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Applications of Chiral Perturbationtheory to lattice QCD (I)
How to extrapolate from what we can simulate to physical answers
Stephen R. Sharpe
University of Washington
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.1/46
Glossary!EFT Effective Field Theory
χPT: Chiral Perturbation Theory
PQQCD: Partially Quenched QCD
PQχPT: Partially Quenched χPT
WχPT: Wilson χPT (including lattice spacing effects)
tmQCD: Twisted mass QCD
tmχPT: Twisted mass χPT (including lattice spacing effects)
SχPT: Staggered χPT (including lattice spacing effects)
(P)GB: (Psuedo) Goldstone Boson
LEC: Low energy coefficient (in chiral Lagrangian)
VEV: Vacuum Expectation Value
LO: leading order
NLO: next-to-leading order, etc.
NP: non-perturbative
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.2/46
Outline of 3 lecturesLecture 1
Overview and aims of 3 lecturesChiral perturbation theory for (continuum) QCD
Lecture 2Incorporating lattice spacing errors
Application to Wilson & twisted mass fermions
Lecture 3Partial quenching and PQχPT
Application to staggered fermions
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.3/46
Outline of Lecture 1Overview and aims
Why LQCD needs other theoretical input
Where is χPT needed
Review of χPT in the continuumEffective field theories in general
Broken chiral symmetry and its implications
Constructing the pionic Lagrangian
Examples of NLO results
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.4/46
Overview and AimsLattice QCD is entering an exciting era
Terascale computers (e.g. PACS-CS, QCDOC, clusters)
Unquenched simulations with mπ → 250 MeV and below
Potential for few percent control over all systematics
Yet LQCD simulations will inevitably require extrapolations
To physical light quark masses, mu, md
from m` = (mu + md)/2 ∼ (2 − 3) × m`,phys [staggered fermions]
To the continuum limit, a = 0from a ∼ 0.05 − 0.1 fm, and from αS ∼ 0.3
To “infinite” box size L 1/mπ
from L ∼ 3 − 5 fm
From αEM = 0 to αEM = 1/137
From finite volume energies of two particles to infinite volumescattering amplitudes
Theoretical input essential for these extrapolations
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.5/46
Overview: Unphysical SimulationsWidespread use of unphysical simulations:
Staggered fermions with the “ 4√
Det” trick
Theory unitary (at best) in continuum limit
“Mixed action” simulations
Overlap/Domain wall valence fermions on staggered sea (using 4√
Det
trick)
Partially quenched QCD
Valence quarks not degenerate with sea quarks
Gives much more information to constrain chiral extrapolations
Matrix elements with unphysical kinematics
e.g. 〈K|OW |ππ〉 with all particles at rest
OW inserts momentum
All require theoretical input to obtain physical results
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.6/46
Overview: PQQCDMost unphysical example is PQQCD
Valence quarks not degenerate with sea quarks
Can easily simulate, but theory is not unitary
Can be described by an extension of χPT (PQχPT)− Requires additional theoretical assumptions
+ Involves few (or no) unphysical “low energy coefficients”
Can be used to significantly improve chiral extrapolations
mSea
mStrange
mValence
mStrange
1/4 1/2 1
1
SimulationsLattice
QCDPQ Chiral
Pert. Theory
Valenc
e
m
= m Sea
Physic
al (U
nque
nche
d) T
heor
ies
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.7/46
Overview: Chiral Perturbation TheoryχPT is the tool for most extrapolations:
Chiral extrapolation of unquenched QCD-like results
Incorporation of leading finite volume effects (pion loops)
Incorporation of operators with momentum insertion (OW )
Extension to partially quenched theories: PQχPT
Incorporation of lattice artefacts, particularly those breaking continuumsymmetries
Wilson fermions (axial symmetry breaking): WχPT
Twisted mass (flavor symmetry breaking): tmχPT
Staggered fermions (taste symmetry breaking, 4√
Det trick): SχPT
Mixed actions: MAχPT
Usually need to simultaneously extrapolate in m, L, and a, and χPTprovides expressions
CAVEAT: need to truncate χPT⇒ additional systematic error
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.8/46
Overview: HighlightsPhase diagram for tmLQCD: tmχPT predicts two possibilities
m/a
µ/a2
2
Aoki-phase along Wilson axis:
apparently holds for quenched theory, and
dynamical fermions at large a(Wilson gauge action)
m/a
µ/a2
2
First-order phase transition:
apparently holds for dynamical
fermions at small a(Wilson or Symanzik gauge action)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.9/46
Overview: Highlights (continued)Comparing [Farchioni et al,hep-lat/0410031] with tmχPT
0
0.2
0.4
0.6
0.8
1
1.2
2.8 2.85 2.9 2.95 3 3.05 3.1
(am
π)2
µκ = (2κ)-1
123x24 latticeβ = 0.67κ = 0.165 - 0.175c1=-1.4088
(amπ)2min=0.0904 -0.06-0.04-0.02 0.02 0.04 0.06m’
0.1
0.2
0.3
0.4
mpi^2 vs. m’, mu=0
Qualitative comparison only
Difference in slopes for positive and negative m from 30% O(a)contribution
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.10/46
Overview: Highlights (continued)Fitting staggered pion properties with SχPT [Aubin et al, hep-lat/0407028]
O(a2) taste-breaking essential for fit
PQ data essential to constrain parameters (e.g. 416 points/48 params)
Determine Ntaste = 1.28(16)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.11/46
Outline of Lecture 1Overview and aims
Why LQCD needs other theoretical input
Where is χPT needed
Review of χPT in the continuumEffective field theories in general
Broken chiral symmetry and its implications
Constructing the pionic Lagrangian
Examples of NLO results
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.12/46
Effective Field Theory/χPT: referencesA selection of books and lecture notes:
H. Georgi, “Weak Interactions and Modern Particle Theory”
J.F. Donoghue, E. Golowich and B.R. Holstein, “Dynamics of the Standard
Model”
A.V. Manohar, “Effective Field Theories”, hep-ph/9606222
G. Ecker, “Chiral Perturbation Theory”, hep-ph/9608226,9805300
A. Pich, “Introduction to Chiral Perturbation Theory”, hep-ph/9502366
D.B. Kaplan, “5 lectures on Effective Field Theory”, nucl-th/0510023
Classic papers:
S.R. Coleman, J. Wess and B. Zumino, “Structure Of Phenomenological
Lagrangians. 1,” Phys. Rev. 177, 2239 (1969).
C.G. Callan, S.R. Coleman, J. Wess and B. Zumino, “Structure Of
Phenomenological Lagrangians. 2,” Phys. Rev. 177, 2247 (1969).
S. Weinberg, “Phenomenological Lagrangians,” PhysicaA 96, 327 (1979).
J. Gasser and H. Leutwyler, “Chiral Perturbation Theory To One Loop,”
Annals Phys. 158, 142 (1984).
J. Gasser and H. Leutwyler, “Chiral Perturbation Theory: Expansions In
The Mass Of The Strange Quark,” Nucl. Phys. B 250, 465 (1985).S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.13/46
Effective Field Theories: RecipeSeparation of energy scales
For χPT: pπ ∼ mπ mρ, mp
For Symanzik effective Lagrangian for LQCD: pq,g ∼ ΛQCD π/a
The only non-analyticities in correlation functions are due to “light”degrees of freedom (d.o.f.)
χPT: PGBs; Symanzik: low momentum quarks and gluons
Write low-energy effective Lagrangian, Leff , in terms of light d.o.f.
LOCAL QFT
Corresponds to integrating out heavy d.o.f.
Constrained by symmetries of underlying theory
Not renormalizable—valid in limited energy range
Terms ordered by power counting in ratio of scales
Justification: gives most general unitary S-matrix consistent withsymmetries [Weinberg]
Gives results valid up to truncation error: (mπ/mp)n, (aΛQCD)n
Unknown LECs (sometimes perturbatively calculable)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.14/46
Effective Field Theories: ExamplesSymanzik: Integrating out q ∼ 1/a gluons for staggered fermions leads to four-fermionoperators
a2
q=1/a
χPT: QCD correlation functions can be represented by PGB contributions
π
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.15/46
Outline of Lecture 1Overview and aims
Why LQCD needs other theoretical input
Where is χPT needed
Review of χPT in the continuumEffective field theories in general
Broken chiral symmetry and its implications
Constructing the pionic Lagrangian
Examples of NLO results
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.16/46
Chiral symmetry of QCD actionFermionic part of Euclidean Lagrangian in matrix notation:
LQCD = QLD/ QL +QRD/ QR +QLMQR +QRM†QL
Qtr = (u, d, s), QL,R = QL,R(1 ± γ5)/2, QL,R = [(1 ∓ γ5)/2]QL,R
In the massless limit, have G = SU(3)L × SU(3)R symmetry:
QL,R → UL,RQL,R and QL,R → QL,RU†L,R, with UL,R ∈ SU(3)L,R
Also have Vector U(1) (quark number) symmetry, but trivial
Axial U(1) broken by anomaly
Add in mass term, e.g. M = diag(mu, md, ms), mq 6= 0
axial transformations UL = U†R broken
vector SU(3) subgroup (UL = UR) also broken, except if masses
degenerate
If treat M as complex “spurion” field then maintain full chiral symmetry
M → ULMU†R, M† → URM†UL
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.17/46
Approximate chiral symmetryChiral symmetry is useful if M is small:
What is small? mq ΛQCD ∼ 300 MeV
More precise criterion in χPT: mπ,K,η Λχ ≡ 4πfπ ≈ 1200 MeV
(mu + md)/2 ≈ 4 MeV ⇒ SU(2)L × SU(2)R is a good approximate
symmetry
ms ≈ 100 MeV or mK,η ≈ Λχ/2 ⇒ SU(3)L × SU(3)R is much less good
Important question for lattice applications of chiral perturbation theoryand thus PQQCD:
Is ms small enough that approximate chiral symmetry is useful to
determine the quark mass dependence when mlats ≈ ms?
If not, then can only use chiral symmetry to guide extrapolations in mu
and md.
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.18/46
Spontaneous breaking of chiral symmetryVacuum breaks chiral symmetry, with order parameter
〈qq〉 = 〈(qLqR + qRqL)〉 ∼ Λ3QCD 6= 0 , q = u, d, s
How do we know this?
No parity doubling in spectrum: mN (P = +) 6= mN (P = −)
Lightness of π, K and η consistent with their being pseudo-Goldstone
bosons (PGBs)
Lattice simulations ⇒ 〈qq〉 6= 0
Success of constituent quark model with mconst. ≈ 300 MeV
Success of chiral perturbation theory
Vector symmetry not spontaneously broken
If mu = md = ms then 〈uu〉 = 〈dd〉 = 〈ss〉
Based on experiment, and [Vafa-Witten] theorem (? [Crompton] )
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.19/46
General description of symmetry breaking (M = 0)Condensate is LR flavor matrix:
Ωij = 〈QL,i,α,cQR,j,α,c〉 −→G
UL Ω U†R
All choices of Ωij are equivalent: “vacuum manifold”
Unbroken vector symmetry ⇒ Ωij = ω δij is in manifold
ω 6= 0 implies chiral symmetry breaking:
SU(3)L × SU(3)R︸ ︷︷ ︸
G
−→ SU(3)︸ ︷︷ ︸
H
“Direction” of condensate depends on M . Conventional choice (M
diagonal and positive) gives, when M → 0:
Ωij = ω δij , ω = −〈qq〉 > 0
Ω = ω −→G
ωULU†R ⇒ H = SU(3)V : UL = UR
Axial tranformations ULU†R = U2
L are broken
Goldstone’s theorem: 8 broken generators ⇒ 8 GBs (π, K, η)
〈πb(p)|Qγµγ5T aQ(0)|0〉 = −ifπpµδba
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.20/46
Building the effective field theoryWe have the correct ingredients:
Separated scales mGB = 0 mp
Maintained even when include quark masses (ms?)
Maintain scale separation by considering pGB mp
Also hold for GB scattering off heavy (almost static) sources phad mhad
Build EFT using only GB fields, static sources, and spurions (M)
Most general local QFT consistent with symmetries (and their breaking)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.21/46
Representing GB fieldsConceptually most non-trivial step of construction:
EFT built from GBs (mesons), while QCD built from quarks
Choice of GB fields not unique (not discussed here)
Use complex scalar field theory as guide (“Mexican hat”):
V = −µ2φ†φ +λ
2(φ†φ)2
Classical minimum |〈φ〉| = v, breaks symmetry G = U(1) −→ H = 1
Spectrum: GB (phase rotations) and massive field (radial excitations):
φ(x) = v exp[ρ(x)] exp[iθ(x)]
To obtain low-energy EFT (GB fields only) can either:
Integrate out ρ(x), leaving Leff in terms of exp[iθ(x)]
Write down most general Leff consistent with symmetries
Leff ∼ c2∂µ(eiθ)∂µ(e−iθ) + c4∂µ(eiθ)∂µ(e−iθ)∂ν(eiθ)∂ν(e−iθ) + . . .
Can determine low-energy constants ci by perturbative matching
No mass term since no invariants without derivatives: (eiθe−iθ)n = 1
Use exponential form as transforms linearly under G: eiθ[x] → eiαeiθ[x]
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.22/46
Exponential Parameterization of GBsLesson for EFT: parameterize excitations using full vacuum manifold
U(1) theory:〈φ〉v
= eiθ −→ eiθ(x)
Note that any choice of 〈θ〉 breaks symmetry:
Symmetry breaking is built in by use of angular variables, so
Goldstones’ theorem guarantees correct spectrum
Corresponding choice for QCD is
Ωij
ω≡
〈QL,i,α,cQR,j,α,c〉|〈qq〉| ≡ Σij −→ Σij(x) ∈ SU(3)
Tranforms under G = SU(3)L × SU(3)R like Ω (i.e. linearly):
Σ(x) −→G
ULΣ(x)U†R
Any VEV of Σ breaks G to H = SU(3) ⇒ desired symmetry breaking
Can decompose into GB (pion) fields. Taking 〈Σ〉 = 1 have:
Σ(x) = exp (2iπa(x)T a/f) , a = 1, 8
GB fields transform non-linearly
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.23/46
Outline of Lecture 1Overview and aims
Why LQCD needs other theoretical input
Where is χPT needed
Review of χPT in the continuumEffective field theories in general
Broken chiral symmetry and its implications
Constructing the pionic Lagrangian
Examples of NLO results
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.24/46
Building blocks for Leff
Ingredients are Σ, Σ†, M , M† and external sources (discussed later)
Σ → ULΣU†R , Σ† → URΣ†U†
L , M → ULMU†R , M† → URM†U†
R ,
Useful building blocks (noting ΣΣ† = 1 = Σ†Σ)
LH: Lµ = Σ∂µΣ† = −∂µΣΣ† = −L†µ −→ ULLµU†
L
LH: MΣ† → UL(MΣ†)U†L , ΣM† → UL(ΣM†)U†
L
RH: Rµ = Σ†∂µΣ = −∂µΣ†Σ = −R†µ −→ URRµU†
R
RH: M†Σ → UR(M†Σ)U†R , Σ†M → UR(Σ†M)U†
R
[Derivatives only act on object immediately to right]
Important property follows from det(Σ) = 1:
0 = ∂µ(det Σ) = ∂µ(exp tr ln Σ) = det Σ tr(Σ−1∂µΣ) = −tr(Lµ)
Thus Lµ, Rµ are elements of Lie algebra su(3)
Can often just use LH building blocks and enforce parity at end
If 〈Σ〉 = 1, parity:
π(x) → −π(xP ), Σ(x) ↔ Σ†(xP ), Lµ(x) ↔ Rµ(xP ), M → M†
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.25/46
Constructing Leff: finally!Rule: local, “Lorentz”, SU(3)L × SU(3)R, C, P and T invariant
No derivatives: no termsOne derivative: no Lorentz scalars
Two derivatives:1. tr(LµLµ) = −tr(∂µΣ∂µΣ†) = tr(RµRµ)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.26/46
Constructing Leff: finally!Rule: local, “Lorentz”, SU(3)L × SU(3)R, C, P and T invariant
No derivatives: no termsOne derivative: no Lorentz scalars
Two derivatives:1. tr(LµLµ) = −tr(∂µΣ∂µΣ†) = tr(RµRµ)
No derivatives and one mass insertion:2. tr(MΣ†) + tr(ΣM†)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.26/46
Constructing Leff: finally!Rule: local, “Lorentz”, SU(3)L × SU(3)R, C, P and T invariant
No derivatives: no termsOne derivative: no Lorentz scalars
Two derivatives:1. tr(LµLµ) = −tr(∂µΣ∂µΣ†) = tr(RµRµ)
No derivatives and one mass insertion:2. tr(MΣ†) + tr(ΣM†)
Four derivatives:3. tr(LµLµ)2
4. tr(LµLν)tr(LµLν)
5. tr(LµLµLνLν) [not independent in SU(2)]
6. tr(LµLνLµLν) [not independent in SU(2) or SU(3)]
7. Wess-Zumino-Witten term involving εµνρσ : not needed here
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.26/46
Constructing Leff: finally!Rule: local, “Lorentz”, SU(3)L × SU(3)R, C, P and T invariant
No derivatives: no termsOne derivative: no Lorentz scalars
Two derivatives:1. tr(LµLµ) = −tr(∂µΣ∂µΣ†) = tr(RµRµ)
No derivatives and one mass insertion:2. tr(MΣ†) + tr(ΣM†)
Four derivatives:3. tr(LµLµ)2
4. tr(LµLν)tr(LµLν)
5. tr(LµLµLνLν) [not independent in SU(2)]
6. tr(LµLνLµLν) [not independent in SU(2) or SU(3)]
7. Wess-Zumino-Witten term involving εµνρσ : not needed here
Two derivatives and one mass insertion:8. tr(LµLµ) tr(MΣ† + ΣM†)9. tr(LµLµ[MΣ† + ΣM†])
Two mass insertions:10. tr(MΣ† + ΣM†)2
11. tr(MΣ† − ΣM†)2
12. tr(MΣ†MΣ† + M†ΣM†Σ)S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.26/46
Leading order LagrangianWill find that expanding in powers of ∂2 ∼ M is appropriate
At leading order have: [recall Σ = exp(2iπaT a/f)]
L(2) =f2
4tr(
∂µΣ∂µΣ†)
− f2B0
2tr(MΣ† + ΣM†)
Two (so far) unknown LECs: f and B0 (both mass dimension 1)
Expect f ∼ B0 ∼ ΛQCD
Up to this stage, M is a complex spurion field. Now set to physical value:
M = diag(mu, md, ms) = M†
Allowed us to expand about massless theory with exact chiral symmetry
Can now determine VEV 〈Σ〉 by minimizing potential:
V(2) = −f2B0
2tr(
M [Σ† + Σ])
If all mq > 0, find 〈Σ〉 = 1
Thus pion fields in Σ are fluctuations about VEV within vacuum manifold
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.27/46
Brief aside on vacuum structure
V(2) = −f2B0
2tr(
M [Σ† + Σ])
For two flavors:
If we use 〈Σ〉 = exp(iθ~n · ~τ), then 〈[Σ† + Σ]〉 = 2 cos θ × 1
Thus V(2) ∝ −tr(M) cos θ
So if trM > 0, 〈Σ〉 = 1, while if trM < 0, 〈Σ〉 = −1
⇒ For degenerate quarks, have first order phase transition at m = 0
For three flavors, Σ = −1 notpossible
Interesting phase structure if some
mq < 0 [Dashen,Creutz]
mu = 0 is not special if md 6= 0:
no subgroup of SU(3)L × SU(3)R
is restored
CP
CP
−1 0 00 −1 00 0 1
( )∼ Σ 1 0 00 −1 00 0 −1
( )∼ Σ
−1 0 00 1 00 0 −1
( )∼ Σ
m
m
u
d
−ms
0 0 1( )∼ Σ
1 0 00 1 0
−ms
ms > 0 fixed
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.28/46
Leading order (P)GB propertiesAssume legitimate to expand in powers of ∂2 ∼ M
Insert Σ = exp(2iπ/f), with π ≡ πaT a, into leading order (LO) L
L(2) =f2
4tr(
∂µΣ∂µΣ†)
− f2B0
2tr(M [Σ† + Σ])
= tr(∂µπ∂µπ) + 2B0 tr(Mπ2)
+1
3f2tr([π, ∂µπ][π, ∂µπ]) − 2B0
3f2tr(Mπ4) + O(π6)
Choosing tr(T aT b) = δab/2, kinetic term normalized correctly (f ’s cancel)
If M = 0, GB interactions all involve derivatives
In general m2PGB ∝ M
For degenerate quarks, m2π = 2B0mq
Sequence of non-renormalizable interactions involving even numbers of
PGBs, size determined by f and B0M
⇒ LO χPT predictive: e.g. 6 pion interactions given by 4 pion term
WZW term leads to interactions involving odd numbers of pions (5, . . . )
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.29/46
LO mass predictions for real QCD
Determine physical particles using U(3)V (π → UV π U†V )
π =
π0
2+ η√
12
π+√
2
K+√
2π−√
2− π0
√2
+ η√12
K0√
2K−√
2
K0√2
− 2η√12
Inserting into −2B0tr(Mπ2) find
Charged particle masses are simple: m2qiqj
= B0(mi + mj), i 6= j
⇒m2
K+ + m2K0
2m2π+
=m` + ms
2m`+ EM ≈ 13
(
m` =mu + md
2
)
π0 and η mix, but with small angle θ ∼ (mu − md)/ms 1
m2π0 = m2
π+ + O(θ2m2K) + EM ,
m2η
︸︷︷︸
(548 MeV)2
= (2[m2K+ + m2
K0 ] − m2π+ )/3
︸ ︷︷ ︸
(566 MeV)2
+O(θ2m2K)
Cannot determine quark masses from χPT since scale dependent
Always appear in combination χ ≡ 2B0M
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.30/46
Lessons for lattice simulations (I)
+ LO χPT works to ∼ 10% in GMOrelation
Indeed, m2π+/mq ∼ const. seen in
all simulations (since 1983)
E.g. quenched Wilson fermions
[Bhattacharya95]
[vertical lines indicate mphyss ]
+ Can vary mq in simulations (more “knobs to turn” than in real QCD), andχPT describes dependence on quark masses in terms of the physicalLECs
− Unfortunately mlattu,d mphys
u,d so we have to rely on χPT to extrapolate
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.31/46
Lessons for lattice simulations (II)+ If simulate isospin limit mu = md then close to real QCD:
mu/md ∼ 1/2 does not lead to large isospin violations
Differences are suppressed by (mu − md)/ms (PGBs) or by
(mu − md)/ΛQCD (other hadrons)
− Calculating isospin breaking effects (e.g. m2π+ − m2
π0) is hard
Quark mass contributions involve disconnected diagrams and are small
u−d u−d
EM contributions are comparable and not easy to calculate (but recent
progress [Namekawa05,Yamada05] using background EM field
[Duncan96] ; also χPT-based method of [Gupta84] )
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.32/46
Power counting in χPT (M = 0)How can a non-renormalizable theory be predictive?
L(2) ∼ f2tr(LµLµ) ∼ (∂π)2 +π2(∂π)2
f2+ . . .
L(4) ∼ LGLtr(LµLµ)2 + · · · ∼ LGL
[(∂π)4
f4+
π2(∂π)4
f6
]
LGL are unknown dimensionless Gasser-Leutwyler coeffs
Consider ππ scattering (with, say, dim. reg. to avoid power divergences):
L(2)tree: p
p∼ p2
f2 L(4)tree: p
ppp ∼ LGL
(p2
f2
)2
L(2)1−loop:
pp p
ppp
pp ∼(
p2
f2
)2ln(p2/µ2)
(4π)2
Straightforward power-counting exercise (counting factors of f) ⇒ haveexpansion in p2/f2 up to logs
LO: L(2)tree (“trivial” to calculate)
NLO: L(4)tree + L
(2)1−loop (“easy” to calculate)
NNLO: L(6)tree + L
(4)1−loop + L
(2)2−loop (hard but done)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.33/46
Power counting (continued)
L(2)tree: p
p∼ p2
f2 L(4)tree: p
ppp ∼ LGL
(p2
f2
)2
L(2)1−loop:
pp p
ppp
pp ∼(
p2
f2
)2ln(p2/µ2)
(4π)2
Theory is predictive up to truncation errors:
E.g. at LO, A(ππ → ππ) predicted in terms of f(= fπ), up to errors of
relative size p2/f2
Only a finite number of diagrams and LECs at each order, so can always
make predictions
Non-analytic behavior (“chiral logs”) does not involve new LECs
Loops renormalize LECs: LGL → LGL(µ)
True expansion parameter? Use “naive dimensional analysis”:
dLGL/d ln(µ) ≈ 1/(4π)2 ⇒ LGL(2µ) − LGL(µ) ≈ 1/(4π)2
So guess: LGL(µ ≈ mρ) ≈ 1/(4π)2
Works well phenomenologically: −1 ∼< LGL(4π)2 ∼
< +1
Implies expansion parameter is p2/Λ2χ, with Λχ = 4πf
For M 6= 0, p2/Λ2χ −→ (p2 or m2
PGB)/Λ2χ
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.34/46
Lessons for lattice simulations (III)+ Use χPT to extend reach of lattice to multiparticle processes
Calculate LECs from lattice simulations using simple physicalquantities (e.g. masses)
Use χPT + LECs to determine multiparticle processes (scatteringamplitudes, ππ → 4π, etc.) that are difficult or impossible todetermine directly using simulations
Using 2L8 − L5 determined from simulations with mu = md to rule
out mu = 0
Determining A(K → ππ) using unphysical, but more accessible, matrix
elements [Rome-Southampton, Laiho-Soni]
− Always have truncation error when using χPT
Need to include NNLO terms (at least approximately) to determine
NLO coefficients (LGL)
Fitting requires (approximate) NNNLO coefficients to work up to
mphyss [MILC]
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.35/46
Technical aside: adding sourcesMatrix elements of Vµ, Aµ, S and P are phenomenologically interesting
Incorporate in QCD using external sources (hermitian matrices)
LQCD = QL(iD/ −γµlµ)QL+QR(iD/ −γµrµ)QR−QL(s+ip)QR−QR(s−ip)QL
Switched to Minkowski space for the moment
s, p not new—rewriting of spurions M = s + ip, M† = s − ip
Obtain correlation functions in QCD by functional derivatives of
ZQCD(lµ, rµ, s, p)
Basic assumption of χPT: ZQCD(lµ, rµ, s, p) = Zχ(lµ, rµ, s, p) forp, mPGB Λχ, up to truncation errors
Functional derivatives of Zχ give χPT result for correlation functions
e.g.δ
δlµ(x)
δ
δp(y)ln Zχ
∣∣∣∣l=r=p=0,s=M
∼ 〈T [Lµ(x)P (y)]〉
⇒ fπ ∝ 〈0|Lµ|π〉
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.36/46
Adding sources (continued)How determine Zχ(lµ, rµ, s, p)?
Generalize spurion trick to local SU(3)L × SU(3)R symmetry
LQCD invariant if l, rµ transform as gauge fields:
lµ → ULlµU†L + iUL∂µU†
L, rµ → URrµU†R + iUR∂µU†
R
s, p transform as before: e.g. (s + ip) → UL(s + ip)U†R
ZQCD invariant (up to anomalies) ⇒ Zχ invariant (up to anomalies)
⇒ Lχ invariant [Gasser-Leutwyler]
Can be accomplished using covariant derivatives: ∂µ → Dµ
e.g. DµΣ = ∂µΣ − ilµΣ + iΣrµ → UL(DµΣ)U†L
Normalization of l, rµ terms fixed
Remainder of enumeration as before (except DµM now allowed)
Convenient to introduce χ = 2B0(s + ip) = 2B0M
In general χ is a matrix source
But also use notation χq = 2B0mq
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.37/46
Final form of chiral LagrangianAt LO (back to Euclidean space):
L(2) =f2
4tr(
DµΣDµΣ†)
− f2
4tr(χΣ† + Σχ†)
Using δ/δlµ(x)|l=r=p=0,s=M can “match” currents with QCD:
QLγµT aQL ' (if2/2)tr(T aΣ∂µΣ†) = −(f/2)∂µπa + . . .
⇒ at LO, f = fπ ≈ 93 MeV
Using δ/δs(x)|l=r=p=0,s=M can relate condensate to B0:
QQ ' −(f2B0/2)tr(Σ + Σ†) = −Nff2B0 + O(π2)
⇒ at LO, 〈qq〉 = −f2B0 [Gell-Mann–Oakes–Renner]
Only using lattice can one determine B0
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.38/46
Final form of chiral Lagrangian (cont)At NLO have 10 LECs and 2 “high-energy coefficients”:
L(4) = −L1 tr(DµΣDµΣ†)2 − L2 tr(DµΣDνΣ†)tr(DµΣDνΣ†)
+L3 tr(DµΣDµΣ†DνΣDνΣ†)
+L4 tr(DµΣ†DµΣ)tr(χ†Σ + Σ†χ) + L5 tr(DµΣ†DµΣ)[χ†Σ + Σ†χ])
−L6
[tr(χ†Σ + Σ†χ)
]2− L7
[tr(χ†Σ − Σ†χ)
]2− L8 tr(χ†Σχ†Σ + p.c.)
+L9 itr(LµνDµΣDνΣ† + p.c.) + L10 tr(LµνΣRµνΣ†)
+H1 tr(LµνLµν + p.c.) + H2 tr(χ†χ)
Possible term ∝ tr(Dµχ†DµΣ) is redundant
Li are “Gasser-Leutwyler coefficients”
Fundamental parameters of QCD, akin to hadron mass ratios
A subset can be determined experimentally to good accuracy
A different subset is straightforward to determine on the lattice
H1,2 give contact terms in correlation functions
At NNLO there are 90 LECs and 4 HECs! [Bijnens et al]
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.39/46
Outline of Lecture 1Overview and aims
Why LQCD needs other theoretical input
Where is χPT needed
Review of χPT in the continuumEffective field theories in general
Broken chiral symmetry and its implications
Constructing the pionic Lagrangian
Examples of NLO results
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.40/46
Results from χPT at NLOCharged PGB masses:
LO: m2PGB,0 = (χq1 + χq2)/2 = 2B0(mq1 + mq2)/2
NLO–tree:
δm2PGB ∼ L(4)
∼ χ L χf2 ∼ χ(16π2L)
m2PGB,0
Λ2χ
NLO–loop:
δm2PGB ∼
L(2)
q∼ χ
f2
∫
q1
q2+m2PGB
∼ χm2
PGB,0
Λ2χ
ln
(m2
PGB,0
µ2
)
m2π± = χ`
1 +8
f2[(2L8 − L5)χ`︸ ︷︷ ︸
valence
+ (2L6 − L4)(2χ` + χs)︸ ︷︷ ︸
sea
] +3Lπ − Lη
6︸ ︷︷ ︸
logs
Lπ =m2
π
Λ2χ
ln
(m2
π
µ2
)
, Lη =m2
η
Λ2χ
ln
(
m2η
µ2
)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.41/46
Lessons for lattice simulations (IV)
m2π± = χ`
1 +8
f2[(2L8 − L5)χ`︸ ︷︷ ︸
valence
+ (2L6 − L4)(2χ` + χs)︸ ︷︷ ︸
sea
] +3Lπ − Lη
6︸ ︷︷ ︸
logs
Non-analytic terms importantat small masses
ms = 0.08 GeV, f = 0.093 GeV,
L5 = 1.45 × 10−3, L8 = 10−3,
L4 = L6 = 0[Bijnens, hep-ph/0409068]
0.1 0.2 0.3 0.4 0.5m_lm_s
6.05
6.15
6.2
6.25
6.3
6.35
mpi^2m_l at NLO
Must see chiral logs to have convincing results
Using PQ simulations allows separation of Li
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.42/46
Further examples of chiral logs
fK
fπ= 1 +
2
f2(L5)(χs − χ`)︸ ︷︷ ︸
valence
+5
8Lπ −
1
4LK −
3
8Lη
︸ ︷︷ ︸
logs
Non-analytic terms importantat small masses
ms = 0.08 GeV, f = 0.085 GeV,
L5 = 1.45 × 10−3, L4 = 0 0.1 0.2 0.3 0.4 0.5m_lm_s
1.051.075
1.1251.15
1.1751.2
1.225fKf_pi at NLO
Good to use “Golden Ratios” in which chiral logs cancel [Becirevic03,04]
Some quantities have enhanced chiral logs, e.g. 〈r2〉π ∼ ln(m2π/µ2)
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.43/46
Volume dependence from χPTFor single particle matrix elements pion (or more generally, PGB) loops
give leading finite volume correction [Gasser+Leutwyler]
Predicted along with coefficient of chiral log:
Replace momentum integral with sum
L(2)
q→∫
q
(
1q2+m2
PGB
)
→∫
q4
∑
~q=2π~n/L
(
1q2+m2
PGB
)
Equivalent to using an infinite set of images
E.g. fπ at a = 0.1 fm, withL = 2.4 fm (thick), 3.2 fmand ∞ms = 0.08 GeV, f = 0.08 GeV,
L5 = 1.45 × 10−3, L4 = 0
0.05 0.1 0.15 0.2 0.25m_lm_s
0.085
0.09
0.095
0.105
0.11
fpi in box at NLO
Formulae extended to higher order for some quantities [Luscher, Colangelo]
Inclusion of volume dependence in χPT fits is now standard
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.44/46
Other quantities involving PGBsSU(2) χPT complete at NNLO, including electroweak interactions
Several predictions despite 53 LECs at NNLO (excluding electroweak)!
Many quantities relevant for lattice simulations, e.g.
Pion scattering amplitude
Form factors of PGBs (vector and scalar)
Semileptonic form factors (K → π)
BK , K → ππ
SU(3) χPT (including electroweak) largely extended to NNLO
Convergence?. [Bijnens, hep-ph/0401039,hep-ph/0409068]
a00(ππ → ππ) = 0.159
︸ ︷︷ ︸
LO
+0.044︸ ︷︷ ︸
NLO
+ 0.016︸ ︷︷ ︸
NNLO
= 0.219±? c.f. 0.220(5)
fK/fπ = 1︸︷︷︸
LO
+0.169︸ ︷︷ ︸
NLO
+ 0.051︸ ︷︷ ︸
NNLO
(fit)
But for m2PGB, NNLO terms larger than NLO
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.45/46
Extension to “heavy” particlesWill not describe χPT technology in these lectures
Heavy-light mesons in 1/mB expansion [Wise, Burdman & Donoghue]
FB ∼ FB,0(1 + m2π
︸︷︷︸
analytic
+ m2π ln(mπ)
︸ ︷︷ ︸
chiral log
+ . . . )
Similar expansion to those for PGB properties
Non-analytic terms involve additional coefficient gπBB∗
Baryons [Jenkins & Manohar] and Vector mesons [Jenkins et al]
MH ∼ M0 + m2π
︸︷︷︸
analytic
+ gπHH′ m3π
︸ ︷︷ ︸
leading loop
+ m4π ln(mπ)
︸ ︷︷ ︸
subleading loop
+ m4π + . . .
Non-analytic terms involve additional coefficients (e.g. gπNN )
Expansion in powers of mπ/Λχ (c.f. (mπ/Λχ)2 for mesons)
⇒ Poorer convergence
(Improve using “finite range regularization”? [Leinweber et al] )
S. Sharpe, “χPT for LQCD (I)”, Nara, 11/7/2005 – p.46/46