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Nucleon Structure with Domain Wall Fermions
at fl = 0.084 fm
S. N. Syritsyn"^, J. D. Bratt, M. F. Lin, H. B. Meyer, J. W. Negele, A. V. Pochinsky and
M. Procura,
Center for TheoreticalPhysics, Massachusetts Institute of Technology, Cambridge, MA021 39, USA
E-mail:
[email protected] jdbratt@mit . edu, meif eng@mit . edu
meyerh@mit . eduj , |neqele@mit . edu[ pvp@mit . edu| |mprocura@mit . edu
R. G. Edwards, K. Orginos and D. G. Richards
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
E-mail: |edwards@ j lab . org dgr@ j lab . org, kostas@ j lab . org
M. Engeihiardt
Department of Physics, New Mexico State University, Las Cruces, NM 88003-8001, USA
E-mail: |engel@physics . nmsu . edu
G. T. Fieming
Sloane Physics Laboratory, Yale University, New Haven, CT 06520, USA
E-mail: [George . Fleming@Yale . edu
Ph. Haglerand B. Musch,
InstitutfUr Theoretische Physik T39, Physik-Department der TU Miinchen
James-Franck-Strasse, D-85747 Garching, Germany
E-mail: |phaegler@ph . turn, de bernhard . musch@ph . turn . de
D. B. Renner
DESY Zeuthen, Theory Group, Platanenallee 6, D-15738 Zeuthen, Germany
E-mail: [drenner@if h . de
W. Schroers
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
E-mail: [AJolf ram. SchroersgField-theory .org
We present initial calculations of nucleon matrix elements of twist-two operators with 2+1 flavors
of domain wall fermions at a lattice spacing a = 0.084 fm for pion masses down to 300 MeV. We
also compare the results with the domain wall calculations on a coarser lattice.
The XXVI International Symposium on Lattice Field Theory
July 14-19, 2008
Williamsburg, Virginia, USA
* Speaker.
© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.
http://pos.sissa.it/
Nucleon Structure with Domain Wall Fermions at a = 0.084 fin
S. N. Syritsyn
1. Introduction
The calculation of nucleon generalized form factors has been performed recently using a mixed
action that combines the computationally economical Asqtad fermion action for sea quarks with
the chirally symmetric domain wall action for valence quarksfjl], |2|].
With the advance of algorithms and computational facilities, use of the domain wall action
for light sea quarks on large, fine lattices has now become possible. Hence, using gauge configu-
rations generated by the RBC and UKQCD collaborations [Q], we investigate nucleon structure in
fully unitary, chirally symmetric lattice QCD. Currently, lattices with two different lattice spacings
are available, a = 0. 114fm and a = 0.084fm, which we will refer to as coarse and fine, respectively.
The lowest pseudoscalar meson mass is m^t 300MeV.
The next section describes the details of our calculation. In the Sect. ^ we study the pion
correlation functions to determine the renormalization constant for the axial vector quark current,
the pion decay constant and the pion mass, and use the pion decay constant and mass to set the scale
of the fine lattices. The results for nucleon form factors are given in Sect. |^ where we focus on the
isovector flavor combination u — d since it has no contributions from disconnected diagrams. We
also show the dependence of the generalized form factors, although we still need to determine
their overall renormalization. Conclusions follow in Sect. ^.
2. Calculation details
The gauge configurations were generated by the RBC and UKQCD collaborations using the
Iwasaki gauge action with A^j = 2 + 1 light and heavy dynamical Domain Wall fermions, as de-
scribed in Ref. [||] and references therein. The extent of the fifth dimension is = 16, which is
large enough to keep the residual quark mass below the bare quark mass as shown in Tab. [T[ The
spatial volume is w (2.7 fm)^ for both lattice spacings.
Before analyzing the gauge configurations, we undertook a systematic search for the optimal
source parameters that provide the best overlap with the nucleon ground state. As in Ref. we use
a gauge-invariant Gaussian-smeared quark source that minimizes the excited state contamination to
the nucleon two-point correlation function. In addition, we apply APE smearing to the gauge field
used to construct the sources to reduce the large variation of the norm of the smeared sources due to
the gauge field noise. The optimization of the source overlap with the nucleon is shown in Fig. |l|a.
With the optimized source, the plateau for the effective nucleon mass starts as early as f = 6 for the
fine lattice (see Fig. |l|b) and ? = 5 for the coarse lattice. This justifies our choice of the source-sink
separation T = 9 and T = 12 for coarse and fine lattices, respectively, which both correspond to
physical separations 1.0 fm, for the calculation of the nucleon three-point correlators.
As described in Ref. [^, to increase the statistics, we use four nucleon sources separated by
T = 16 and calculate the forward quark propagators. The backward propagators are calculated for
the sum of four nucleon sinks on each lattice. The cross-contributions between different sources
and sinks average to zero due to the gauge invariance, provided there is no temporal link gauge
fixing. Similarly, four antinucleon sinks are also treated analogously to obtain a total of eight
measurements per lattice, which have been verified to be independent by jackknife binning [^].
2
Nucleon Structure with Domain Wall Fermions at a = 0.084 fin
S. N. Syritsyn
For each source, we construct sinks with momenta P' = (0,0,0) and P' = (—1,0,0). Since the
three-point correlators quickly become noisy with growing initial and final state momenta, we have
limited the source momenta to <4 for the non-zero sink momentum P'.
Table 1: Dynamic DW fermion gauge configurations calculated by the RBC/UKQCD collaboration. The
total number of nucleon correlator measurements, #, includes eight measurements per gauge field.
a[fm\
#
ami /am/i
anires X 10^
mj[[MeV]
24^ ■ 64
0.114
3208
0.005/0.04
3.15(1)
329(5)
32^ • 64
0.084
1568
0.008/0.03
0.668(3)
406(7)
4208
0.006/0.03
0.663(2)
356(6)
2392
0.004/0.03
0.665(3)
298(5)
3. Pion correlation functions and the current renormalization
The operators calculated on a lattice must be renormalized in order to compare the results
with other lattice studies and phenomenology. In the case of the vector quark current, the renor-
malization constant is determined by the total charge measured as gv = Fi{0). The axial vector
quark current renormalization constant Za can be determined from the relation between the local
axial vector current Aq and the true (partially conserved) axial vector current £/q associated with
the axial transformation of the DW fermion integral ^ :
{n\^o\0)=ZA{n\Ao\0), ^^^j)^ ^ ZA,t ^ ^, (3.1)
\M[t}J5 (0))
where is the smeared pseudoscalar density operator. Averaging the ratio (|3JJ) over the plateau
region 10 < f < 54, we extract Za with high precision as shown in Tab. ^ We determine the pion
mass, the residual mass lUres and the pion decay constant from the simultaneous fit of the following
correlators of local operators:
(Ao(0/5(0)) =c_A5 (^-^'-^-^(^-0
(75(0-/5(0))=c_B5 {e-"'^' + e
where J^q is the DW mid-point contribution to the divergence of the axial vector current and
Csmear is the factor due to the source smearing, which is evaluated separately. Constants A5 =
f^m^ /4ZA{mq + nires) and B5 = f^m\/%{mq + mresf' provide us with two ways to extract the pion
decay constant, /^j. The obtained values agree within errors. Results are summarized in Tab. ^.
At the time of the talk, the lattice scale had only been set for the coarse lattice, using the ;^PT
extrapolated Q. baryon mass Hence, we set the fine lattice scale by comparing the lattice values
of the pion decay constant afj^ and the nucleon mass amj^ on the coarse lattice to that on the fine
lattices, linearly interpolated in {mjt/ fj^Y to points* = {f^n / fnY\^.^jaj.^g-
iafn)* l{afnT"''' = 0.7369(15), (amw)7(amA,)" = 0.7530(54)
3
Nucleon Structure with Domain Wall Fermions at a = 0.084 fin
S. N. Syritsyn
32 x64, m, = 298 MeV
1.2
1
0.8
0.6
0.4
0.2
t=l .
t=2,-
1 2 3 4 5 6 7
sqrt(<r2>)
(a)
8 9
1.2
1
0.8
0.6
0.4
0.2
average
eff. mass ' — ^
^ ^ <&
10
t
15
(b)
20
Figure 1: Source is optimized studying the overlap with the ground state (a). The effective mass plateau
starts at f = 6 (b).
Since the discrepancy between these ratios is smaller than the uncertainty in a*^"""*^, we set a^""^ =
0.0841(14).
Table 2: Pseudoscalar meson quantities. The residual mass is shown in Tab. 1. The fact that Z^gy is so
close to unity shows the close agreement between the vector and axial current renormalization constants.
a[fm]
ami 1 anih
Za
ZAgV
0.114
0.005/0.04
0.1900(1)
0.08615(13)
0.71722(4)
0.086
0.008/0.03
0.006/0.03
0.004/0.03
0.1729(1)
0.1516(1)
0.1269(1)
0.06707(11)
0.06460(8)
0.06229(11)
0.74530(4)
0.74523(3)
0.74494(4)
0.988(4)
0.999(4)
1.000(4)
0.5338(25)
0.5048(24)
0.4758(12)
4. Nucleon form factors
To extract the quark current matrix elements between nucleon states, we use the standard ratio
of the momentum projected correlation functions [|I|] :
^ff,rr. ^1 „^ ^ff ( C2nf combination \ ., ^
/?^(r,T;P',P) =Cf„, • \^{P'W\P), withr^oo
where the quantity in brackets represents the appropriate combination of two-point functions to
cancel out normalization factors at the source and sink. We calculate the following operators,
4
Nucleon Structure with Domain Wall Fermions at a = 0.084 fin
S. N. Syritsyn
<
N
0.748
0.747
0.746
0.745
0.744
0.743
0.742
32 x64, = 298 MeV
10
average
no blocking
block=8
15
20
(a)
25
30
1.4
1.35
1.3
3
cn
1.25
0)
-2
1.2
■5.
<
60
1.15
1.1
1.05
1
32 x64, m„=298 MeV
ff-ff-ft ?t TT
average
no blocking -s
block=16 -»
10
12
(b)
Figure 2: Ratio determining the renormalization constant Za (a) and the plateau for the axial charge gA (b)
for m„ ~ 298 MeV. Both graphs show the average and the points with/without Jackknife binning.
which we use to extract the corresponding (generalized) form factors
{P'\qY^q\P) = U{P')
{P'\q'fr'q\P)=U{P')
2^f^^V
2^ r
{P'\^^-^-^"\P)=U{P'[
U{P),
where
If]
qj
.{Ml
M2 ,
On Fig. ^, we show the results for the isovector form factors of the the vector current F^^'^{Q^)
and F2^'^{Q^), where both form factors are fitted with the dipole formula. The form factors F\ {Q^)
and F2{Q^) are renormalized with gy = F/'"''^(0), so that F\ (0) = 1.
These initial results for both form factors, using only a small fraction of the full set of planned
domain wall ensembles, are already of high quality and consistent on coarse and fine lattices. As
expected, decreasing the pion mass leads to a larger Dirac mean squared radius and correspondingly
to a steeper form factor F\{Q^). Since the intermediate pion mass is nearly halfway between the
light and heavy masses, and the form factors differ only by a very small amount, expanding it
to leading order one would expect, in the absence of any lattice spacing dependence, that the
intermediate curve would also be halfway between the upper and lower curve. The fact that the
coarse lattice result at the intermediate mass indeed lies halfway between the two fine lattice results
is a clear signature that lattice artifacts associated with the lattice spacing are very small for these
form factors.
The only generalized form factors whose dependency on the transferred momentum can be
extracted reliably with the present statistics are the leading ones, A„o and A„o- In Fig. ^ we show the
results for these form factors, normalized to unity at the zero momentum transfer = 0. As one
goes to higher moments, involving correspondingly more derivatives in the twist- two operators, the
statistical errors increase as expected. However, when the statistics are eventually increased by up
to an order of magnitude, these higher generahzed form factors will also be well determined.
5
Nucleon Structure with Domain Wall Fermions at a = 0.084 fin
S. N. Syritsyn
m„ = 298MeV -
: 329 MeV (coarse) ^
m„ = 356 MeV
m, = 406MeV
a
4 r-
3.5
3
2.5
2
1.5 -
1
0.5 -
-
m„ = 298 MeV
: 329 MeV (coarse)
m„ = 356 MeV
m„ = 406 MeV
0.2 0.4 0.6
[GeV^]
0.8
1.2
Figure 3: The vector form factors F" ^'(Q^) and ''(2^) for three different niji. The curves correspond
to the dipole fit in range < < 0.4GeV2 and range 0.2 <Q^ < 0.8 GeV^, respectively. On the "''(Q^)
plot, the errorbars to the left of 2^ = line show the uncertainty in the determination of F2^^'{Q).
5. Conclusions
By virtue of performing eight measurements per fine domain wall lattice configuration, our
initial calculations on ensembles of roughly 300 to 500 configurations show a good statistical pre-
cision for nucleon generalized form factors. Comparison of results with coarse and fine lattices
indicates that the errors in the form factors arising from lattice spacing artifacts are quite small
for domain wall fermions. The study of other generalized form factors, such as those related to
the quark angular momentum, will require calculation of renormalization constants for the corre-
sponding operators, which is in progress.
Acknowledgements
This work was supported in part by U.S. DOE Contract No. DE-AC05-06OR23177 under
which JS A operates Jefferson Laboratory, by the DOE Office of Nuclear Physics under grants DE-
FG02-94ER40818, DE-FG02-04ER41302, DE-FG02-96ER40965, by the DFG (Forschergruppe
Gitter-Hadronen-Phanomenologie), and the EU Integrated Infrastructure Initiative Hadron Physics
(I3HP) under contract RII3-CT-2004-506078. W.S. acknowledges support by the National Science
Council of Taiwan under grants NSC96-2112-M002-020-MY3 and NSC96-2811-M002-026, K.O.
acknowledges support from the Jeffress Memorial Trust grant J-813, Ph. H. and B. M. acknowledge
support by the Emmy-Noether program of the DFG, and Ph. H., M.P and W. S. acknowledge
support by the A. v. Humboldt-foundation through the Feodor-Lynen program. Ph. H. thanks the
Excellence Cluster Universe at the TU Munich for support. This research used resources under the
INCITE and ESP programs of the Argonne Leadership Computing Facility at Argonne National
Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under
contract DE-AC02-06CH11357, resources provided by the DOE through the USQCD project at
Jefferson Lab and through its support of the MIT Blue Gene/L under grant DE-FG02-05ER25681,
resources provided by the William and Mary Cyclades Cluster, and resources provided by the New
Mexico Computing Applications Center (NMCAC) on Encanto. We are indebted to members of
6
Nucleon Structure with Domain Wall Fermions at a = 0.084 fin
S. N. Syritsyn
[GeV^]
(a)
1.4
1.2
1 ^^.^
0.8
0.6
0.4
0.2
m.
298 MeV
= 356 MeV
m„ = 406 MeV
0.2 0.4
J.
\
J
:
0.6
0.8
[GeV^]
1.2
1 K
0.8
0.6
0.4
0.2
= 298 MeV
= 356 MeV
= 406 MeV
0.2
0.4
[GeV
(b)
0.6 O.i
2,
1.2
1.2
Figure 4: Helicity-even (a) and helicity-odd (b) generalized form factors on fine lattices. All the form
factors are normalized to one at = 0. Lines represent one-parameter dipole fits.
the MILC, RBC, and UKQCD Collaborations for providing the dynamical quark configurations
that made our full QCD calculations possible.
References
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