Chiral Effective Field Theory Beyond the Power Counting Regime · Introduction EFT for Nucleons...

Jonathan HallSupervisors:

DerekLeinweber &Ross Young

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Chiral Effective Field Theory Beyond the Power

Counting Regime

Jonathan HallSupervisors: Derek Leinweber & Ross Young

CSSM, University of Adelaide

14th of April 2010

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Overview

Introduction

Effective Field Theory for nucleons

Loop integralsRenormalization

Ideal ‘pseudodata’

Intrinsic energy scale

EvidenceStatistical uncertaintyHigher chiral order

Quenched ρ meson case

Conclusion & future directions

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Lattice QCD can rarely be evaluated at physical quarkmasses. We want to be able to extrapolate current resultsto this physical point.

Chiral Perturbation Theory gives insight into this lowenergy region, but is limited to use over a very small rangeof quark masses.

We will discover that using more of the available dataoften entails model-dependence. But the extent of themodel-dependence can be quantified and thus removed.

This will lead us to realizing the presence of an ‘intrinsicenergy scale’, embedded in such lattice QCD data.

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Why use χEFT?

Chiral Effective Field Theory (χEFT) complements latticeQCD.

It assists in understanding the consequences of dynamicalchiral symmetry breaking.

It provides a scheme-independent approach forinvestigating the properties of hadrons.

In particular, it can be used in conjunction with latticeQCD data to extrapolate results:

to physical quark masses,

to infinite lattice volume and continuum limit.

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Overview

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Why use χEFT?

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Why use χEFT?

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Overview

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Why use χEFT?

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Overview

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Why use χEFT?

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Overview

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Why use χEFT?

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Why use χEFT?

Chiral Perturbation Theory (χPT) is a low energy theorywhere gluons and quarks can be replaced by effectivedegrees of freedom.

χPT provides a formal expansion in terms of low energymomenta and quark masses.

The expansion is convergent if the quark mass is small sothat higher order terms are negligible. This is called thePower Counting Regime (PCR).

Within the PCR, χPT is scheme-independent, and can beused to connect lattice simulations to the real world.

Outside the PCR, χPT is scheme-dependent, and shouldnot be used.

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Why use χEFT?

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Why use χEFT?

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Overview

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Why use χEFT?

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Why use χEFT?

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Why use χEFT?

The PCR is small (mπ . 200 MeV); lattice resultsinvariably extend outside the PCR.

...enter Effective Field Theory, which provides novelmethods for describing results beyond the PCR.

EFT can be used to search for the possible presence of an‘intrinsic energy scale’ embedded in lattice QCD results.

But first...

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Why use χEFT?

But first...

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EFT forNucleons

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Conclusion

Why use χEFT?

But first...

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EFT forNucleons

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Conclusion

Why use χEFT?

But first...

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Formal Background

For an effective field theory, one writes out a low energyeffective Lagrangian.

The terms of the Lagrangian are ordered in powers ofmomenta and mass.

For nucleons (fermions) written as an SU(2) doubletΨ = (p n)T, the first order (lowest energy) Lagrangiantakes the form:

L(1)πN = Ψ̄

/∂−◦

2fπγµγ5~τ · ∂µ~π

The circle ◦ denotes a “bare” quantity: it getsrenormalized by chiral loops from the field theory. Let’slook at the nucleon mass MN ...

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Formal Background

L(1)πN = Ψ̄

/∂−◦

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Formal Background

L(1)πN = Ψ̄

/∂−◦

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Formal Background

L(1)πN = Ψ̄

/∂−◦

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Formal Background

L(1)πN = Ψ̄

/∂−◦

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Chiral Effective Field Theory

The nucleon mass MN is renormalized by:

an analytic polynomial associated with the quark massesmq.chiral loop integrals Σloops .

The low energy expansion formula about the chiral limit(small mq) is expressed using theGell-Mann−Oakes−Renner Relation mq ∝ m2

MN = {terms analytic in m2π } + {chiral loop corrections}

= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

The analytic terms will be collectively called the ‘residualseries’, and their coefficients ai will be determined byfitting to lattice QCD data.

The chiral loops have known, scheme-independentcoefficients, but given rise to non-analytic behaviour.

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= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

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= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

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= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

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= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

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= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

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= {a0 + a2m2π + a4m

4π + O(m6

π)} + {Σloops} .

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Chiral Loops

The integral form of the chiral loops are obtained usingthe Feynman Rules for χPT, and can then be solved.

Each loop, when evaluated from its integral form,produces a non-analytic term.

To finite chiral order (O(m4π log mπ)), the leading order

chiral loops are:

the 1−pion loop (ΣN ∼ m3π),

the pion loop decuplet transition (Σ∆ ∼ m4π log mπ),

and the ‘tadpole’ loop (Σtad ∼ m4π log mπ).

In general, each loop integral also produces an analyticpolynomial in m2

π of its own.

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Chiral Loops

chiral loops are:

π of its own.

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Chiral Loops

chiral loops are:

π of its own.

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Chiral Loops

chiral loops are:

π of its own.

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Chiral Loops

chiral loops are:

π of its own.

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Chiral Loops

chiral loops are:

π of its own.

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Chiral Loops

chiral loops are:

π of its own.

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Chiral Loops

= ΣN = bN0 + bN

2 m2π + χNm3

π + bN4 m4

π + O(m5π)

= Σ∆ = b∆0 + b∆

2 m2π + b∆

4 m4π +

4π∆χ∆m4

π logmπ

µ+ O(m5

= Σtad = bt′

2 m2π + bt′

4 m4π + χ′

tm4π log

µ+ O(m5

Note: ∆ is the mass splitting◦

M∆ −◦

MN , and µ is themass scale associated with renormalization.

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Chiral Loops

= ΣN = bN0 + bN

2 m2π + χNm3

π + bN4 m4

π + O(m5π)

= Σ∆ = b∆0 + b∆

2 m2π + b∆

4 m4π +

4π∆χ∆m4

π logmπ

µ+ O(m5

= Σtad = bt′

2 m2π + bt′

4 m4π + χ′

tm4π log

µ+ O(m5

M∆ −◦

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Chiral Loops

= ΣN = bN0 + bN

2 m2π + χNm3

π + bN4 m4

π + O(m5π)

= Σ∆ = b∆0 + b∆

2 m2π + b∆

4 m4π +

4π∆χ∆m4

π logmπ

µ+ O(m5

= Σtad = bt′

2 m2π + bt′

4 m4π + χ′

tm4π log

µ+ O(m5

M∆ −◦

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Chiral Loops

= ΣN = bN0 + bN

2 m2π + χNm3

π + bN4 m4

π + O(m5π)

= Σ∆ = b∆0 + b∆

2 m2π + b∆

4 m4π +

4π∆χ∆m4

π logmπ

µ+ O(m5

= Σtad = bt′

2 m2π + bt′

4 m4π + χ′

tm4π log

µ+ O(m5

M∆ −◦

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Chiral Loops

= ΣN = bN0 + bN

2 m2π + χNm3

π + bN4 m4

π + O(m5π)

= Σ∆ = b∆0 + b∆

2 m2π + b∆

4 m4π +

4π∆χ∆m4

π logmπ

µ+ O(m5

= Σtad = bt′

2 m2π + bt′

4 m4π + χ′

tm4π log

µ+ O(m5

M∆ −◦

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Expansion Formulae

When the integrals are evaluated, the expansion becomesan expansion in increasing powers of mπ, withrenormalized analytic and non-analytic terms occuring attheir respective orders:

MN = c0 + c2m2π + χNm3

π + c4m4π

4π∆χ∆ + χ′

m4π log

µ+ O(m5

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Expansion Formulae

When the integrals are evaluated, the expansion becomesan expansion in increasing powers of mπ, withrenormalized analytic and non-analytic terms occuring attheir respective orders:

MN = c0 + c2m2π + χNm3

π + c4m4π

4π∆χ∆ + χ′

m4π log

µ+ O(m5

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Renormalization Issues

The coefficients χN , χ∆ & χ′

t are known,

scheme-independent parameters (related to◦

gA, fπ, etc).

The coefficients bi however, are scheme-dependent, butthey occur at the relevant chiral orders to renormalize theresidual series:

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

These renormalized coefficients ci arescheme-independent, and of phenomenological interest.

c0 is the nucleon mass in the chiral limit (m2π = 0), and c2

is related to the ‘sigma term’ σπN of explicit chiralsymmetry breaking.

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t are known,

gA, fπ, etc).

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

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t are known,

gA, fπ, etc).

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

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t are known,

gA, fπ, etc).

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

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t are known,

gA, fπ, etc).

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

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t are known,

gA, fπ, etc).

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

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t are known,

gA, fπ, etc).

c0 = a0 + bN0 + b∆

c2 = a2 + bN2 + b∆

2 + bt′

c4 = a4 + bN4 + b∆

4 + bt′

4 , etc.

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How does the renormalization take place? Consider the1−pion loop integral as a test example:

ΣN =2χN

k2 + m2π

(k2 + m2π)(k2 − m2

π) + m4π

k2 + m2π

0dk k2 − m2

+ χNm3π .

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ΣN =2χN

k2 + m2π

(k2 + m2π)(k2 − m2

π) + m4π

k2 + m2π

0dk k2 − m2

+ χNm3π .

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ΣN =2χN

k2 + m2π

(k2 + m2π)(k2 − m2

π) + m4π

k2 + m2π

0dk k2 − m2

+ χNm3π .

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ΣN =2χN

k2 + m2π

(k2 + m2π)(k2 − m2

π) + m4π

k2 + m2π

0dk k2 − m2

+ χNm3π .

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ΣN =2χN

k2 + m2π

(k2 + m2π)(k2 − m2

π) + m4π

k2 + m2π

0dk k2 − m2

+ χNm3π .

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In Dimensional Regularization (DR), there is no explicitmomentum cutoff, so each of the ai coefficients undergoesan infinite renormalization or none at all:

c0 = a0 +2χN

0dk k2 ,

c2 = a2 −2χN

c4 = a4 + 0 , etc.

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c0 = a0 +2χN

0dk k2 ,

c2 = a2 −2χN

c4 = a4 + 0 , etc.

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c0 = a0 +2χN

0dk k2 ,

c2 = a2 −2χN

c4 = a4 + 0 , etc.

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c0 = a0 +2χN

0dk k2 ,

c2 = a2 −2χN

c4 = a4 + 0 , etc.

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In Finite Range Regularization (FRR), a momentum cutoffΛ is introduced (via a regulator function), and the chiralexpansion is resummed.

For a sharp cutoff regulator:

ΣN (Λ) =2χN

∫ Λ

k2 + m2π

3− Λm2

π + m3πarctan

πΛm2

π + χNm3π −

π + · · · .

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ΣN (Λ) =2χN

∫ Λ

k2 + m2π

3− Λm2

π + m3πarctan

πΛm2

π + χNm3π −

π + · · · .

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ΣN (Λ) =2χN

∫ Λ

k2 + m2π

3− Λm2

π + m3πarctan

πΛm2

π + χNm3π −

π + · · · .

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ΣN (Λ) =2χN

∫ Λ

k2 + m2π

3− Λm2

π + m3πarctan

πΛm2

π + χNm3π −

π + · · · .

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ΣN (Λ) =2χN

∫ Λ

k2 + m2π

3− Λm2

π + m3πarctan

πΛm2

π + χNm3π −

π + · · · .

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The DR result is recovered as Λ → ∞.

c0 = a0 +2χN

3Λ3 ,

c2 = a2 −2χN

πΛ ,

c4 = a4 −2χN

Λ, etc.

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c0 = a0 +2χN

3Λ3 ,

c2 = a2 −2χN

πΛ ,

c4 = a4 −2χN

Λ, etc.

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c0 = a0 +2χN

3Λ3 ,

c2 = a2 −2χN

πΛ ,

c4 = a4 −2χN

Λ, etc.

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c0 = a0 +2χN

3Λ3 ,

c2 = a2 −2χN

πΛ ,

c4 = a4 −2χN

Λ, etc.

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Loop Integrals

We would like to define the loop integrals so that therenormalization of a0 and a2 happen automatically.

This simply means that, by convention, relevant b0 and b2

terms will be subtracted from each integral.

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Loop Integrals

We would like to define the loop integrals so that therenormalization of a0 and a2 happen automatically.

This simply means that, by convention, relevant b0 and b2

terms will be subtracted from each integral.

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Loop Integrals

Taking the heavy-baryon limit and performing the k0

integration, the loop integrals take the following forms:

Σ̃N =χN

d3kk2u2(k; Λ)

ω2(k)− bΛ,N

0 − bΛ,N2 m2

Σ̃∆ =χ∆

d3kk2u2(k; Λ)

ω(k)(∆ + ω(k))− bΛ,∆

0 − bΛ,∆2 m2

Σ̃tad = c2m2π

d3k2u2(k; Λ)

ω(k)− bΛ,t

= c2m2πσ̃tad , pulling out c2 as a factor .

The pion energy is ω(k) ≡√

k2 + m2π and the FRR

regulator function is denoted by u(k; Λ).

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Loop Integrals

Σ̃N =χN

d3kk2u2(k; Λ)

ω2(k)− bΛ,N

0 − bΛ,N2 m2

Σ̃∆ =χ∆

d3kk2u2(k; Λ)

ω(k)(∆ + ω(k))− bΛ,∆

0 − bΛ,∆2 m2

Σ̃tad = c2m2π

d3k2u2(k; Λ)

ω(k)− bΛ,t

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Loop Integrals

Σ̃N =χN

d3kk2u2(k; Λ)

ω2(k)− bΛ,N

0 − bΛ,N2 m2

Σ̃∆ =χ∆

d3kk2u2(k; Λ)

ω(k)(∆ + ω(k))− bΛ,∆

0 − bΛ,∆2 m2

Σ̃tad = c2m2π

d3k2u2(k; Λ)

ω(k)− bΛ,t

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Loop Integrals

Σ̃N =χN

d3kk2u2(k; Λ)

ω2(k)− bΛ,N

0 − bΛ,N2 m2

Σ̃∆ =χ∆

d3kk2u2(k; Λ)

ω(k)(∆ + ω(k))− bΛ,∆

0 − bΛ,∆2 m2

Σ̃tad = c2m2π

d3k2u2(k; Λ)

ω(k)− bΛ,t

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Loop Integrals

Σ̃N =χN

d3kk2u2(k; Λ)

ω2(k)− bΛ,N

0 − bΛ,N2 m2

Σ̃∆ =χ∆

d3kk2u2(k; Λ)

ω(k)(∆ + ω(k))− bΛ,∆

0 − bΛ,∆2 m2

Σ̃tad = c2m2π

d3k2u2(k; Λ)

ω(k)− bΛ,t

18 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Loop Integrals

Σ̃N =χN

d3kk2u2(k; Λ)

ω2(k)− bΛ,N

0 − bΛ,N2 m2

Σ̃∆ =χ∆

d3kk2u2(k; Λ)

ω(k)(∆ + ω(k))− bΛ,∆

0 − bΛ,∆2 m2

Σ̃tad = c2m2π

d3k2u2(k; Λ)

ω(k)− bΛ,t

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Expansion Formulae

Note that the tadpole integral has a coefficient χ′

t = c2χt,which involves c2 (obtained from the Lagrangian

L(2),tadπN = c2Trf [Mq]Ψ̄Ψ).

Thus the nucleon mass expansion formula can beconveniently factorized:

MN = {a0 + a2m2π + a4m

4π + O(m6

π)} + {ΣN + Σ∆ + Σtad}

= c0 + c2m2π(1 + σ̃tad) + aΛ

4 m4π + Σ̃N + Σ̃∆ .

This formula can be used for extrapolations, with fitcoefficients c0, c2 and aΛ

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Expansion Formulae

MN = {a0 + a2m2π + a4m

4π + O(m6

π)} + {ΣN + Σ∆ + Σtad}

= c0 + c2m2π(1 + σ̃tad) + aΛ

4 m4π + Σ̃N + Σ̃∆ .

19 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Expansion Formulae

MN = {a0 + a2m2π + a4m

4π + O(m6

π)} + {ΣN + Σ∆ + Σtad}

= c0 + c2m2π(1 + σ̃tad) + aΛ

4 m4π + Σ̃N + Σ̃∆ .

19 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Expansion Formulae

MN = {a0 + a2m2π + a4m

4π + O(m6

π)} + {ΣN + Σ∆ + Σtad}

= c0 + c2m2π(1 + σ̃tad) + aΛ

4 m4π + Σ̃N + Σ̃∆ .

19 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Finite Volume Corrections

Lattice QCD is done on a finite volume box.

Our ideal infinite volume expansion formula should bemodified to include finite volume corrections.

Each integral can be converted into a discrete summation,and then the difference is taken to achieve the correction:

δFVCi =

[ (2π)3

LxLyLz

kx,ky ,kz

The tadpole finite volume corrections are subtle and willnot be dealt with in this talk.

20 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

δFVCi =

[ (2π)3

LxLyLz

kx,ky ,kz

20 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

δFVCi =

[ (2π)3

LxLyLz

kx,ky ,kz

20 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

δFVCi =

[ (2π)3

LxLyLz

kx,ky ,kz

20 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

δFVCi =

[ (2π)3

LxLyLz

kx,ky ,kz

20 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The finite volume corrections are easily incorporated intoour expansion formula:

MVN = c0 + c2m

2π(1 + σ̃tad) + aΛ

4 m4π

+ (Σ̃N + δFVCN ) + (Σ̃∆ + δFVC

∆ ) + O(m5π) .

We are almost ready to try an extrapolation from latticeQCD data. But what form ought the regulator u(k; Λ) totake?

21 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

MVN = c0 + c2m

2π(1 + σ̃tad) + aΛ

4 m4π

∆ ) + O(m5π) .

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

MVN = c0 + c2m

2π(1 + σ̃tad) + aΛ

4 m4π

∆ ) + O(m5π) .

21 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Finite-Range Regulators

All forms of u(k; Λ) are equivalent within the PCR, aslong as they are normalized to 1, and are suppressed to 0for large momenta k. DR corresponds to Λ → ∞.

The step function θ(Λ − k) is acceptable, but isunfavorable for use with the finite volume of the lattice.

Consider the family of smooth n−tuple dipole attenuators:

un(k; Λ) =

1 +k2n

The dipole corresponds to n = 1. We shall also considerthe cases n = 2, 3, the double and triple dipole forms,respectively.

We shall analyze data using these three different regulatorsto demonstrate the model-independence of this approach.

22 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

un(k; Λ) =

1 +k2n

22 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

un(k; Λ) =

1 +k2n

22 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

un(k; Λ) =

1 +k2n

22 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

un(k; Λ) =

1 +k2n

22 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

un(k; Λ) =

1 +k2n

22 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here are the three dipole-like forms plotted for Λ = 1.0GeV:

23 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Trial Extrapolations

Consider the behaviour of MN as a function of m2π.

Here is an extrapolation of data from JLQCD, using adipole regulator with Λdip = 1.0 GeV.

The JLQCD data uses Nf = 2 overlap fermions atL = 1.9 fm.

24 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

24 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

24 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is an extrapolation of data from PACS-CS, using adipole regulator with Λdip = 1.0 GeV.The PACS-CS data uses non-perturbativelyO(a)-improved Wilson quark action at L = 2.9 fm.

25 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

25 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

25 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is an extrapolation of data from CP-PACS, using adipole regulator with Λdip = 1.0 GeV.The CP-PACS data uses a mean field improved cloverquark action at L = 2.2 → 2.8 fm.

26 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

26 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

26 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

There is nothing special about Λdip = 1.0 GeV.

What happens to the extrapolation as Λdip is changed?Consider the CP-PACS data:

27 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

There is nothing special about Λdip = 1.0 GeV.

What happens to the extrapolation as Λdip is changed?Consider the CP-PACS data:

27 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

Different choices of regulator give different results! But isthere an optimal choice?

Also, if we want to stay close to the PCR, how many datapoints should we use? Does it matter?

Let’s do a test: Using the extrapolation formula for MN ,generate some ideal ‘pseudodata’.

Generate one set of 100 closely spaced, low energypseudodata points entirely within the PCR, created atΛc

dip = 1.0 GeV.

Generate two more sets, at different upper values m2π,max,

thus progressing outside the PCR.

Choose infinite volume, thus avoiding any finite volumesubtleties.

28 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

dip = 1.0 GeV.

28 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

dip = 1.0 GeV.

28 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

dip = 1.0 GeV.

28 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

dip = 1.0 GeV.

28 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

dip = 1.0 GeV.

28 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata

dip = 1.0 GeV.

28 / 64

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Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata - Renormalization Flow

We can use these pseudodata sets for our analysis ofregulator dependence.

The regulator dependence is characterized by thebehaviour of the renormalized constants ci with respect toΛdip.

Let’s plot our fit coefficients c0 and c2 over a range ofΛdip values, for each of the three data sets. We havechosen m2

π,max = 0.04, 0.25, 0.5 GeV2.

29 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max = 0.04, 0.25, 0.5 GeV2.

29 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max = 0.04, 0.25, 0.5 GeV2.

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Pseudodata- Renormalization Flow

Here is the result for c0 .

Notice that the correct value of c0 is recovered exactlywhen Λdip = Λc

30 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Notice that the correct value of c0 is recovered exactlywhen Λdip = Λc

30 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Though it is tempting to read off the value of any ci asΛ → ∞, recovering the DR result, it is wrong!

31 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Though it is tempting to read off the value of any ci asΛ → ∞, recovering the DR result, it is wrong!

31 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

This intersection point is not trivial. To demonstrate this,we can analyze the pseudodata using a triple dipole.

Here is the result for c0:

32 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

This intersection is no longer a clear point, but a cluster atΛdip ≈ 0.5 − 0.6 GeV. This is the preferred value of Λtrip.

33 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

This intersection is no longer a clear point, but a cluster atΛdip ≈ 0.5 − 0.6 GeV. This is the preferred value of Λtrip.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The regulator dependence increased as the pseudodataextended outside the PCR.

We also see that FRR breaks down if Λ is too small.

This makes sense mathematically, as bΛi ∝ Λ3−i, and so

for i = 4, 6, .. these higher order coefficients blow up forsmall Λ.

This also makes sense physically, as any ultravioletregulator Λ must be large enough to allow inclusion of thechiral physics being studied. Otherwise we essentiallydestroy the non-analytic behaviour by making the integrals≈ 0.

Thus there is a lowest suitable value Λlower below whichwe cannot ensure consistent renormalization.

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

34 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

34 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

34 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

34 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

34 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Evidence for an Intrinsic Scale

In the pseudodata test example, the optimal cutoff (byconstruction) was obtained from the pseudodatathemselves.

But do actual lattice QCD data have an intrinsic scaleembedded in them?

If so, it would indicate that the data contain informationregarding an optimal FRR regulator.

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

35 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

35 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Let us repeat our analysis of c0 and c2 for the JLQCD,PACS-CS and CP-PACS data sets.

We will obtain each one using the lightest 4 data points,and increase m2

π,max by one data point at a time.

Each time we add a new data point, we increase thedistance the data set extends outside the PCR, thusincreasing the scheme-dependence. This helps identify theintrinsic scale.

Since actual lattice QCD data is not ideal like ourpseudodata, we can’t expect that the renormalization flowcurves will cross at exactly the same value of Λ.

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

36 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

36 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

36 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

36 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is the result for c0 using JLQCD data, working tochiral order O(m3

π) and using a dipole regulator:

37 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π) and using a double dipole regulator:

38 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π) and using a triple dipole regulator:

39 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

40 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is the result for c0 using PACS-CS data, working tochiral order O(m3

41 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is the result for c0 using CP-PACS data, working tochiral order O(m3

42 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

There is a reasonably well-defined intersection pointindicating the intrinsic scale.

For each regulator, the intersection occurs at the samevalue of Λ for both c0 and c2. This is a highly significantresult.

The value of the intrinsic scale differs between regulatortypes. The regulators are different shapes and a differentcutoff is required to achieve a similar suppression of thelarge loop momenta.

To obtain a systematic uncertainty in the intrinsic scale,apply a kind of χ2

dof analysis...

43 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

dof analysis...

43 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

dof analysis...

43 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

dof analysis...

43 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

dof analysis...

43 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Statistical Uncertainty

On each of these renormalization flow plots, differentcurves correspond to different values of m2

π,max.

To what extent do the curves match?

Construct χ2dof , where dof equals the number of m2

π,max

values:

χ2dof =

n − 1

(ci(Λ) − cav(Λ))2

(δci(Λ))2,

where cav(Λ) =

∑ni=1 ci(Λ)/(δci(Λ))2∑n

j=1 1/(δcj(Λ))2.

We shall construct χ2dof for c0 and c2 seperately, and plot

against Λ.

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max.

π,max

values:

χ2dof =

n − 1

(δci(Λ))2,

where cav(Λ) =

j=1 1/(δcj(Λ))2.

against Λ.

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max.

π,max

values:

χ2dof =

n − 1

(δci(Λ))2,

where cav(Λ) =

j=1 1/(δcj(Λ))2.

against Λ.

44 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max.

π,max

values:

χ2dof =

n − 1

(δci(Λ))2,

where cav(Λ) =

j=1 1/(δcj(Λ))2.

against Λ.

44 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max.

π,max

values:

χ2dof =

n − 1

(δci(Λ))2,

where cav(Λ) =

j=1 1/(δcj(Λ))2.

against Λ.

44 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

π,max.

π,max

values:

χ2dof =

n − 1

(δci(Λ))2,

where cav(Λ) =

j=1 1/(δcj(Λ))2.

against Λ.

44 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Example plot: here is the result for χ2dof obtained from c0

using PACS-CS data, working to chiral order O(m3π) and

using a dipole regulator:

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Results

The central values of Λ (GeV) are tabulated below:

regulator form

optimal scale dipole double triple

Λscalec0,JLQCD 1.44 1.08 0.96

Λscalec2,JLQCD 1.40 1.05 0.94

Λscalec0,PACS−CS 1.21 0.93 0.83

Λscalec2,PACS−CS 1.21 0.93 0.83

Λscalec0,CP−PACS 1.20 0.98 0.88

Λscalec2,CP−PACS 1.19 0.97 0.87

46 / 64

Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

We found strong scheme-dependence when working tochiral order O(m3

π) outside the PCR.

What happens if we try the higher chiral orderO(m4

π log mπ)?

47 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

We found strong scheme-dependence when working tochiral order O(m3

π) outside the PCR.

What happens if we try the higher chiral orderO(m4

π log mπ)?

47 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ) and using a dipole regulator:

48 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ) and using a dipole regulator:

49 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

At higher chiral order, there are no clear intersectionpoints. We are unable to identify an intrinsic scale.

This means that the scheme-dependence is weakened byworking to higher chiral order.

This systematic error in c0 and c2 is larger than theirstatistical errors, thus indicating that the data is outsidethe PCR.

There are now at least two ways of assessing thesystematic uncertainty in Λ:

from the χ2dof analysis at O(m3

from the systematic error over Λ from the plots atO(m4

π log mπ).

50 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ).

50 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ).

50 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ).

50 / 64

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ).

50 / 64

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Higher Chiral Order

π log mπ).

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Summary for the Nucleon

We are now able to extrapolate MN,phys and obtain c0 andc2 by using FRR χEFT and selecting the intrinsic scale.

We are also able to provide a realistic systematic error inthe result.

Examples using the dipole regulator:

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

MPACS−CSN,phys = 0.967(45)(43)(2 − 3) (GeV).

re: PACS-CS data uses non-perturbatively O(a)-improvedWilson quark action at L = 2.9 fm.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

51 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

51 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

51 / 64

Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

cPACS−CS0 = 0.900(51)(15)(5 − 6) (GeV),

cPACS−CS2 = 3.06(32)(15)(18 − 25) (GeV−1),

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

’The Challenge’

Consider the quenched ρ meson.

We want to predict the mass of the quenched ρ meson atphysical pion mass (mπ,phys = 140 MeV)

We have quenched lattice QCD (QQCD) results from theKentucky Group, but we are blinded to the lowest energydata.

QQCD observables are an important testing ground, sincethere are no experimentally known values that canintroduce a prejudice in the final result.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

’The Challenge’

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

’The Challenge’

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

’The Challenge’

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

QQCD Data from the Lattice

The following data from Kentucky Group (L = 3.06 fm)are missing points close to the chiral limit (mq = 0).

The available data lie in the range 380 < mπ < 720 MeV,

The unavailable data lie in the range 200 < mπ < 380MeV.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Chiral Extrapolation Formulae

The quenched ρ meson mass mρ,Q has a similar chiralexpansion to the nucleon.

The expansion similarly contains a residual series and loopintegrals. We will work to chiral order O(m4

The renormalization of the low energy constants takesplace just as before. The fit parameters are c0, c2 and c4.

The triple dipole is chosen as the regulator form.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Test for an Intrinsic Scale

Here is the result for c0 using Kentucky Group data,working to chiral order O(m4

π) and using a triple dipoleregulator:

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The crossings are much harder to identify, so we will relyon our χ2

dof method.

Here is the result for χ2dof obtained from the same c0:

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The crossings are much harder to identify, so we will relyon our χ2

dof method.

Here is the result for χ2dof obtained from the same c0:

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The central, upper and lower values of Λ (GeV) aretabulated below:

scale (GeV) for c0 for c2 for c4

Λcentral 0.64 0.64 0.64

Λupper 0.76 0.70 0.68

Λlower 0.52 0.58 0.59

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Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The Intrinsic Scale

By averaging the result for the central value, the upperand the lower limits among c0, c2, and c4, the optimalregulator scale Λscale

trip for the quenched ρ meson mass canbe calculated for this data set.

Using the triple dipole regulator, Λscaletrip = 0.64 GeV

(+0.08 − 0.07) GeV.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

The Intrinsic Scale

By averaging the result for the central value, the upperand the lower limits among c0, c2, and c4, the optimalregulator scale Λscale

trip for the quenched ρ meson mass canbe calculated for this data set.

Using the triple dipole regulator, Λscaletrip = 0.64 GeV

(+0.08 − 0.07) GeV.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Completing ’The Challenge’

The extrapolation of the quenched ρ meson mass can nowbe completed.

Treating the various coupling constants and Λscaletrip as

independent, their errors can be added in quadrature.

We shall plot an inner error bar corresponding to thesystematic error coming from the choice in parametersonly.

We shall plot an outer error bar corresponding to thesystematic and statistical errors of each point added inquadrature.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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Overview

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is the result of the extrapolation, filling in for themissing Kentucky Group data points.

At the physical point, we find mρ,Q(m2π,phys) = 0.915 GeV

(± 0.036) GeV, an error just under 4%.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here is the result of the extrapolation, filling in for themissing Kentucky Group data points.

At the physical point, we find mρ,Q(m2π,phys) = 0.915 GeV

(± 0.036) GeV, an error just under 4%.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Now, the lattice results are added to the plot:

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here, the error bars are correlated relative to the lightestdata point in the original set, m2

π = 0.143 GeV2.

Our extrapolation error bars are smaller than for thenumerically evaluated data.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Here, the error bars are correlated relative to the lightestdata point in the original set, m2

π = 0.143 GeV2.

Our extrapolation error bars are smaller than for thenumerically evaluated data.

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Overview

Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

We have been able to extrapolate current lattice QCDresults to the physical point, using Chiral Effective FieldTheory.

We have discovered that Finite-Range Regularization isinstrumental for the analysis of data extending outside thechiral Power Counting Regime.

We have developed a robust procedure for quantifying thedegree of scheme-dependence, through the search for anintrinsic scale Λscale.

In quenched QCD, we have shown that the extrapolationscheme can make quantifiable predictions withoutphenomenologically motivated assumptions.

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EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

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EFT forNucleons

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Quenched ρ

Conclusion

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EFT forNucleons

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Quenched ρ

Conclusion

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EFT forNucleons

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Quenched ρ

Conclusion

Future Directions

An alternative technique for propagation of uncertainty inthe scale-dependence would be to consider marginalizationof the scale.

The extrapolation scheme can be applied to otherobservables such as magnetic moment and charge radii ofoctet baryons, which have large chiral curvature.

Finite volume corrections are of particular interest whenconsidering such observables.

The extrapolation scheme will also be useful forcalculating the Roper resonance, which is difficult toevaluate in lattice QCD.

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Introduction

EFT forNucleons

Pseudodata

Intrinsic Scale

Quenched ρ

Conclusion

Future Directions

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EFT forNucleons

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Quenched ρ

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Future Directions

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Future Directions

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Chiral Effective Field Theory Beyond the Power Counting Regime · Introduction EFT for Nucleons...

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