1. Lattice 2008 @ College of William and Mary
Nucleon sigma term and strange quark content from dynamical overlap simulations
arXiv:  [hep-lat]
Lattice College of William and Mary
Hiroshi Ohki (YITP and Kyoto University)
for JLQCD Collaboration
Thank you.
I would like to talk about Nucleon sigma term and stragne quark content from dynaimical overlap simulations.
This talk is based on the recent work with JLQCD collaborations.

2. Outline Introduction and motivation Methods Simulation details Results
Summary
Here this is outline of my talk.
Introduction and Explanation of our methods
and simulation details
and results.
Finally we summarize my talk

3. JLQCD collaboration KEK BlueGene (10 racks, 57.3 TFlops)
KEK 　 　　　　　 S. Hashimoto, H. Ikeda, T. Kaneko, H. Matsufuru,
J. Noaki, E. Shintani, N. Yamada
Niels Bohr H. Fukaya
Tsukuba 　　　 S. Aoki, T. Kanaya, N. Ishizuka, K.Takeda,
Y. Taniguchi,　A. Ukawa, T. Yoshie
Hiroshima 　 K.-I. Ishikawa, M. Okawa
YITP H. Ohki, T. Onogi, T. Yamazaki
This is amember of JLQCD collabolations.
Numerical simulation is carried out by Blue Gene at KEK.
KEK BlueGene (10 racks, 57.3 TFlops)

4. Introduction What is sigma term? Strange quark content of Nucelon
scalar form factor of the nucleon at zero recoil
and
Low energy parameter of ChPT
O.K. let me start introduction.
I show the definition of sigma term and related quantity.
First, Sigma term is scalar form factor of the nucleon at zero recoil
and characterize the quark mass effect in the nucleon field.
Up to higher order, sigma term is arising as Nucleon mass parameter
In the chiral perturbation theory.
Phenomenologically, sigma term can be rerated to the pi N scattering amplitude at the certain kinematics.
Moreover, several related parameters can be defined, such as y or f parameter which are characterize sea quark effect from the strange quark.

5. strange quark contribution is dominant and important.
Motivation
Y and f parameters are quite important phenomenologically.
Neutralino Dark matter search
The interaction with nucleon is mediated by the higgs boson exchange
in the t-channel.
K. Griest, Phys.Rev.Lett.62,666(1988)
Phys,Rev,D38, 2375(1988)
Baltz, Battaglia, Peskin, Wizanksy
Phys. Rev. D74, (2006).
H, h
Here I mention the implication of sigma term.
This is one of the motivations of calculating Sigma term.
Y and f parameters plays an important role to determine the detection rate of possible nuetralalino.
effective higgs yukawa interaction can be represent as this form.
In this sence, strange quark contributions is dominant.
If neutralino Dark matter can be scatted by higgs particle,
It can be detected by quark field within atoms.
So Presice determination of strange quark contribution is very useful for future experiment of neutralino dark matter search.
heavy quark loop
strange quark
strange quark contribution is dominant and important.

6. Introduction
y (and f) parameters are calculated by connected and disconnected contributions.
While the up and down quarks contributions to sigma piN both as valence and sea quarks which means connected and disconnected contributions, strange quark appears only as a sea quark contributions.
It is very important to calculate both the contributions separetely.
It is necessary for the determination of y parameter to calculate each contribution separately.

7. How well are the parameters known?
Here shows the recent date for determination of sigma term.
Sigma term is about 30 tp 50 MeV from ChPT prediction
Which is consistent for lattice calculations.
But y paramter is large error in both sides.
So the strange quark content has almost 100% unvertainty.
c.f. Recent work of nucleon mass for plenary talk of Walker-Loud
The strange quark content has an almost 100% uncertainty.

8. Uncertainties in y parameter
ChPT: Low Energy Constants (higher order).
c.f. C. Michael et.al. Nucl. Phys. Proc. Suppl.
106, 293 (2002)
Previous lattice calculations (Wilson type fermion).
Mixing of connected and disconnected contributions
(Matrix methods and spectrum methods) due to lattice artifact.
The most crucial uncertainty is the additive mass shift.
Spectrum methods with Wilson type fermions
Sea quark mass derivative with fixed bare valence quark mass is contaminated by physical valence quark mass derivative Which is unwanted lattice artifact ( red arrow).
Here we consider the uncertainties in y parameter.
In the chiral perturbations, such problems arise from the significant uncertainties of the low energy constant in especially analysis including more higher order.
On the other hands, In lattice calculations, in principle it is possible to calculate the nucleon sigma term as well as the sea and valence quark contributions separately.
But lattice calculations have several uncertainty from mixing due to lattice artifact. This problems arise the both methods of calculating sigma terms.
In especially, the sae and valence mass contirubtions have additive mass renormalization and lattice spacing dependence. This contaminations have pointed out by these authors at first.
The most crucial uncertainty is additive mass shift from Wilson type fermions.
Here I explain how these mixing come from lattice simulations in spectrum methods.
In this figure, virticl line shows bare sea quark mass,
In the case of wilson type fermions, additive mass shift depends on the sea quark mass, Sea quark mass derivative with fixed bare valence quark mass is contaminated by physical valence quark mass derivative Which is unwanted lattice artifact ( red arrow).
In previous lattice calculations, after subtracting this contaminations,
The unquenched results have huge statistical errors.

9. Our strategy The advantage of the exact chiral symmetry
Determine the nucleon sigma term in unquenched QCD
using the dynamical quark (overlap fermion),
which has an exact chiral symmetry on the lattice.
The advantage of the exact chiral symmetry
No mixing of connected and disconnected contributions
In this study, we work in nf=2 unquenched QCD.
Result for nf=2+1 QCD will also soon appear.
So our strategy is Determination the nucleon sigma term in unquenched QCD using the dynamilac quark, which has ans exzt chiral symmetery on the lattice.
We exploit mass specturm methods, which is explain in the next.

10. the sigma term from the nucleon mass spectrum.
Our method
the sigma term from the nucleon mass spectrum.
Feynman - Hellman theorem
partial derivatives with respect to the valence and sea quark masses give contributions from ‘connected’ and ‘disconnected’ diagrams.
This is our using formula called Feynman-Hellman theorem,
Which can be extract the nucelons sigma term from nucelons mass spectrum.
Moreover, partial dericatices with respect tie the vakesnce and sea quark masses give contiruiosn from conn.l and disconn. Diagmras.
Please note that there are no subtractions from additive mass shift due to partial derivative respect to the sea quark mass because of the exact chiral symmetry.
no additive mass shift which causes dangerous lattice artifact
Exact chiral symmetry

11. Numerical simulation Measurement of the nucleon 2pt function
6pts(sea) and 9pts(valence) for quark masses
Low mode averaging is employed (#eigenmodes=100)
Nf=2 overlap fermion configurations
16^3 x 32, a=0.12 fm, L=1.9 fm
6 values of sea quark mass
fixed topology
At Q=0 accumulated 10,000
trajectories
O.K.
Here we explain our simulation parameters.
We carry out the 2 flavor overlap fermions,
Volume is 16*32 physical volume is 1.9 fm.
We generate the 500 configuration with 6laues of sea quark mass
Which correspondind to about 300MeV to 800MeV
And We employ the measurement of 2pt function of nucleon.
We have 9 valence quark masses.
Therefore we get the total 6times 9 configuration of nucleon mass data.

12. Results Nucleon masses from 2-pt functions
This graph is numerical data of effective nucelon mass.
We get the nice plateau.
We extract the effective mass from the fit with single exponent.
Effective mass plot for amq=0.035 Solid lines are the mass from the fit

13. Chiral extrap. (unitary point)
extraction of nucleon sigma term
Fit without lightest quark mass data(5pts)
several fit forms to study chiral extrapolation errors
Fit with finite volume correction
(5 and 6pts)
fits including finite volume effects estimated by ChPT.
Next, we show the analysis for unitary point.
First, we shows the chiral perturbation fit of the nucelon mass in diagonal point without lightest quark mass data.
In this analysis, our purpose is studying chiral extrapolation error from several fit forms.
Next, we also try fits including finite volume effect.
Because Box size of L=1.9 fm is rather small for baryon with light quarks.
We can estimate the Finite Size Effect(FSE) using ChPT
Finally, we extract the sigma term from the fit.

14. ChPT Fit of nucleon mass spectrum
Fit formula with Heavy Baryon chiral perturbation theory
c.f. E. E. Jenkins et. al., PLB255,558 (1991)
M. Procura et. al. PRD69, (2004)
I :
II : with input 　　 　　　　　　　　　　 ,　
III : with input ,
(0 : simplified version of Fit I)
We use these fit forms beyond the Heavy baryon chiral perturbation theory.
Fit I is BChPt formula of p to third.
Fit II and III is p to fourth formula with fixed parameter of c2 and c3.
And up to log and higher terms, one can simplfy the fit 0 from these functions.

15. Fit results with and without finite volume corrections
raw data
Finite volume corrected(Fit 0)
Solid …fit 0
dot…fit I
Dashed…Fit II
dot-dashed…Fit III
Raw data
Finite volume
corrected data
The latticed data fit to the CHPT formulaa without laithgt poiint.
As you can see from this left figure, chiral extrapolation error is estimated as
Order ten percent.
Moreover, including finite volume correction, Fit is very sucesesful with all the lattice data point.
Nicely fit to the ChPT formula without lightest point.　
Fit uncertainty is O(10)%.
Successful with all data point

16. Results of sigma term
1. The systematic error is mainly the chiral extrap. error.
2. Finite volume effect (FVE) is sub-leading (~ 9%).
3. We quate final results from Fit 0(FVE uncorrected).
From the fit, we can extract the nucelon sigma term.
Although there are finite volume effect, this is sub-leading effect which is about 9 %.
We take ChPT fit of Fit 0 as our best values,
Results of Sigma term is like this.

17. PQChPT fit (partially quenched data points)
extraction of y parameter
Fit with partially quenched ChPT
(5 X 8 data points)
consistency check of the unitary point fit
interpolation to the strange quark mass.
Separate extraction of connected and disconnected contributions
Next, We employ the fit of Partially quenched data points.
We use partially quenched chiral perturbation theory and
Data points of 5 sea quark mass times 8 valence quark mass,
Because we want to interpolate to the strange quark mass and
Extract the connected and disconnected diagrams.

18. PQChPT fit function Fit a: 6 parameters Fit b: 7 parameters
J.W. Chen et al.,PRD65,094001(2002)
S.R. Beane et al.NPA709,319 (2002)
Fit a: 6 parameters
Fit b: 7 parameters
Fit b: 8 parameters
This is partially qudenched nucelon mass formlua.
There are 8 fit parameters, B00,b01,b02,b11,b10,b20

19. Fit results (PQChPT) PQChPT fit works very well.
It gives consistent results
with the unitary point fit.

20. Connected and disconnected contributions at
valence
Sea
The disconnected contribution (sea quark content) is always smaller than the connected contribution (valence quark content).

21. Connected and disconnected contributions at
Strictly speaking, it is not possible to extract the strange quark content within two-flavor QCD.
For the final result, we should wait for 2+1-flavor QCD result (coming soon).
We present semi-quenched estimate of the y parameter
Semi quenched estimate of y

22. Comparison with other results
Our results of is consistent with ChPT .
Finite Volume correction is controllable.
Previous lattice result of sea/valence is larger than 1.
Our result is 0~0.3.
ChPT predicts
Previous lattice results due to large sea quark contribution without removing lattice artifact.
After removing lattice artifact previous y is -0.3(3)
Our result gives

23. 6. summary
We studied the nucleon mass spectrum for nf=2 unquenched QCD using exactly chiral symmetric fermions.
Our calculation is free from the dangerous lattice artifacts (mixing of connected and disconnected contributions)
Our result of sigma term is consistent with the ChPT prediction.
We found that the disconnected (strange quark content) part is tiny.

24. Thank you

25. Backup slide

26. Fit of the quark mass dependence (5pt)
a means fit with input gA
B means fit with gA free
The solid, dot, dashed, dot-dashed
curves represent the Fit 0a, Ia, II, and III, respectively.

27. Fit of the quark mass dependence (5 and 6pt)
Box size of L=1.9 fm is rather small for baryon with light quarks.
We can estimate the Finite Size Effect(FSE) using ChPT.
ＦＶＣ　Ｒｅｆする
After correcting the lattice data including FSE, we can redo the ChPT fit.
Raw data
Finite volume corrected data

28. Results of sigma term 5ｐｔと６ｐｔ
Although there are finite size effects (FSE), sigma term gets only about 5% change.
ChPT fit (gA fixed) without considering FSE gives reasonable result.
We take ChPT fit (gA fixed, FSE uncorrected) as our best fit, assuming possible 10% finite size error.

29. JLQCD’s Simulations
Overlap fermion ( explicit construction by Neuberger)
Exact chiral symmetry on the lattice　(index theorem)
Hasenfratz, Laliena and Niedermayer, Phys.Lett. B427(1998) 125
Luscher, Phys.Lett.B428(1998)342.
We use the dynamical overlap fermions
like this operator

30. Our analysis Fit 5 pt data without finite volume correction
Fit 0a and 0b: 3 and 4 parameters fit
Fit Ia and Ib: 3 and 4 parameters fit
Fit II : 3 parameters with c_2=3.2[GeV-1] and c_3=-3.4[GeV-1]
Fit III : 3 parameters with c_2=3.2[GeV-1] and c_3=-4.7[GeV-1]
a means fit with input gA=1.267,
B means fit with gA free
Fit 5 and 6 pt data with finite volume correction
Fit 0a: 3 parameters fit
Fit Ia: 3 parameters fit
Fit II : 3 parameters with c_2=3.2[GeV-1] and c_3=-3.4[GeV-1]
Fit III : 3 parameters with c_2=3.2[GeV-1] and c_3=-4.7[GeV-1]

31. Fit of the quark mass dependence (5pt)
The solid, dot, dashed, dot-dashed
curves represent the Fit 0a, Ia, II, and III, respectively.

32. Fit the finite volume corrected data (5 and 6pt)
Box size of L=1.9 fm is rather small for baryon with light quarks.
We can estimate the Finite Size Effect(FSE) using ChPT.
c.f. A. Ali Khan et al.,Nucl, Phys.B 689,175(2004)
For the input parameter, we use the nominal values of gA, c1, c2 and c3, after correcting the lattice data including FSE, we carry out the ChPT fit.
Raw data
Finite volume corrected data