Pion vector and scalar form factors with dynamical overlap quarks

Abstract

We calculate the pion vector and scalar form factors in two-flavor QCD. Gauge configurations are generated with dynamical overlap quarks on a 16^3 x 32 lattice at a lattice spacing of 0.12 fm with sea quark masses down to a sixth of the physical strange quark mass. Contributions of disconnected diagrams to the scalar form factor is calculated employing the all-to-all quark propagators. We present a detailed comparison of the vector and scalar radii with chiral perturbation theory to two loops. Comment: 7 pages; Talk presented at the XXVI International Symposium on Lattice Field Theory, July 14 - 19 2008, Williamsburg, Virginia, USA; v2: a reference added

Full text

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arXiv:0810.2590v2 [hep-lat] 22 Oct 2008
KEK-CP-216
Pion vector and scalar form factors with dynamical
overlap quarks
JLQCD and TWQCD collaborations: T. Kaneko∗a,b†, S. Aokic,d, T. W. Chiue,
H. Fukayaa,f, S. Hashimotoa,b, T. H. Hsiehg, H. Matsufurua, J. Noakia, T. Onogih,
E. Shintaniaand N. Yamadaa,b
aHigh Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan
bSchool of High Energy Accelerator Science, The Graduate University for Advanced Studies
(Sokendai), Ibaraki 305-0801,Japan
cGraduate School of Pure and Applied Sciences, University of Tsukuba, Ibaraki 305-8571, Japan
dRiken BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
ePhysics Department, Center for Theoretical Sciences, and Center for Quantum Science and
Engineering, National Taiwan University, Taipei, 10617, Taiwan
fThe Niels Bohr Institute, The Niels Bohr International Academy, Blegdamsvej 17 DK-2100
Copenhagen Ø, Denmark
gResearch Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan
hYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We calculate the pion vector and scalar form factors in two-flavor QCD. Gauge configurationsare
generated with dynamical overlap quarks on a 163×32 lattice at a lattice spacing of 0.12 fm with
sea quark masses down to a sixth of the physical strange quark mass. Contributions of discon-
nected diagrams to the scalar form factor is calculated employingthe all-to-all quark propagators.
We present a detailed comparison of the vector and scalar radii with chiral perturbation theory to
two loops.
The XXVI International Symposium on Lattice Field Theory
July 14-19 2008
Williamsburg, Virginia, USA
∗Speaker.
†E-mail: takashi.kaneko@kek.jp
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/

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Pion vector and scalar form factors with dynamical overlap quarks
T. Kaneko
1. Introduction
Pion electromagnetic form factor FV(q2) is one of the fundamental observables in hadron
physics. An analysis of experimental data based on chiral perturbation theory (ChPT) at two loops
leads to a precise estimate of the charge radius ?r2?V[1]. A detailed comparison of ?r2?Vbetween
ChPT and non-perturbative calculations on the lattice may provide a good testing ground for recent
lattice simulations in the chiral regime as well as a better understanding of the chiral behavior of
FV(q2).
While there is no experimental processes directly related to the scalar form factor FS(q2), the
chiral behavior of the scalar radius ?r2?Sis interesting, as it provides a determination of the LEC l4
and has a 6 times enhanced chiral logarithm compared to ?r2?V. A non-perturbative determination
on the lattice is challenging, because we need to evaluate disconnected three-point functions.
In this article, we update our analysis of FV(q2)reported atthe last conference [2]with doubled
statistics, and present newly obtained results for FS(q2). These quantities are measured on gauge
configurations oftwo-flavor QCDon a163×32lattice generated withthe overlap quark action along
the fixed topology strategy [3]. The lattice spacing determined from the Sommer scale r0=0.49 fm
is a=0.1184(21) fm. We refer the reader to Refs.[4, 5] for detailed setup and overviews of our
production simulations.
2. Measurement of pion correlation functions
We measure pion correlators through all-to-all quark propagators [6]. Contributions of 100
low-lying modes (λ(k), u(k)) (k=1,...,Nep;Nep=100) of the overlap operator D are evaluated ex-
actly, whereas the remaining high modes are taken into account stochastically by the Z2noise
method. We prepare a single noise vector for each configuration, and dilute [6] it into Nd=
3×4×Nt/2 vectors η(k)(k=1,...,Nd) with support on a single value for color and spinor in-
dices and at two time-slices. The all-to-all propagator can then be expressed in a simple form
D−1=∑Nvec
k=1v(k)w(k)†(Nvec= Nep+Nd) with two set of vectors
v(k)=
?
u(1)
λ(1),...,u(Nep)
λ(Nep),x(1),...,x(Nd)
?
, w(k)=
?
u(1),...,u(Nep),η(1),...,η(Nd)?
, (2.1)
where x(d)=D−1(1−∑ku(k)u(k)†)η(d).
From the v and w vectors, we may construct meson fields at a temporal coordinate t with the
Dirac matrix Γ and spatial momentum p
O(k,l)
Γ,φ(t;p) =∑
x,r
φ(r)w(x+r,t)(k)†Γv(x,t)(l)e−ipx.
(2.2)
For the smearing function φ(r), we choose the local φl(r)=δr,0and exponential function φs(r)=
exp[−0.4|r|]. Connected and disconnected three-point functions as well as the subtraction term of
the vev contribution to the scalar form factor, shown in Fig. 1, are calculated from these meson
fields as
C(conn)
πΓπ(∆t,∆t′;p,p′) =
1
Nt∑
t
Nvec
∑
k,l,m=1
O(m,l)
γ5,φs(t +∆t+∆t′;p′)O(l,k)
Γ,φl(t +∆t;p−p′)
×O(k,m)
γ5,φs(t;−p),
(2.3)
2

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Pion vector and scalar form factors with dynamical overlap quarks
T. Kaneko
Figure 1: Connected (left-most diagram) and disconnected (middle diagram) three point functions. Note
that we have the contribution to FS(0) from the right-most diagram due to the non-zero vacuum expectation
value of the scalar operator S.
C(disc)
πΓπ(∆t,∆t′;p,p′) =
1
Nt∑
t
Nvec
∑
k,l=1
O(k,l)
γ5,φs(t +∆t+∆t′;p′)O(l,k)
γ5,φs(t;−p)
×
Nvec
∑
m=1
O(m,m)
Γ,φl(t +∆t;p−p′),
(2.4)
C(vev)
πΓπ(∆t,∆t′;p,p′) =
1
Nt∑
t
Nvec
∑
k,l=1
O(k,l)
γ5,φs(t +∆t+∆t′;p′)O(l,k)
γ5,φs(t;−p)
×
?
1
Nt∑
t′
Nvec
∑
m=1
O(m,m)
Γ,φl(t′;p−p′)
?
conf
,
(2.5)
where ?···?confrepresents a Monte Carlo average. We denote the temporal separation and spatial
momentum for the initial (final) meson by ∆t and p (∆t′and p′), respectively.
Our measurements are carried out at four values of the quark mass mudin the range 290?
Mπ[MeV]?520. We explore the region of the momentum transfer −1.7?q2[GeV2]≤0 by taking
the meson momentum p with |p|≤2. Note that the spatial meson momentum is shown in units of
2πa/L in this article. While we have simulated only the trivial topological sector, the effect of the
fixed global topology is suppressed by the inverse of the space-time volume ∼ 1/V [3].
3. Determination of pion form factors
We calculate effective value of the vector form factor from a ratio
FV(∆t,∆t′;q2) =
2Mπ
Eπ(|p|)+Eπ(|p′|)
C(conn)
Cππ,φsφl(∆t;p)Cππ,φlφs(∆t′;p′).
RV(∆t,∆t′;|p|,|p′|,q2)
RV(∆t,∆t′;0,0,0)
,
(3.1)
RV(∆t,∆t′;|p|,|p′|,q2) =
πγ4π (∆t,∆t′;p,p′)
(3.2)
Here Cππ,φφ′ is the pion two-point function with the smearing function φ (φ′) for the source (sink)
operator, and it can also be calculated from the meson field Eq. (2.2). We take the average of RV
over momentum configurations corresponding to the same value of q2. This average as well as
that over the location of the source operator in Eqs. (2.3)–(2.5) leads to an accurate estimate of
FV(∆t,∆t′;q2) as shown in Fig. 2. The vector form factor Fπ(q2) is determined from a constant fit
in a range of (∆t,∆t′), where FV(∆t,∆t′;q2) shows a reasonable plateau. We include the leading
finite volume correction (FVC) [7] to FV(q2).
The scalar form factor normalized at a certain momentum transfer q2
a similar ratio
refcan be calculated from
3

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Pion vector and scalar form factors with dynamical overlap quarks
T. Kaneko
0.6
0.8
1.0
FV(∆t,∆t′;q2)
∆t = 5
∆t = 6
∆t = 7
∆t = 8
0
5
10
15
∆t′
0.0
0.2
0.4
p = (-1,0,0), p′ = (0,1,0), q = (1,1,0)
p = (-1,0,0), p′ = (0,0,0), q = (1,0,0)
1.0
1.5
FS(∆t,∆t′;q2) / FS(∆t,∆t′;qref
2)
∆t = 5
∆t = 6
∆t = 7
∆t = 8
0
5
10
15
∆t′
0.5
1.0
p = (-1,0,0), p′ = (0,1,0), q = (1,1,0)
p = (0,0,0), p′ = (0,0,0), q = (0,0,0)
Figure 2: Effective value of FV(∆t,∆t′;q2) (left panels) and FS(∆t,∆t′;q2)/FS(∆t,∆t′;q2
mud∼ms,phys/2, where ms,physis the physical strange quark mass.
ref) (right panels) at
FS(∆t,∆t′;q2)
FS(∆t,∆t′;q2
ref)=RS(∆t,∆t′;q2)
RS(∆t,∆t′;q2
ref),
RS(∆t,∆t′;q2) =
Cπ1π(∆t,∆t′;p,p′)
Cππ,φsφl(∆t;p)Cππ,φlφs(∆t′;p′), (3.3)
whereCπ1π=C(conn)
large statistical error than those at q2?=0 due to the severe cancellation between C(disc)
We therefore use FS(q2) normalized at the smallest non-zero momentum transfer with |qref|=1 in
the following analysis. The normalized form factor FS(q2)/FS(q2
while FVC to FS(q2) is not available so far and is not taken into account.
π1π
−C(disc)
π1π+C(vev)
π1π. As Fig. 2 indicates, FS(q2) at q2=0 suffers from a relatively
π1π
and C(vev)
π1π.
ref) is determined by a constant fit,
4. Parametrization of q2dependence
The vector and scalar form factors are plotted as a function of q2in Fig. 3. We observe that
FV(q2) is close to the pole dependence 1/(1−q2/M2
q2dependence is therefore parametrized by the following form of the ρ pole with a polynomial
correction to determine the charge radius ?r2?Vand the curvature cV
ρ) with Mρmeasured at simulated mud. Its
FV(q2) =
1
1−q2/M2
ρ
+c1q2+c2(q2)2+c3(q2)3= 1+1
6?r2?Vq2+cV(q2)2+··· . (4.1)
Because the deviation from the ρ pole is small, we obtain a reasonable χ2/dof∼1, and results for
?r2?Vand cVare stable against the inclusion of the cubic correction term.
-0.6
-0.4 -0.20.0
q2
0.2
0.4
0.6
0.8
1.0
FV(q2)
fit: ρ pole + cubic
VMD
mud = 0.050
-0.6
-0.4 -0.20
q2
0.4
0.6
0.8
1.0
1.2
1.4
FS(q2) / FS(qref
2)
fit: quadratic
mud = 0.050
Figure 3: Vector(left panel)and normalizedscalar formfactors (rightpanel)at mud∼ms,phys/2as a function
of q2. Solid lines show our parametrization and its statistical error. In the left panel, we also plot ρ pole
contribution expected from the vector meson dominance hypothesis by the dashed line.
4

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Pion vector and scalar form factors with dynamical overlap quarks
T. Kaneko
0.00.1 0.2
Mπ
0.3 0.4
0.5
2 [GeV2]
0.2
0.3
0.4
0.5
<r2>V [fm2]
expr’t + ChPT
0.0 0.10.2 0.3
Mπ
2 [GeV2]
0.0
0.2
0.4
0.6
0.8
<r2>S [fm2]
expr’t + ChPT
Figure 4: Chiral extrapolation of ?r2?V (left panel) and ?r2?S(right panel) based on one-loop ChPT. Star
symbols show the experimental value for ?r2?V[1] and an indirect determination of ?r2?Sthrough ππ scat-
tering [10].
Such pole contribution in the scalar channel is not clear within our statistical accuracy. Our
data can be fitted to a simple quadratic form
FS(q2) = FS(0)
?
1+1
6?r2?Sq2+cS(q2)2
?
.
(4.2)
We confirm that the result for the scalar radius ?r2?Sis stable if we switch to the cubic or single
pole form FS(0)/(1−q2/M2
depends on the choice of the parametrization form, and hence is not used in the following analysis.
fit) with Mfitas a fit parameter. The curvature cSis however strongly
5. Chiral extrapolation
In one-loop ChPT, the radii ?r2?Vand ?r2?Sare given by [8]
?r2?V = −
1
NF2(1+6Nlr
1
NF2
6)−
1
NF2ln
?
?M2
π
µ2
?
,
(5.1)
?r2?S=
?
−13
2+6Nlr
4
−
6
NF2ln
?M2
π
µ2
?
,
(5.2)
where N =(4π)2. We set the renormalization scale µ to 4πF, and fix F to the value determined
from our study of the pion mass and decay constant [9]. These fits are however not quite suc-
cessful as seen in Fig. 4. While our data of ?r2?V are fitted with reasonable χ2/dof∼0.3, the
value extrapolated to the physical quark mass 0.362(4) fm2is significantly smaller than experi-
ment 0.437(16) fm2[1]. On the other hand, the one-loop formula for ?r2?Swith the enhanced
chiral log fails to reproduce our data and results in large χ2/dof∼16. We note that similar mild
quark mass dependence of the radii is also observed by the ETM Collaboration with a different
discretization on a slightly finer lattice [11]. It is unlikely that the failure of the fits within one-loop
ChPT is caused by systematic uncertainties due to the fixed topology and the finite lattice spacing.
We then extend our analysis to two loops. The higher order contributions to the radii are given
by [1, 12]
∆?r2?V =
1
N2F4
?13N
192−181
48+6N2rr
V,1
?
M2
π+
1
N2F4
?19
6−12Nlr
1,2
?
M2
πln
?M2
π
µ2
?
, (5.3)
5

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Pion vector and scalar form factors with dynamical overlap quarks
T. Kaneko
0.0 0.1 0.2 0.3
Mπ
2 [GeV2]
-0.1
0.0
0.1
0.2
0.3
0.4
<r2>V [fm2]
expr’t + ChPT
total
at one loop
at two loops
0.00.10.2 0.3
Mπ
2 [GeV2]
0.0
1.0
2.0
3.0
4.0
5.0
cV [GeV-4]
expr’t + ChPT
total
at one loop
at two loops
Figure 5: Simultaneous chiral extrapolation of ?r2?V (left panel) and cV (right panel) based on two-loop
ChPT (solid lines). We also plot one- and two-loop contributions by dashed and dotted lines, respectively.
∆?r2?S=
1
N2F4
?
−23N
192+869
108+88Nlr
1,2+80Nlr
2+5Nlr
3−24N2lr
3lr
4+6N2rr
S,1
?
M2
π
+
1
N2F4
?
−323
36+124Nlr
1,2+130Nlr
2
?
M2
πln
?M2
π
µ2
?
−
65
3N2F4M2
πln
?M2
π
µ2
?2
. (5.4)
At two loops, the curvature cVhas non-trivial contributions and can be included in our analysis
1
60NF2
M2
π
N2F4
1
N2F4
cV =
1
+
1
?N
1,2+N
720−8429
25920+N
?M2
3lr
1,2+N
6lr
6+N2rr
V,2
?
?2
+
?
1
108+N
3lr
6lr
6
?
ln
π
µ2
?
+
1
72N2F4ln
?M2
π
µ2
.
(5.5)
The analytic terms containing rr
Lagrangian. We denote the linear combination lr
lr
While the two-loop formulae involve many
LECs, the simultaneous fit to ?r2?V and cV has
only four free parameters lr
fit plotted in Fig. 5 shows that the two-loop contri-
butions are significant in our simulated region of
mud. Weobtain areasonable value of χ2/dof∼0.5,
and the extrapolated values of ?r2?V and cV are
consistent with experiment [1].
The inclusion of ?r2?Sinto the simultaneous
chiral fit introduces additional four free parame-
ters. In order to stabilize this fit, we fix lr
which appear only in the two-loop terms, to a phe-
nomenological estimate¯l2=4.31 [10] and a lattice
result¯l3=3.44 from our analysis of the pion spectroscopy [9]1. The extrapolation of ?r2?Vand cV
turns out to be consistent with those in Fig. 5. The extrapolation of ?r2?Sis shown in Fig.6. From
this simultaneous fit, we obtain
X,i(X =V,S, i=1,2) represent contributions from O(p6) chiral
1−lr
2/2 appearing commonly in ?r2?Vand cVas
1,2.
0.00.10.2 0.3
Mπ
2 [GeV2]
-0.2
0.0
0.2
0.4
0.6
0.8
<r2>S [fm2]
expr’t + ChPT
total
at one loop
at two loops
Figure 6: Chiral extrapolation of ?r2?Sfrom si-
multaneous to ?r2?V,Sand cV based on two-loop
formulae.
6, lr
1,2, rr
V,1and rr
V,2. This
2and lr
3,
?r2?V= 0.404(22)(22) fm2, ?r2?S= 0.578(69)(46) fm2, cV= 3.11(14)(86) GeV−4. (5.6)
1The µ independent convention¯liis defined from lr
i=γi(¯li+ln[M2
π/µ2])/2N with γ3=−1/2, γ4=2 and γ6=−1/3.
6

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Pion vector and scalar form factors with dynamical overlap quarks
T. Kaneko
The first error is statistical. The second is systematic error estimated by changing the inputs for
lr
radii (?r2?Vand ?r2?S) or those in the vector channel (?r2?Vand cV). We also test ?r2?Sfrom the
cubic parametrization for the q2dependence of FS(q2). Note that all the extrapolated values in
Eq. (5.6) are consistent with experiment.
We obtain¯l6=11.8(0.7)(1.3),¯l4= 4.06(44)(99), and lr
O(p4) LECs. Our estimate of¯l6is slightly smaller than 16.0(0.9) obtained in Ref.[1] partly due
to a deviation of F between our lattice determination [9] and two-loop ChPT [13]. We note that
¯l4is consistent with our determination¯l4=4.12(56) from Fπ[9] and a phenomenological estimate
4.39(22) [10]. Our results for the O(p6) LECs are rr
rr
2and lr
3to different phenomenological estimates in Ref.[1], and by limiting the fitting data to the
1,2=−2.9(0.8)(2.4)×10−3for the
V,1=−1.1×10−5, rr
V,2=−4.0×10−5and
S,1=1.3×10−4with substantial uncertainty of 50–100%.
6. Conclusions
In this article, we report on our calculation of the pion form factors with two flavors of dynam-
ical overlap quarks. By employing the all-to-all quark propagators, FS(q2) is calculated including
contributions from the disconnected diagrams for the first time. The one-loop ChPT formulae fail
to reproduce our data of FS(q2). In our analysis extended to two loops, we observe significant
two-loop contributions at our simulated quark masses, and obtain ?r2?V,Sand cV consistent with
experiment. Further investigations of systematics due to the fixed global topology and quenching
of strange quarks are in progress by direct simulations in the non-trivial topological sectors and in
three-flavor QCD.
Numerical simulations are performed on Hitachi SR11000 and IBM System Blue Gene Solu-
tion at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale
Simulation Program (No. 08-05). Thiswork issupported in part bythe Grant-in-Aid ofthe Ministry
of Education (No. 18340075, 18740167, 19540286, 19740160, 20025010, 20039005, 20340047,
and 20740156), the National Science Council of Taiwan (No. NSC96-2112-M-002-020-MY3,
NSC96-2112-M-001-017-MY3, NSC97-2119-M-002-001), and NTU-CQSE (No. 97R0066-69).
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