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La#ce QCD : Part 1
Tetsuya Onogi (Osaka University) December 7, 2010 @Kyoto Sangyo Univ.
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Contents
Part 1 １．Overview ２．La#ce formalism： ３．Gauge acKon ４. La#ce fermions: ５．Numerical methods:
Part 2 6．Recent developments • Formalism：Ginsparg-‐Wilson fermion • Algorithm： improved simulaKon methods
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1. Overview Why la#ce QCD? • Quarks are confined so that the elementary process are dressed by nonperturbaKve QCD effects.
• QuanKtaKve understanding of the QCD correcKons is indispensible for the test of the Standard Model and the physics beyond.
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La#ce field theory • A construcKve definiKon of the field theory on a discreKzed spaceKme.
• Enables the nonperturbaKve calculaKon by analyKcal methods (e.g. strong coupling expansion) or by numerical simulaKon.
• Advantage: dynamics from 1st principles
• Disadvantage: symmteries are parKally destroyed due to the discreKzaKon
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２．La#ce formalism
• DiscreKze the space Kme and put the fields on sites or links. The acKons are also discreKzed.
DerivaKve Difference
Covariant derivaKve Covariant difference
• Symmetries can be destroyed by discreKzaKon
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• Lorentz Symmetry
Hyper-‐Cubic symmetries are preserved. Lorentz symmetry is violated only by irrelevant terms.
• Gauge symmetry Must be preserved. Otherwise, unphysical degrees of freedom
(longitudinal modes) cannot be decoupled.
• In general, vector-‐like chiral symmetry and supersymmetry are destroyed by discreKzaKon.
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RenormalizaKon and conKnuum limit • : bare la#ce coupling
: correlaKon length in la#ce units. • To get a conKnuum limit, must diverge at criKcal
coupling as
• ConKnuum limit can be taken by assigning a physical la#ce length for a given in such a way that the physical correlaKon length is kept fixed.
• In othere words,
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In the case of QCD, the criKcal coupling is g=0：
Therefore, requiring
The conKnuum limit is given by
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One can compute the following quanKKes in la#ce units. • String tension： • Nucleon masses, meson masses • Meson decay constants
For example, using as inputs , one can determine the la#ce spacing for each bare couplings .
Any ther quanKtes X can be predicted as
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How to compute the matrix elements? To obtain the matrix elements involving an operator, one needs to match the bare la#ce operator with the one in the conKnuum renormalizaKon scheme.
Matching coefficients (normally computed by perturbaKon theory)
Bare matrix elements (computable numerically)
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3. Gauge acKon • Guiding principle: gauge invariance • Compact variable is used:
• AcKon should be invariant under the gauge tr.
• The Wilson-‐loops are gauge invariant. 1x1 Wilson loop = plaqueie
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• Naïve conKnuum limit
Using Baker-‐Hausdorf formula
The plaqueie reads
Then a gauge invariant ‘plaqueie acKon’ can be constructed as
SomeKmes the gauge coupling is parametrized as
ConKnuum limit at the quantum level Determine the la#ce spacing by keeping a physical quanKty contant. e.g. The quark potenKal from the Wilson loop
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• PerturbaKve analysis
As a result, the bare coupling saKsfies the following renormalizaKon group equaKon.
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The soluKon to the perturbaKve RG equaKon
RelaKon to other schemes A.Hasenfratz and P.Hasenfratz , Phys.Lei.93B(1990)165
H.Kawai, R,Nakayama, K.Seo, Nucl.Phys.B189(1981)40.
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• Analysis from the strong couplin expansion The Wilson loop for TxL loop C
, where
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The Haar measure for the link variable saKsfies
Therefore the leading contribuKon to the Wilson loop comes from the term of the expasion of the gauge acKon, in which the plaqueies fill the area surrounded by the TxL .
Pure gauge theory saKsfies the area law in the strong coupling limit.
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• NonperturbaKve calculaKon by Monte Carlo M.Creutz Phys.Rev.D２１（１９８０）２３０８
The string tension in SU(2) gauge theory
Wilson ｌoop receives contribuKons from the area law（Linear PotenKal) and perimeter law (self-‐energy) etc.
To extract the string tension one considers the following raKo（Creutz RaKo)
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0 1 2
10-2
100
a2
- log( /4)
exp(-6 2/11 ( -2) )
The numerical result reproduces the strong coupling expansion in the strong coupling regime, and the perturbaKve behavior in the weak coupling regime.
The confinement phase is retained in the conKnuum limit.
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4. La#ce fermion • Naïve fermion
Free fermion
light fermion degrees of freedom appear at
Species doubling problem
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• Wilson fermion（add a ‘Wilson term’ to kill the doublers)
AcKon
Free fermion
The doulbers at acquire mass term.
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• Wilson fermion
Advantage:
It can describe a single flavor fermion at low energy. Very useful for its simplicity.
Disadvantage： Chiral symmetry is explicitly violated. AddiKve renormalizaKon to the mass can arise, which requires fine-‐tuning. Small mass region can be numerically unstable due to the fluctuaKon.
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• Staggered fermion A new basis
where
This saKsfies
Then the spin degrees of freedom can be diagonalized as
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Then take only one components out of four components.
Instead, re-‐interpret the space degrees of in the unit cell as spin.
Advantage: Exact U(1) axial symmetry in the SU(4) is preserved. The mass in only mulKplicaKvely renormalized. No fine-‐tuning is needed. Small numerical cost due to small degrees of freedom
Disadvantage: It cannot describe a single fermion at low energy.
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5．ComputaKonal method for physical quanKKes
• Decay constant Define a correlator
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• Here, is an average over the gauge configuraKon
which are generated by the Monte-‐Carlo methos with probability
• is a soluKon to ,
which can be solved by matrix inversion.
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• InserKng complete set of hadron states
to the same correlator as
• Then an exponenKal fit at large Kme t can give the decay constant and the mass .
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More general matrix elements • Semileptonic form factor
3-‐point funcKon
Where is a convoluted propagator,
This can be obtained by the matrix inversion of
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How to obtain gauge configutaions ? pseudo-‐fermion (fermion determinant as bosonic integral)
Used for even number of flavors
odd flavor massive fermion can also be described by bosonic integral
can approxmate the determinant within arbitrary precision ( ex. smaller numerical roundoff errors) if we take sufficiently large degree of the polynomial.
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Hybrid Monte Carlo method StocasKc quanKzaKon+moelcuar dynamics + Metropolis method
EvoluKon by equaKon of moKon with “ficKKous Kme” combined with diffusion by random noise
Errors from discrete ‘ficKous Kme’ can be removed
by the Metropolis accept/reject test
Compute inverse of Dirac operator
in each ‘ficKKous Kme’ step Most Kme consuming part!!
Molecular Dynamics
Conjugate momentum (refreshed using random noise)
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ExtrapolaKon in light quark mass
• In pracKcal Monte-‐Carlo simulaKons, the light quark masses are typically
where is the strange quark mass • However, the physical ud quark mass is
we need chiral extrapolaKon for all physical quanKKes.
What kind of funcKonal form is adequate?
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• Example： PCAC relaKon
• SystemaKc method: Chiral PerturbaKon theory（ChPT)
An effecKve theory for the Nambu-‐Goldstone boson which appears from the spontaneous chiral symmetry breaking in QCD.
A systemaKc expansion based on the non-‐linear realizaKon of the chiral symmetry.
Although it is a non-‐renormalizable theory, it has only a finite number of terms to a given order in the expansion with the quark mass and momentu
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• Example of ChPT at 1-‐loop quark mass dependence of the pseudo-‐scalar mass
Non-‐analyKc part is predictable, which are important in the light quark mass regime. It can affect the result of the chiral extrapolaKon by 10%.
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ChPT Lagrangian
where
１．Each term of this acKon is a free parameter. One cannot predict the physical quanKty at a given mass. But once the parameters are determined from some input (experiment, la#ce results) then the quark mass or momentum dependences can be predicted.
２．One must be careful at which mass and momentum regime the expansion is under control.
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Even if one understand the funcKonal form of the chiral extrapolaKon, one can someKmes get into trouble if the la#ce calculaKon lacks an exact chiral symmetry.
e.g. K meson Bag parameter
The chiral symmetry predicts that the matrix element should behave as
However, without chiral symmetry the radiaKve correcKons mixes V-‐A type 4-‐fermion operator with scalar type operator .
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.
This arKfact cannot be extrapolated away in the conKnuum limit.
In order to avoid it
１．One needs to fine-‐tune the counter term nonperturbaKvely. ２．Use a fermion acKon with exact chiral symmetry.
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Lesson：
In order to carry out correct chiral extrapolaKon
（1） One needs to use the ChPT for the correct fit form. （2） The quark mass must be small enough so that the ChPT
can be reliably applied.
（3） ViolaKon of symmetry can be a serious problem.
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How to compute matrix elements?
1. Path integral with sea quark effects 2. RenormalizaKon of la#ce operators
3. ConKnuum limit and chiral limit
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Summary of the computaKonal steps • Generate the gauge configuraKon
• Obtain the quark propagator and construct the hadron correlaKon funcKons
• Extract the matrix elements from exponenKal fit at large Kme.
• Extrapolate the light quark mass towards chiral limit
• Determine the la#ce spacing using one input.
• Renormalize the operator.
• Take the conKnuum limit.
La#ce QCD : Part 2
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Major devolopments in la#ce QCD
I Exact chiral symmetry on the la#ce:
Ginsparg-‐Wilson fermion
II Improved algorithm in dynamical fermion simulaKon falldown of the ‘Berlin wall’
I. Exact chiral symmetry on the la#ce
QCD with overlap fermion 41
QCD with overlap fermion 42
Ginsparg-‐Wilson fermion
QCD with overlap fermion 43
DerivaKon of the Ginsparg-‐Wilson relaKon
Define an effecKve acKon with finite la#ce spacing from the conKnuum theory using block-‐spin transformaKon.
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We perform a naïve chiral transformaKon for the la#ce field.
Then the equaKon in the previous page gives
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Ginsparg-‐Wilson relaKon
In parKcular, choosing one obtains the standard form
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From one can derive
Explicit construcKon by Neuberger（Overlap Dirac Operator)
where
Free fermion case:
In small momentum limit, this reduces to the Dirac operator.
The denominator in the second term never becomes zero, which guarantees the locality of the Dirac operator.
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Locality in the interacKng case If holds, by expanding the kernel as
where
Then in order to connect (x,y) with
one needs at least operaKons of . Therefore, the overlap Dirac operator exponenKally damps at long distance
in la#ce units as
Locality in a broad sense ( although not ultra-‐local).
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c.f. It has been shown that
a sufficient condiKon for the locality is the admissibiKlity condiKon given by
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• Chiral anomaly can be reproduced. Peforming chiral transformaKon in the path integral
An anomaly term arises from the measure, which reduces to the known chiral anomaly in the conKnnum limit.
Moreover, index theorem also holds at finite la#ce spacing.
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JLQCD+TWQCD collaborations • JLQCD
– SH, H. Ikeda, T. Kaneko, H. Matsufuru, J. Noaki, N. Yamada (KEK) – H. Fukaya (Nagoya) – T. Onogi, E. Shintani (Osaka) – H. Ohki (Kyoto) – S. Aoki, N. Ishizuka, K. Kanaya, Y. Kuramashi, K. Takeda, Y.
Taniguchi, A. Ukawa, T. Yoshie (Tsukuba) – K. Ishikawa, M. Okawa (Hiroshima)
• TWQCD – T.W. Chiu, T.H. Hsieh, K. Ogawa (National Taiwan Univ)
• Machines at KEK (since 2006) – BlueGene/L (10 racks, 57.3 Tflops)
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topics from the project
Topological susceptibility Aoki et al., Phys. Lett. B665, 294 (2008).
S-parameter and pseudo NG boson mass from QCD Shintani et al. Phys. Rev. Lett. 101, 242001 (2008); arXiv:
0806.4222 [hep-lat].
Nucleon sigma term Ohki et al., Phys. Rev. D 78, 054502 (2008); arXiv:
0806.4744 [hep-lat] .
Vacuum polarization functions Shintani et al., Phys. Rev. D 79, 074510 (2009); arXiv:0807.0556
[hep-lat].
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II. Improved algorithm in dynamical fermion simulaKon
Monte-‐Carlo method
We generate gauge configuraKons
with probability
Then, take the average of the physical quanKty
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How to obtain gauge configutaions ? pseudo-‐fermion (fermion determinant as bosonic integral)
Used for even number of flavors
odd flavor massive fermion can also be described by bosonic integral
can approxmate the determinant within arbitrary precision ( ex. smaller numerical roundoff errors) if we take sufficiently large degree of the polynomial.
55
Hybrid Monte Carlo method StocasKc quanKzaKon+moelcuar dynamics + Metropolis method
EvoluKon by equaKon of moKon with “ficKKous Kme” combined with diffusion by random noise
Errors from discrete ‘ficKous Kme’ can be removed
by the Metropolis accept/reject test
Compute inverse of Dirac operator
in each ‘ficKKous Kme’ step Most Kme consuming part!!
Molecular Dynamics
Conjugate momentum (refreshed using random noise)
SimulaKon cost for hybrid Monte-‐Carlo • Time for matrix inversion is proporKonal to the raKo of
the highet eigen value
to the lowest eigen value
• Molecular dynamics Kme step size should be sufficiently fine in proporKonal to the pseudo-‐fermion force size
• Total Molecular dyamics Kme should be taken sufficiently long considering the auto-‐correlaKon. Due to the criKal slowing down we have extra factor
As we decrease the quark mass, the simulaKon cost grows as
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‘The Berlin Wall’ Ukawa (la#ce2001@Berlin)
Empirical formula of the CPU Kme for the 2-‐flavor la#ce QCD simulaKon With O(a)-‐improved Wilson fermion
24 years in TeraFlops machine
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0 0.1 0.2 0.3 0.4 0.5 0
m (GeV)0
1
2
3
4
5
Tfl
op
s y
ear
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Falldown of the Berlin wall 1. SeperaKon of the low mode and High mode
-‐ Hasenbush trick
-‐ Domain decomposiKon method(Luscher et al.)
seperaKon of the long-‐ and short-‐ distance mode in real spaceKme
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2. MulK-‐Kme scale in Molecular Dynamics
• Observed raKo of forces are typically
gauge : pf 1 : pf2 = 20 : 5 : 1
The most Kme-‐consuming long distance mode gives the smallest contribuKon to the force!
• Therefore, one can reduce the frequency of the force measurement as follows （Sexton-‐Weingarten）
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! " # $ %
&'!
!
!&
((F (( FG
F
FR! !"!# !"!$ !"!%!
!"!$
!"!&
!"!'
!"!(
(am )$m ')'MeV
&(&
%(#
$*&
quark mass dep. of the force
Horizontal axis: quark mass VerKcal axis:
Dynamical light fermion simulaKon has become possible with PC-‐cluster(64node)x 0.5yea
DrasKc improvement
Results from the Domain DecomposiKon method （Luscher et al. ）
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Unquenched simulations in chiral regime
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Applications
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PACS-CS O(a)-improved Wilson fermion
JLQCD overlap fermion
Large Scale 2+1-Flavor Lattice QCD Simulations
Large volume is feasible with reasonable numerical cost. No need for chiral extrapolation. Direct test of the hardon spectrum is possible.
Complementary to each other
“Physical point simulation” “Exact chiral symmetry”
Large numerical cost but the chiral effective theory can be reliably applied.
It is found that both approaches are theoretically under control as long as they are applied to appropriate problems.
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Physical point simulation in 2+1 lattice QCD
• O(a)-improved Wilson fermion with PACS-CS and T2K • (Large volume) • Simulations with physical quark masses
tuning of quark masses with reweighting
S. Aoki et al., (PACS-CS collaboration) arXiv:0911.2561[hep-lat]
Comparison of the hadron spectrum : lattice vs experiment Good agreement within a few percent level.
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Nf=2+1 simulation with exact chiral symmetry • Nf=2+1 lattice QCD with overlap fermion • a=0.11 fm, L=1.8, 2.6 fm • p-regime : mud= ms/6～ms • ε-regime : mud=0.002 (mq ～3 MeV)
Comparison with the chiral perturbation theory
ε-regime p-regime
Lattice v.s. NNLO ChPT Lattice v.s. NLO ChPT
Good Agreements
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Physics Applications
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Highlights from PACS-CS
“Rho meson decay from lattice”, N. Ishizuka et al
Preliminary 2+1-flavor QCD result
can be determined!
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Highlights from PACS-CS “Nonperturbative renormalization” , Y. Taniguchi et al
• First systematic study with SF-scheme for 2+1 flavor • Precise determination of quark mass and gauge coupling • Very important on its own • Also useful for further application: flavor physics in general
including heavy quark physics
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Highlights from PACS-CS “Helium Atom from lattice” talk by T. Yamazaki
• Quenched QCD simulation • Study of volume dependence • Significant cost reduction in Wick
contraction • Good agreement with experiment
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• Another testing ground of ChPT – Vector and scalar
– Charge and scalar radius
– Calculation using the all-to-all technique.
q2 dependence well described by a vector meson pole + corrections.
Vector form factor
Results from JLQCD “Pion form factors”, T. Kaneko et al.
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Results from JLQCD “Strange quark content of the nucleon”, Takeda et al.
• Important parameter for WIMP dark matter detection rate
• exact chiral symmetry is crucial for reliable calculation
