions

Figure 27: Bare momentum fraction as a function of the operator insertion
time. Results are averaged over all sources. The top figure depicts the
$t_{\rm sep}=12$ calculation at the lightest quark mass, $m_{f}=0.02$. The
other three figures are obtained from $t_{\rm sep}=10$ calculations at all
three quark masses (in order of increasing mass from top to bottom). The fit
ranges are shown by horizontal lines.

Let us now turn our attention from the elastic form factors to the deep
inelastic structure functions. As is well known and has been summarized in the
introduction, what are calculable on the lattice in regard to the structure
functions are their low-order moments. In this work we limit ourselves to the
lowest order non-trivial moments that are calculable with zero momentum
transfer. As in previous RBC reports, the operator renormalizations at the
chiral limit are obtained nonperturbatively using the RI/MOM scheme as
described in Sec. II.3.

We first discuss the quark momentum fraction, $\langle x\rangle_{u-d}$, or the
first moment of the unpolarized structure function. The bare lattice three-
point correlators are shown in Fig. 27 and listed in Table 9. The results for
$t_{\rm sep}=12$ and $t_{\rm sep}=10$ at the lightest pion mass differ by
about two standard deviations. This may suggest excited state contamination in
this quantity, even though its plateau looks very nice and flat over the range
$3\leq t\leq 7$ for $t_{\rm sep}=10$. From the top panel in Fig. 27, the
plateau for $t_{\rm sep}=12$ is not so clear or flat, owing to the much larger
statistical fluctuations in the data for this larger source-sink separation.
However, since it is low on the left and high on the right, the average value
is insensitive to the choice of (a symmetric) fit range. In the earlier
calculation, using a box source on a single time slice and a source-sink
separation of 12 Ohta and Orginos (2005); Orginos (2003), the momentum
fraction was consistent, within large statistical errors, with the $t_{\rm
sep}=10$ result. In this study, we can not conclude whether this is a true
systematic effect or just statistics, especially given the quality of the
$t_{\rm sep}=12$ plateau.

We show the results for all the measurements but only use the result from
$t_{\rm sep}=10$ at the lightest pion mass for extrapolation. Similar results
would be obtained using the $t_{\rm sep}=12$ result at the lightest mass since
the larger error on this point does not tightly constrain the fit. Using the
fit ranges indicated in the figures and the renormalizations presented in
subsection II.3, we arrive at the values of $\langle x\rangle_{u-d}$ shown in
Fig. 28.

Figure 28: Renormalized first moment of the unpolarized structure functions or
the quark momentum fraction, $\langle x\rangle_{u-d}$. Note that the circles
indicate the data at each pion mass with two errorbars: the inner are
statistical errorbars and the outer are errorbars propagated with
renormalization factors. The solid square is the extrapolated point; the star
is the experimental value; and the band shows the chiral extrapolation with
jackknife-calculated uncertainty.

Note that there is only small pion-mass dependence; therefore, we use a simple
linear fit. The quality of this fit in terms of $\chi^{2}$ per degree of
freedom is reasonable at 0.28. The chiral extrapolated value is $\langle
x\rangle_{u-d}=0.282(19)$, overshooting the experimental value, 0.154(3) Lai
et al. (1997); Gluck et al. (1998); Martin et al. (2002), by five to six
standard deviations. Here, the trend of the $t_{\rm sep}=12$ data point is
suggestive, but again, not conclusive because of the relatively large
statistical error.

The quark helicity fraction, $\langle x\rangle_{\Delta u-\Delta d}$, is the
first moment of the polarized structure function. The bare three-point
correlator values are shown in Fig. 29 and the values are listed in Table 9.
Similar remarks and conclusions about the plateaus for $t_{\rm sep}=10$ and 12
as for the momentum fraction hold here as well.

Figure 29: The bare helicity fraction; symbols as in Fig. 27. Figure 30:
Renormalized first moment of the polarized structure functions or quark
helicity fraction, $\langle x\rangle_{\Delta u-\Delta d}$, as a function of
$m_{\pi}^{2}$. (Similar graphics description as in Fig. 28.)

The renormalization factors at the chiral limit for the helicity fraction can
be found in Sec. II.3. Fig. 30 shows the renormalized quantities at each pion
mass point. A linear fit to these values versus $m_{\pi}^{2}$ gives a chiral
extrapolation of 0.286(25), overshooting the experimental value of 0.196(4)
Gluck et al. (1996); Gehrmann and Stirling (1996); Dolgov et al. (2002) by
almost four standard deviations. The quality of the fit is poor, as $\chi^{2}$
per degree of freedom is 6.78. In contrast to the momentum fraction, the
result at the light mass deviates from the heavier pion mass points, even at
$t_{\rm sep}=10$, a situation similar to the deviation observed in the axial
charge. This trend is welcome since the experimental value apparently must be
approached from above and suggests trying fits different than linear, to which
we turn below, after discussing the ratio of the momentum fraction to the
helicity fraction.

In an earlier quenched calculation Orginos et al. (2006) we reported that the
ratio of the momentum and helicity fractions, $\langle x\rangle_{u-d}/\langle
x\rangle_{\Delta u-\Delta d}$, agrees well with the experimental value,
0.78(2). The quark mass dependence of this ratio is presented in Fig. 31.
Unlike the quenched case where the ratio is only weakly dependent on the quark
mass and is in rough agreement with experiment, there is a noticeable
deviation for the lightest pion mass. A naive linear extrapolation gives very
poor fit quality, $\chi^{2}/{\rm dof}=15.7$, and a value at the physical pion
mass of $0.93(7)$, which is about two standard deviations above the
experimental one. This may be due to systematic effects like finite volume, or
excited state contamination; the $t_{\rm sep}=12$ point is consistent with
experiment and shows the weak mass dependence observed in the quenched case.

Figure 31: Renormalized ratio $\langle x\rangle_{\Delta u-\Delta d}/\langle x\rangle_{u-d}$ of the quark momentum and helicity fraction. A linear fit is shown with an error estimate at the physical pion mass. Experiment gives 0.78(2). Table 9:  Bare quark momentum and helicity fractions and their naturally renormalized ratio at each sea quark mass. $m_{f}$ | $t_{\rm sep}$ | $\langle x\rangle_{u-d}$ | $\langle x\rangle_{\Delta u-\Delta d}$ | $\langle x\rangle_{u-d}/\langle x\rangle_{\Delta u-\Delta d}$
---|---|---|---|---
0.02 | 12 | 0.195(17) | 0.22(3) | 0.99(14)
0.02 | 10 | 0.236(9) | 0.255(11) | 0.93(4)
0.03 | 10 | 0.242(6) | 0.306(6) | 0.790(17)
0.04 | 10 | 0.253(4) | 0.306(6) | 0.827(12)

Since our pion masses are rather heavy (700, 600 and 500 MeV), it is not
surprising that the chiral extrapolation to the physical pion mass (140 MeV)
sometimes misses the experimental values. Also our heavy pion masses probably
invalidate the application of a leading-order chiral form. The chiral
extrapolation should ultimately be studied with more realistic lattice QCD
ensembles such as the 2+1-flavor ones being generated jointly by RBC and UKQCD
CollaborationsAntonio et al. (2007); Allton et al. (2007), a work which is in
progress. Nevertheless, it is a worthwhile exercise to try to extend the
extrapolations beyond linear forms. We now discuss how some such attempts fair
with our data.

We begin with a parametrization by Chen, et al. Chen and Savage (2002); Chen
and Ji (2001); Detmold et al. (2002) (the relevant equations are summarized in
appendix .2.3.) Their prescription is to fix three parameters, the axial
charge $g_{A,{\rm exp}}$, pion decay constant $F_{\pi,{\rm exp}}$ and the
scale $\mu$ (set to the pion mass), to their experimental values and fit the
coefficients $C$ and $e(\mu^{2})$. The results are summarized in Fig. 32. The
fit yields momentum fraction 0.147(9) after extrapolation, which is consistent
with the experimental value. However, the quality of the fit is poor,
$\chi^{2}/{\rm dof}=2.19$; this is caused by a strong downward curvature in
the fit form due to log terms. The corresponding helicity fraction
extrapolation results in a value of 0.170(14), again, consistent with the
experimental value. The fit for the helicity fraction, while still poor, is
improved over the linear one, with a $\chi^{2}$ per degree of freedom of 3.57.

Figure 32: Renormalized quark momentum fraction, $\langle x\rangle_{u-d}$, and
helicity fraction, $\langle x\rangle_{\Delta u-\Delta d}$. The
phenomenological fit (brown band) is according to Refs. Chen and Savage
(2002); Chen and Ji (2001); Detmold et al. (2002). The dashed lines are the
linear extrapolation for comparison. (Graphics conventions as in Fig. 28.)

Another alternative is to consider the dependence in quantities on the
dimensionless ratio $m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$, as shown in Fig. 33.
Similar analyses have been done before, for example, in Refs. Beane et al.
(2006); Edwards et al. (2006). (Note that $F_{\pi,{\rm lat}}$ is replaced by
$f_{\pi,{\rm lat}}(m_{f})/\sqrt{2}$ at each $m_{f}$ to be consistent with the
chiral extrapolation formulation used in Refs. Chen and Savage (2002); Chen
and Ji (2001); Detmold et al. (2002).) Since this is a dimensionless quantity,
there is no systematic error coming from the lattice scale determination;
therefore, it is a good way to find out whether we have control over lattice
artifacts.

For the momentum fraction, $\langle x\rangle_{u-d}$, the fit yields an
extrapolated value of 0.260(30) with an acceptable $\chi^{2}/{\rm dof}=0.48$.
That this extrapolation is consistent with the earlier ones indicates no
further uncontrollable systematics other than the extrapolation itself. The
value is about three standard deviations above experiment. Note the larger
error on the extrapolated point is caused by the “re-scaling” of the
extrapolation range. Similar conclusions are drawn for the helicity fraction,
$\langle x\rangle_{\Delta u-\Delta d}$: the updated extrapolation brings the
value to 0.224(41), within two standard deviations of experiment. However, the
$\chi^{2}$ per degree of freedom for this fit is still large at 5.17.

The fit results from the non-linear chiral extrapolations (see Appendix .2.3
for more details) are summarized in Fig. 33. The momentum fraction is
0.171(19) at the physical point and agrees with experiment within one standard
deviation. The fit has an acceptable $\chi^{2}$ per degree of freedom of 0.85.
The helicity fraction extrapolates to 0.163(28) and is consistent both with
the extrapolation in terms of $m_{\pi}$ and the experimental number. However
the fit quality is poor: $\chi^{2}/{\rm dof}=4.57$. These values are
consistent with the earlier fits.

Figure 33: Renormalized quark momentum fraction, $\langle x\rangle_{u-d}$, and
helicity fraction, $\langle x\rangle_{\Delta u-\Delta d}$, in terms of
$m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$. The blue band indicates the linear
extrapolation and the brown band is the phenomenological fit as described in
Refs. Chen and Savage (2002); Chen and Ji (2001); Detmold et al. (2002).
(Graphics conventions as in Fig. 28.)

The trial fits and extrapolations are summarized in Table 10. In the cases of
both the quark momentum and helicity fraction, the extrapolation with the
chiral form seems to give us consistent values with the experiments. However,
these extrapolations require a 50% drop from the magnitude of our measured
points and do not always give the smallest $\chi^{2}/{\rm dof}$. We simply
must measure at lower pion masses before such extrapolations can be deemed
trustworthy. Also note that the extrapolations in terms of $m_{\pi}^{2}$ or
the dimensionless quantity $m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$ are consistent
with each other.

Table 10:  Summary of the extrapolations of the renormalized first moment of the unpolarized distribution. The chiral perturbation theory formulation can be found in Ref. Chen and Savage (2002); Chen and Ji (2001); Detmold et al. (2002) or appendix .2.3. The fit parameters are in the order of fit slope, intersection for the linear fits and $C$, $e$ for the chiral formulation.  | $\langle x\rangle_{u-d}$ | $\chi^{2}/{\rm d.o.f.}$ | fit parameters
---|---|---|---
Linear vs. $m_{\pi}^{2}$ | 0.282(19) | 0.28 | {0.09(5), 0.280(19)}
Linear vs. $m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$ | 0.260(30) | 0.48 | {0.0024(12), 0.26(3)}
ChPT vs. $m_{\pi}^{2}$ | 0.147(9) | 2.19 | {0.109(8), 2.64(11)}
ChPT vs. $m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$ | 0.171(19) | 0.85 | {0.139(17), 3.74(22)}
experiment | 0.154(3) |  |
| $\langle x\rangle_{\Delta u-\Delta d}$ | $\chi^{2}/{\rm d.o.f.}$ | fit parameters
Linear vs. $m_{\pi}^{2}$ | 0.286(25) | 6.78 | {0.21(6), 0.281(27)}
Linear vs. $m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$ | 0.224(41) | 5.17 | {0.0061(17), 0.22(4)}
ChPT vs. $m_{\pi}^{2}$ | 0.170(14) | 3.57 | {0.134(12),0.25(12)}
ChPT vs. $m_{\pi}^{2}/F_{\pi,{\rm lat}}^{2}$ | 0.163(28) | 4.57 | { 0.135(27), 3.31(20)}
experiment | 0.196(4) |  |

We emphasize that the data in general is not consistent with a simple linear
form, but that it does trend toward experiment as the quark mass is reduced.

###  III.4 Transversity

Table 11: Bare transversity. $m_{f}$ | $t_{\rm sep}$ | $\langle 1\rangle_{\delta u-\delta d}$
---|---|---
0.02 | 12 | 1.29(11)
0.02 | 12 | 1.25(4)
0.03 | 10 | 1.43(3)
0.04 | 10 | 1.41(4)

Another interesting quantity regarding the spin structure of the nucleon is
its transversityJaffe and Ji (1993); Grosse Perdekamp (2002), $\langle
1\rangle_{\delta q}$. Here again we have calculated only the isovector
quantity. The raw data are shown in Fig. 34, together with the fit range we
use; Table 11 summarizes the bare values. Again, at the $m_{f}=0.02$ point, we
use two different source-sink separations to control possible excited state
contributions; in this case the two choices give results that are comfortably
consistent. Therefore, we use the value obtained from $t_{\rm sep}=10$ for all
further analysis and discussion.

Figure 34: The bare values of the transversity; symbols as in Fig. 27.

Using these fit values and the renormalization at the chiral limit listed in
Table 3 we obtain the quark-mass dependence as shown in Fig. 35.

Figure 35: Nonperturbatively renormalized transversity with linear
extrapolation to the physical pion mass. (Graphic conventions as in Fig. 28.)

Similar to the axial charge radius, we observe a significant deviation at the
lightest quark mass. We extrapolate linearly and obtain $\langle
1\rangle_{\delta u-\delta d}=0.93(6)+0.37(17)m_{\pi}^{2}$ with a $\chi^{2}$
per degree of freedom of 7.3. This translates into a value at the physical
point of 0.93(6). It will be interesting to compare this result with
experiment when the latter becomes available.

###  III.5 Twist-3 moment

We also calculated the twist-3 first moment of the polarized structure
function, $d_{1}$. The good chiral symmetry of DWF prevents lower-dimensional
operators from spoiling the calculation, unlike in calculations with fermion
discretizations that violate chiral symmetry, such as Wilson fermions. The
calculated bare values are summarized in Fig. 36 and Table 12.

Figure 36: The bare twist-3 moment $d_{1}$; symbols as in Fig. 27.

The quark-mass dependence of the bare values is plotted in Fig. 37. (Note that
we do not intend to compare these calculations with experiment so do not
renormalize them.) A linear fit yields $-0.0059(16)+0.126(4)m_{\pi}^{2}$ and
an extrapolated value $d_{1}^{\rm bare}=-0.0035(13)$ at the physical pion
mass, which is about three standard deviations away from zero, but small
compared to the values at non-zero mass. This suggests the Wandzura-Wilczek
relation Wandzura and Wilczek (1977), which need not hold in a confining
theory, nevertheless holds in QCD.

Figure 37: Bare values of the twist-3 moment $d_{1}$, with linear extrapolation to the physical pion point. Up quark (up triangles), down quark (down triangles), and the isovector combination (circles) are shown. Table 12: Bare values of the twist-3 moment $d_{1}$ of the polarized structure function. $m_{f}$ | $t_{\rm sep}$ | $d_{1}^{u-d}$ | $d_{1}^{u}$ | $d_{1}^{d}$
---|---|---|---|---
0.02 | 12 | 0.0240(25) | 0.0185(19) |  $-$0.0055(13)
0.02 | 10 | 0.0246(8) | 0.0189(7) |  $-$0.0056(4)
0.03 | 10 | 0.0411(8) | 0.0325(6) |  $-$0.0085(3)
0.04 | 10 | 0.0544(7) | 0.0428(6) |  $-$0.0117(3)

##  IV Conclusion and Outlook

In this work, we have presented numerical lattice QCD calculations using two
degenerate flavors of dynamical (DWF) quarks. We calculated isovector vector,
axial, tensor and pseudoscalar form factors and some low-order moments of the
unpolarized and polarized structure functions.

We found that the ratio of the axial charge to vector charge, $g_{A}/g_{V}$,
has a significant drop at the lightest pion mass point (about 500 MeV) which
may be a sign of finite-volume effect; the heavier pion mass points (around
600 and 700 MeV) are in rough agreement with experiment and show small
dependence on the quark mass. A linear extrapolation in pion mass-squared
gives 0.89(6), which deviates from experiment by more than five standard
deviations; leaving out the lightest point gives 1.23(12). If the downward
trend at the lightest point, away from experiment, is indeed a finite-volume
effect, then even $m_{\pi}L\approx 5$ is not large enough for nucleon
calculations with dynamical fermions, in contrast to quenched calculations
Sasaki et al. (2003)222This effect may in fact be illusory due to the presence
of quenched chiral logs as recently argued in Yamazaki et al. (2008).

We studied the momentum-transfer dependence of the Dirac, Pauli, axial and
induced pesudoscalar form factors. Using a conventional dipole extrapolation,
we found masses $M_{V}$, $M_{T}$ and $M_{A}$ at each pion mass points that are
more than 50% higher than the experimental ones, while the mean-square charge
radii, $\langle r_{V}^{2}\rangle$, $\langle r_{T}^{2}\rangle$, $\langle
r_{A}^{2}\rangle$ are far below experiment. Similar effects have been observed
in the past dynamical calculations Edwards et al. (2006). Most
phenomenological studies expect to see a dramatic increase at lower pion mass;
although this trend is not observed in our current study, it may be resolved
in the future with lighter pion calculations.

The magnetic moments of nucleon were calculated in two different ways: first,
directly from extrapolation of the Pauli form factor and second, by looking at
the ratio of the isovector electric and magnetic moments
$G^{v}_{M}(0)/G_{E}^{v}(0)$ in the forward limit; the latter expression has
milder momentum dependence. Using the ratio method, we see decreasing
$m_{\pi}^{2}$ dependence, and with a simple linear extrapolation we find
$\mu_{p}-\mu_{n}=3.4(7)$, roughly consistent with the experimental value.

In studying the axial current form factors, we find that the renormalized
axial form factor at finite momenta appears to be in broad agreement with
experimental values from neutrino scattering. The Goldberger-Treiman relation
appears to hold for low momentum transfer, $q^{2}\leq 0.5\mbox{ GeV}^{2}$.
Assuming the relation, we obtain an estimate for the pion-nucleon coupling of
$g_{\pi NN}=15.5(1.4)$ and
$g_{P}=\frac{m_{\mu}}{2m_{N}}G_{P}(0.88m_{\mu}^{2})=7.68\pm 1.03$, which is
consistent with experiment.

For the structure functions, the renormalized values of $\langle
x\rangle_{u-d}$ and $\langle x\rangle_{\Delta u-\Delta d}$ show a trend toward
their respective experimental values at the lightest quark mass point in our
study. The quark momentum fraction, $\langle x\rangle_{u-d}(2\mbox{ GeV})$, is
0.282(19) by simple linear chiral extrapolation, which overshoots the
experimental value by more than three standard deviations. The quark helicity
fraction, $\langle x\rangle_{\Delta u-\Delta d}(2\mbox{ GeV})$, is 0.286(25)
by simple linear extrapolation, which overshoots the experimental value by
almost four standard deviations. The former does not deviate significantly
from a linear ansatz when the source-sink separation is set to 10 time slices.
When increased to 12, the lightest point lies below the straight line fit,
suggesting possible excited state contamination for $t_{\rm sep}=10$, and
perhaps the desired physical quark mass dependence for $t_{\rm sep}=12$.
However, the statistics for the larger separation are relatively poor, and
definitive conclusions can not be drawn. A similar pattern holds for the
helicity fraction, except that at $t_{\rm sep}=10$ a non-linear quark mass
dependence is already observed. Non-linear fits motivated by chiral
perturbation theory were also used to fit the data. Again, these indicated
favorable trends in some cases, but definitive results require simulations
with lighter quark masses and multiple volumes, work that is now well
underway.

We also calculated transversity and twist-3 matrix elements. The transversity
renormalized at 2 GeV, $\langle 1\rangle_{\delta u-\delta d}(2\mbox{ GeV})$,
is 0.93(6) by simple linear extrapolation in $m_{\pi}^{2}$. However, there is
noticeable non-linearity in the data. A linear extrapolation to the physical
point yields $d_{1}^{\rm bare}=-0.0035(13)$ for the twist-3 operator, which is
about three standard deviations away from zero, indicating only small breaking
of the Wandzura-Wilczek relation Wandzura and Wilczek (1977).

Unfortunately, even with our dynamical calculations, we cannot resolve the
long-standing differences between the predictions of QCD and experimental
measurements, in such quantities as the charge radii, and quark momentum and
helicity fractions. If both QCD and the relevant experiments are correct, we
expect to see a dramatic shift in these values as the constraints of high
quark mass, small volume and coarse lattice spacing are lifted. The
limitations of the current gauge ensembles do not allow such a study at this
stage. However, we are currently studying these quantities with a new series
of ensembles produced by the RBC and UKQCD Collaborations Antonio et al.
(2007); Allton et al. (2007). And another ensemble is being generated by these
groups and LHPC. They are respectively at $a^{-1}=1.7$ and 2.1 GeV lattice
cutoff with up to ($2.75\mbox{ fm})^{3}$ volumes; they feature physical
dynamical strange quark mass and two degenerate up and down quarks as light as
1/7 the strange mass. This will give us better control of the systematic
errors in a calculation with much lighter pion masses in the valence and the
sea sectors; whether we will start to see the curvature suggested by the
chiral perturbation theory or other phenomenological models will be very
interesting to discover.

## ACKNOWLEDGMENTS

The authors would like to acknowledge S. Choi for his private communication
providing actual values of the form factor $g_{P}(q^{2})$ in his experiment
and Kostas Orginos for his useful discussions on this work. We thank RBC
members for physics discussions and RIKEN, Brookhaven National Laboratory and
the U.S. Department of Energy for providing the facilities essential for the
completion of this work. In addition, TB and TY were partially funded by US
Department of Energy Outstanding Junior Investigator Grant DE-FG02-92ER40716;
SS is supported by the JSPS for a Grant-in-Aid for Scientific Research (C)
(No. 19540265); HWL is supported by DOE contract DE-AC05-06OR23177 under which
the Jefferson Science Associates, LLC operates the Thomas Jefferson National
Accelerator Facility.

## Appendices

###  .1 Nonperturbative renormalization

In this appendix, we describe the formulation we used in the nonperturbative
renormalization in more detail. In subsection II.3, we obtain renormalization
constants in RI/MOM scheme. In order to compare our results with the
experimental ones, we need to further to convert the renormalization constant
to $\overline{\rm MS}$ scheme. In the case of the first moments of the
momentum fraction and helicity operators, one finds the conversion equations
in the continuum via Gockeler et al. (1999b)

$\displaystyle\frac{Z^{{\rm\overline{MS}}}}{Z^{{\rm RI}}}$ $\displaystyle=$
$\displaystyle 1+\frac{g^{2}}{16\pi^{2}}C_{F}\Bigg{[}G_{n}+(1-\xi)S_{n-1}$
$\displaystyle+$
$\displaystyle\left(-\frac{4}{n+1}+(1-\xi)\frac{2}{n}\right)\frac{\left(\sum_{\mu}p_{\mu}h_{\mu}(p)\right)^{2}}{p^{2}\sum_{\mu}h_{\mu}(p)^{2}}\Bigg{]}$

with

$\displaystyle G_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{n(n+1)}\left(-3-S_{n-1}+2S_{n+1}\right)$ (42)
$\displaystyle+$
$\displaystyle\frac{2}{n+1}-4\sum_{j=2}^{n}\frac{1}{j}\left(2S_{j}-S_{j-1}\right)-1\,,$
$\displaystyle S_{n}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\frac{1}{j}\,,$ (43) $\displaystyle h_{\mu}(p)$
$\displaystyle=$
$\displaystyle\sum_{\mu_{2},\ldots,\mu_{n}}c_{\mu\mu_{2}\ldots\mu_{n}}p_{\mu_{2}}\cdots
p_{\mu_{n}}\,,$ (44)

where $\xi=0$ in Landau gauge. $c_{\mu\mu_{2}\ldots\mu_{n}}=c_{\mu\nu}$ are
$\delta_{\mu 4}\delta_{\nu 4}-\frac{1}{3}\sum_{k=1}^{3}\delta_{\mu
k}\delta_{\nu k}$ for momentum fraction and $\delta_{\mu 3}\delta_{\nu
4}+\delta_{\mu 4}\delta_{\nu 3}$ for helicity operators. In the tensor-current
case, we use the matching factor calculated from Ref. Aoki et al. (2007):

$\displaystyle\frac{Z^{{\rm\overline{MS}}}}{Z^{{\rm RI}}}$ $\displaystyle=$
$\displaystyle
1+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\frac{1}{81}(-4866+1656\zeta(3)+259n_{f}).$

We use the three-loop solution for the running of $\alpha_{s}$ Gimenez et al.
(1998):

$\displaystyle\frac{\alpha_{s}}{4\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{\beta_{0}\ln\left(\mu^{2}/\Lambda^{2}_{\rm
QCD}\right)}-\frac{\beta_{1}\ln\ln\left(\mu^{2}/\Lambda^{2}_{\rm
QCD}\right)}{\beta_{0}^{3}\ln^{2}\left(\mu^{2}/\Lambda^{2}_{\rm QCD}\right)}$
(46) $\displaystyle+$
$\displaystyle\frac{1}{\beta_{0}^{5}\ln^{3}\left(\mu^{2}/\Lambda^{2}_{\rm
QCD}\right)}\left\\{\beta_{1}^{2}\ln^{2}\ln\left(\mu^{2}/\Lambda^{2}_{\rm
QCD}\right)\right.$ $\displaystyle-$
$\displaystyle\left.\beta_{1}^{2}\ln\ln\left(\mu^{2}/\Lambda^{2}_{\rm
QCD}\right)+\beta_{2}\beta_{0}-\beta_{1}^{2}\right\\}$

where $\alpha=\frac{g^{2}}{4\pi}\,$ and $C_{F}=\frac{N_{C}^{2}-1}{2N_{C}}$,
$\Lambda_{\rm QCD}=300$ MeVIzubuchi (2004). The various $\beta_{i}$ are

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{11N_{c}-2n_{f}}{3}$ $\displaystyle\beta_{1}$
$\displaystyle=$
$\displaystyle\frac{34N_{c}^{2}}{3}-\frac{10}{3}N_{c}n_{f}-\frac{(N_{c}^{2}-1)n_{f}}{N_{c}}$
$\displaystyle\beta_{2}$ $\displaystyle=$
$\displaystyle\frac{2857}{54}N_{c}^{3}+\frac{(N_{c}^{2}-1)^{2}}{4N_{c}^{2}}n_{f}-\frac{205}{36}(N_{c}^{2}-1)n_{f}$
(47) $\displaystyle-$ $\displaystyle\frac{1415}{54}N_{c}^{2}n_{f}$
$\displaystyle+$
$\displaystyle\frac{11}{18}\frac{N_{c}^{2}-1}{N_{c}}n_{f}^{2}+\frac{79}{54}N_{c}n_{f}^{2}\quad\mbox{in
$\overline{\rm MS}$ scheme},$

where $N_{c}$ is the number of colors and $n_{f}$ is the number of quark
flavors. In the quenched approximation, $n_{f}=0$; in this work, $n_{f}=2$.

The running of the $Z$-factors to two loops Buras (1998) is

$\displaystyle
C(\mu^{2})=\alpha_{s}(\mu)^{\overline{\gamma}_{0}}\left\\{1+\frac{\alpha_{s}(\mu)}{4\pi}\left(\overline{\gamma}_{1}-\overline{\beta}_{1}\overline{\gamma}_{0}\right)\right\\},$
(48)

where $\overline{\gamma}_{i}$ is defined by the anomalous dimensions
$\gamma_{i}$ divided by $2\beta_{0}$. The anomalous dimensions for twist-two
operators of interest were calculated by Floratos et al. Floratos et al.
(1977):

$\displaystyle\gamma_{0}$ $\displaystyle=$ $\displaystyle
2C_{F}\left(1-\frac{2}{n(n+1)}+4\sum_{j=2}^{n}\frac{1}{j}\right)$
$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle(-8.30C_{F}^{2}+27.85C_{F}C_{A}-9.48C_{F}T_{F}),$ (49)

for twist-${n}$ operator. (Note that where $T_{F}(n_{f})=\frac{n_{f}}{2}$,
$C_{F}=\frac{N_{c}-1}{2N_{c}}$ and $C_{A}=N_{c}$.) The anomalous dimensions
for the operator, such as ${\cal O}_{34}^{\sigma q}$, can be found in Ref.
Broadhurst and Grozin (1995)

$\displaystyle\gamma_{0}$ $\displaystyle=$ $\displaystyle 2C_{F}$
$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle\frac{C_{F}}{9}(257C_{A}-171C_{F}-52T_{F}).$ (50)

###  .2 Chiral extrapolation models

In this appendix, we collect some relevant formulae used in this paper from
various models proposed to conduct chiral extrapolations for baryonic
observables.

####  .2.1 Finite-volume correction on $g_{A}$

The small-scale expansion scheme Hemmert et al. (1998) tries to use explicit
degrees of freedom from the pion, nucleon and $\Delta(1232)$ resonance,
expanding in terms of $\Delta_{0}$, the mass splitting between the $N$ and
$\Delta$ in the chiral limit. This splitting is treated as a small parameter,
of the order $O(\epsilon)$. They define $g_{A}^{0}$ and $F_{\pi}^{0}$ as the
value of the axial charge $g_{A}$ and the pion decay constant $F_{\pi}$ in the
chiral limit, $c_{A}$ the $N\Delta$ and $g_{1}$ the $\Delta\Delta$ axial
coupling constants respectively. We simply abbreviate $\Delta_{0}$,
$g_{A}^{0}$ and $F_{\pi}^{0}$ as $\Delta$, $g_{A}$ and $F_{\pi}^{0}$ in the
following formula.

At the leading order $O(\epsilon^{3})$, the finite-volume correction to the
nucleon axial charge, $g_{A}$, was proposed in Ref. Beane and Savage (2004)
within the SSE scheme:

$\displaystyle\delta g_{A}$ $\displaystyle=$ $\displaystyle{m_{\pi}^{2}\over
3\pi^{2}F_{\pi}^{2}}\left[\ g_{A}^{3}{\bf
F_{1}}+2c_{A}^{2}\left(g_{A}-{25\over 81}g_{1}\right){\bf F_{2}}\right.$ (51)
$\displaystyle+$ $\displaystyle\left.g_{A}{\bf F_{3}}+2c_{A}^{2}g_{A}{\bf
F_{4}}\right],$

where

$\displaystyle{\bf F_{1}}(m,L)$ $\displaystyle=$ $\displaystyle\sum_{{\bf
n}\neq{\bf 0}}\left[\ K_{0}(mL|{\bf n}|)-{K_{1}(mL|{\bf n}|)\over mL|{\bf
n}|}\ \right];$ $\displaystyle{\bf F_{2}}(m,\Delta,L)$ $\displaystyle=$
$\displaystyle-\sum_{{\bf n}\neq{\bf 0}}\left[\ {K_{1}(mL|{\bf n}|)\over
mL|{\bf n}|}+{\Delta^{2}-m^{2}\over m^{2}}K_{0}(mL|{\bf n}|)-\ {\Delta\over
m^{2}}\int_{m}^{\infty}d\beta\;{2\beta K_{0}(\beta L|{\bf
n}|)+(\Delta^{2}-m^{2})L|{\bf n}|K_{1}(\beta L|{\bf
n}|)\over\sqrt{\beta^{2}+\Delta^{2}-m^{2}}}\right];$ $\displaystyle{\bf
F_{3}}(m,L)$ $\displaystyle=$ $\displaystyle-{3\over 2}\sum_{{\bf n}\neq{\bf
0}}{K_{1}(mL|{\bf n}|)\over mL|{\bf n}|};$ $\displaystyle{\bf
F_{4}}(m,\Delta,L)$ $\displaystyle=$ $\displaystyle{8\over 9}\sum_{{\bf
n}\neq{\bf 0}}\left[\ {K_{1}(mL|{\bf n}|)\over mL|{\bf n}|}-{\pi e^{-mL|{\bf
n}|}\over 2\Delta L|{\bf n}|}-{\Delta^{2}-m^{2}\over
m^{2}\Delta}\int_{m}^{\infty}d\beta\ {\beta\ K_{0}(\beta L|{\bf
n}|)\over\sqrt{\beta^{2}+\Delta^{2}-m^{2}}}\right],$ (52)

with modified Bessel function of the second kind $K_{\alpha}(z)$. (Note that
the $m$ in the above equations are short for $m_{\pi}$.) This formula can be
reduced to that of LO HBChPT with $c_{A}=0$.

####  .2.2 The mean-squared Dirac radius in HBChPT

Heavy baryon chiral perturbation theory (HBChPT)Bernard et al. (1992) involves
only the pion and nucleon, expanding in terms of the momentum $p$. At leading
order (LO), once we use the physical values of $g_{A}$, $F_{\pi}$ and $\langle
r_{V}^{2}\rangle$ instead of their values in the chiral limit, the HBChPT
formula provides a closed form for the $m_{\pi}^{2}$ dependence of the Dirac
mean-squared radius without unknown parameters Beg and Zepeda (1972); Bernard
et al. (1992):

$\langle r_{V}^{2}\rangle(m_{\pi,{\rm lat}})=\langle r_{V}^{2}\rangle_{\rm
exp}-\frac{1+5g^{2}_{A,{\rm exp}}}{(4\pi F_{\pi,{\rm exp}})^{2}}{\rm
ln}\left(\frac{m^{2}_{\pi,{\rm lat}}}{\mu^{2}}\right)$ (53)

where the scale $\mu$ should be set by the physical value of the pion mass as
$\mu=0.139$ GeV 333A missing factor of 2 in the prefactor of $\ln m_{\pi}^{2}$
term, which can be found in Ref. Leinweber and Cohen (1993), causes an
accidental agreement between this leading one-loop form and lattice data
obtained in the heavy quark mass region. .

####  .2.3 Chen, et al.

Chen et al. Chen and Ji (2001); Chen and Savage (2002); Detmold et al. (2002)
calculate a chiral perturbation expression in the continuum for the moments of
the unpolarized and helicity distributions:

$\displaystyle\langle x\rangle_{u-d}$ $\displaystyle=$ $\displaystyle
C\left[1-\frac{3g_{A,{\rm exp}}^{2}+1}{(4\pi F_{\pi,{\rm
exp}})^{2}}m_{\pi,{\rm lat}}^{2}\ln\left(\frac{m_{\pi,{\rm
lat}}^{2}}{\mu^{2}}\right)\right]+e(\mu^{2})\frac{m_{\pi,{\rm lat}}^{2}}{(4\pi
F_{\pi,{\rm exp}})^{2}}$ (54) $\displaystyle\langle x\rangle_{\Delta u-\Delta
d}$ $\displaystyle=$ $\displaystyle\tilde{C}\left[1-\frac{2g_{A,{\rm
exp}}^{2}+1}{(4\pi F_{\pi,{\rm exp}})^{2}}m_{\pi,{\rm
lat}}^{2}\ln\left(\frac{m_{\pi,{\rm
lat}}^{2}}{\mu^{2}}\right)\right]+\tilde{e}(\mu^{2})\frac{m_{\pi,{\rm
lat}}^{2}}{(4\pi F_{\pi,{\rm exp}})^{2}}.$ (55)

We use $\mu=0.139$ MeV and $F_{\pi,{\rm exp}}=92.4$ MeV in our extrapolation.

There is a variation of this formulation replacing the $F_{\pi,{\rm exp}}$
with $F_{\pi,{\rm lat}}$ and changing the scale $\mu$ to $F_{\pi,{\rm lat}}$.
This provides a dimensionless quantity $\frac{m_{\pi,{\rm
lat}}^{2}}{F_{\pi,{\rm lat}}^{2}}$ in the extrapolation, which is independent
of the scale setting in the lattice calculation.

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