New Applications of Chiral Perturbation theory to lattice QCD (I) · 2005. 10. 28. · E ective...

$: New Applications of Chiral Perturbation theory to lattice QCD (I) · 2005. 10. 28. · E ective Field Theory/PT: references A selection of books and lecture notes: H. Georgi, \Weak$
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


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


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


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


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


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


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


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)


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


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)


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)


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


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


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.


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] )


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


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)


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]


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


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†


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






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


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


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, . . . )


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


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


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] )


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)


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


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]


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〉


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


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


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]


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

)


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


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)


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


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


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] )


New Applications of Chiral Perturbation theory to lattice QCD (I) · 2005. 10. 28. · E ective...

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Transcript of New Applications of Chiral Perturbation theory to lattice QCD (I) · 2005. 10. 28. · E ective...